Session 21 — Tier ii′ planning + launch: close SfBoundProp in Lean (D-064)

{
  "pattern": "SfBoundProp",
  "path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1",
  "output_mode": "content",
  "-n": true
}

{
  "pattern": "research/derivations/D-0{37,56,62}*.md",
  "path": "D:\\Python\\Quantum"
}

{
  "pattern": "SfBoundProp|contour|hypotheses|remaining|modulo",
  "path": "D:\\Python\\Quantum\\research\\paper\\paperA.tex",
  "output_mode": "content",
  "-n": true,
  "head_limit": 60
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{
  "file_path": "D:\\Python\\Quantum\\research\\derivations\\D-037-finite-height-theorem.md",
  "limit": 80
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{
  "command": "ls \"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\.lake\\packages\" 2>/dev/null; ls \"D:\\Python\\Quantum\\research\\lean\\zeta-23-lean\\.lake\\packages\" 2>/dev/null",
  "description": "Locate vendored Mathlib package path"
}

{
  "todos": [
    {
      "content": "Verify paper/Lean baseline is clean and frozen",
      "status": "in_progress",
      "activeForm": "Verifying paper/Lean baseline"
    },
    {
      "content": "Inventory vendored residue/meromorphic API (agent)",
      "status": "pending",
      "activeForm": "Inventorying residue/meromorphic API"
    },
    {
      "content": "Inventory Zeta23 zero-counting + Mathlib zeta API + liCoeff (agent)",
      "status": "pending",
      "activeForm": "Inventorying Zeta23/Mathlib zeta API"
    },
    {
      "content": "Read D-036, D-056, LiSkeleton tier-(ii) defs",
      "status": "pending",
      "activeForm": "Reading D-036/D-056/LiSkeleton"
    },
    {
      "content": "Write D-064 design doc (routes + lemma graph)",
      "status": "pending",
      "activeForm": "Writing D-064 design doc"
    },
    {
      "content": "Create LiContour.lean sorry skeleton, compiles",
      "status": "pending",
      "activeForm": "Creating LiContour.lean skeleton"
    },
    {
      "content": "Session-014 closure (journal, STATE, commit)",
      "status": "pending",
      "activeForm": "Closing session 014"
    }
  ]
}

{
  "oldTodos": [],
  "newTodos": [
    "content='Verify paper/Lean baseline is clean and frozen' status='in_progress' active_form='Verifying paper/Lean baseline'",
    "content='Inventory vendored residue/meromorphic API (agent)' status='pending' active_form='Inventorying residue/meromorphic API'",
    "content='Inventory Zeta23 zero-counting + Mathlib zeta API + liCoeff (agent)' status='pending' active_form='Inventorying Zeta23/Mathlib zeta API'",
    "content='Read D-036, D-056, LiSkeleton tier-(ii) defs' status='pending' active_form='Reading D-036/D-056/LiSkeleton'",
    "content='Write D-064 design doc (routes + lemma graph)' status='pending' active_form='Writing D-064 design doc'",
    "content='Create LiContour.lean sorry skeleton, compiles' status='pending' active_form='Creating LiContour.lean skeleton'",
    "content='Session-014 closure (journal, STATE, commit)' status='pending' active_form='Closing session 014'"
  ],
  "cleared": false
}

{
  "description": "Inventory residue/meromorphic API",
  "subagent_type": "Explore",
  "prompt": "Research task (read-only inventory, NO code changes). Context: Lean 4 project at D:\\Python\\Quantum (toolchain v4.33.0-rc2). We need a precise capability inventory of the vendored complex-analysis API to plan a formalization of a rectangle-contour residue argument for \u03b6\u2032/\u03b6 with a higher-order pole.\n\nSurvey THOROUGHLY and report with exact declaration names + file paths + full signatures (hypotheses and conclusions) for:\n\n1. In D:\\Python\\Quantum\\zeta-23-lean\\Zeta23\\FromPNTPlus\\ResidueCalcOnRectangles.lean (1153 lines):\n   - Full signature of `RectangleIntegral'_eq_sumResiduesIn` (ALL hypotheses \u2014 especially whether HasSimplePolesOn is among them, and what RectangleIntegral' vs RectangleIntegral are)\n   - The `residue` placeholder def, `sumResiduesIn`, `HasSimplePolesOn`, and any lemmas for computing residues (residue_eq_of_tendsto, ResidueMult, logDerivResidue family \u2014 full signatures)\n   - `RectanglePullToNhdOfPole*` signatures\n   - Any higher-order-pole residue machinery (search for \"order\", \"principalPart\", \"MeromorphicNF\", \"iterated deriv\")\n2. In the same directory: Rectangle.lean and ZetaBounds.lean \u2014 list the main public theorems (one line each: name + what it proves). Especially anything about \u03b6\u2032/\u03b6 bounds, zeta residue at 1, logDeriv.\n3. In the vendored Mathlib at D:\\Python\\Quantum\\zeta-23-lean\\.lake\\packages\\mathlib\\Mathlib:\n   - Does Mathlib have its own `residue` API? Search Analysis/Complex/ and Geometry/ for \"residue\" (case-insensitive, def/theorem names). Report what exists in THIS vendored version.\n   - `MeromorphicAt` / `meromorphicOrderAt` API: which file, and list key lemmas (especially: order of product/sum, meromorphicOrderAt_eq_int_iff, normal form MeromorphicNFAt, any Laurent-series or principal-part API, anything relating the order to derivatives).\n   - Digamma: file Mathlib/Analysis/SpecialFunctions/Gamma/Digamma.lean \u2014 list main declarations.\n   - Search for \"Stieltjes\" and \"laurent\" (case-insensitive) related to riemannZeta \u2014 any Laurent expansion of \u03b6 at s=1?\n   - riemannZeta API: which files define it, the functional equation theorem names, trivial zeros theorem names (\u03b6(\u22122n)=0), pole at 1 API.\n   - Jensen's formula / Borel\u2013Carath\u00e9odory / Hadamard factorization / Weierstrass products: anything? Search Complex analysis dirs for \"Jensen\", \"Borel\", \"Carath\", \"Hadamard\", \"Weierstrass\", \"factorization\".\n4. Check what `Zeta23/FromPNTPlus/MediumPNT.lean` and `StrongPNTPrefix.lean` export at a high level (2-3 sentences each).\n\nReport as a structured list with exact names and signatures where I asked. Be thorough \u2014 this is the foundation of a multi-session design document. No need to read our project files (research/), only zeta-23-lean and the vendored Mathlib."
}

{
  "status": "completed",
  "prompt": "Research task (read-only inventory, NO code changes). Context: Lean 4 project at D:\\Python\\Quantum (toolchain v4.33.0-rc2). We need a precise capability inventory of the vendored complex-analysis API to plan a formalization of a rectangle-contour residue argument for \u03b6\u2032/\u03b6 with a higher-order pole.\n\nSurvey THOROUGHLY and report with exact declaration names + file paths + full signatures (hypotheses and conclusions) for:\n\n1. In D:\\Python\\Quantum\\zeta-23-lean\\Zeta23\\FromPNTPlus\\ResidueCalcOnRectangles.lean (1153 lines):\n   - Full signature of `RectangleIntegral'_eq_sumResiduesIn` (ALL hypotheses \u2014 especially whether HasSimplePolesOn is among them, and what RectangleIntegral' vs RectangleIntegral are)\n   - The `residue` placeholder def, `sumResiduesIn`, `HasSimplePolesOn`, and any lemmas for computing residues (residue_eq_of_tendsto, ResidueMult, logDerivResidue family \u2014 full signatures)\n   - `RectanglePullToNhdOfPole*` signatures\n   - Any higher-order-pole residue machinery (search for \"order\", \"principalPart\", \"MeromorphicNF\", \"iterated deriv\")\n2. In the same directory: Rectangle.lean and ZetaBounds.lean \u2014 list the main public theorems (one line each: name + what it proves). Especially anything about \u03b6\u2032/\u03b6 bounds, zeta residue at 1, logDeriv.\n3. In the vendored Mathlib at D:\\Python\\Quantum\\zeta-23-lean\\.lake\\packages\\mathlib\\Mathlib:\n   - Does Mathlib have its own `residue` API? Search Analysis/Complex/ and Geometry/ for \"residue\" (case-insensitive, def/theorem names). Report what exists in THIS vendored version.\n   - `MeromorphicAt` / `meromorphicOrderAt` API: which file, and list key lemmas (especially: order of product/sum, meromorphicOrderAt_eq_int_iff, normal form MeromorphicNFAt, any Laurent-series or principal-part API, anything relating the order to derivatives).\n   - Digamma: file Mathlib/Analysis/SpecialFunctions/Gamma/Digamma.lean \u2014 list main declarations.\n   - Search for \"Stieltjes\" and \"laurent\" (case-insensitive) related to riemannZeta \u2014 any Laurent expansion of \u03b6 at s=1?\n   - riemannZeta API: which files define it, the functional equation theorem names, trivial zeros theorem names (\u03b6(\u22122n)=0), pole at 1 API.\n   - Jensen's formula / Borel\u2013Carath\u00e9odory / Hadamard factorization / Weierstrass products: anything? Search Complex analysis dirs for \"Jensen\", \"Borel\", \"Carath\", \"Hadamard\", \"Weierstrass\", \"factorization\".\n4. Check what `Zeta23/FromPNTPlus/MediumPNT.lean` and `StrongPNTPrefix.lean` export at a high level (2-3 sentences each).\n\nReport as a structured list with exact names and signatures where I asked. Be thorough \u2014 this is the foundation of a multi-session design document. No need to read our project files (research/), only zeta-23-lean and the vendored Mathlib.",
  "agentId": "a35f4584ec1204acf",
  "agentType": "Explore",
  "content": [
    "type='text' text=\"Below is a read-only capability inventory of the vendored complex-analysis API, organized by your four questions. Exact declaration names, file paths, and full signatures are given where requested.\\n\\n---\\n\\n## 1. `Zeta23/FromPNTPlus/ResidueCalcOnRectangles.lean`\\n\\nFile: `D:/Python/Quantum/zeta-23-lean/Zeta23/FromPNTPlus/ResidueCalcOnRectangles.lean`.\\n\\n### Rectangle integrals\\n\\n```lean\\nnoncomputable def HIntegral (f : \u2102 \u2192 E) (x\u2081 x\u2082 y : \u211d) : E\\nnoncomputable def VIntegral (f : \u2102 \u2192 E) (x y\u2081 y\u2082 : \u211d) : E\\n\\nnoncomputable def RectangleIntegral (f : \u2102 \u2192 E) (z w : \u2102) : E\\n-- = HIntegral f z.re w.re z.im - HIntegral f z.re w.re w.im\\n--   + VIntegral f w.re z.im w.im - VIntegral f z.re z.im w.im\\n\\nnoncomputable abbrev RectangleIntegral' (f : \u2102 \u2192 E) (z w : \u2102) : E\\n-- = (1 / (2 * \u03c0 * I)) \u2022 RectangleIntegral f z w\\n```\\n\\nSo `RectangleIntegral` is the raw boundary integral; `RectangleIntegral'` is normalized by `2\u03c0i`.\\n\\n### Pull-to-pole lemmas\\n\\n```lean\\nlemma RectanglePullToNhdOfPole' [CompleteSpace E] {z\u2080 z\u2081 z\u2082 z\u2083 p : \u2102}\\n    (h_orientation : z\u2080.re \u2264 z\u2083.re \u2227 z\u2080.im \u2264 z\u2083.im \u2227\\n      z\u2081.re \u2264 z\u2082.re \u2227 z\u2081.im \u2264 z\u2082.im)\\n    (hp : Rectangle z\u2081 z\u2082 \u2208 \ud835\udcdd p)\\n    (hz : Rectangle z\u2081 z\u2082 \u2286 Rectangle z\u2080 z\u2083)\\n    (fHolo : HolomorphicOn f (Rectangle z\u2080 z\u2083 \\\\ {p})) :\\n    RectangleIntegral f z\u2080 z\u2083 = RectangleIntegral f z\u2081 z\u2082\\n\\nlemma RectanglePullToNhdOfPole [CompleteSpace E] {z w p : \u2102}\\n    (zRe_lt_wRe : z.re \u2264 w.re) (zIm_lt_wIm : z.im \u2264 w.im)\\n    (hp : Rectangle z w \u2208 \ud835\udcdd p)\\n    (fHolo : HolomorphicOn f (Rectangle z w \\\\ {p})) :\\n    \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\\n      RectangleIntegral f z w =\\n      RectangleIntegral f (-c - I * c + p) (c + I * c + p)\\n\\nlemma RectanglePullToNhdOfPole'' [CompleteSpace E] {z w p : \u2102}\\n    (zRe_le_wRe : z.re \u2264 w.re) (zIm_le_wIm : z.im \u2264 w.im)\\n    (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\\n    (fHolo : HolomorphicOn f (Rectangle z w \\\\ {p})) :\\n    \u2200\u1da0 (c : \u211d) in \ud835\udcdd[>]0,\\n      RectangleIntegral' f z w =\\n      RectangleIntegral' f (-c - I * c + p) (c + I * c + p)\\n```\\n\\n### Simple-pole residue scaffold\\n\\n```lean\\ndef HasSimplePolesOn (f : \u2102 \u2192 \u2102) (s : Set \u2102) : Prop :=\\n  \u2200 z \u2208 s, (-1 : \u2124) \u2264 meromorphicOrderAt f z\\n```\\n\\n```lean\\nnoncomputable def residue (f : \u2102 \u2192 \u2102) (z\u2080 : \u2102) : \u2102 :=\\n  Filter.limUnder (nhdsWithin z\u2080 {z\u2080}\u1d9c) (fun z \u21a6 (z - z\u2080) * f z)\\n\\nnoncomputable def sumResiduesIn (f : \u2102 \u2192 \u2102) (S : Set \u2102) : \u2102 :=\\n  \u2211' z : S, residue f z\\n```\\n\\n```lean\\nlemma residue_eq_of_tendsto {f : \u2102 \u2192 \u2102} {p c : \u2102}\\n    (h : Filter.Tendsto (fun z \u21a6 (z - p) * f z)\\n       (nhdsWithin p {p}\u1d9c) (nhds c)) :\\n    residue f p = c\\n\\nlemma residue_analyticAt_eq_zero {f : \u2102 \u2192 \u2102} {p : \u2102}\\n    (hf : AnalyticAt \u2102 f p) :\\n    residue f p = 0\\n\\nlemma simplePole_sub_residue_isBigO_one {f : \u2102 \u2192 \u2102} {p : \u2102}\\n    (hf : MeromorphicAt f p)\\n    (hord : meromorphicOrderAt f p = (-1 : \u2124)) :\\n    (f - (fun z \u21a6 residue f p / (z - p))) =O[nhdsWithin p {p}\u1d9c] (1 : \u2102 \u2192 \u2102)\\n\\nlemma residue_eq_zero_of_not_pole_of_meromorphicAt {F : \u2102 \u2192 \u2102} {s : \u2102}\\n    (hs_mero : MeromorphicAt F s)\\n    (hs_not_pole : 0 \u2264 meromorphicOrderAt F s) :\\n    residue F s = 0\\n\\nlemma sumResiduesIn_inter_eq_of_set_eq {F : \u2102 \u2192 \u2102} {Rn S2 P : Set \u2102}\\n    (h_set_eq : Rn \u2229 P = S2 \u2229 P)\\n    (h_residue_zero : \u2200 s \u2208 S2, s \u2209 P \u2192 residue F s = 0) :\\n    sumResiduesIn F (Rn \u2229 P) = sumResiduesIn F S2\\n```\\n\\n### Rectangle residue theorem for simple poles\\n\\n```lean\\nlemma RectangleIntegral'_eq_sumResiduesIn {f : \u2102 \u2192 \u2102} {z w : \u2102}\\n    (zRe_le_wRe : z.re \u2264 w.re) (zIm_le_wIm : z.im \u2264 w.im)\\n    (f_mero : MeromorphicOn f (Rectangle z w))\\n    (f_no_poles_boundary : Disjoint (RectangleBorder z w)\\n       {z | meromorphicOrderAt f z < 0})\\n    (f_poles_finite : (Rectangle z w \u2229 {z | meromorphicOrderAt f z < 0}).Finite)\\n    (f_simple_poles : HasSimplePolesOn f (Rectangle z w)) :\\n    RectangleIntegral' f z w =\\n    sumResiduesIn f (Rectangle z w \u2229 {z | meromorphicOrderAt f z < 0})\\n```\\n\\nSo **`HasSimplePolesOn is indeed a hypothesis`**, and the conclusion is a `tsum` over the set of points where `meromorphicOrderAt f z < 0`.\\n\\n### Other simple-pole residue lemmas in the same file\\n\\n```lean\\nlemma ResidueTheoremAtOrigin' {z w c : \u2102}\\n    (h1 : z.re < 0) (h2 : z.im < 0) (h3 : 0 < w.re) (h4 : 0 < w.im) :\\n    RectangleIntegral (fun s => c / s) z w = 2 * I * \u03c0 * c\\n\\ntheorem ResidueTheoremInRectangle\\n    (zRe_le_wRe : z.re \u2264 w.re) (zIm_le_wIm : z.im \u2264 w.im)\\n    (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p) :\\n    RectangleIntegral' (fun s => c / (s - p)) z w = c\\n\\nlemma ResidueTheoremOnRectangleWithSimplePole {f g : \u2102 \u2192 \u2102} {z w p A : \u2102}\\n    (zRe_le_wRe : z.re \u2264 w.re) (zIm_le_wIm : z.im \u2264 w.im)\\n    (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\\n    (gHolo : HolomorphicOn g (Rectangle z w))\\n    (principalPart : Set.EqOn\\n       (f - fun s \u21a6 A / (s - p)) g (Rectangle z w \\\\ {p})) :\\n    RectangleIntegral' f z w = A\\n\\ntheorem ResidueTheoremOnRectangleWithSimplePole' {f : \u2102 \u2192 \u2102} {z w p A : \u2102}\\n    (zRe_le_wRe : z.re \u2264 w.re) (zIm_le_wIm : z.im \u2264 w.im)\\n    (pInRectInterior : Rectangle z w \u2208 \ud835\udcdd p)\\n    (fHolo : HolomorphicOn f (Rectangle z w \\\\ {p}))\\n    (near_p : (f - (fun s \u21a6 A / (s - p))) =O[\ud835\udcdd[\u2260] p] (1 : \u2102 \u2192 \u2102)) :\\n    RectangleIntegral' f z w = A\\n```\\n\\n### Higher-order-pole / normal-form machinery used by the proof\\n\\nThere is **no** general higher-order residue formula or `iteratedDeriv`-based residue computation. The file works around this by assuming `HasSimplePolesOn`. The main normal-form/order lemmas used are:\\n\\n| Name | Kind | Statement summary |\\n|------|------|-------------------|\\n| `meromorphicOrderAt_eq_neg_one_of_simplePole` | private | under `HasSimplePolesOn`, any negative order is exactly `-1` |\\n| `residue_toMeromorphicNFOn_eq_residue` | private | residue is unchanged by `toMeromorphicNFOn` |\\n| `principalPart_meromorphicOn` | private | finite sum `\u2211 c p / (s-p)` is meromorphic |\\n| `sub_principalPart_analyticAt_of_not_mem_poles` | private | subtracting the principal part is analytic away from the chosen poles |\\n| `meromorphicOrderAt_sub_principalPart_nonneg` | private | after subtracting the principal part, order at each pole becomes `\u2265 0` |\\n| `holoPart_holomorphicOn` | private | the \u201cholomorphic part\u201d after subtracting principal part is holomorphic on the rectangle |\\n| `principalPart_borderIntegrable` | private | the principal part is integrable on the rectangle boundary |\\n| `rectangleIntegral'_toMeromorphicNFOn_eq` | private | boundary integral unchanged by `toMeromorphicNFOn` |\\n| `toMeromorphicNFOn_add_integral` | private | splits `RectangleIntegral'` of `fNF` into holomorphic part + principal part |\\n| `rectangleIntegral'_sum_div_sub` | private | integral of a finite sum of simple-pole terms equals sum of coefficients |\\n| `sumResiduesIn_inter_eq_of_set_eq` | public | re-indexes `sumResiduesIn` using a set equality |\\n| `residue_eq_zero_of_not_pole_of_meromorphicAt` | public | residue is `0` at non-negative-order points |\\n\\nNo `iteratedDeriv` residue formula appears in this file.\\n\\n---\\n\\n## 2. `Rectangle.lean` and `ZetaBounds.lean` (main public theorems)\\n\\n### `Zeta23/FromPNTPlus/Rectangle.lean`\\n\\nKey public declarations:\\n\\n- `RectangleBorder z w` \u2014 the boundary of the axis-parallel rectangle with corners `z,w`.\\n- `Square p c` \u2014 axis-parallel square centered at `p` with half-side `c`.\\n- `segment_reProdIm_segment_eq_convexHull` \u2014 rectangle = convex hull of its four corners.\\n- `rectangle_in_convex` \u2014 if the four corners lie in a convex set, the whole rectangle lies in it.\\n- `mem_Rect` \u2014 membership in a rectangle in terms of real/imag inequalities.\\n- `rect_subset_iff` \u2014 `Rectangle z' w' \u2286 Rectangle z w` iff both corners are inside.\\n- `RectSubRect` / `RectSubRect'` \u2014 nested rectangles under coordinate inequalities.\\n- `rectangle_disjoint_singleton` / `rectangleBorder_disjoint_singleton` \u2014 conditions for a point to be disjoint from a rectangle/border.\\n- `rectangle_subset_punctured_rect` / `rectangleBorder_subset_punctured_rect` \u2014 shrink a rectangle around a pole.\\n- `rectangle_mem_nhds_iff` \u2014 a rectangle is a neighborhood of `p` iff `p` is in the open product interval.\\n- `Complex.nhds_hasBasis_square` \u2014 neighborhoods of `p` have a basis of squares `Square p c`.\\n- `square_mem_nhds` / `square_subset_square` / `SmallSquareInRectangle` \u2014 square neighborhoods can be chosen inside a given rectangle.\\n- `not_mem_rectangleBorder_of_rectangle_mem_nhds` \u2014 interior points are not on the border.\\n- Many `mapsTo_rectangle*_NoP` lemmas \u2014 boundary parametrizations map into punctured rectangles.\\n\\n### `Zeta23/FromPNTPlus/ZetaBounds.lean`\\n\\nMain public theorems (one line each):\\n\\n| Theorem | What it proves |\\n|---------|----------------|\\n| `ResidueOfTendsTo` | If `(s-p)f(s) \u2192 A` and `f` is holomorphic on a punctured neighborhood of `p`, then `f - A/(s-p)` is bounded near `p`. |\\n| `analyticAt_riemannZeta` | `\u03b6` is analytic at every `s \u2260 1`. |\\n| `differentiableAt_deriv_riemannZeta` | `\u03b6'` is differentiable at every `s \u2260 1`. |\\n| `riemannZetaResidue` | `\u03b6` has simple-pole residue `1` at `s = 1` (boundedness form). |\\n| `logDerivResidue'` / `logDerivResidue` / `logDerivResidue''` | If `f` has a simple pole with residue `A \u2260 0` and `f` is nonzero on a punctured neighborhood, then `f'/f + 1/(s-p) =O[\ud835\udcdd[\u2260]p] 1`, and a boundedness corollary. |\\n| `ResidueMult` | If `f - A/(s-p) =O 1` near `p` and `g` is holomorphic at `p`, then `f*g - A*g(p)/(s-p) =O 1`. |\\n| `riemannZetaLogDerivResidue` | `-\u03b6'/\u03b6 - 1/(s-1)` is bounded near `s = 1`. |\\n| `riemannZetaLogDerivResidueBigO` | `-\u03b6'/\u03b6 - 1/(s-1) =O[\ud835\udcdd[\u2260]1] 1`. |\\n| `ZetaUpperBnd` | `\u2016\u03b6(\u03c3+it)\u2016 \u2264 C log |t|` for `3 < |t|` and `1 - A/log|t| \u2264 \u03c3 \u2264 2`. |\\n| `ZetaDerivUpperBnd` | `\u2016\u03b6'(\u03c3+it)\u2016 \u2264 C (log |t|)\u00b2` in the same region. |\\n| `ZetaLowerBound3` | Lower bound `c(\u03c3-1)^{3/4}/(log|t|)^{1/4} \u2264 \u2016\u03b6(\u03c3+it)\u2016` for `1 < \u03c3 \u2264 2`. |\\n| `ZetaInvBnd` | `1/\u2016\u03b6(\u03c3+it)\u2016 \u2264 C (log|t|)^7` in a narrow strip around `Re s = 1`. |\\n| `ZetaLowerBnd` | `c/(log|t|)^7 \u2264 \u2016\u03b6(\u03c3+it)\u2016` for `\u03c3 < 1` in a narrow strip. |\\n| `ZetaZeroFree` | `\u03b6(\u03c3+it) \u2260 0` for `3 < |t|` and `\u03c3` in a narrow strip to the left of `1`. |\\n| `LogDerivZetaBnd` | `\u2016\u03b6'/\u03b6(\u03c3+it)\u2016 \u2264 C (log|t|)^9` in a narrow strip around `1`. |\\n| `LogDerivZetaBndUnif` | Uniform version of the above for `\u03c3 \u2265 1 - A/(log|t|)^9`. |\\n| `ZetaNoZerosOn1Line` | `\u03b6(1+it) \u2260 0` for all real `t`. |\\n| `ZetaNoZerosInBox` | For every `T`, there is `\u03c3\u2082 < 1` such that `\u03b6` has no zeros in `[[\u03c3\u2082,2]] \u00d7\u2102 [[-T,T]]`. |\\n| `LogDerivZetaHoloOn` | `\u03b6'/\u03b6` is holomorphic on any set not containing `1` and where `\u03b6` has no zeros. |\\n| `LogDerivZetaHolcSmallT` | `\u03b6'/\u03b6` is holomorphic on a small box around `1` minus `{1}`. |\\n| `LogDerivZetaHolcLargeT` | `\u03b6'/\u03b6` is holomorphic on a large box `[[1-A/log^9 T,2]] \u00d7\u2102 [[-T,T]] \\\\ {1}`. |\\n| `dlog_riemannZeta_bdd_on_vertical_lines_generalized` | `\u2016-\u03b6'(\u03c3\u2081+it)/\u03b6(\u03c3\u2081+it)\u2016 \u2264 \u2016\u03b6'(\u03c3\u2080)/\u03b6(\u03c3\u2080)\u2016` for `1 < \u03c3\u2080 \u2264 \u03c3\u2081`. |\\n| `triv_bound_zeta` | `\u2016-\u03b6'(\u03c3\u2080+it)/\u03b6(\u03c3\u2080+it)\u2016 \u2264 (\u03c3\u2080-1)\u207b\u00b9 + C` for `\u03c3\u2080 > 1`. |\\n\\n---\\n\\n## 3. Vendored Mathlib inventory\\n\\n### Complex / contour `residue` API\\n\\n**There is no native complex-contour `residue` API in this vendored Mathlib.** Searching `Mathlib/Analysis/Complex` and `Mathlib/Geometry` for \u201cresidue\u201d yields only:\\n\\n- `Mathlib/Geometry/RingedSpace/LocallyRingedSpace/ResidueField.lean` \u2014 algebraic-geometry residue fields (`X.residueField x`, `X.residue x`), unrelated to contour integration.\\n\\nNo file `Mathlib/Analysis/Complex/Residue.lean` or similar exists.\\n\\n### Meromorphic order / normal-form API\\n\\nThe core API is in:\\n\\n- `Mathlib/Analysis/Meromorphic/Basic.lean` \u2014 `MeromorphicAt`.\\n- `Mathlib/Analysis/Meromorphic/Order.lean` \u2014 `meromorphicOrderAt`.\\n- `Mathlib/Analysis/Meromorphic/NormalForm.lean` \u2014 `MeromorphicNFAt`, `toMeromorphicNFAt`, `toMeromorphicNFOn`.\\n- `Mathlib/Analysis/Meromorphic/FactorizedRational.lean` \u2014 factorized rational functions, zeros/poles extraction.\\n- `Mathlib/Analysis/Meromorphic/TrailingCoefficient.lean` \u2014 `meromorphicTrailingCoeffAt`.\\n\\nKey definitions and lemmas:\\n\\n```lean\\n-- Basic.lean\\ndef MeromorphicAt (f : \ud835\udd5c \u2192 E) (x : \ud835\udd5c) :=\\n  \u2203 (n : \u2115), AnalyticAt \ud835\udd5c (fun z \u21a6 (z - x) ^ n \u2022 f z) x\\n```\\n\\n```lean\\n-- Order.lean\\nnoncomputable def meromorphicOrderAt (f : \ud835\udd5c \u2192 E) (x : \ud835\udd5c) : WithTop \u2124\\n\\nlemma meromorphicOrderAt_eq_top_iff :\\n    meromorphicOrderAt f x = \u22a4 \u2194 \u2200\u1da0 z in \ud835\udcdd[\u2260] x, f z = 0\\n\\nlemma meromorphicOrderAt_eq_int_iff {n : \u2124} (hf : MeromorphicAt f x) :\\n    meromorphicOrderAt f x = n \u2194\\n    \u2203 g : \ud835\udd5c \u2192 E, AnalyticAt \ud835\udd5c g x \u2227 g x \u2260 0 \u2227\\n      \u2200\u1da0 z in \ud835\udcdd[\u2260] x, f z = (z - x) ^ n \u2022 g z\\n\\nlemma meromorphicOrderAt_ne_top_iff {f : \ud835\udd5c \u2192 E} {z\u2080 : \ud835\udd5c} (hf : MeromorphicAt f z\u2080) :\\n    meromorphicOrderAt f z\u2080 \u2260 \u22a4 \u2194\\n    \u2203 (g : \ud835\udd5c \u2192 E), AnalyticAt \ud835\udd5c g z\u2080 \u2227 g z\u2080 \u2260 0 \u2227\\n      f =\u1da0[\ud835\udcdd[\u2260] z\u2080] fun z \u21a6 (z - z\u2080) ^ ((meromorphicOrderAt f z\u2080).untop\u2080) \u2022 g z\\n```\\n\\nOrder under arithmetic:\\n\\n```lean\\ntheorem meromorphicOrderAt_mul {f g : \ud835\udd5c \u2192 \ud835\udd5c'} (hf : MeromorphicAt f x)\\n    (hg : MeromorphicAt g x) :\\n    meromorphicOrderAt (f * g) x =\\n    meromorphicOrderAt f x + meromorphicOrderAt g x\\n\\ntheorem meromorphicOrderAt_prod {x : \ud835\udd5c} {\u03b9 : Type*} {s : Finset \u03b9} {f : \u03b9 \u2192 \ud835\udd5c \u2192 \ud835\udd5c'}\\n    (hf : \u2200 i \u2208 s, MeromorphicAt (f i) x) :\\n    meromorphicOrderAt (\u220f i \u2208 s, f i) x =\\n    \u2211 i \u2208 s, meromorphicOrderAt (f i) x\\n\\ntheorem meromorphicOrderAt_add (hf\u2081 : MeromorphicAt f\u2081 x) (hf\u2082 : MeromorphicAt f\u2082 x) :\\n    min (meromorphicOrderAt f\u2081 x) (meromorphicOrderAt f\u2082 x) \u2264\\n    meromorphicOrderAt (f\u2081 + f\u2082) x\\n\\ntheorem meromorphicOrderAt_add_of_ne\\n    (hf\u2081 : MeromorphicAt f\u2081 x) (hf\u2082 : MeromorphicAt f\u2082 x)\\n    (h : meromorphicOrderAt f\u2081 x \u2260 meromorphicOrderAt f\u2082 x) :\\n    meromorphicOrderAt (f\u2081 + f\u2082) x =\\n    min (meromorphicOrderAt f\u2081 x) (meromorphicOrderAt f\u2082 x)\\n```\\n\\nOrder and derivatives:\\n\\n```lean\\nlemma meromorphicOrderAt_deriv_eq_sub_one [CompleteSpace E] {f : \ud835\udd5c \u2192 E} {x : \ud835\udd5c} {n : \u2124}\\n    (hn : (n : \ud835\udd5c) \u2260 0) (hf : meromorphicOrderAt f x = \u2191n) :\\n    meromorphicOrderAt (deriv f) x = \u2191(n - 1)\\n\\nlemma meromorphicOrderAt_deriv [CompleteSpace E] {f : \ud835\udd5c \u2192 E} {x : \ud835\udd5c} {n : \u2124}\\n    (hn : (\u2191(n + 1) : \ud835\udd5c) \u2260 0) (hf : meromorphicOrderAt f x = \u2191(n + 1)) :\\n    meromorphicOrderAt (deriv f) x = \u2191n\\n```\\n\\nNormal form:\\n\\n```lean\\n-- NormalForm.lean\\ndef MeromorphicNFAt :=\\n  f =\u1da0[\ud835\udcdd x] 0 \u2228\\n    \u2203 (n : \u2124) (g : \ud835\udd5c \u2192 E), AnalyticAt \ud835\udd5c g x \u2227 g x \u2260 0 \u2227\\n      f =\u1da0[\ud835\udcdd x] (\u00b7 - x) ^ n \u2022 g\\n\\nnoncomputable def toMeromorphicNFAt (f : \ud835\udd5c \u2192 E) (x : \ud835\udd5c) : \ud835\udd5c \u2192 E\\nnoncomputable def toMeromorphicNFOn (f : \ud835\udd5c \u2192 E) (U : Set \ud835\udd5c) : \ud835\udd5c \u2192 E\\n\\ntheorem meromorphicNFAt_toMeromorphicNFAt :\\n    MeromorphicNFAt (toMeromorphicNFAt f x) x\\n\\ntheorem meromorphicNFOn_toMeromorphicNFOn :\\n    MeromorphicNFOn (toMeromorphicNFOn f U) U\\n\\ntheorem toMeromorphicNFOn_eqOn_codiscrete (hf : MeromorphicOn f U) :\\n    f =\u1da0[Filter.codiscreteWithin U] toMeromorphicNFOn f U\\n\\ntheorem MeromorphicOn.toMeromorphicNFOn_eq_self_on_nhdsNE\\n    (hf : MeromorphicOn f U) (hx : x \u2208 U) :\\n    toMeromorphicNFOn f U =\u1da0[\ud835\udcdd[\u2260] x] f\\n\\ntheorem meromorphicOrderAt_toMeromorphicNFOn (hf : MeromorphicOn f U) (hx : x \u2208 U) :\\n    meromorphicOrderAt (toMeromorphicNFOn f U) x = meromorphicOrderAt f x\\n\\ntheorem toMeromorphicNFOn_eq_toMeromorphicNFAt_on_nhds\\n    (hf : MeromorphicOn f U) (hx : x \u2208 U) :\\n    toMeromorphicNFOn f U =\u1da0[\ud835\udcdd x] toMeromorphicNFAt f x\\n```\\n\\nPrincipal-part-like factorization:\\n\\n```lean\\n-- FactorizedRational.lean\\ntheorem MeromorphicOn.extract_zeros_poles {f : \ud835\udd5c \u2192 E} (h\u2081f : MeromorphicOn f U)\\n    (h\u2082f : \u2200 u : U, meromorphicOrderAt f u \u2260 \u22a4)\\n    (h\u2083f : (divisor f U).support.Finite) :\\n    \u2203 g : \ud835\udd5c \u2192 E, AnalyticOnNhd \ud835\udd5c g U \u2227 (\u2200 u : U, g u \u2260 0) \u2227\\n      f =\u1da0[codiscreteWithin U] (\u220f\u1da0 u, (\u00b7 - u) ^ divisor f U u) \u2022 g\\n\\ntheorem MeromorphicOn.extract_zeros_poles_log {f g : \ud835\udd5c \u2192 E} ...\\n    (h : f =\u1da0[codiscreteWithin U] (\u220f\u1da0 u, (\u00b7 - u) ^ D u) \u2022 g) :\\n    (log \u2016f \u00b7\u2016) =\u1da0[codiscreteWithin U]\\n      \u2211\u1da0 u, (D u * log \u2016\u00b7 - u\u2016) + (log \u2016g \u00b7\u2016)\\n```\\n\\nThere is **no** dedicated Laurent-series expansion or principal-part coefficient API beyond this normal-form / factorized-rational machinery.\\n\\n### Digamma\\n\\nFile: `Mathlib/Analysis/SpecialFunctions/Gamma/Digamma.lean`.\\n\\n```lean\\nnoncomputable def Complex.digamma : \u2102 \u2192 \u2102 := logDeriv Gamma\\n\\ntheorem digamma_def : digamma = logDeriv Gamma\\n\\n@[simp] theorem digamma_zero : digamma 0 = 0\\n\\ntheorem digamma_one : digamma 1 = - Real.eulerMascheroniConstant\\n\\ntheorem digamma_one_half : digamma (1 / 2) = - 2 * log 2 - Real.eulerMascheroniConstant\\n\\ntheorem digamma_apply_add_one (s : \u2102) (hs : \u2200 m : \u2115, s \u2260 - m) :\\n    digamma (s + 1) = digamma s + s\u207b\u00b9\\n\\ntheorem meromorphic_digamma : Meromorphic digamma\\n```\\n\\n### Stieltjes / Laurent related to `riemannZeta`\\n\\n- **Stieltjes**: `Mathlib` has a general theory of Stieltjes functions/measures (`MeasureTheory/Measure/Stieltjes.lean`, `Probability/Kernel/Disintegration/MeasurableStieltjes.lean`, etc.), but **nothing named \u201cStieltjes\u201d is tied to `riemannZeta`** in this vendored version.\\n- **Laurent**: generic Laurent polynomial/series files exist (`Algebra/Polynomial/Laurent.lean`, `Algebra/Vertex/...`), and `Analysis/Complex/ValueDistribution/Cartan.lean` mentions a \u201cJensen-type identity.\u201d There is **no Laurent expansion of `\u03b6` at `s = 1`**. The closest result is the asymptotic expansion in `NumberTheory/Harmonic/ZetaAsymp.lean`:\\n\\n```lean\\ntheorem tendsto_riemannZeta_sub_one_div :\\n    Tendsto (fun s : \u2102 \u21a6 riemannZeta s - 1 / (s - 1))\\n      (\ud835\udcdd[\u2260] 1) (\ud835\udcdd Real.eulerMascheroniConstant)\\n```\\n\\ni.e. the principal part is `1/(s-1)` and the constant term is `\u03b3`.\\n\\n### `riemannZeta` API\\n\\nDefinitions and main results are in `Mathlib/NumberTheory/LSeries/RiemannZeta.lean`.\\n\\n```lean\\ndef completedRiemannZeta\u2080 (s : \u2102) : \u2102\\ndef completedRiemannZeta  (s : \u2102) : \u2102\\ndef riemannZeta            : \u2102 \u2192 \u2102 := hurwitzZetaEven 0\\n```\\n\\nKey theorems:\\n\\n```lean\\ntheorem differentiable_completedZeta\u2080 : Differentiable \u2102 completedRiemannZeta\u2080\\n\\ntheorem differentiableAt_completedZeta {s : \u2102} (hs : s \u2260 0) (hs' : s \u2260 1) :\\n    DifferentiableAt \u2102 completedRiemannZeta s\\n\\ntheorem differentiableAt_riemannZeta {s : \u2102} (hs' : s \u2260 1) :\\n    DifferentiableAt \u2102 riemannZeta s\\n\\ntheorem completedRiemannZeta\u2080_one_sub (s : \u2102) :\\n    completedRiemannZeta\u2080 (1 - s) = completedRiemannZeta\u2080 s\\n\\ntheorem completedRiemannZeta_one_sub (s : \u2102) :\\n    completedRiemannZeta (1 - s) = completedRiemannZeta s\\n\\ntheorem riemannZeta_one_sub {s : \u2102} (hs : \u2200 n : \u2115, s \u2260 -n) (hs' : s \u2260 1) :\\n    riemannZeta (1 - s) =\\n    2 * (2 * \u03c0) ^ (-s) * Gamma s * cos (\u03c0 * s / 2) * riemannZeta s\\n\\ntheorem riemannZeta_neg_two_mul_nat_add_one (n : \u2115) :\\n    riemannZeta (-2 * (n + 1)) = 0\\n\\ntheorem riemannZeta_residue_one :\\n    Tendsto (fun s \u21a6 (s - 1) * riemannZeta s) (\ud835\udcdd[\u2260] 1) (\ud835\udcdd 1)\\n\\ntheorem zeta_eq_tsum_one_div_nat_cpow {s : \u2102} (hs : 1 < re s) :\\n    riemannZeta s = \u2211' n : \u2115, 1 / (n : \u2102) ^ s\\n```\\n\\nSpecial values and pole data are further in:\\n\\n- `Mathlib/NumberTheory/Harmonic/ZetaAsymp.lean`:\\n  - `tendsto_riemannZeta_sub_one_div` (principal part + \u03b3),\\n  - `riemannZeta_one` (`\u03b6 1 = (\u03b3 - log(4\u03c0))/2`),\\n  - `riemannZeta_one_ne_zero`,\\n  - `deriv_riemannZeta_zero`.\\n\\n- `Mathlib/NumberTheory/LSeries/HurwitzZetaValues.lean`:\\n  - `riemannZeta_two_mul_nat`,\\n  - `riemannZeta_neg_nat_eq_bernoulli`,\\n  - `riemannZeta_two`, `riemannZeta_four`.\\n\\n- `Mathlib/NumberTheory/LSeries/Nonvanishing.lean`:\\n  - `riemannZeta_ne_zero_of_one_le_re`,\\n  - `riemannZeta_ne_zero_of_one_lt_re`.\\n\\n- `Mathlib/NumberTheory/LSeries/ZetaZeros.lean`:\\n  - `riemannZetaZeros : Set \u2102`,\\n  - `isClosed_riemannZetaZeros`,\\n  - `isDiscrete_riemannZetaZeros`,\\n  - `IsCompact.inter_riemannZetaZeros_finite`.\\n\\n### Jensen / Borel\u2013Carath\u00e9odory / Hadamard / Weierstrass\\n\\n- **Borel\u2013Carath\u00e9odory**: `Mathlib/Analysis/Complex/BorelCaratheodory.lean`\\n  - `Complex.borelCaratheodory_zero`\\n  - `Complex.borelCaratheodory`\\n\\n- **Jensen\u2019s formula**: `Mathlib/Analysis/Complex/JensenFormula.lean`\\n  - `MeromorphicOn.circleAverage_log_norm`\\n  - `AnalyticOnNhd.circleAverage_log_norm`\\n  - `AnalyticOnNhd.sum_divisor_le`\\n\\n- **Hadamard three-lines theorem**: `Mathlib/Analysis/Complex/Hadamard.lean`\\n  - Several variants of `HadamardThreeLines` interpolation (not Hadamard factorization).\\n\\n- **Weierstrass theorem on uniform limits of holomorphic functions**: `Mathlib/Analysis/Complex/LocallyUniformLimit.lean`\\n  - `TendstoLocallyUniformlyOn.differentiableOn`\\n  - `TendstoLocallyUniformlyOn.deriv`\\n  - `differentiableOn_tsum_of_summable_norm`\\n  - `hasSum_deriv_of_summable_norm`\\n\\nThere is **no** Weierstrass factorization theorem or Hadamard factorization theorem in this vendored Mathlib. The project\u2019s `StrongPNTPrefix.lean` builds a finite Blaschke-factor argument by hand.\\n\\n---\\n\\n## 4. `MediumPNT.lean` and `StrongPNTPrefix.lean` exports\\n\\n### `Zeta23/FromPNTPlus/MediumPNT.lean`\\n\\nThis file implements the **smoothed explicit formula** step of the medium PNT proof. It defines the smoothed Chebyshev integral `SmoothedChebyshev` as a vertical Mellin inversion of `-\u03b6'/\u03b6` times a smooth cutoff, proves the Dirichlet-series identity `SmoothedChebyshevDirichlet` (the smoothed integral equals `\u2211 \u039b(n) \u00b7 Smooth1(n/X)`), and gives the approximation estimate `SmoothedChebyshevClose` showing that this smoothed sum is within `O(\u03b5 X log X)` of the Chebyshev psi function `\u03c8 X`.\\n\\n### `Zeta23/FromPNTPlus/StrongPNTPrefix.lean`\\n\\nThis is a sorry-free prefix of a Strong Prime Number Theorem development. It contains a self-contained complex-analysis toolkit: a local Borel\u2013Carath\u00e9odory derivative bound (`BorelCaratheodoryDeriv`), construction of a logarithm `J_B` of a nonvanishing analytic function (`LogOfAnalyticFunction`), a finite zero-set / Blaschke-factor setup (`ZeroFactor`, `Cf`, `BlaschkeB`), and culminates in `ZerosBound` \u2014 a bound on the weighted number of zeros of an analytic function in a smaller disk in terms of its sup norm on a larger disk. The remainder of the original `StrongPNT.lean` (after `ZerosBound`) is not ported.\""
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  "prompt": "Research task (read-only inventory, NO code changes). Context: Lean 4 project at D:\\Python\\Quantum (toolchain v4.33.0-rc2). We are planning to formalize an explicit contour bound for the Li coefficients of \u03b6. I need a precise inventory of zero-counting infrastructure and the current Li-coefficient definitions.\n\nSurvey and report with exact declaration names + file paths + signatures:\n\n1. Zeta23 zero infrastructure, in D:\\Python\\Quantum\\zeta-23-lean\\Zeta23\\:\n   - Look in Defs/ and Defs.lean: how are nontrivial zeros of riemannZeta defined? Is there a zero-counting function (N(T) or similar)? Any API for: zeros are symmetric (\u03c1\u0304, 1\u2212\u03c1), zeros in critical strip, zero-free at Re \u2265 1?\n   - Search all of Zeta23/ for theorems about riemannZeta zeros (grep \"riemannZeta.*= 0\" or \"zeros\" in theorem/def names) \u2014 report the useful ones with signatures.\n   - Anything formalizing an explicit formula or zero-sum over \u03c1 (WeilEF/, ExplicitFormula/) \u2014 brief summary of what exists (we may reuse zero-enumeration techniques).\n2. In our package D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\:\n   - Find the definition of `liCoeff` (probably in LiSkeleton.lean or a Defs file): full definition, what zero set it sums over, how the tsum is set up, any summability lemmas already proved. Also `SInfty`, `sInftyLower`, `C0hat`, `Hhat`, `IIMaj`, `FAlt`, `Bn`, `marginTilde` \u2014 report each def's exact Lean code (they're short) and the file/line.\n   - Report the exact statements of `sInfty_lower_bound`, `marginTilde_convex`, `pos_of_convex_of_two_points` (signatures only).\n3. In the vendored Mathlib (D:\\Python\\Quantum\\zeta-23-lean\\.lake\\packages\\mathlib\\Mathlib):\n   - Does Mathlib have von Mangoldt \u039b (ArithmeticFunction vonMangoldt)? The Chebyshev \u03c8? Brief note.\n   - Search for theorems about `riemannZeta` being nonzero (e.g., Re s \u2265 1) \u2014 report names.\n   - Any API for enumerating roots/zeros of analytic functions with counting (e.g., \"AnalyticOn... zeros finite on compacts\" \u2014 the identity theorem / discreteness of zeros)? Report names/files.\n\nReport as a structured list. Be thorough; exact names matter. Read-only."
}

{
  "status": "completed",
  "prompt": "Research task (read-only inventory, NO code changes). Context: Lean 4 project at D:\\Python\\Quantum (toolchain v4.33.0-rc2). We are planning to formalize an explicit contour bound for the Li coefficients of \u03b6. I need a precise inventory of zero-counting infrastructure and the current Li-coefficient definitions.\n\nSurvey and report with exact declaration names + file paths + signatures:\n\n1. Zeta23 zero infrastructure, in D:\\Python\\Quantum\\zeta-23-lean\\Zeta23\\:\n   - Look in Defs/ and Defs.lean: how are nontrivial zeros of riemannZeta defined? Is there a zero-counting function (N(T) or similar)? Any API for: zeros are symmetric (\u03c1\u0304, 1\u2212\u03c1), zeros in critical strip, zero-free at Re \u2265 1?\n   - Search all of Zeta23/ for theorems about riemannZeta zeros (grep \"riemannZeta.*= 0\" or \"zeros\" in theorem/def names) \u2014 report the useful ones with signatures.\n   - Anything formalizing an explicit formula or zero-sum over \u03c1 (WeilEF/, ExplicitFormula/) \u2014 brief summary of what exists (we may reuse zero-enumeration techniques).\n2. In our package D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\:\n   - Find the definition of `liCoeff` (probably in LiSkeleton.lean or a Defs file): full definition, what zero set it sums over, how the tsum is set up, any summability lemmas already proved. Also `SInfty`, `sInftyLower`, `C0hat`, `Hhat`, `IIMaj`, `FAlt`, `Bn`, `marginTilde` \u2014 report each def's exact Lean code (they're short) and the file/line.\n   - Report the exact statements of `sInfty_lower_bound`, `marginTilde_convex`, `pos_of_convex_of_two_points` (signatures only).\n3. In the vendored Mathlib (D:\\Python\\Quantum\\zeta-23-lean\\.lake\\packages\\mathlib\\Mathlib):\n   - Does Mathlib have von Mangoldt \u039b (ArithmeticFunction vonMangoldt)? The Chebyshev \u03c8? Brief note.\n   - Search for theorems about `riemannZeta` being nonzero (e.g., Re s \u2265 1) \u2014 report names.\n   - Any API for enumerating roots/zeros of analytic functions with counting (e.g., \"AnalyticOn... zeros finite on compacts\" \u2014 the identity theorem / discreteness of zeros)? Report names/files.\n\nReport as a structured list. Be thorough; exact names matter. Read-only.",
  "agentId": "a6939341f0a284148",
  "agentType": "Explore",
  "content": [
    "type='text' text=\"Below is a read-only inventory of the requested zero-counting and Li-coefficient infrastructure, with exact declaration names, file paths, and signatures.\\n\\n---\\n\\n## 1. Zeta23 zero infrastructure (`D:/Python/Quantum/zeta-23-lean/Zeta23/`)\\n\\n### 1.1 Definitions of nontrivial zeros and counting functions\\n\\nAll of these are in `Zeta23/Statement.lean`.\\n\\n- **`IsNontrivialZero`** (line 38)\\n  ```lean\\n  def IsNontrivialZero (\u03c1 : \u2102) : Prop := riemannZeta \u03c1 = 0 \u2227 0 < \u03c1.re \u2227 \u03c1.re < 1\\n  ```\\n- **`zeroMult`** (line 43) \u2014 multiplicity as the analytic order:\\n  ```lean\\n  def zeroMult (\u03c1 : \u2102) : \u2115 := (analyticOrderAt riemannZeta \u03c1).toNat\\n  ```\\n- **`zerosIn`** (line 46) \u2014 zeros with ordinate in a window:\\n  ```lean\\n  def zerosIn (T\u2081 T\u2082 : \u211d) : Set \u2102 :=\\n    {\u03c1 | IsNontrivialZero \u03c1 \u2227 T\u2081 < \u03c1.im \u2227 \u03c1.im \u2264 T\u2082}\\n  ```\\n- **`Ncount`** (line 49) \u2014 counted with multiplicity:\\n  ```lean\\n  def Ncount (T\u2081 T\u2082 : \u211d) : \u2115 := \u2211\u1da0 \u03c1 \u2208 zerosIn T\u2081 T\u2082, zeroMult \u03c1\\n  ```\\n- **`Ndist`** (line 52) \u2014 distinct zeros:\\n  ```lean\\n  def Ndist (T\u2081 T\u2082 : \u211d) : \u2115 := (zerosIn T\u2081 T\u2082).ncard\\n  ```\\n- **`N0`** (line 56) \u2014 critical-line zeros with multiplicity:\\n  ```lean\\n  def N0 (T\u2081 T\u2082 : \u211d) : \u2115 :=\\n    \u2211\u1da0 \u03c1 \u2208 zerosIn T\u2081 T\u2082 \u2229 {\u03c1 | \u03c1.re = 1 / 2}, zeroMult \u03c1\\n  ```\\n- **`N0star`** (line 59) \u2014 critical-line zeros without multiplicity:\\n  ```lean\\n  def N0star (T\u2081 T\u2082 : \u211d) : \u2115 :=\\n    (zerosIn T\u2081 T\u2082 \u2229 {\u03c1 | \u03c1.re = 1 / 2}).ncard\\n  ```\\n- **`N0simple`** (line 62) \u2014 simple critical-line zeros:\\n  ```lean\\n  def N0simple (T\u2081 T\u2082 : \u211d) : \u2115 :=\\n    (zerosIn T\u2081 T\u2082 \u2229 {\u03c1 | \u03c1.re = 1 / 2} \u2229 {\u03c1 | zeroMult \u03c1 = 1}).ncard\\n  ```\\n- **`Nsimple`** (line 65) \u2014 simple zeros anywhere:\\n  ```lean\\n  def Nsimple (T\u2081 T\u2082 : \u211d) : \u2115 :=\\n    (zerosIn T\u2081 T\u2082 \u2229 {\u03c1 | zeroMult \u03c1 = 1}).ncard\\n  ```\\n\\nThere is **no single-argument `N(T)`**; the formalization uses the dyadic/window counts above and an abstract version in `ZeroConfig`.\\n\\n### 1.2 Abstract zero configuration and counting (`Zeta23/Defs.lean`)\\n\\n- **`ZeroConfig`** (lines 120\u2013130) \u2014 an abstract locally finite multiset of strip zeros, closed under `reflect`:\\n  ```lean\\n  structure ZeroConfig where\\n    carrier : Set \u2102\\n    mult : \u2102 \u2192 \u2115\\n    one_le_mult : \u2200 \u03c1 \u2208 carrier, 1 \u2264 mult \u03c1\\n    strip : \u2200 \u03c1 \u2208 carrier, 0 \u2264 \u03c1.re \u2227 \u03c1.re \u2264 1\\n    reflect_mem : \u2200 \u03c1 \u2208 carrier, reflect \u03c1 \u2208 carrier\\n    mult_reflect : \u2200 \u03c1 \u2208 carrier, mult (reflect \u03c1) = mult \u03c1\\n    finite_window : \u2200 T\u2081 T\u2082 : \u211d, (carrier \u2229 {\u03c1 | T\u2081 < \u03c1.im \u2227 \u03c1.im \u2264 T\u2082}).Finite\\n  ```\\n- **`reflect`** (line 108):\\n  ```lean\\n  def reflect (\u03c1 : \u2102) : \u2102 := 1 - (starRingEnd \u2102) \u03c1\\n  ```\\n- Counting functions on `ZeroConfig` (lines 145\u2013161): `ZeroConfig.N`, `ZeroConfig.Nd`, `ZeroConfig.N0`, `ZeroConfig.N0star`, `ZeroConfig.N0s`, `ZeroConfig.Ns`.\\n- The zero-side matrix / Weil sum (lines 164\u2013170):\\n  ```lean\\n  def Wsummand (f g : \u211d \u2192 \u2102) (\u03c1 : \u2102) : \u2102 := ...\\n  def W (f g : \u211d \u2192 \u2102) : \u2102 := \u2211' \u03c1 : Z.carrier, Z.Wsummand f g \u03c1\\n  ```\\n\\n### 1.3 \u03b6 zeros packaged as a `ZeroConfig`\\n\\nIn `Zeta23/Statement/SeamClosed.lean`:\\n\\n- **`zetaSeam`** (line 22): discharges all four seam obligations.\\n- **`zetaZeroConfig`** (line 26):\\n  ```lean\\n  def zetaZeroConfig : ZeroConfig := zetaZeros zetaSeam\\n  ```\\n  so `carrier = {\u03c1 | IsNontrivialZero \u03c1}` and `mult = zeroMult`.\\n- Simp lemmas at lines 28\u201338 relate `zetaZeroConfig.N` to `Ncount`, etc.\\n- **`zerosIn_finite`** (line 47): every ordinate window is finite.\\n\\n### 1.4 Symmetry (\u03c1 \u21a6 1 \u2212 \u03c1\u0304)\\n\\nIn `Zeta23/ZetaReflect.lean`:\\n\\n- **`zeta_reflect_zero`** (line 278)\\n  ```lean\\n  theorem zeta_reflect_zero : \u2200 \u03c1, IsNontrivialZero \u03c1 \u2192 IsNontrivialZero (reflect \u03c1)\\n  ```\\n- **`zeta_mult_reflect`** (line 314)\\n  ```lean\\n  theorem zeta_mult_reflect : \u2200 \u03c1, IsNontrivialZero \u03c1 \u2192 zeroMult (reflect \u03c1) = zeroMult \u03c1\\n  ```\\n\\nUseful reflection geometry lemmas appear in `Zeta23/ZeroSide.lean` (lines 593\u2013611):\\n```lean\\nlemma reflect_re (\u03c1 : \u2102) : (reflect \u03c1).re = 1 - \u03c1.re\\nlemma reflect_im (\u03c1 : \u2102) : (reflect \u03c1).im = \u03c1.im\\nlemma reflect_eq_self_iff (\u03c1 : \u2102) : reflect \u03c1 = \u03c1 \u2194 \u03c1.re = 1 / 2\\n```\\n\\n### 1.5 Finiteness / discreteness / zero-free facts\\n\\nIn `Zeta23/Statement/Seam.lean`:\\n\\n- **`riemannZeta_analyticOnNhd_compl_one`** (line 30)\\n  ```lean\\n  lemma riemannZeta_analyticOnNhd_compl_one :\\n      AnalyticOnNhd \u2102 riemannZeta ({1}\u1d9c : Set \u2102)\\n  ```\\n- **`analyticOrderAt_riemannZeta_ne_top`** (line 40)\\n  ```lean\\n  lemma analyticOrderAt_riemannZeta_ne_top {s : \u2102} (hs : s \u2260 1) :\\n      analyticOrderAt riemannZeta s \u2260 \u22a4\\n  ```\\n- **`ZetaSeam.one_le_mult_holds`** (line 50)\\n  ```lean\\n  theorem ZetaSeam.one_le_mult_holds :\\n      \u2200 \u03c1, IsNontrivialZero \u03c1 \u2192 1 \u2264 zeroMult \u03c1\\n  ```\\n- **`riemannZeta_zeros_locallyFinite`** (line 63)\\n  ```lean\\n  lemma riemannZeta_zeros_locallyFinite (z : \u2102) :\\n      \u2203 t \u2208 \ud835\udcdd z, (t \u2229 {\u03c1 : \u2102 | \u03c1 \u2260 1 \u2227 riemannZeta \u03c1 = 0}).Finite\\n  ```\\n- **`ZetaSeam.finite_window_holds`** (line 91)\\n  ```lean\\n  theorem ZetaSeam.finite_window_holds (T\u2081 T\u2082 : \u211d) :\\n      ({\u03c1 | IsNontrivialZero \u03c1} \u2229 {\u03c1 | T\u2081 < \u03c1.im \u2227 \u03c1.im \u2264 T\u2082}).Finite\\n  ```\\n\\nZero-free on `Re s \u2265 1` is imported from Mathlib (see Part 3) and used heavily; in Zeta23 it is invoked as `riemannZeta_ne_zero_of_one_le_re`.\\n\\n### 1.6 Riemann\u2013von Mangoldt / zero counting\\n\\nIn `Zeta23/Hypotheses.lean` (lines 67\u201370) the abstract hypothesis is:\\n```lean\\nstructure RiemannVonMangoldt (Z : ZeroConfig) : Prop where\\n  main : \u2203 C T\u2080 : \u211d, \u2200 T : \u211d, T\u2080 \u2264 T \u2192\\n      |(Z.N T (2 * T) : \u211d) - T / (2 * Real.pi) * ell1 T| \u2264 C * Real.log T\\n  local_count : \u2203 A\u2080 : \u211d, 1 \u2264 A\u2080 \u2227 \u2200 t : \u211d,\\n      (Z.N t (t + 1) : \u211d) \u2264 A\u2080 * Real.log (|t| + 3)\\n```\\n\\nFor \u03b6 itself, in `Zeta23/RvM/LocalCount.lean`:\\n\\n- **`zeta_local_zero_count`** (line 266)\\n  ```lean\\n  theorem zeta_local_zero_count : \u2203 A\u2080 : \u211d, 1 \u2264 A\u2080 \u2227 \u2200 t : \u211d,\\n      (Ncount t (t + 1) : \u211d) \u2264 A\u2080 * Real.log (|t| + 3)\\n  ```\\n- **`zetaZeroConfig_local_count`** (line 294)\\n  ```lean\\n  theorem zetaZeroConfig_local_count : \u2203 A\u2080 : \u211d, 1 \u2264 A\u2080 \u2227 \u2200 t : \u211d,\\n      (zetaZeroConfig.N t (t + 1) : \u211d) \u2264 A\u2080 * Real.log (|t| + 3)\\n  ```\\n\\nAnd the full Riemann\u2013von Mangoldt theorem is in `Zeta23/RvM/Statement.lean` (line 20):\\n```lean\\ntheorem riemannVonMangoldt (h\u0393 : GammaFacts) : RiemannVonMangoldt zetaZeroConfig\\n```\\n\\n### 1.7 Explicit formula / zero-sum infrastructure\\n\\nIn `Zeta23/ExplicitFormula.lean`:\\n\\n- **`EF.literatureRHS`** (line 70) \u2014 the right-hand side of the literature explicit formula.\\n- **`EF.EF_lit`** (line 81) \u2014 literature-form hypothesis:\\n  ```lean\\n  def EF_lit (Z : ZeroConfig) : Prop :=\\n    \u2200 k : \u211d \u2192 \u2102, ContDiff \u211d 2 k \u2192 HasCompactSupport k \u2192\\n      Summable (fun \u03c1 : Z.carrier => (Z.mult \u03c1 : \u2102) * paperFT k (gammaOf \u03c1)) \u2227\\n      \u2211' \u03c1 : Z.carrier, (Z.mult \u03c1 : \u2102) * paperFT k (gammaOf \u03c1) = literatureRHS k\\n  ```\\n- **`EF.EF_paper`** (line 88) \u2014 paper spectral form `W(f,g) = \u222b h_f conj(h_g) \u03bd_X`.\\n- **`EF.prop_EF_of_lit`** (line 703) \u2014 proves paper form from literature form.\\n\\nThe Weil-EF directory (`Zeta23/WeilEF/`) builds `EF_lit` for \u03b6 unconditionally:\\n\\n- `Zeta23/WeilEF/ZeroSummability.lean`:\\n  - `WeilEF.zero_sum_inv_sq_gen`\\n  - `WeilEF.EF_zero_sum_summable_gen`\\n  - `WeilEF.zero_sum_inv_sq`\\n  - `WeilEF.EF_zero_sum_summable`\\n\\n- `Zeta23/WeilEF/ZeroSumLimit.lean` (line 26):\\n  ```lean\\n  theorem zero_sum_limit (hs : ZetaSeam) {k : \u211d \u2192 \u2102} (hk : ContDiff \u211d 2 k)\\n      (hkc : HasCompactSupport k) {R : \u2115 \u2192 \u211d} ... :\\n      Tendsto (fun j : \u2115 => \u2211 \u03c1 \u2208 Z j, (zeroMult \u03c1 : \u2102) * Hfn k \u03c1) atTop\\n        (\ud835\udcdd (\u2211' \u03c1 : (zetaZeros hs).carrier, ((zetaZeros hs).mult \u03c1 : \u2102) * Hfn k \u03c1))\\n  ```\\n\\n- `Zeta23/WeilEF/Contour.lean` (line 31): rectangle identity / weighted argument principle.\\n- `Zeta23/WeilEF/FullLineAssembly.lean` (line 39): `full_line_identity`.\\n- `Zeta23/WeilEF/Main.lean` (lines 51, 270):\\n  ```lean\\n  theorem EF_lit_zeta (hs : ZetaSeam) : Zeta23.EF.EF_lit (zetaZeros hs)\\n  theorem EF_lit_zetaZeroConfig : Zeta23.EF.EF_lit zetaZeroConfig\\n  ```\\n\\nThese give absolutely convergent zero-summation over `{\u03c1 : \u2102 // IsNontrivialZero \u03c1}` weighted by `zeroMult`, exactly the enumeration technique reusable for a contour-bound on Li coefficients.\\n\\n---\\n\\n## 2. NguyenRH1 Li-coefficient definitions (`D:/Python/Quantum/research/lean/NguyenRH1/NguyenRH1/LiSkeleton.lean`)\\n\\n### 2.1 The Keiper\u2013Li coefficient `liCoeff`\\n\\n- **`liCoeff`** (line 39)\\n  ```lean\\n  def liCoeff (n : \u2115) : \u2102 :=\\n    tsum fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} =>\\n      (zeroMult \u03c1 : \u2102) * (1 - (1 - 1 / (\u03c1 : \u2102)) ^ n)\\n  ```\\n  It sums over the **subtype of nontrivial zeros** `{\u03c1 : \u2102 // IsNontrivialZero \u03c1}` with weight `zeroMult \u03c1`.  \\n  **Summability note:** the file contains an \u201chonesty note\u201d that `tsum` of a non-summable family is `0` in Lean; no `liCoeff`-specific summability lemma is proved in this file. The summability is intended to follow from `\u03a3 |\u03c1|\u207b\u00b2 < \u221e`.\\n\\n### 2.2 Supporting definitions (exact code)\\n\\n- **`lagariasA1`** (line 68)\\n  ```lean\\n  def lagariasA1 : \u211d := (Real.log (4 * Real.pi) + Real.eulerMascheroniConstant) / 2\\n  ```\\n\\n- **`SInfty`** (line 73)\\n  ```lean\\n  def SInfty (n : \u2115) : \u211d :=\\n    (1 - lagariasA1) * n - 1 +\\n      tsum fun m : {m : \u2115 // Odd m \u2227 3 \u2264 m} =>\\n        ((1 - 1 / (m : \u211d)) ^ n - 1 + (n : \u211d) / (m : \u211d))\\n  ```\\n\\n- **`lagariasC1`** (line 79)\\n  ```lean\\n  def lagariasC1 : \u211d :=\\n    (Real.eulerMascheroniConstant - 1 - Real.log (2 * Real.pi)) / 2\\n  ```\\n\\n- **`sInftyLower`** (line 84)\\n  ```lean\\n  def sInftyLower (n : \u211d) : \u211d :=\\n    (1 / 2) * n * Real.log n + (lagariasC1 - Real.eulerMascheroniConstant / 2) * n - 4\\n  ```\\n\\n- **`C0hat`** (line 429)\\n  ```lean\\n  def C0hat (t : \u211d) : \u211d :=\\n    0.433 * Real.log (t + 2) + 0.886 * Real.log (Real.log (t + 2)) + 2.9\\n  ```\\n\\n- **`Hhat`** (line 433)\\n  ```lean\\n  def Hhat (T : \u211d) : \u211d :=\\n    3 * C0hat T ^ 2 + 10.5 * Real.log (T + 2) + 48\\n  ```\\n\\n- **`IIMaj`** (line 438)\\n  ```lean\\n  def IIMaj (n : \u211d) : \u211d :=\\n    2 * (Real.sqrt n + 1) * (Real.exp (Real.sqrt n / 2) + 1) * 0.57 *\\n      2 ^ (-(2 * Real.sqrt n - 2))\\n  ```\\n\\n- **`marginTilde`** (line 447)\\n  ```lean\\n  def marginTilde (F : \u211d \u2192 \u211d) (n : \u211d) : \u211d :=\\n    let T := Real.sqrt n + 1\\n    ((1 / 2) * n * Real.log n +\\n        (lagariasC1 - Real.eulerMascheroniConstant / 2) * n - 4) -\\n      (2 * F T + 4 * T * ((2 / 3) * C0hat T + 10.5 * Real.log (T + 2) + 48) +\\n        40 * 1.3352 + 2 * (Real.exp 4 + 1) * (7 * Hhat T + 3.3) +\\n        IIMaj n + 1) + 1\\n  ```\\n\\n- **`FAlt`** (line 2097)\\n  ```lean\\n  def FAlt (T : \u211d) : \u211d :=\\n    (1 / (2 * Real.pi)) * (T * Real.log T) + 0.137 * Real.log T +\\n      0.443 * Real.log (Real.log T) + (7 / 8 + 1.588)\\n  ```\\n\\n- **`Bn`** (line 2106)\\n  ```lean\\n  def Bn (F : \u211d \u2192 \u211d) (n : \u211d) : \u211d :=\\n    let T := Real.sqrt n + 1\\n    2 * F T + 4 * T * ((2 / 3) * C0hat T + 10.5 * Real.log (T + 2) + 48) +\\n      40 * 1.3352 + 2 * (Real.exp 4 + 1) * (7 * Hhat T + 3.3) + IIMaj n + 1\\n  ```\\n\\n### 2.2 Requested theorem signatures\\n\\n- **`sInfty_lower_bound`** (line 320)\\n  ```lean\\n  theorem sInfty_lower_bound (n : \u2115) (hn : 20 \u2264 n) :\\n      sInftyLower n \u2264 SInfty n\\n  ```\\n\\n- **`marginTilde_convex`** (line 1849)\\n  ```lean\\n  theorem marginTilde_convex (F : \u211d \u2192 \u211d) (hF : ContDiffOn \u211d 2 F (Set.Ioi 1))\\n      (h\u03c8 : \u2200 T : \u211d, Real.sqrt 20 + 1 \u2264 T \u2192\\n        deriv (deriv F) T - deriv F T / (T - 1) < 0) :\\n      ConvexOn \u211d (Set.Ici 20) (marginTilde F)\\n  ```\\n\\n- **`pos_of_convex_of_two_points`** (line 2409)\\n  ```lean\\n  theorem pos_of_convex_of_two_points {f : \u211d \u2192 \u211d} {a N\u2080 N\u2081 x : \u211d}\\n      (hconv : ConvexOn \u211d (Set.Ici a) f)\\n      (hN : a < N\u2080) (hN01 : N\u2080 < N\u2081) (hx : N\u2081 \u2264 x)\\n      (h0 : 0 < f N\u2080) (h01 : f N\u2080 < f N\u2081) :\\n      0 < f x\\n  ```\\n\\n---\\n\\n## 3. Mathlib vendored API (`D:/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/`)\\n\\n### 3.1 von Mangoldt and Chebyshev \u03c8\\n\\n- **`ArithmeticFunction.vonMangoldt`**  \\n  File: `Mathlib/NumberTheory/ArithmeticFunction/VonMangoldt.lean` (line 65)\\n  ```lean\\n  noncomputable def vonMangoldt : ArithmeticFunction \u211d :=\\n    \u27e8fun n => if IsPrimePow n then Real.log (minFac n) else 0, if_neg not_isPrimePow_zero\u27e9\\n  ```\\n  Notation `\u039b` is available in the `ArithmeticFunction` locale.\\n\\n- **`Chebyshev.psi`**  \\n  File: `Mathlib/NumberTheory/Chebyshev.lean` (line 74)\\n  ```lean\\n  noncomputable def psi (x : \u211d) : \u211d :=\\n    \u2211 n \u2208 Ioc 0 \u230ax\u230b\u208a, \u039b n\\n  ```\\n  Notation `\u03c8` is available in the `Chebyshev` scope. The same file also defines `Chebyshev.theta` (\u03b8) and many explicit bounds.\\n\\n### 3.2 \u03b6 non-vanishing on / to the right of the 1-line\\n\\n- **`riemannZeta_ne_zero_of_one_le_re`**  \\n  File: `Mathlib/NumberTheory/LSeries/Nonvanishing.lean` (line 411)\\n  ```lean\\n  lemma _root_.riemannZeta_ne_zero_of_one_le_re \u2983s : \u2102\u2984 (hs : 1 \u2264 s.re) :\\n      riemannZeta s \u2260 0\\n  ```\\n\\n- **`riemannZeta_ne_zero_of_one_lt_re`**  \\n  File: `Mathlib/NumberTheory/LSeries/Dirichlet.lean` (line 328)\\n  ```lean\\n  lemma riemannZeta_ne_zero_of_one_lt_re {s : \u2102} (hs : 1 < s.re) : riemannZeta s \u2260 0\\n  ```\\n\\n### 3.3 Identity theorem / discreteness of zeros / finite zero sets\\n\\nRelevant general theorems are in `Mathlib/Analysis/Analytic/IsolatedZeros.lean` and `Mathlib/Analysis/Analytic/Order.lean`.\\n\\n- **`AnalyticAt.eventually_eq_zero_or_eventually_ne_zero`**  \\n  `Mathlib/Analysis/Analytic/IsolatedZeros.lean` (line 125)\\n  ```lean\\n  theorem eventually_eq_zero_or_eventually_ne_zero (hf : AnalyticAt \ud835\udd5c f z\u2080) :\\n      (\u2200\u1da0 z in \ud835\udcdd z\u2080, f z = 0) \u2228 \u2200\u1da0 z in \ud835\udcdd[\u2260] z\u2080, f z \u2260 0\\n  ```\\n\\n- **`AnalyticOnNhd.eqOn_zero_of_preconnected_of_frequently_eq_zero`**  \\n  `Mathlib/Analysis/Analytic/IsolatedZeros.lean` (line 214)\\n  ```lean\\n  theorem eqOn_zero_of_preconnected_of_frequently_eq_zero (hf : AnalyticOnNhd \ud835\udd5c f U)\\n      (hU : IsPreconnected U) (h\u2080 : z\u2080 \u2208 U) (hfw : \u2203\u1da0 z in \ud835\udcdd[\u2260] z\u2080, f z = 0) :\\n      EqOn f 0 U\\n  ```\\n\\n- **`AnalyticOnNhd.eqOn_of_preconnected_of_frequently_eq`**  \\n  `Mathlib/Analysis/Analytic/IsolatedZeros.lean` (line 238)\\n  ```lean\\n  theorem eqOn_of_preconnected_of_frequently_eq (hf : AnalyticOnNhd \ud835\udd5c f U)\\n      (hg : AnalyticOnNhd \ud835\udd5c g U) (hU : IsPreconnected U) (h\u2080 : z\u2080 \u2208 U)\\n      (hfg : \u2203\u1da0 z in \ud835\udcdd[\u2260] z\u2080, f z = g z) :\\n      EqOn f g U\\n  ```\\n\\n- **`AnalyticOnNhd.eqOn_zero_or_eventually_ne_zero_of_preconnected`**  \\n  `Mathlib/Analysis/Analytic/IsolatedZeros.lean` (line 219)\\n  ```lean\\n  theorem eqOn_zero_or_eventually_ne_zero_of_preconnected (hf : AnalyticOnNhd \ud835\udd5c f U)\\n      (hU : IsPreconnected U) :\\n      EqOn f 0 U \u2228 \u2200\u1da0 x in codiscreteWithin U, f x \u2260 0\\n  ```\\n\\n- **`AnalyticOnNhd.preimage_zero_mem_codiscreteWithin`**  \\n  `Mathlib/Analysis/Analytic/IsolatedZeros.lean` (line 346)\\n  ```lean\\n  theorem AnalyticOnNhd.preimage_mem_codiscreteWithin {U : Set \ud835\udd5c} {s : Set E} {f : \ud835\udd5c \u2192 E}\\n      (hfU : AnalyticOnNhd \ud835\udd5c f U) (h\u2082f : \u2200\u1da0 x in codiscreteWithin U, f x \u2260 f x)\\n      (hs : s \u2208 codiscreteWithin (f '' U)) :\\n      f \u207b\u00b9' s \u2208 codiscreteWithin U\\n  ```\\n\\n- **`Filter.codiscreteWithin_iff_locallyFiniteComplementWithin`**  \\n  `Mathlib/Topology/DiscreteSubset.lean` (line 317)\\n  ```lean\\n  theorem codiscreteWithin_iff_locallyFiniteComplementWithin [T1Space X] {s U : Set X} :\\n      s \u2208 codiscreteWithin U \u2194 \u2200 z \u2208 U, \u2203 t \u2208 \ud835\udcdd z, Set.Finite (t \u2229 (U \\\\ s))\\n  ```\\n  This is the bridge from \u201ccodiscrete\u201d to \u201clocally finite complement,\u201d i.e. zeros are isolated in the domain.\\n\\n- **`analyticOrderAt_ne_top_of_isPreconnected`**  \\n  `Mathlib/Analysis/Analytic/Order.lean` (line 627)\\n  ```lean\\n  theorem analyticOrderAt_ne_top_of_isPreconnected {x y : \ud835\udd5c} (hf : AnalyticOnNhd \ud835\udd5c f U)\\n      (hU : IsPreconnected U) (h\u2081x : x \u2208 U) (hy : y \u2208 U) (h\u2082x : analyticOrderAt f x \u2260 \u22a4) :\\n      analyticOrderAt f y \u2260 \u22a4\\n  ```\\n\\n- **Concrete finiteness on compacts for \u03b6 zeros**  \\n  `Mathlib/NumberTheory/LSeries/ZetaZeros.lean` defines:\\n  ```lean\\n  noncomputable def riemannZetaZeros : Set \u2102 := riemannZeta \u207b\u00b9' {0}\\n  lemma isClosed_riemannZetaZeros\\n  lemma isDiscrete_riemannZetaZeros\\n  lemma IsCompact.inter_riemannZetaZeros_finite {S : Set \u2102} (hS : IsCompact S) :\\n      (S \u2229 riemannZetaZeros).Finite\\n  ```\\n\\nThese are the main tools available for proving that an analytic function\u2019s zero set is discrete and hence finite on compact subsets, and for controlling zero-counting via `codiscreteWithin`.\\n\\n---\""
  ],
  "totalDurationMs": 209645.17700195312,
  "totalTokens": 152014,
  "totalToolUseCount": 66,
  "usage": {
    "input_tokens": 743,
    "output_tokens": 5863,
    "cache_creation_input_tokens": 0,
    "cache_read_input_tokens": 145408
  }
}

{
  "todos": [
    {
      "content": "Verify paper/Lean baseline is clean and frozen",
      "status": "completed",
      "activeForm": "Verifying paper/Lean baseline"
    },
    {
      "content": "Inventory vendored residue/meromorphic API (agent)",
      "status": "completed",
      "activeForm": "Inventorying residue/meromorphic API"
    },
    {
      "content": "Inventory Zeta23 zero-counting + Mathlib zeta API + liCoeff (agent)",
      "status": "completed",
      "activeForm": "Inventorying Zeta23/Mathlib zeta API"
    },
    {
      "content": "Read liCoeff/Sf/perimeter defs + check summability encoding",
      "status": "in_progress",
      "activeForm": "Reading liCoeff/Sf/perimeter defs"
    },
    {
      "content": "Read D-036 contour computation + D-056 R-bound proof",
      "status": "pending",
      "activeForm": "Reading D-036/D-056"
    },
    {
      "content": "Write D-064 design doc (routes + lemma graph)",
      "status": "pending",
      "activeForm": "Writing D-064 design doc"
    },
    {
      "content": "Create LiContour.lean sorry skeleton, compiles",
      "status": "pending",
      "activeForm": "Creating LiContour.lean skeleton"
    },
    {
      "content": "Session-014 closure (journal, STATE, commit)",
      "status": "pending",
      "activeForm": "Closing session 014"
    }
  ]
}

{
  "oldTodos": [
    "content='Verify paper/Lean baseline is clean and frozen' status='in_progress' active_form='Verifying paper/Lean baseline'",
    "content='Inventory vendored residue/meromorphic API (agent)' status='pending' active_form='Inventorying residue/meromorphic API'",
    "content='Inventory Zeta23 zero-counting + Mathlib zeta API + liCoeff (agent)' status='pending' active_form='Inventorying Zeta23/Mathlib zeta API'",
    "content='Read D-036, D-056, LiSkeleton tier-(ii) defs' status='pending' active_form='Reading D-036/D-056/LiSkeleton'",
    "content='Write D-064 design doc (routes + lemma graph)' status='pending' active_form='Writing D-064 design doc'",
    "content='Create LiContour.lean sorry skeleton, compiles' status='pending' active_form='Creating LiContour.lean skeleton'",
    "content='Session-014 closure (journal, STATE, commit)' status='pending' active_form='Closing session 014'"
  ],
  "newTodos": [
    "content='Verify paper/Lean baseline is clean and frozen' status='completed' active_form='Verifying paper/Lean baseline'",
    "content='Inventory vendored residue/meromorphic API (agent)' status='completed' active_form='Inventorying residue/meromorphic API'",
    "content='Inventory Zeta23 zero-counting + Mathlib zeta API + liCoeff (agent)' status='completed' active_form='Inventorying Zeta23/Mathlib zeta API'",
    "content='Read liCoeff/Sf/perimeter defs + check summability encoding' status='in_progress' active_form='Reading liCoeff/Sf/perimeter defs'",
    "content='Read D-036 contour computation + D-056 R-bound proof' status='pending' active_form='Reading D-036/D-056'",
    "content='Write D-064 design doc (routes + lemma graph)' status='pending' active_form='Writing D-064 design doc'",
    "content='Create LiContour.lean sorry skeleton, compiles' status='pending' active_form='Creating LiContour.lean skeleton'",
    "content='Session-014 closure (journal, STATE, commit)' status='pending' active_form='Closing session 014'"
  ],
  "cleared": false
}

{
  "file_path": "D:\\Python\\Quantum\\research\\derivations\\D-036-explicit-contour-constants.md"
}

{
  "command": "grep -n \"zero_sum_inv_sq\\b\\|zero_sum_inv_sq_gen\\|theorem zero_sum_inv_sq\" /d/Python/Quantum/zeta-23-lean/Zeta23/WeilEF/ZeroSummability.lean | head; echo ===; sed -n '1,60p' /d/Python/Quantum/zeta-23-lean/Zeta23/WeilEF/ZeroSummability.lean",
  "description": "Check zero-sum summability theorem statements"
}

{
  "command": "grep -rn \"def gammaOf\" /d/Python/Quantum/zeta-23-lean/Zeta23/ | head -3",
  "description": "Find gammaOf definition"
}

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\Scratch.lean",
  "content": "/- Session 014 scratch: axiom check of planned tier-ii\u2032 dependencies. -/\nimport Mathlib\nimport Zeta23.WeilEF.ZeroSummability\nimport Zeta23.RvM.LocalCount\nimport Zeta23.RvM.Statement\nimport Zeta23.FromPNTPlus.ResidueCalcOnRectangles\nimport Zeta23.ZetaReflect\n\nopen Zeta23\n\n#print axioms WeilEF.zero_sum_inv_sq\n#print axioms RvM.zeta_local_zero_count\n#print axioms zeta_reflect_zero\n#print axioms zeta_mult_reflect\n#print axioms RectangleIntegral'_eq_sumResiduesIn\n#print axioms ResidueMult\n#print axioms logDerivResidue\n#print axioms riemannZetaLogDerivResidue\n"
}

{
  "file_path": "D:\\Python\\Quantum\\research\\derivations\\D-037-finite-height-theorem.md",
  "offset": 80,
  "limit": 130
}

{
  "command": "grep -n \"S_f\\|S_\u221e\\|BL Thm\\|Bombieri\" /d/Python/Quantum/research/derivations/D-033*.md | head -30",
  "description": "Find S_f definition discussion in D-033"
}

{
  "command": "wc -l /d/Python/Quantum/research/paper/BL-1999-fulltext.txt /d/Python/Quantum/research/paper/Theorem-2.txt 2>/dev/null; grep -n \"S_\\\\\\\\infty\\|S_f\\|Theorem 2\" /d/Python/Quantum/research/paper/Theorem-2.txt | head -20",
  "description": "Check BL text files and Theorem 2 content"
}

{
  "command": "cd /d/Python/Quantum/research/paper && python -c \"\nimport subprocess, sys\ntry:\n    from pypdf import PdfReader\nexcept ImportError:\n    from PyPDF2 import PdfReader\nr = PdfReader('lagarias-li-coefficients-automorphic-0404394.pdf')\nprint('pages:', len(r.pages))\nfor i, p in enumerate(r.pages):\n    t = p.extract_text() or ''\n    if 'Theorem 6.1' in t or '6.1' in t[:200]:\n        print(i, t[:100].replace(chr(10),' '))\n\" 2>&1 | head -20",
  "description": "Locate Theorem 6.1 in Lagarias PDF"
}

{
  "command": "cd /d/Python/Quantum && ./.venv/Scripts/python.exe -c \"\nimport mpmath\nmpmath.mp.mp.dps = 50\nz = mpmath.zeta\n# Laurent coefficients of -zeta'/zeta at s=1: -z'/z(1+u) = 1/u + c0 + c1 u + c2 u^2 + ...\ndef negld(u):\n    return -mpmath.diff(z, 1+u)/z(1+u)\n# extract c0..c3 by fitting at geometric points (Richardson-ish)\nus = [mpmath.mpf(10)**(-k) for k in [2,3,4,5]]\n# series: negld(u) - 1/u = c0 + c1 u + c2 u^2 + c3 u^3\nimport mpmath as m\ndef A(u): return negld(u) - 1/u\n# extrapolate A(u) to u->0 via polynomial fit on 4 points\ncs = []\nvals = [(u, A(u)) for u in us]\n# fit c0 + c1 u + c2 u^2 + c3 u^3\nM = m.matrix([[u**j for j in range(4)] for u,_ in vals])\nb = m.matrix([v for _,v in vals])\nsol = m.lu_solve(M, b)\nc0,c1,c2,c3 = [sol[i] for i in range(4)]\nprint('c0 =', m.nstr(c0,30))\nprint('c1 =', m.nstr(c1,30))\nprint('c2 =', m.nstr(c2,20))\ngamma = m.euler\ng1 = m.stieltjes(1)\nprint('euler =', m.nstr(gamma,20), ' stieltjes1 =', m.nstr(g1,20))\nprint('candidate c1 = g1 + gamma^2 =', m.nstr(g1 + gamma**2,20))\nprint('candidate c1 = -(g1 + gamma^2) =', m.nstr(-(g1 + gamma**2),20))\n# S_f via archimedean: S_inf(n) + 1 - lambda_n needs lambda_n; use known small-n lambda\n# lambda_1, lambda_2 from zero side is costly; instead check BL identity consistency:\n# S_inf(n) for n=1,2\na1 = (m.log(4*m.pi)+gamma)/2\ndef Sinf(n):\n    tot = (1-a1)*n - 1\n    s = m.mpf(0)\n    for mm in range(3, 20001, 2):\n        s += (1-1/m)**n - 1 + n/m\n    # tail ~ C(n,2)/2 * sum_{m>20000 odd} 1/m^2 ~ C(n,2)/(4*10000)\n    return tot + s\nfor n in [1,2]:\n    print('Sinf(%d) ='%n, m.nstr(Sinf(n),20))\n\"\n",
  "description": "Compute Laurent coefficients of -zeta'/zeta at 1 numerically",
  "timeout": 180000
}

{
  "command": "cd /d/Python/Quantum && ls research/numerics/data/ | grep -i \"zeros\" | head; ./.venv/Scripts/python.exe -c \"\nimport mpmath as m\nm.mp.dps = 50\n# Laurent coefficients c0..c6 of -zeta'/zeta at s=1 via mpmath taylor\nz = m.zeta\nf = lambda u: -m.diff(z, 1+u)/z(1+u) - 1/u\nts = m.taylor(f, 0, 6)\nprint('taylor of -z\\'/z(1+u)-1/u:')\nfor j,c in enumerate(ts):\n    print(' c%d ='%j, m.nstr(c, 20))\n\"",
  "description": "High-precision Laurent coefficients via mpmath taylor",
  "timeout": 180000
}

{
  "command": "cd /d/Python/Quantum && ./.venv/Scripts/python.exe -c \"\nimport mpmath as m\nm.mp.dps = 40\nz = m.zeta\nld = lambda w: -m.diff(z, w)/z(w)   # -zeta'/zeta\ngamma = m.euler\n# Laurent c0..c3 (from prior fit, dps-50 verified): c0=-gamma, c1=0.18754623283535631...\n# refit at dps 40 with 5 unknowns for safety\nus = [m.mpf(10)**(-k) for k in [3,4,5,6,7]]\nA = lambda u: ld(1+u) - 1/u\nM = m.matrix([[u**j for j in range(5)] for u in us])\nb = m.matrix([A(u) for u in us])\nsol = m.lu_solve(M, b)\ncs = [sol[i] for i in range(5)]\nprint('c0..c4:', [m.nstr(c,18) for c in cs])\ndef kn(s, n): return (1+1/s)**n - 1\ndef Res1(n):\n    return m.nsum(lambda j: 0, [1,0]) + sum(m.binomial(n,j)*cs[j-1] for j in range(1,n+1))\ndef triv(n): return 1 - (m.mpf(2)/3)**n\na1 = (m.log(4*m.pi)+gamma)/2\ndef Sinf(n):\n    return (1-a1)*n - 1 + m.nsum(lambda k: (1-1/(2*k+1))**n - 1 + n/(2*k+1), [1, m.inf], tol=m.mpf('1e-30'))\n# zeros from cache\nimport numpy as np\ngam = np.load('research/numerics/data/E-021-zeros-cache.npy')\nprint('zeros:', len(gam), 'max:', gam[-1])\ndef lam_n(n, T=None):\n    tot = m.mpf(0)\n    for g in gam:\n        if T is not None and g > T: break\n        rho = m.mpc(m.mpf('0.5'), m.mpf(float(g)))\n        # pair: rho and conj(rho)\n        tot += (1 - (1-1/rho)**n).real*2\n    return tot\ndef sides(n, T):\n    J = lambda w: kn(w-1, n)*ld(w)\n    s0, s1 = m.mpf('-2.5'), 2*m.sqrt(n)\n    I1 = m.quad(lambda t: J(m.mpc(s1,t)), [-T, T])          # right vertical\n    I3 = m.quad(lambda t: J(m.mpc(s0,t)), [-T, T])          # left vertical\n    I2 = m.quad(lambda x: J(m.mpc(x,T)), [s0, s1])          # top\n    I4 = m.quad(lambda x: J(m.mpc(x,-T)), [s0, s1])         # bottom\n    # CCW: bottom (left->right), right (bottom->top), top (right->left), left (top->bottom)\n    total = I4 + 1j*I1 - I2 - 1j*I3\n    return total/(2*m.pi*1j)\nfor n in [2, 3]:\n    T = m.sqrt(n) + m.mpf('0.5')\n    S = sides(n, T)\n    lamT = lam_n(n, T)\n    print(f'n={n}: sides={m.nstr(S,16)}')\n    print(f'  lam_n(T)={m.nstr(lamT,16)} triv={m.nstr(triv(n),10)} Res1={m.nstr(Res1(n),16)}')\n    print(f'  lam_n(T)+triv+Res1 = {m.nstr(lamT+triv(n)+Res1(n),16)}')\n    print(f'  sides - (lam+triv+Res1) = {m.nstr(S-(lamT+triv(n)+Res1(n)),8)}')\n    Si = Sinf(n); lamF = lam_n(n)\n    Sf = Si + 1 - lamF\n    B0 = Si + 1 + triv(n) + Res1(n)\n    print(f'  Sinf={m.nstr(Si,14)} lambda_full(approx)={m.nstr(lamF,12)} Sf={m.nstr(Sf,12)}')\n    print(f'  B0 = Sinf+1+triv+Res1 = {m.nstr(B0,14)}')\n    print(f'  check MASTER: B0 - sides - tail = {m.nstr(B0 - S - (lamF-lamT),14)}  vs Sf')\n\"",
  "description": "Numerically verify the full contour identity chain at n=2,3",
  "timeout": 300000
}

{
  "command": "cd /d/Python/Quantum && ./.venv/Scripts/python.exe -c \"\nimport mpmath as m\nimport numpy as np\nm.mp.dps = 40\ngam = np.load('research/numerics/data/E-021-zeros-cache.npy')\n# sigma_2 = sum 1/rho^2, absolutely convergent; pair rho, conj(rho)\ns2 = m.mpf(0)\nfor g in gam:\n    rho = m.mpc(m.mpf('0.5'), m.mpf(float(g)))\n    s2 += 2*(1/rho**2).real\nprint('sigma_2 (4624 zeros) =', m.nstr(s2, 15))\n# tail estimate: sum_{g>5096} 2|1/rho^2| ~ 2*log(5096)/(2*pi*5096)\nprint('tail bound ~', m.nstr(2*m.log(5096)/(2*m.pi*5096), 6))\n# (4.10) j=1: -sigma_2 = tau_1 - eta_1 - 1\ntau1 = m.mpf(3)/4*m.zeta(2)\neta1 = m.mpf('0.187546232840365225')\nprint('tau1 - eta1 - 1 =', m.nstr(tau1 - eta1 - 1, 15))\nprint('=> -sigma_2 should equal that; diff =', m.nstr(-s2 - (tau1-eta1-1), 8))\n\"",
  "description": "Verify Lagarias (4.10) at j=1 via sigma_2",
  "timeout": 120000
}

{
  "command": "cd /d/Python/Quantum && head -5 research/numerics/data/E-015*.csv 2>/dev/null; ls research/numerics/data/ | grep \"E-015\"",
  "description": "Check E-015 certified lambda_2 value"
}

{
  "todos": [
    {
      "content": "Verify paper/Lean baseline is clean and frozen",
      "status": "completed",
      "activeForm": "Verifying paper/Lean baseline"
    },
    {
      "content": "Inventory vendored residue/meromorphic API (agent)",
      "status": "completed",
      "activeForm": "Inventorying residue/meromorphic API"
    },
    {
      "content": "Inventory Zeta23 zero-counting + Mathlib zeta API + liCoeff (agent)",
      "status": "completed",
      "activeForm": "Inventorying Zeta23/Mathlib zeta API"
    },
    {
      "content": "Read liCoeff/Sf/perimeter defs + check summability encoding",
      "status": "completed",
      "activeForm": "Reading liCoeff/Sf/perimeter defs"
    },
    {
      "content": "Read D-036 contour + D-056 R-bound + Lagarias Thm 6.1 source",
      "status": "completed",
      "activeForm": "Reading D-036/D-056/Lagarias"
    },
    {
      "content": "Settle exact identity chain numerically (crux: Sf = Res1)",
      "status": "completed",
      "activeForm": "Settling identity chain numerically"
    },
    {
      "content": "Write D-064 design doc (routes + lemma graph)",
      "status": "in_progress",
      "activeForm": "Writing D-064 design doc"
    },
    {
      "content": "Session-014 closure (journal, STATE, commit)",
      "status": "pending",
      "activeForm": "Closing session 014"
    }
  ]
}

{
  "oldTodos": [
    "content='Verify paper/Lean baseline is clean and frozen' status='completed' active_form='Verifying paper/Lean baseline'",
    "content='Inventory vendored residue/meromorphic API (agent)' status='completed' active_form='Inventorying residue/meromorphic API'",
    "content='Inventory Zeta23 zero-counting + Mathlib zeta API + liCoeff (agent)' status='completed' active_form='Inventorying Zeta23/Mathlib zeta API'",
    "content='Read liCoeff/Sf/perimeter defs + check summability encoding' status='in_progress' active_form='Reading liCoeff/Sf/perimeter defs'",
    "content='Read D-036 contour computation + D-056 R-bound proof' status='pending' active_form='Reading D-036/D-056'",
    "content='Write D-064 design doc (routes + lemma graph)' status='pending' active_form='Writing D-064 design doc'",
    "content='Create LiContour.lean sorry skeleton, compiles' status='pending' active_form='Creating LiContour.lean skeleton'",
    "content='Session-014 closure (journal, STATE, commit)' status='pending' active_form='Closing session 014'"
  ],
  "newTodos": [
    "content='Verify paper/Lean baseline is clean and frozen' status='completed' active_form='Verifying paper/Lean baseline'",
    "content='Inventory vendored residue/meromorphic API (agent)' status='completed' active_form='Inventorying residue/meromorphic API'",
    "content='Inventory Zeta23 zero-counting + Mathlib zeta API + liCoeff (agent)' status='completed' active_form='Inventorying Zeta23/Mathlib zeta API'",
    "content='Read liCoeff/Sf/perimeter defs + check summability encoding' status='completed' active_form='Reading liCoeff/Sf/perimeter defs'",
    "content='Read D-036 contour + D-056 R-bound + Lagarias Thm 6.1 source' status='completed' active_form='Reading D-036/D-056/Lagarias'",
    "content='Settle exact identity chain numerically (crux: Sf = Res1)' status='completed' active_form='Settling identity chain numerically'",
    "content='Write D-064 design doc (routes + lemma graph)' status='in_progress' active_form='Writing D-064 design doc'",
    "content='Session-014 closure (journal, STATE, commit)' status='pending' active_form='Closing session 014'"
  ],
  "cleared": false
}

{
  "command": "tail -15 /d/Python/Quantum/research/derivations/index.md",
  "description": "Check derivations index format"
}

{
  "file_path": "D:\\Python\\Quantum\\research\\journal\\session-014.md",
  "content": "# Session 014 \u2014 2026-08-21 \u2014 Tier ii\u2032 launched: design + two discoveries\n\n## Directive\nUser chose option (a): do tier ii\u2032 (close SfBoundProp in Lean) BEFORE\nsubmission, with Paper A frozen and submittable at all times. Freeze\nprotocol in force: no edits to paperA.tex/paperB.tex/draft.tex/\nreferences.bib until ii\u2032-7; no edits to existing Lean proofs except the\nliCoeff correctness fix (below, flagged); commit at every sub-stage.\n\n## Work done (sub-stage ii\u2032-0: design \u2192 D-064)\n1. Baseline verified clean (git: only untracked LaTeX artifacts).\n2. Two parallel inventory agents surveyed: (i) vendored residue/\n   meromorphic API; (ii) Zeta23 zero infrastructure + NguyenRH1 defs +\n   Mathlib zeta/digamma/Jensen/Borel\u2013Carath\u00e9odory surface.\n3. Read liCoeff/Sf/perimeter defs, D-036, D-056, D-037 (full), and the\n   primary source (Lagarias math/0404394 \u00a74\u20136, archived PDF).\n4. Axiom-checked all planned Zeta23 dependencies LIVE: zero_sum_inv_sq,\n   zeta_local_zero_count, zeta_reflect_zero, zeta_mult_reflect,\n   RectangleIntegral'_eq_sumResiduesIn, ResidueMult, logDerivResidue,\n   riemannZetaLogDerivResidue \u2014 ALL 3 standard axioms only.\n5. Settled the exact identity chain NUMERICALLY (dps 40\u201350, mpmath +\n   E-021 zero cache; see D-064 \u00a73 table).\n6. Wrote D-064 (design doc: discoveries, routes R1\u2013R5, lemma graph,\n   revised sub-stage map ii\u2032-1a..7). Registered in index.md + db.\n\n## DISCOVERY 1 \u2014 liCoeff encoding defect (correctness, not cosmetic)\n`liCoeff` (LiSkeleton.lean:39) is an unordered tsum whose family is NOT\nSummable for any n \u2265 1 (summand ~ n/|\u03b3|, \u03a3 mult/|\u03c1| = \u221e by RvM).\nTherefore liCoeff n = 0 (junk), making `LiVerifiedUpTo` FALSE and\n`SfBoundProp` FALSE as encoded (at n = 10\u2075: |Sf| \u2265 432 501 > 347 000\n\u2265 Bn by the certified LiBasePoints bounds). The session-012 theorems\nare vacuously true; the session-013 audit could not see this (it checks\naxioms, not hypothesis truth). Paper A's mathematics is unaffected \u2014\npure encoding defect. FIX DESIGNED (D-064 \u00a71): paired symmetric form\n(manifestly real, absolutely summable via pair bound |pair| \u2264\nC(n)\u00b7mult/|\u03c1|\u00b2 + Zeta23's proved `zero_sum_inv_sq`). Scheduled as\nsub-stage ii\u2032-1a (first executable item).\n\n## DISCOVERY 2 \u2014 the crux identity is Sf = Res\u2081 EXACTLY\nThe D-036/D-037 bookkeeping, settled against Lagarias's paper and\nverified numerically:\n- **Sf(n) := S_\u221e(n) + 1 \u2212 \u03bb_n = Res\u2081(n) = \u03a3_{j=1}^n C(n,j)\u00b7c_{j\u22121}**\n  (c_k = Laurent coefficients of \u2212\u03b6\u2032/\u03b6 at 1). Follows from Lagarias\n  (4.10) [\u03c4_j = \u03b7_j + (\u22121)\u02b2\u03c3_{j+1} \u2212 (\u22121)\u02b2\u03b4] summed binomially.\n  Verified: (4.10) at j=0 exact to 15 digits (\u03bb\u2081 = 0.02309570896612103);\n  crux consequence \u03bb\u2082 = 0.09234573523 matches E-015's certified\n  0.09234573522804667 exactly.\n- **Residue theorem**: sides(T) = \u03bb_n(T) + triv + Res\u2081(n), verified to\n  2e-21 (n=2) by direct contour integration.\n- **Assembly is TAIL-FREE**: |Sf| \u2264 |\u03bb_n(T)| + |sides| + |triv| \u2264\n  2N(T) + [side pieces] + 1 = B(n) \u2014 no tail term anywhere (the crux\n  absorbs it). This is why D-037's B(n) has none.\n- The w=1 pole (order n+1) residue computation reduces to a Leibniz\n  rule: (w\u22121)^{n+1}J = [w\u207f \u2212 (w\u22121)\u207f]h(w), h = (w\u22121)(\u2212\u03b6\u2032/\u03b6) analytic \u2014\n  Res = \u03a3_j C(n,j)c_{j\u22121} confirmed algebraically.\n- SIDE EFFECT: D-037's line-63 compressed identity needs an erratum\n  pointer to D-064 (correct reading: Sf = Res\u2081 = sides \u2212 \u03bb_n(T) \u2212 triv).\n\n## Inventory highlights (why tier ii\u2032 is feasible)\n- Rectangle residue theorem EXISTS (vendored FromPNTPlus\n  `RectangleIntegral'_eq_sumResiduesIn`, axiom-clean) \u2014 needs the\n  simple-pole scaffold extended to finite order (bounded, R1).\n- `logDerivResidue`, `ResidueMult`, `riemannZetaLogDerivResidue`\n  vendored and clean.\n- Mathlib HAS Borel\u2013Carath\u00e9odory, Jensen's formula, digamma,\n  HadamardThreeLines, MeromorphicAt/Order/NormalForm/FactorizedRational.\n- Zeta23 HAS proved \u03a3 mult/(1+\u2016\u03b3_\u03c1\u2016\u00b2) < \u221e (zero_sum_inv_sq), local\n  zero count, RvM, reflect/conj symmetry, window finiteness.\n- Mathlib has NO Weierstrass/Hadamard factorization; FromPNTPlus\n  StrongPNTPrefix has the local Blaschke/Borel toolkit.\n\n## Route decisions (D-064 \u00a74)\nR1 build higher-order residue machinery (1\u20132 sess); R2 crux via (4.10)\n\u2014 investigate zeta-23 EF reuse (\u03b22) first, Hadamard fallback (\u03b21)\n(2\u20134 sess, the long pole); R3 R-bound via LOCAL route\n(Borel\u2013Carath\u00e9odory/Jensen/Blaschke \u2014 machinery exists) (2\u20133 sess);\nR4 the old \"w=1 \u2194 S_\u221e\" monster is KILLED by Discovery 2 (factors into\nR1's Leibniz computation + R2's (4.10)); R5 perimeter unchanged.\n\n## State at stop\nD-064 complete; sub-stage map ii\u2032-1a..7 (total ~9\u201316 sessions).\nNEXT: ii\u2032-1a (liCoeff pairing fix + summability). Paper A untouched\n(compiles at session-013 state; submission-ready fallback intact \u2014\nD-064 \u00a78).\n"
}

{
  "file_path": "D:\\Python\\Quantum\\NGUYEN.md",
  "old_string": "- [~] 5.1 Lean 4 formalization of all new lemmas (build on zeta-23-lean).\n      \u2192 SKELETON STAGE COMPLETE (session 010, D-061): NguyenRH1 package\n      (research/lean/NguyenRH1/) \u2014 9 statements compile against\n      Mathlib + Zeta23 types. **TIER (i) COMPLETE (session 011)**:\n      chain_rule_sqrt_shift, sInfty_lower_bound, marginTilde_convex\n      PROVED (~1100 lines; full support library \u2014 see journal-011).\n      **TIER (ii) STEPS 1+2 COMPLETE (session 012, D-062)**:\n      lambda_pos_finite_height_verified PROVED modulo the single named\n      analytic hypothesis SfBoundProp + the declared computational\n      perimeter (assembly-first split per session-012-notes; FAlt\n      majorant swap; secant two-point base; base points discharged by\n      certified interval bounds in LiBasePoints.lean \u2014 margin\u0303(8\u00d710\u2074)\n      \u2265 8001 certified, margin\u0303(10\u2075) \u2265 85 501 certified).\n      Remaining: tier ii\u2032 (SfBoundProp contour layers L0\u2013L3 \u2014\n      POST-PUBLICATION per session-012-end-notes review; Trudgian\n      N(T)/S(T) = declared perimeter, D-062 \u00a74), (iii) valley,\n      (iv) Suzuki.",
  "new_string": "- [~] 5.1 Lean 4 formalization of all new lemmas (build on zeta-23-lean).\n      \u2192 SKELETON STAGE COMPLETE (session 010, D-061): NguyenRH1 package\n      (research/lean/NguyenRH1/) \u2014 9 statements compile against\n      Mathlib + Zeta23 types. **TIER (i) COMPLETE (session 011)**:\n      chain_rule_sqrt_shift, sInfty_lower_bound, marginTilde_convex\n      PROVED (~1100 lines; full support library \u2014 see journal-011).\n      **TIER (ii) STEPS 1+2 COMPLETE (session 012, D-062)**:\n      lambda_pos_finite_height_verified PROVED modulo the single named\n      analytic hypothesis SfBoundProp + the declared computational\n      perimeter (assembly-first split per session-012-notes; FAlt\n      majorant swap; secant two-point base; base points discharged by\n      certified interval bounds in LiBasePoints.lean \u2014 margin\u0303(8\u00d710\u2074)\n      \u2265 8001 certified, margin\u0303(10\u2075) \u2265 85 501 certified).\n      **TIER ii\u2032 LAUNCHED (session 014, user directive \u2014 BEFORE\n      submission, option (a); design D-064)**: TWO DISCOVERIES \u2014\n      (1) liCoeff encoding defect (unordered tsum NOT Summable \u27f9 junk;\n      LiVerifiedUpTo/SfBoundProp FALSE as encoded; paired-form fix =\n      ii\u2032-1a); (2) crux identity Sf(n) = Res\u2081(n) EXACT (numerically\n      verified vs E-015/E-021) \u27f9 assembly TAIL-FREE. Routes R1\u2013R5;\n      sub-stage map ii\u2032-1a..7 (~9\u201316 sessions); perimeter unchanged\n      (Trudgian N(T)/S(T) declared, D-062 \u00a74). Freeze protocol: paper\n      files + existing proofs untouched until ii\u2032-7 (liCoeff fix\n      excepted). Remaining after ii\u2032: (iii) valley, (iv) Suzuki."
}

{
  "file_path": "D:\\Python\\Quantum\\NGUYEN.md",
  "offset": 401,
  "limit": 45
}

File does not exist. Note: your current working directory is D:\Python\Quantum\research\lean\NguyenRH1\NguyenRH1.

File does not exist. Note: your current working directory is D:\Python\Quantum\research\lean\NguyenRH1\NguyenRH1.

{
  "todos": [
    {
      "content": "Check zetaZeros carrier def + zeroMult/gammaOf coercion paths",
      "status": "in_progress",
      "activeForm": "Checking zetaZeros defs"
    },
    {
      "content": "Develop summability proof in Scratch.lean (pair bound + comparison)",
      "status": "pending",
      "activeForm": "Developing summability proof"
    },
    {
      "content": "Replace liCoeff def in LiSkeleton.lean + prove liCoeff_summable",
      "status": "pending",
      "activeForm": "Replacing liCoeff def"
    },
    {
      "content": "Full rebuild + axiom audit (LiSkeleton/LiBasePoints unchanged downstream)",
      "status": "pending",
      "activeForm": "Rebuilding + auditing"
    },
    {
      "content": "Session-015 closure (journal, STATE, NGUYEN checkoff, commit)",
      "status": "pending",
      "activeForm": "Closing session 015"
    }
  ]
}

{
  "oldTodos": [
    "content='Read current web/ files (settings, urls, models, views, healthcheck)' status='completed' active_form='Reading current web/ files'",
    "content='Add VisitorLog model + tracking middleware' status='completed' active_form='Adding VisitorLog model + middleware'",
    "content='Wire Django admin (settings, urls, admin.py)' status='completed' active_form='Wiring Django admin'",
    "content='Add feedback email notification hook' status='completed' active_form='Adding feedback email hook'",
    "content='Migrate + test locally (admin, visitor rows, feedback)' status='completed' active_form='Migrating + testing locally'",
    "content='Deploy to server via SSH (files, prod settings, migrate, superuser, collectstatic)' status='in_progress' active_form='Deploying to server via SSH'",
    "content='Verify live: /admin/ login, visitor log, feedback email' status='pending' active_form='Verifying live site'",
    "content='Update RH-WEBSITE.md + memory' status='pending' active_form='Updating RH-WEBSITE.md + memory'"
  ],
  "newTodos": [
    "content='Read current web/ files (settings, urls, models, views, healthcheck)' status='completed' active_form='Reading current web/ files'",
    "content='Add VisitorLog model + tracking middleware' status='completed' active_form='Adding VisitorLog model + middleware'",
    "content='Wire Django admin (settings, urls, admin.py)' status='completed' active_form='Wiring Django admin'",
    "content='Add feedback email notification hook' status='completed' active_form='Adding feedback email hook'",
    "content='Migrate + test locally (admin, visitor rows, feedback)' status='completed' active_form='Migrating + testing locally'",
    "content='Deploy to server via SSH (files, prod settings, migrate, superuser, collectstatic)' status='completed' active_form='Deploying to server via SSH'",
    "content='Verify live: /admin/ login, visitor log, feedback email' status='completed' active_form='Verifying live site'",
    "content='Update RH-WEBSITE.md + memory' status='completed' active_form='Updating RH-WEBSITE.md + memory'"
  ],
  "cleared": true
}

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\Scratch.lean",
  "content": "/- Session 015 scratch: ii\u2032-1a \u2014 paired liSummand + summability (D-064 \u00a71).\n   When green, merge into LiSkeleton.lean \u00a71 replacing the defective liCoeff. -/\nimport Mathlib\nimport Zeta23.WeilEF.ZeroSummability\nimport Zeta23.Statement.SeamClosed\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-- The paired (symmetrized) Li summand: \u00bd\u00b7mult\u00b7(f \u03c1 + f \u03c1\u0304),\nf \u03c1 = 1 \u2212 (1\u22121/\u03c1)\u207f. Manifestly real-valued (`liSummand_eq`), absolutely\nsummable (`liSummable`) \u2014 the correct Keiper\u2013Li coefficient (D-064 \u00a71:\nthe plain unordered sum of f over the zeros is not absolutely\nsummable; the pairing is the classical Bombieri\u2013Lagarias convention). -/\ndef liSummand (n : \u2115) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) : \u2102 :=\n  (zeroMult \u03c1 : \u2102) / 2 * ((1 - (1 - 1 / (\u03c1 : \u2102)) ^ n) + (1 - (1 - 1 / (star (\u03c1 : \u2102))) ^ n))\n\nlemma summand_star (n : \u2115) (z : \u2102) :\n    (1 : \u2102) - (1 - 1 / (star z)) ^ n = star ((1 - (1 - 1 / z) ^ n)) := by\n  rw [map_sub, map_one, map_pow, map_sub, map_one, map_div\u2080, map_one]\n\nlemma liSummand_eq (n : \u2115) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) :\n    liSummand n \u03c1 = (zeroMult \u03c1 : \u2102) * (((1 - (1 - 1 / (\u03c1 : \u2102)) ^ n).re) : \u2102) := by\n  rw [liSummand, summand_star, Complex.add_conj]\n  ring\n\n/-! ### The geometric-sum estimate: |Re(1 \u2212 w\u207f)| \u2264 2n\u00b2/\u2016z\u2016\u00b2 for w = 1\u22121/z -/\n\n/-- (1 + 1/(2n))\u207f \u2264 2 for n \u2265 1. Via (1+x)\u207f \u2264 exp(nx) and exp(1/2) \u2264 2. -/\nlemma pow_one_add_inv_two_mul_le (n : \u2115) (hn : 1 \u2264 n) :\n    ((1 : \u211d) + 1 / (2 * n)) ^ n \u2264 2 := by\n  have hn0 : (0 : \u211d) < n := by exact_mod_cast hn\n  have h1 : ((1 : \u211d) + 1 / (2 * n)) ^ n \u2264 Real.exp (n * (1 / (2 * n))) := by\n    have h := Real.add_one_le_exp (1 / (2 * (n : \u211d)))\n    calc ((1 : \u211d) + 1 / (2 * n)) ^ n\n        \u2264 (Real.exp (1 / (2 * n))) ^ n := by\n          gcongr\n          \u00b7 linarith\n          \u00b7 exact h\n      _ = Real.exp (n * (1 / (2 * n))) := by\n          rw [\u2190 Real.exp_nat_mul]\n  have h2 : (n : \u211d) * (1 / (2 * n)) = 1 / 2 := by field_simp\n  rw [h2] at h1\n  have h3 : Real.exp (1 / 2 : \u211d) \u2264 2 := by\n    have hsq : (Real.exp (1 / 2 : \u211d)) ^ 2 \u2264 (2 : \u211d) ^ 2 := by\n      rw [\u2190 Real.exp_nat_mul]\n      have : Real.exp (2 * (1 / 2 : \u211d)) = Real.exp 1 := by ring_nf\n      rw [this]\n      have h := Real.exp_one_lt_d9\n      norm_num at h \u22a2; linarith\n    nlinarith [Real.exp_pos (1 / 2 : \u211d)]\n  exact h1.trans h3\n\n/-- Key estimate: for n \u2265 1, z with re z \u2208 [0,1], 2n \u2264 \u2016z\u2016:\n|Re(1 \u2212 (1\u22121/z)\u207f)| \u2264 2n\u00b2/\u2016z\u2016\u00b2. -/\nlemma re_li_summand_le {n : \u2115} (hn : 1 \u2264 n) {z : \u2102}\n    (hzre0 : 0 \u2264 z.re) (hzre1 : z.re \u2264 1) (hz : 2 * n \u2264 \u2016z\u2016) :\n    |(1 - (1 - 1 / z) ^ n).re| \u2264 2 * n ^ 2 / \u2016z\u2016 ^ 2 := by\n  have hn0 : (0 : \u211d) < n := by exact_mod_cast hn\n  have hz0 : z \u2260 0 := by\n    intro h; rw [h, norm_zero] at hz; linarith\n  have hzn : (0 : \u211d) < \u2016z\u2016 := norm_pos_iff.mpr hz0\n  set w : \u2102 := 1 - 1 / z with hw\n  -- kernel identity: 1 \u2212 w\u207f = (1/z)\u00b7S, S = \u03a3_{j<n} w^j\n  have hgeom : w ^ n - 1 = (w - 1) * \u2211 j \u2208 Finset.range n, w ^ j := geom_sum_mul w n\n  have hw1 : w - 1 = -(1 / z) := by rw [hw]; ring\n  have hf : (1 : \u2102) - w ^ n = (1 / z) * \u2211 j \u2208 Finset.range n, w ^ j := by\n    have : (1 : \u2102) - w ^ n = -(w - 1) * \u2211 j \u2208 Finset.range n, w ^ j := by\n      rw [\u2190 hgeom]; ring\n    rw [this, hw1]; ring\n  set S : \u2102 := \u2211 j \u2208 Finset.range n, w ^ j\n  -- S = n + (S \u2212 n); Re(f) = n\u00b7Re(1/z) + Re((S\u2212n)/z)\n  have hS : S = (n : \u2102) + (S - n) := by ring\n  -- |S \u2212 n| \u2264 n\u00b2/\u2016z\u2016\n  have hwnorm : \u2016w\u2016 \u2264 1 + 1 / \u2016z\u2016 := by\n    have : w = 1 - z\u207b\u00b9 := by rw [hw, div_eq_mul_inv, one_mul]\n    rw [this]\n    calc \u2016(1 : \u2102) - z\u207b\u00b9\u2016 \u2264 \u2016(1 : \u2102)\u2016 + \u2016z\u207b\u00b9\u2016 := norm_sub_le _ _\n      _ = 1 + 1 / \u2016z\u2016 := by rw [norm_one, norm_inv]\n  have hwmax : max 1 \u2016w\u2016 \u2264 1 + 1 / (2 * n) := by\n    have h1 : 1 / \u2016z\u2016 \u2264 1 / (2 * n) := by\n      apply one_div_le_one_div_of_le (by positivity : (0:\u211d) < 2 * n)\n      exact_mod_cast hz\n    have h2 : \u2016w\u2016 \u2264 1 + 1 / (2 * n) := hwnorm.trans (by linarith)\n    apply max_le\n    \u00b7 linarith\n    \u00b7 exact h2\n  have hSn : \u2016S - n\u2016 \u2264 n ^ 2 / \u2016z\u2016 := by\n    have hSn1 : S - n = \u2211 j \u2208 Finset.range n, (w ^ j - 1) := by\n      rw [Finset.sum_sub_distrib]\n      simp [Finset.sum_const, Finset.card_range]\n    rw [hSn1]\n    have hbound : \u2200 j \u2208 Finset.range n,\n        \u2016w ^ j - 1\u2016 \u2264 (1 / \u2016z\u2016) * (j * (1 + 1 / (2 * n)) ^ (n - 1)) := by\n      intro j hj\n      have hgeomj : w ^ j - 1 = (w - 1) * \u2211 i \u2208 Finset.range j, w ^ i := geom_sum_mul w j\n      rw [hgeomj, norm_mul, hw1, norm_neg, norm_div, norm_one]\n      have hsum : \u2016\u2211 i \u2208 Finset.range j, w ^ i\u2016 \u2264 j * (1 + 1 / (2 * n)) ^ (n - 1) := by\n        calc \u2016\u2211 i \u2208 Finset.range j, w ^ i\u2016\n            \u2264 \u2211 i \u2208 Finset.range j, \u2016w ^ i\u2016 := norm_sum_le _ _\n          _ \u2264 \u2211 i \u2208 Finset.range j, (1 + 1 / (2 * n)) ^ (n - 1) := by\n              apply Finset.sum_le_sum\n              intro i hi\n              rw [norm_pow]\n              have hjn : i \u2264 n - 1 := by\n                have hij : i < j := Finset.mem_range.mp hi\n                have hjn' : j \u2264 n := le_of_lt (Finset.mem_range.mp hj)\n                omega\n              exact pow_le_pow_left\u2080 (by positivity)\n                (le_trans (le_max_right 1 \u2016w\u2016) hwmax) i (by omega)\n          _ = j * (1 + 1 / (2 * n)) ^ (n - 1) := by\n              rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul]\n      exact mul_le_mul_of_nonneg_left hsum (by positivity)\n    calc \u2016\u2211 j \u2208 Finset.range n, (w ^ j - 1)\u2016\n        \u2264 \u2211 j \u2208 Finset.range n, \u2016w ^ j - 1\u2016 := norm_sum_le _ _\n      _ \u2264 \u2211 j \u2208 Finset.range n, (1 / \u2016z\u2016) * (j * (1 + 1 / (2 * n)) ^ (n - 1)) := by\n          exact Finset.sum_le_sum hbound\n      _ = (1 / \u2016z\u2016) * (1 + 1 / (2 * n)) ^ (n - 1) * \u2211 j \u2208 Finset.range n, j := by\n          rw [Finset.mul_sum]; congr 1; rw [Finset.mul_sum]\n          apply Finset.sum_congr rfl; intro j _; ring\n      _ \u2264 n ^ 2 / \u2016z\u2016 := by\n          have hsumj : (\u2211 j \u2208 Finset.range n, j : \u211d) = n * (n - 1) / 2 := by\n            have h := Finset.sum_range_id_mul_two n\n            have h2 : ((\u2211 j \u2208 Finset.range n, j : \u2115) : \u211d) * 2 = n * (n - 1) := by\n              exact_mod_cast h\n            linarith\n          rw [hsumj]\n          have hpow : (1 + 1 / (2 * n : \u211d)) ^ (n - 1) \u2264 2 := by\n            have h := pow_one_add_inv_two_mul_le n hn\n            calc (1 + 1 / (2 * n : \u211d)) ^ (n - 1)\n                \u2264 (1 + 1 / (2 * n : \u211d)) ^ n := by\n                  apply pow_le_pow_right\u2080 (by positivity)\n                  \u00b7 have : 1 \u2264 (1 : \u211d) + 1 / (2 * n) := by positivity\n                    exact this\n                  \u00b7 omega\n              _ \u2264 2 := h\n          have h3 : (1 / \u2016z\u2016) * (1 + 1 / (2 * n : \u211d)) ^ (n - 1) * (n * (n - 1) / 2)\n              \u2264 (1 / \u2016z\u2016) * 2 * (n ^ 2 / 2) := by\n            apply mul_le_mul_of_nonneg_left _ (by positivity)\n            apply mul_le_mul_of_nonneg_left hpow (by positivity)\n            \u00b7 have : (n : \u211d) * (n - 1) / 2 \u2264 n ^ 2 / 2 := by\n                have hn1 : (1 : \u211d) \u2264 n := by exact_mod_cast hn\n                nlinarith\n              linarith\n          calc (1 / \u2016z\u2016) * (1 + 1 / (2 * n : \u211d)) ^ (n - 1) * (n * (n - 1) / 2)\n              \u2264 (1 / \u2016z\u2016) * 2 * (n ^ 2 / 2) := h3\n            _ = n ^ 2 / \u2016z\u2016 := by ring\n  -- assemble\n  have hfre : (1 - w ^ n).re = n * z.re / \u2016z\u2016 ^ 2 + (((S - n) / z).re) := by\n    rw [hf, hS]\n    have h1 : ((1 / z) * ((n : \u2102) + (S - n))) = (n : \u2102) * (1 / z) + (S - n) / z := by\n      field_simp; ring\n    rw [h1]\n    have hre1 : ((n : \u2102) * (1 / z)).re = n * z.re / \u2016z\u2016 ^ 2 := by\n      have hinv : ((1 : \u2102) / z).re = z.re / \u2016z\u2016 ^ 2 := by\n        rw [one_div, Complex.inv_def]\n        simp [Complex.div_re, Complex.normSq_apply, Complex.norm_eq_abs,\n          Complex.sq_abs]\n      rw [Complex.add_re] at *\n      simp [hinv]\n    rw [Complex.add_re, hre1]\n  rw [hfre]\n  have hA : |n * z.re / \u2016z\u2016 ^ 2| \u2264 n / \u2016z\u2016 ^ 2 := by\n    rw [abs_div, abs_mul, abs_of_nonneg hn0.le, abs_pow]\n    apply div_le_div_of_nonneg_right (by positivity)\n    calc |z.re| \u2264 1 := abs_le.mpr \u27e8by linarith, hzre1\u27e9\n      _ \u2264 _ := by linarith\n  have hB : |((S - n) / z).re| \u2264 n ^ 2 / \u2016z\u2016 ^ 2 := by\n    have h1 : |((S - n) / z).re| \u2264 \u2016(S - n) / z\u2016 := Complex.abs_re_le_norm _\n    have h2 : \u2016(S - n) / z\u2016 \u2264 (n ^ 2 / \u2016z\u2016) / \u2016z\u2016 := by\n      rw [norm_div]\n      apply div_le_div_of_nonneg_right hSn (by positivity)\n    rw [div_pow] at *\n    calc |((S - n) / z).re| \u2264 \u2016(S - n) / z\u2016 := h1\n      _ \u2264 (n ^ 2 / \u2016z\u2016) / \u2016z\u2016 := h2\n      _ = n ^ 2 / \u2016z\u2016 ^ 2 := by ring\n  calc |n * z.re / \u2016z\u2016 ^ 2 + ((S - n) / z).re|\n      \u2264 n / \u2016z\u2016 ^ 2 + n ^ 2 / \u2016z\u2016 ^ 2 := abs_add _ _ |>.trans (add_le_add hA hB)\n    _ \u2264 2 * n ^ 2 / \u2016z\u2016 ^ 2 := by\n        have hn1 : (n : \u211d) \u2264 n ^ 2 := by\n          have : (1 : \u211d) \u2264 n := by exact_mod_cast hn\n          nlinarith\n        field_simp at *\n        nlinarith\n\nend NguyenRH1\n"
}

{
  "command": "cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem star_div\u2080\\|theorem star_div'\\|lemma star_div\u2080\\|lemma star_div'\" Mathlib/Algebra/Star/ | head -4; grep -rn \"theorem Complex.add_conj\\|lemma Complex.add_conj\\|add_conj \" Mathlib/Data/Complex/Basic.lean Mathlib/Analysis/SpecialFunctions/*.lean 2>/dev/null | head -4; grep -rn \"theorem inv_def\\|inv_re \" Mathlib/Data/Complex/Basic.lean | head -4; grep -rn \"theorem geom_sum_mul \\|lemma geom_sum_mul \" Mathlib/Algebra/GeomSum.lean | head -3",
  "description": "Check exact lemma names in vendored Mathlib"
}

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\Scratch.lean",
  "content": "/- Session 015 scratch: ii\u2032-1a \u2014 paired liSummand + summability (D-064 \u00a71).\n   When green, merge into LiSkeleton.lean \u00a71 replacing the defective liCoeff. -/\nimport Mathlib\nimport Zeta23.WeilEF.ZeroSummability\nimport Zeta23.Statement.SeamClosed\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-- The paired (symmetrized) Li summand: \u00bd\u00b7mult\u00b7(f \u03c1 + f \u03c1\u0304),\nf \u03c1 = 1 \u2212 (1\u22121/\u03c1)\u207f. Manifestly real-valued (`liSummand_eq`), absolutely\nsummable (`liSummable`) \u2014 the correct Keiper\u2013Li coefficient (D-064 \u00a71:\nthe plain unordered sum of f over the zeros is not absolutely\nsummable; the pairing is the classical Bombieri\u2013Lagarias convention). -/\ndef liSummand (n : \u2115) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) : \u2102 :=\n  (zeroMult \u03c1 : \u2102) / 2 * ((1 - (1 - 1 / (\u03c1 : \u2102)) ^ n) + (1 - (1 - 1 / (conj (\u03c1 : \u2102))) ^ n))\n\nlemma summand_conj (n : \u2115) (z : \u2102) :\n    (1 : \u2102) - (1 - 1 / (conj z)) ^ n = conj ((1 - (1 - 1 / z) ^ n)) := by\n  rw [map_sub, map_one, map_pow, map_sub, map_one, map_div\u2080, map_one]\n\nlemma liSummand_eq (n : \u2115) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) :\n    liSummand n \u03c1 = (zeroMult \u03c1 : \u2102) * (((1 - (1 - 1 / (\u03c1 : \u2102)) ^ n).re) : \u2102) := by\n  rw [liSummand, summand_conj, Complex.add_conj]\n  ring\n\n/-! ### The geometric-sum estimate: |Re(1 \u2212 w\u207f)| \u2264 2n\u00b2/\u2016z\u2016\u00b2 for w = 1\u22121/z -/\n\n/-- (1 + 1/(2n))\u207f \u2264 2 for n \u2265 1. Via (1+x)\u207f \u2264 exp(nx) and exp(1/2) \u2264 2. -/\nlemma pow_one_add_inv_two_mul_le (n : \u2115) (hn : 1 \u2264 n) :\n    ((1 : \u211d) + 1 / (2 * n)) ^ n \u2264 2 := by\n  have hn0 : (0 : \u211d) < n := by exact_mod_cast hn\n  have hbase : (1 : \u211d) + 1 / (2 * n) \u2264 Real.exp (1 / (2 * n)) :=\n    Real.add_one_le_exp _\n  have h1 : ((1 : \u211d) + 1 / (2 * n)) ^ n \u2264 Real.exp (1 / 2) := by\n    calc ((1 : \u211d) + 1 / (2 * n)) ^ n\n        \u2264 (Real.exp (1 / (2 * n))) ^ n := pow_le_pow_left\u2080 (by positivity) hbase n\n      _ = Real.exp (\u2191n * (1 / (2 * n))) := by rw [Real.exp_nat_mul]\n      _ = Real.exp (1 / 2) := by congr 1; field_simp\n  have h2 : Real.exp (1 / 2 : \u211d) \u2264 2 := by\n    have h3 : (1 : \u211d) / 2 \u2264 Real.log 2 := by\n      have h := Real.log_two_gt_d9\n      linarith\n    calc Real.exp (1 / 2 : \u211d) \u2264 Real.exp (Real.log 2) := Real.exp_le_exp.mpr h3\n      _ = 2 := Real.exp_log (by norm_num)\n  exact h1.trans h2\n\n/-- Key estimate: for n \u2265 1, z with re z \u2208 [0,1], 2n \u2264 \u2016z\u2016:\n|Re(1 \u2212 (1\u22121/z)\u207f)| \u2264 2n\u00b2/\u2016z\u2016\u00b2. -/\nlemma re_li_summand_le {n : \u2115} (hn : 1 \u2264 n) {z : \u2102}\n    (hzre0 : 0 \u2264 z.re) (hzre1 : z.re \u2264 1) (hz : 2 * n \u2264 \u2016z\u2016) :\n    |(1 - (1 - 1 / z) ^ n).re| \u2264 2 * n ^ 2 / \u2016z\u2016 ^ 2 := by\n  have hn0 : (0 : \u211d) < n := by exact_mod_cast hn\n  have hz0 : z \u2260 0 := by\n    intro h; rw [h, norm_zero] at hz; linarith\n  have hzn : (0 : \u211d) < \u2016z\u2016 := norm_pos_iff.mpr hz0\n  have hnsq : Complex.normSq z = \u2016z\u2016 ^ 2 := by\n    rw [Complex.normSq_eq_abs, Complex.norm_eq_abs]\n  set w : \u2102 := 1 - 1 / z with hw\n  -- kernel identity: 1 \u2212 w\u207f = (1/z)\u00b7S, S = \u03a3_{j<n} w^j\n  have hgeom : (w - 1) * \u2211 j \u2208 Finset.range n, w ^ j = w ^ n - 1 := mul_geom_sum w n\n  have hw1 : w - 1 = -(1 / z) := by rw [hw]; ring\n  have hf : (1 : \u2102) - w ^ n = (1 / z) * \u2211 j \u2208 Finset.range n, w ^ j := by\n    have h := hgeom\n    rw [hw1] at h\n    have e := congr_arg Neg.neg h\n    simp only [neg_sub, sub_eq_add_neg] at e \u22a2\n    have e2 : (1 : \u2102) - w ^ n = -(w ^ n - 1) := by ring\n    rw [e2, h]; ring\n  set S : \u2102 := \u2211 j \u2208 Finset.range n, w ^ j\n  have hS : S = (n : \u2102) + (S - n) := by ring\n  -- \u2016w\u2016 \u2264 1 + 1/\u2016z\u2016\n  have hwnorm : \u2016w\u2016 \u2264 1 + 1 / \u2016z\u2016 := by\n    have e : w = 1 - z\u207b\u00b9 := by rw [hw, div_eq_mul_inv, one_mul]\n    rw [e]\n    calc \u2016(1 : \u2102) - z\u207b\u00b9\u2016 \u2264 \u2016(1 : \u2102)\u2016 + \u2016z\u207b\u00b9\u2016 := norm_sub_le _ _\n      _ = 1 + 1 / \u2016z\u2016 := by rw [norm_one, norm_inv, inv_eq_one_div]\n  have h1z : 1 / \u2016z\u2016 \u2264 1 / (2 * n) :=\n    one_div_le_one_div_of_le (by positivity) (by exact_mod_cast hz)\n  have hwmax : max 1 \u2016w\u2016 \u2264 1 + 1 / (2 * n) := by\n    apply max_le (by positivity)\n    exact hwnorm.trans (by linarith)\n  -- |S \u2212 n| \u2264 n\u00b2/\u2016z\u2016\n  have hSn : \u2016S - n\u2016 \u2264 n ^ 2 / \u2016z\u2016 := by\n    have hSn1 : S - n = \u2211 j \u2208 Finset.range n, (w ^ j - 1) := by\n      rw [Finset.sum_sub_distrib]\n      simp [Finset.sum_const, Finset.card_range]\n    have hbound : \u2200 j \u2208 Finset.range n,\n        \u2016w ^ j - 1\u2016 \u2264 (1 / \u2016z\u2016) * (j * (1 + 1 / (2 * n)) ^ (n - 1)) := by\n      intro j hj\n      have hgeomj : (w - 1) * \u2211 i \u2208 Finset.range j, w ^ i = w ^ j - 1 := mul_geom_sum w j\n      have hjn : j \u2264 n := le_of_lt (Finset.mem_range.mp hj)\n      have h2 : w ^ j - 1 = -(1 / z) * \u2211 i \u2208 Finset.range j, w ^ i := by\n        have h := hgeomj; rw [hw1] at h\n        rw [\u2190 h]; ring\n      rw [h2, norm_mul, norm_neg, norm_div, norm_one]\n      have hsum : \u2016\u2211 i \u2208 Finset.range j, w ^ i\u2016 \u2264 j * (1 + 1 / (2 * n)) ^ (n - 1) := by\n        calc \u2016\u2211 i \u2208 Finset.range j, w ^ i\u2016\n            \u2264 \u2211 i \u2208 Finset.range j, \u2016w ^ i\u2016 := norm_sum_le _ _\n          _ \u2264 \u2211 i \u2208 Finset.range j, (1 + 1 / (2 * n)) ^ (n - 1) := by\n              apply Finset.sum_le_sum\n              intro i hi\n              rw [norm_pow]\n              have hij : i < j := Finset.mem_range.mp hi\n              have h1 : \u2016w\u2016 ^ i \u2264 (1 + 1 / (2 * n)) ^ i :=\n                pow_le_pow_left\u2080 (norm_nonneg _)\n                  (le_trans (le_max_right 1 \u2016w\u2016) hwmax) i\n              have h2 : (1 + 1 / (2 * n)) ^ i \u2264 (1 + 1 / (2 * n)) ^ (n - 1) := by\n                apply pow_le_pow_right\u2080 (by positivity)\n                omega\n              exact h1.trans h2\n          _ = j * (1 + 1 / (2 * n)) ^ (n - 1) := by\n              rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul]\n      exact mul_le_mul_of_nonneg_left hsum (by positivity)\n    calc \u2016\u2211 j \u2208 Finset.range n, (w ^ j - 1)\u2016\n        \u2264 \u2211 j \u2208 Finset.range n, \u2016w ^ j - 1\u2016 := norm_sum_le _ _\n      _ \u2264 \u2211 j \u2208 Finset.range n, (1 / \u2016z\u2016) * (j * (1 + 1 / (2 * n)) ^ (n - 1)) :=\n          Finset.sum_le_sum hbound\n      _ = ((1 / \u2016z\u2016) * (1 + 1 / (2 * n)) ^ (n - 1)) * \u2211 j \u2208 Finset.range n, j := by\n          rw [\u2190 Finset.mul_sum]\n          apply Finset.sum_congr rfl\n          intro j _; ring\n      _ \u2264 n ^ 2 / \u2016z\u2016 := by\n          have hsumj : (\u2211 j \u2208 Finset.range n, j : \u211d) = n * (n - 1) / 2 := by\n            have h := Finset.sum_range_id_mul_two n\n            have h2 : ((\u2211 j \u2208 Finset.range n, j : \u2115) : \u211d) * 2 = n * (n - 1) := by\n              exact_mod_cast h\n            linarith\n          rw [hsumj]\n          have hpow : (1 + 1 / (2 * n : \u211d)) ^ (n - 1) \u2264 2 :=\n            le_trans (pow_le_pow_right\u2080 (by positivity) (Nat.sub_le n 1))\n              (pow_one_add_inv_two_mul_le n hn)\n          have hb1 : (n : \u211d) * (n - 1) / 2 \u2264 n ^ 2 / 2 := by nlinarith [hn0]\n          have hb2 : (1 + 1 / (2 * n : \u211d)) ^ (n - 1) * (n * (n - 1) / 2)\n              \u2264 2 * (n ^ 2 / 2) :=\n            mul_le_mul_of_nonneg hpow hb1 (by positivity) (by positivity)\n          have hb3 : ((1 : \u211d) / \u2016z\u2016) * (1 + 1 / (2 * n)) ^ (n - 1) * (n * (n - 1) / 2)\n              \u2264 (1 / \u2016z\u2016) * (2 * (n ^ 2 / 2)) :=\n            mul_le_mul_of_nonneg_left hb2 (by positivity)\n          exact hb3.trans (by ring_nf)\n  -- Re(f) = n\u00b7z.re/\u2016z\u2016\u00b2 + ((S\u2212n)/z).re\n  have hre1 : ((n : \u2102) * (1 / z)).re = n * z.re / \u2016z\u2016 ^ 2 := by\n    have hinv : ((1 : \u2102) / z).re = z.re / \u2016z\u2016 ^ 2 := by\n      rw [one_div, Complex.inv_re, hnsq]\n    rw [Complex.mul_re]\n    simp [hinv]\n  have hfre : (1 - w ^ n).re = n * z.re / \u2016z\u2016 ^ 2 + ((S - n) / z).re := by\n    rw [hf, hS, Complex.add_re]\n    congr 1\n    rw [mul_comm, div_eq_mul_inv]\n    exact hre1\n  rw [hfre]\n  have hA : |n * z.re / \u2016z\u2016 ^ 2| \u2264 n / \u2016z\u2016 ^ 2 := by\n    rw [abs_div, abs_mul, abs_of_nonneg hn0.le,\n      abs_of_nonneg (sq_nonneg \u2016z\u2016)]\n    apply (div_le_div_right (by positivity)).mpr\n    apply mul_le_mul_of_nonneg_left _ hn0.le\n    exact abs_le.mpr \u27e8hzre0, hzre1\u27e9\n  have hB : |((S - n) / z).re| \u2264 n ^ 2 / \u2016z\u2016 ^ 2 := by\n    have h1 : |((S - n) / z).re| \u2264 \u2016(S - n) / z\u2016 := Complex.abs_re_le_norm _\n    have h2 : \u2016(S - n) / z\u2016 \u2264 (n ^ 2 / \u2016z\u2016) / \u2016z\u2016 := by\n      rw [norm_div]\n      exact (div_le_div_right hzn).mpr hSn\n    exact h1.trans (h2.trans (by rw [div_div, \u2190 pow_two]))\n  have hnn : (n : \u211d) \u2264 n ^ 2 := by\n    have h : (1 : \u211d) \u2264 n := by exact_mod_cast hn\n    nlinarith\n  calc |n * z.re / \u2016z\u2016 ^ 2 + ((S - n) / z).re|\n      \u2264 n / \u2016z\u2016 ^ 2 + n ^ 2 / \u2016z\u2016 ^ 2 := (abs_add _ _).trans (add_le_add hA hB)\n    _ = (n + n ^ 2) / \u2016z\u2016 ^ 2 := by ring\n    _ \u2264 2 * n ^ 2 / \u2016z\u2016 ^ 2 := by\n        apply (div_le_div_right (by positivity)).mpr\n        linarith\n\nend NguyenRH1\n"
}

{
  "type": "update",
  "filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\Scratch.lean",
  "content": "/- Session 015 scratch: ii\u2032-1a \u2014 paired liSummand + summability (D-064 \u00a71).\n   When green, merge into LiSkeleton.lean \u00a71 replacing the defective liCoeff. -/\nimport Mathlib\nimport Zeta23.WeilEF.ZeroSummability\nimport Zeta23.Statement.SeamClosed\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-- The paired (symmetrized) Li summand: \u00bd\u00b7mult\u00b7(f \u03c1 + f \u03c1\u0304),\nf \u03c1 = 1 \u2212 (1\u22121/\u03c1)\u207f. Manifestly real-valued (`liSummand_eq`), absolutely\nsummable (`liSummable`) \u2014 the correct Keiper\u2013Li coefficient (D-064 \u00a71:\nthe plain unordered sum of f over the zeros is not absolutely\nsummable; the pairing is the classical Bombieri\u2013Lagarias convention). -/\ndef liSummand (n : \u2115) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) : \u2102 :=\n  (zeroMult \u03c1 : \u2102) / 2 * ((1 - (1 - 1 / (\u03c1 : \u2102)) ^ n) + (1 - (1 - 1 / (conj (\u03c1 : \u2102))) ^ n))\n\nlemma summand_conj (n : \u2115) (z : \u2102) :\n    (1 : \u2102) - (1 - 1 / (conj z)) ^ n = conj ((1 - (1 - 1 / z) ^ n)) := by\n  rw [map_sub, map_one, map_pow, map_sub, map_one, map_div\u2080, map_one]\n\nlemma liSummand_eq (n : \u2115) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) :\n    liSummand n \u03c1 = (zeroMult \u03c1 : \u2102) * (((1 - (1 - 1 / (\u03c1 : \u2102)) ^ n).re) : \u2102) := by\n  rw [liSummand, summand_conj, Complex.add_conj]\n  ring\n\n/-! ### The geometric-sum estimate: |Re(1 \u2212 w\u207f)| \u2264 2n\u00b2/\u2016z\u2016\u00b2 for w = 1\u22121/z -/\n\n/-- (1 + 1/(2n))\u207f \u2264 2 for n \u2265 1. Via (1+x)\u207f \u2264 exp(nx) and exp(1/2) \u2264 2. -/\nlemma pow_one_add_inv_two_mul_le (n : \u2115) (hn : 1 \u2264 n) :\n    ((1 : \u211d) + 1 / (2 * n)) ^ n \u2264 2 := by\n  have hn0 : (0 : \u211d) < n := by exact_mod_cast hn\n  have hbase : (1 : \u211d) + 1 / (2 * n) \u2264 Real.exp (1 / (2 * n)) :=\n    Real.add_one_le_exp _\n  have h1 : ((1 : \u211d) + 1 / (2 * n)) ^ n \u2264 Real.exp (1 / 2) := by\n    calc ((1 : \u211d) + 1 / (2 * n)) ^ n\n        \u2264 (Real.exp (1 / (2 * n))) ^ n := pow_le_pow_left\u2080 (by positivity) hbase n\n      _ = Real.exp (\u2191n * (1 / (2 * n))) := by rw [Real.exp_nat_mul]\n      _ = Real.exp (1 / 2) := by congr 1; field_simp\n  have h2 : Real.exp (1 / 2 : \u211d) \u2264 2 := by\n    have h3 : (1 : \u211d) / 2 \u2264 Real.log 2 := by\n      have h := Real.log_two_gt_d9\n      linarith\n    calc Real.exp (1 / 2 : \u211d) \u2264 Real.exp (Real.log 2) := Real.exp_le_exp.mpr h3\n      _ = 2 := Real.exp_log (by norm_num)\n  exact h1.trans h2\n\n/-- Key estimate: for n \u2265 1, z with re z \u2208 [0,1], 2n \u2264 \u2016z\u2016:\n|Re(1 \u2212 (1\u22121/z)\u207f)| \u2264 2n\u00b2/\u2016z\u2016\u00b2. -/\nlemma re_li_summand_le {n : \u2115} (hn : 1 \u2264 n) {z : \u2102}\n    (hzre0 : 0 \u2264 z.re) (hzre1 : z.re \u2264 1) (hz : 2 * n \u2264 \u2016z\u2016) :\n    |(1 - (1 - 1 / z) ^ n).re| \u2264 2 * n ^ 2 / \u2016z\u2016 ^ 2 := by\n  have hn0 : (0 : \u211d) < n := by exact_mod_cast hn\n  have hz0 : z \u2260 0 := by\n    intro h; rw [h, norm_zero] at hz; linarith\n  have hzn : (0 : \u211d) < \u2016z\u2016 := norm_pos_iff.mpr hz0\n  have hnsq : Complex.normSq z = \u2016z\u2016 ^ 2 := by\n    rw [Complex.normSq_eq_abs, Complex.norm_eq_abs]\n  set w : \u2102 := 1 - 1 / z with hw\n  -- kernel identity: 1 \u2212 w\u207f = (1/z)\u00b7S, S = \u03a3_{j<n} w^j\n  have hgeom : (w - 1) * \u2211 j \u2208 Finset.range n, w ^ j = w ^ n - 1 := mul_geom_sum w n\n  have hw1 : w - 1 = -(1 / z) := by rw [hw]; ring\n  have hf : (1 : \u2102) - w ^ n = (1 / z) * \u2211 j \u2208 Finset.range n, w ^ j := by\n    have h := hgeom\n    rw [hw1] at h\n    have e := congr_arg Neg.neg h\n    simp only [neg_sub, sub_eq_add_neg] at e \u22a2\n    have e2 : (1 : \u2102) - w ^ n = -(w ^ n - 1) := by ring\n    rw [e2, h]; ring\n  set S : \u2102 := \u2211 j \u2208 Finset.range n, w ^ j\n  have hS : S = (n : \u2102) + (S - n) := by ring\n  -- \u2016w\u2016 \u2264 1 + 1/\u2016z\u2016\n  have hwnorm : \u2016w\u2016 \u2264 1 + 1 / \u2016z\u2016 := by\n    have e : w = 1 - z\u207b\u00b9 := by rw [hw, div_eq_mul_inv, one_mul]\n    rw [e]\n    calc \u2016(1 : \u2102) - z\u207b\u00b9\u2016 \u2264 \u2016(1 : \u2102)\u2016 + \u2016z\u207b\u00b9\u2016 := norm_sub_le _ _\n      _ = 1 + 1 / \u2016z\u2016 := by rw [norm_one, norm_inv, inv_eq_one_div]\n  have h1z : 1 / \u2016z\u2016 \u2264 1 / (2 * n) :=\n    one_div_le_one_div_of_le (by positivity) (by exact_mod_cast hz)\n  have hwmax : max 1 \u2016w\u2016 \u2264 1 + 1 / (2 * n) := by\n    apply max_le (by positivity)\n    exact hwnorm.trans (by linarith)\n  -- |S \u2212 n| \u2264 n\u00b2/\u2016z\u2016\n  have hSn : \u2016S - n\u2016 \u2264 n ^ 2 / \u2016z\u2016 := by\n    have hSn1 : S - n = \u2211 j \u2208 Finset.range n, (w ^ j - 1) := by\n      rw [Finset.sum_sub_distrib]\n      simp [Finset.sum_const, Finset.card_range]\n    have hbound : \u2200 j \u2208 Finset.range n,\n        \u2016w ^ j - 1\u2016 \u2264 (1 / \u2016z\u2016) * (j * (1 + 1 / (2 * n)) ^ (n - 1)) := by\n      intro j hj\n      have hgeomj : (w - 1) * \u2211 i \u2208 Finset.range j, w ^ i = w ^ j - 1 := mul_geom_sum w j\n      have hjn : j \u2264 n := le_of_lt (Finset.mem_range.mp hj)\n      have h2 : w ^ j - 1 = -(1 / z) * \u2211 i \u2208 Finset.range j, w ^ i := by\n        have h := hgeomj; rw [hw1] at h\n        rw [\u2190 h]; ring\n      rw [h2, norm_mul, norm_neg, norm_div, norm_one]\n      have hsum : \u2016\u2211 i \u2208 Finset.range j, w ^ i\u2016 \u2264 j * (1 + 1 / (2 * n)) ^ (n - 1) := by\n        calc \u2016\u2211 i \u2208 Finset.range j, w ^ i\u2016\n            \u2264 \u2211 i \u2208 Finset.range j, \u2016w ^ i\u2016 := norm_sum_le _ _\n          _ \u2264 \u2211 i \u2208 Finset.range j, (1 + 1 / (2 * n)) ^ (n - 1) := by\n              apply Finset.sum_le_sum\n              intro i hi\n              rw [norm_pow]\n              have hij : i < j := Finset.mem_range.mp hi\n              have h1 : \u2016w\u2016 ^ i \u2264 (1 + 1 / (2 * n)) ^ i :=\n                pow_le_pow_left\u2080 (norm_nonneg _)\n                  (le_trans (le_max_right 1 \u2016w\u2016) hwmax) i\n              have h2 : (1 + 1 / (2 * n)) ^ i \u2264 (1 + 1 / (2 * n)) ^ (n - 1) := by\n                apply pow_le_pow_right\u2080 (by positivity)\n                omega\n              exact h1.trans h2\n          _ = j * (1 + 1 / (2 * n)) ^ (n - 1) := by\n              rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul]\n      exact mul_le_mul_of_nonneg_left hsum (by positivity)\n    calc \u2016\u2211 j \u2208 Finset.range n, (w ^ j - 1)\u2016\n        \u2264 \u2211 j \u2208 Finset.range n, \u2016w ^ j - 1\u2016 := norm_sum_le _ _\n      _ \u2264 \u2211 j \u2208 Finset.range n, (1 / \u2016z\u2016) * (j * (1 + 1 / (2 * n)) ^ (n - 1)) :=\n          Finset.sum_le_sum hbound\n      _ = ((1 / \u2016z\u2016) * (1 + 1 / (2 * n)) ^ (n - 1)) * \u2211 j \u2208 Finset.range n, j := by\n          rw [\u2190 Finset.mul_sum]\n          apply Finset.sum_congr rfl\n          intro j _; ring\n      _ \u2264 n ^ 2 / \u2016z\u2016 := by\n          have hsumj : (\u2211 j \u2208 Finset.range n, j : \u211d) = n * (n - 1) / 2 := by\n            have h := Finset.sum_range_id_mul_two n\n            have h2 : ((\u2211 j \u2208 Finset.range n, j : \u2115) : \u211d) * 2 = n * (n - 1) := by\n              exact_mod_cast h\n            linarith\n          rw [hsumj]\n          have hpow : (1 + 1 / (2 * n : \u211d)) ^ (n - 1) \u2264 2 :=\n            le_trans (pow_le_pow_right\u2080 (by positivity) (Nat.sub_le n 1))\n              (pow_one_add_inv_two_mul_le n hn)\n          have hb1 : (n : \u211d) * (n - 1) / 2 \u2264 n ^ 2 / 2 := by nlinarith [hn0]\n          have hb2 : (1 + 1 / (2 * n : \u211d)) ^ (n - 1) * (n * (n - 1) / 2)\n              \u2264 2 * (n ^ 2 / 2) :=\n            mul_le_mul_of_nonneg hpow hb1 (by positivity) (by positivity)\n          have hb3 : ((1 : \u211d) / \u2016z\u2016) * (1 + 1 / (2 * n)) ^ (n - 1) * (n * (n - 1) / 2)\n              \u2264 (1 / \u2016z\u2016) * (2 * (n ^ 2 / 2)) :=\n            mul_le_mul_of_nonneg_left hb2 (by positivity)\n          exact hb3.trans (by ring_nf)\n  -- Re(f) = n\u00b7z.re/\u2016z\u2016\u00b2 + ((S\u2212n)/z).re\n  have hre1 : ((n : \u2102) * (1 / z)).re = n * z.re / \u2016z\u2016 ^ 2 := by\n    have hinv : ((1 : \u2102) / z).re = z.re / \u2016z\u2016 ^ 2 := by\n      rw [one_div, Complex.inv_re, hnsq]\n    rw [Complex.mul_re]\n    simp [hinv]\n  have hfre : (1 - w ^ n).re = n * z.re / \u2016z\u2016 ^ 2 + ((S - n) / z).re := by\n    rw [hf, hS, Complex.add_re]\n    congr 1\n    rw [mul_comm, div_eq_mul_inv]\n    exact hre1\n  rw [hfre]\n  have hA : |n * z.re / \u2016z\u2016 ^ 2| \u2264 n / \u2016z\u2016 ^ 2 := by\n    rw [abs_div, abs_mul, abs_of_nonneg hn0.le,\n      abs_of_nonneg (sq_nonneg \u2016z\u2016)]\n    apply (div_le_div_right (by positivity)).mpr\n    apply mul_le_mul_of_nonneg_left _ hn0.le\n    exact abs_le.mpr \u27e8hzre0, hzre1\u27e9\n  have hB : |((S - n) / z).re| \u2264 n ^ 2 / \u2016z\u2016 ^ 2 := by\n    have h1 : |((S - n) / z).re| \u2264 \u2016(S - n) / z\u2016 := Complex.abs_re_le_norm _\n    have h2 : \u2016(S - n) / z\u2016 \u2264 (n ^ 2 / \u2016z\u2016) / \u2016z\u2016 := by\n      rw [norm_div]\n      exact (div_le_div_right hzn).mpr hSn\n    exact h1.trans (h2.trans (by rw [div_div, \u2190 pow_two]))\n  have hnn : (n : \u211d) \u2264 n ^ 2 := by\n    have h : (1 : \u211d) \u2264 n := by exact_mod_cast hn\n    nlinarith\n  calc |n * z.re / \u2016z\u2016 ^ 2 + ((S - n) / z).re|\n      \u2264 n / \u2016z\u2016 ^ 2 + n ^ 2 / \u2016z\u2016 ^ 2 := (abs_add _ _).trans (add_le_add hA hB)\n    _ = (n + n ^ 2) / \u2016z\u2016 ^ 2 := by ring\n    _ \u2264 2 * n ^ 2 / \u2016z\u2016 ^ 2 := by\n        apply (div_le_div_right (by positivity)).mpr\n        linarith\n\nend NguyenRH1\n",
  "structuredPatch": [
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      "lines": [
        " the plain unordered sum of f over the zeros is not absolutely",
        " summable; the pairing is the classical Bombieri\u2013Lagarias convention). -/",
        " def liSummand (n : \u2115) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) : \u2102 :=",
        "-  (zeroMult \u03c1 : \u2102) / 2 * ((1 - (1 - 1 / (\u03c1 : \u2102)) ^ n) + (1 - (1 - 1 / (star (\u03c1 : \u2102))) ^ n))",
        "+  (zeroMult \u03c1 : \u2102) / 2 * ((1 - (1 - 1 / (\u03c1 : \u2102)) ^ n) + (1 - (1 - 1 / (conj (\u03c1 : \u2102))) ^ n))",
        " ",
        "-lemma summand_star (n : \u2115) (z : \u2102) :",
        "-    (1 : \u2102) - (1 - 1 / (star z)) ^ n = star ((1 - (1 - 1 / z) ^ n)) := by",
        "+lemma summand_conj (n : \u2115) (z : \u2102) :",
        "+    (1 : \u2102) - (1 - 1 / (conj z)) ^ n = conj ((1 - (1 - 1 / z) ^ n)) := by",
        "   rw [map_sub, map_one, map_pow, map_sub, map_one, map_div\u2080, map_one]",
        " ",
        " lemma liSummand_eq (n : \u2115) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) :",
        "     liSummand n \u03c1 = (zeroMult \u03c1 : \u2102) * (((1 - (1 - 1 / (\u03c1 : \u2102)) ^ n).re) : \u2102) := by",
        "-  rw [liSummand, summand_star, Complex.add_conj]",
        "+  rw [liSummand, summand_conj, Complex.add_conj]",
        "   ring",
        " ",
        " /-! ### The geometric-sum estimate: |Re(1 \u2212 w\u207f)| \u2264 2n\u00b2/\u2016z\u2016\u00b2 for w = 1\u22121/z -/"
      ]
    },
    {
      "oldStart": 33,
      "oldLines": 26,
      "newStart": 33,
      "newLines": 20,
      "lines": [
        " lemma pow_one_add_inv_two_mul_le (n : \u2115) (hn : 1 \u2264 n) :",
        "     ((1 : \u211d) + 1 / (2 * n)) ^ n \u2264 2 := by",
        "   have hn0 : (0 : \u211d) < n := by exact_mod_cast hn",
        "-  have h1 : ((1 : \u211d) + 1 / (2 * n)) ^ n \u2264 Real.exp (n * (1 / (2 * n))) := by",
        "-    have h := Real.add_one_le_exp (1 / (2 * (n : \u211d)))",
        "+  have hbase : (1 : \u211d) + 1 / (2 * n) \u2264 Real.exp (1 / (2 * n)) :=",
        "+    Real.add_one_le_exp _",
        "+  have h1 : ((1 : \u211d) + 1 / (2 * n)) ^ n \u2264 Real.exp (1 / 2) := by",
        "     calc ((1 : \u211d) + 1 / (2 * n)) ^ n",
        "-        \u2264 (Real.exp (1 / (2 * n))) ^ n := by",
        "-          gcongr",
        "-          \u00b7 linarith",
        "-          \u00b7 exact h",
        "-      _ = Real.exp (n * (1 / (2 * n))) := by",
        "-          rw [\u2190 Real.exp_nat_mul]",
        "-  have h2 : (n : \u211d) * (1 / (2 * n)) = 1 / 2 := by field_simp",
        "-  rw [h2] at h1",
        "-  have h3 : Real.exp (1 / 2 : \u211d) \u2264 2 := by",
        "-    have hsq : (Real.exp (1 / 2 : \u211d)) ^ 2 \u2264 (2 : \u211d) ^ 2 := by",
        "-      rw [\u2190 Real.exp_nat_mul]",
        "-      have : Real.exp (2 * (1 / 2 : \u211d)) = Real.exp 1 := by ring_nf",
        "-      rw [this]",
        "-      have h := Real.exp_one_lt_d9",
        "-      norm_num at h \u22a2; linarith",
        "-    nlinarith [Real.exp_pos (1 / 2 : \u211d)]",
        "-  exact h1.trans h3",
        "+        \u2264 (Real.exp (1 / (2 * n))) ^ n := pow_le_pow_left\u2080 (by positivity) hbase n",
        "+      _ = Real.exp (\u2191n * (1 / (2 * n))) := by rw [Real.exp_nat_mul]",
        "+      _ = Real.exp (1 / 2) := by congr 1; field_simp",
        "+  have h2 : Real.exp (1 / 2 : \u211d) \u2264 2 := by",
        "+    have h3 : (1 : \u211d) / 2 \u2264 Real.log 2 := by",
        "+      have h := Real.log_two_gt_d9",
        "+      linarith",
        "+    calc Real.exp (1 / 2 : \u211d) \u2264 Real.exp (Real.log 2) := Real.exp_le_exp.mpr h3",
        "+      _ = 2 := Real.exp_log (by norm_num)",
        "+  exact h1.trans h2",
        " ",
        " /-- Key estimate: for n \u2265 1, z with re z \u2208 [0,1], 2n \u2264 \u2016z\u2016:",
        " |Re(1 \u2212 (1\u22121/z)\u207f)| \u2264 2n\u00b2/\u2016z\u2016\u00b2. -/"
      ]
    },
    {
      "oldStart": 63,
      "oldLines": 41,
      "newStart": 57,
      "newLines": 46,
      "lines": [
        "   have hz0 : z \u2260 0 := by",
        "     intro h; rw [h, norm_zero] at hz; linarith",
        "   have hzn : (0 : \u211d) < \u2016z\u2016 := norm_pos_iff.mpr hz0",
        "+  have hnsq : Complex.normSq z = \u2016z\u2016 ^ 2 := by",
        "+    rw [Complex.normSq_eq_abs, Complex.norm_eq_abs]",
        "   set w : \u2102 := 1 - 1 / z with hw",
        "   -- kernel identity: 1 \u2212 w\u207f = (1/z)\u00b7S, S = \u03a3_{j<n} w^j",
        "-  have hgeom : w ^ n - 1 = (w - 1) * \u2211 j \u2208 Finset.range n, w ^ j := geom_sum_mul w n",
        "+  have hgeom : (w - 1) * \u2211 j \u2208 Finset.range n, w ^ j = w ^ n - 1 := mul_geom_sum w n",
        "   have hw1 : w - 1 = -(1 / z) := by rw [hw]; ring",
        "   have hf : (1 : \u2102) - w ^ n = (1 / z) * \u2211 j \u2208 Finset.range n, w ^ j := by",
        "-    have : (1 : \u2102) - w ^ n = -(w - 1) * \u2211 j \u2208 Finset.range n, w ^ j := by",
        "-      rw [\u2190 hgeom]; ring",
        "-    rw [this, hw1]; ring",
        "+    have h := hgeom",
        "+    rw [hw1] at h",
        "+    have e := congr_arg Neg.neg h",
        "+    simp only [neg_sub, sub_eq_add_neg] at e \u22a2",
        "+    have e2 : (1 : \u2102) - w ^ n = -(w ^ n - 1) := by ring",
        "+    rw [e2, h]; ring",
        "   set S : \u2102 := \u2211 j \u2208 Finset.range n, w ^ j",
        "-  -- S = n + (S \u2212 n); Re(f) = n\u00b7Re(1/z) + Re((S\u2212n)/z)",
        "   have hS : S = (n : \u2102) + (S - n) := by ring",
        "+  -- \u2016w\u2016 \u2264 1 + 1/\u2016z\u2016",
        "+  have hwnorm : \u2016w\u2016 \u2264 1 + 1 / \u2016z\u2016 := by",
        "+    have e : w = 1 - z\u207b\u00b9 := by rw [hw, div_eq_mul_inv, one_mul]",
        "+    rw [e]",
        "+    calc \u2016(1 : \u2102) - z\u207b\u00b9\u2016 \u2264 \u2016(1 : \u2102)\u2016 + \u2016z\u207b\u00b9\u2016 := norm_sub_le _ _",
        "+      _ = 1 + 1 / \u2016z\u2016 := by rw [norm_one, norm_inv, inv_eq_one_div]",
        "+  have h1z : 1 / \u2016z\u2016 \u2264 1 / (2 * n) :=",
        "+    one_div_le_one_div_of_le (by positivity) (by exact_mod_cast hz)",
        "+  have hwmax : max 1 \u2016w\u2016 \u2264 1 + 1 / (2 * n) := by",
        "+    apply max_le (by positivity)",
        "+    exact hwnorm.trans (by linarith)",
        "   -- |S \u2212 n| \u2264 n\u00b2/\u2016z\u2016",
        "-  have hwnorm : \u2016w\u2016 \u2264 1 + 1 / \u2016z\u2016 := by",
        "-    have : w = 1 - z\u207b\u00b9 := by rw [hw, div_eq_mul_inv, one_mul]",
        "-    rw [this]",
        "-    calc \u2016(1 : \u2102) - z\u207b\u00b9\u2016 \u2264 \u2016(1 : \u2102)\u2016 + \u2016z\u207b\u00b9\u2016 := norm_sub_le _ _",
        "-      _ = 1 + 1 / \u2016z\u2016 := by rw [norm_one, norm_inv]",
        "-  have hwmax : max 1 \u2016w\u2016 \u2264 1 + 1 / (2 * n) := by",
        "-    have h1 : 1 / \u2016z\u2016 \u2264 1 / (2 * n) := by",
        "-      apply one_div_le_one_div_of_le (by positivity : (0:\u211d) < 2 * n)",
        "-      exact_mod_cast hz",
        "-    have h2 : \u2016w\u2016 \u2264 1 + 1 / (2 * n) := hwnorm.trans (by linarith)",
        "-    apply max_le",
        "-    \u00b7 linarith",
        "-    \u00b7 exact h2",
        "   have hSn : \u2016S - n\u2016 \u2264 n ^ 2 / \u2016z\u2016 := by",
        "     have hSn1 : S - n = \u2211 j \u2208 Finset.range n, (w ^ j - 1) := by",
        "       rw [Finset.sum_sub_distrib]",
        "       simp [Finset.sum_const, Finset.card_range]",
        "-    rw [hSn1]",
        "     have hbound : \u2200 j \u2208 Finset.range n,",
        "         \u2016w ^ j - 1\u2016 \u2264 (1 / \u2016z\u2016) * (j * (1 + 1 / (2 * n)) ^ (n - 1)) := by",
        "       intro j hj",
        "-      have hgeomj : w ^ j - 1 = (w - 1) * \u2211 i \u2208 Finset.range j, w ^ i := geom_sum_mul w j",
        "-      rw [hgeomj, norm_mul, hw1, norm_neg, norm_div, norm_one]",
        "+      have hgeomj : (w - 1) * \u2211 i \u2208 Finset.range j, w ^ i = w ^ j - 1 := mul_geom_sum w j",
        "+      have hjn : j \u2264 n := le_of_lt (Finset.mem_range.mp hj)",
        "+      have h2 : w ^ j - 1 = -(1 / z) * \u2211 i \u2208 Finset.range j, w ^ i := by",
        "+        have h := hgeomj; rw [hw1] at h",
        "+        rw [\u2190 h]; ring",
        "+      rw [h2, norm_mul, norm_neg, norm_div, norm_one]",
        "       have hsum : \u2016\u2211 i \u2208 Finset.range j, w ^ i\u2016 \u2264 j * (1 + 1 / (2 * n)) ^ (n - 1) := by",
        "         calc \u2016\u2211 i \u2208 Finset.range j, w ^ i\u2016",
        "             \u2264 \u2211 i \u2208 Finset.range j, \u2016w ^ i\u2016 := norm_sum_le _ _"
      ]
    },
    {
      "oldStart": 105,
      "oldLines": 22,
      "newStart": 104,
      "newLines": 25,
      "lines": [
        "               apply Finset.sum_le_sum",
        "               intro i hi",
        "               rw [norm_pow]",
        "-              have hjn : i \u2264 n - 1 := by",
        "-                have hij : i < j := Finset.mem_range.mp hi",
        "-                have hjn' : j \u2264 n := le_of_lt (Finset.mem_range.mp hj)",
        "+              have hij : i < j := Finset.mem_range.mp hi",
        "+              have h1 : \u2016w\u2016 ^ i \u2264 (1 + 1 / (2 * n)) ^ i :=",
        "+                pow_le_pow_left\u2080 (norm_nonneg _)",
        "+                  (le_trans (le_max_right 1 \u2016w\u2016) hwmax) i",
        "+              have h2 : (1 + 1 / (2 * n)) ^ i \u2264 (1 + 1 / (2 * n)) ^ (n - 1) := by",
        "+                apply pow_le_pow_right\u2080 (by positivity)",
        "                 omega",
        "-              exact pow_le_pow_left\u2080 (by positivity)",
        "-                (le_trans (le_max_right 1 \u2016w\u2016) hwmax) i (by omega)",
        "+              exact h1.trans h2",
        "           _ = j * (1 + 1 / (2 * n)) ^ (n - 1) := by",
        "               rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul]",
        "       exact mul_le_mul_of_nonneg_left hsum (by positivity)",
        "     calc \u2016\u2211 j \u2208 Finset.range n, (w ^ j - 1)\u2016",
        "         \u2264 \u2211 j \u2208 Finset.range n, \u2016w ^ j - 1\u2016 := norm_sum_le _ _",
        "-      _ \u2264 \u2211 j \u2208 Finset.range n, (1 / \u2016z\u2016) * (j * (1 + 1 / (2 * n)) ^ (n - 1)) := by",
        "-          exact Finset.sum_le_sum hbound",
        "-      _ = (1 / \u2016z\u2016) * (1 + 1 / (2 * n)) ^ (n - 1) * \u2211 j \u2208 Finset.range n, j := by",
        "-          rw [Finset.mul_sum]; congr 1; rw [Finset.mul_sum]",
        "-          apply Finset.sum_congr rfl; intro j _; ring",
        "+      _ \u2264 \u2211 j \u2208 Finset.range n, (1 / \u2016z\u2016) * (j * (1 + 1 / (2 * n)) ^ (n - 1)) :=",
        "+          Finset.sum_le_sum hbound",
        "+      _ = ((1 / \u2016z\u2016) * (1 + 1 / (2 * n)) ^ (n - 1)) * \u2211 j \u2208 Finset.range n, j := by",
        "+          rw [\u2190 Finset.mul_sum]",
        "+          apply Finset.sum_congr rfl",
        "+          intro j _; ring",
        "       _ \u2264 n ^ 2 / \u2016z\u2016 := by",
        "           have hsumj : (\u2211 j \u2208 Finset.range n, j : \u211d) = n * (n - 1) / 2 := by",
        "             have h := Finset.sum_range_id_mul_two n"
      ]
    },
    {
      "oldStart": 128,
      "oldLines": 63,
      "newStart": 130,
      "newLines": 50,
      "lines": [
        "               exact_mod_cast h",
        "             linarith",
        "           rw [hsumj]",
        "-          have hpow : (1 + 1 / (2 * n : \u211d)) ^ (n - 1) \u2264 2 := by",
        "-            have h := pow_one_add_inv_two_mul_le n hn",
        "-            calc (1 + 1 / (2 * n : \u211d)) ^ (n - 1)",
        "-                \u2264 (1 + 1 / (2 * n : \u211d)) ^ n := by",
        "-                  apply pow_le_pow_right\u2080 (by positivity)",
        "-                  \u00b7 have : 1 \u2264 (1 : \u211d) + 1 / (2 * n) := by positivity",
        "-                    exact this",
        "-                  \u00b7 omega",
        "-              _ \u2264 2 := h",
        "-          have h3 : (1 / \u2016z\u2016) * (1 + 1 / (2 * n : \u211d)) ^ (n - 1) * (n * (n - 1) / 2)",
        "-              \u2264 (1 / \u2016z\u2016) * 2 * (n ^ 2 / 2) := by",
        "-            apply mul_le_mul_of_nonneg_left _ (by positivity)",
        "-            apply mul_le_mul_of_nonneg_left hpow (by positivity)",
        "-            \u00b7 have : (n : \u211d) * (n - 1) / 2 \u2264 n ^ 2 / 2 := by",
        "-                have hn1 : (1 : \u211d) \u2264 n := by exact_mod_cast hn",
        "-                nlinarith",
        "-              linarith",
        "-          calc (1 / \u2016z\u2016) * (1 + 1 / (2 * n : \u211d)) ^ (n - 1) * (n * (n - 1) / 2)",
        "-              \u2264 (1 / \u2016z\u2016) * 2 * (n ^ 2 / 2) := h3",
        "-            _ = n ^ 2 / \u2016z\u2016 := by ring",
        "-  -- assemble",
        "-  have hfre : (1 - w ^ n).re = n * z.re / \u2016z\u2016 ^ 2 + (((S - n) / z).re) := by",
        "-    rw [hf, hS]",
        "-    have h1 : ((1 / z) * ((n : \u2102) + (S - n))) = (n : \u2102) * (1 / z) + (S - n) / z := by",
        "-      field_simp; ring",
        "-    rw [h1]",
        "-    have hre1 : ((n : \u2102) * (1 / z)).re = n * z.re / \u2016z\u2016 ^ 2 := by",
        "-      have hinv : ((1 : \u2102) / z).re = z.re / \u2016z\u2016 ^ 2 := by",
        "-        rw [one_div, Complex.inv_def]",
        "-        simp [Complex.div_re, Complex.normSq_apply, Complex.norm_eq_abs,",
        "-          Complex.sq_abs]",
        "-      rw [Complex.add_re] at *",
        "-      simp [hinv]",
        "-    rw [Complex.add_re, hre1]",
        "+          have hpow : (1 + 1 / (2 * n : \u211d)) ^ (n - 1) \u2264 2 :=",
        "+            le_trans (pow_le_pow_right\u2080 (by positivity) (Nat.sub_le n 1))",
        "+              (pow_one_add_inv_two_mul_le n hn)",
        "+          have hb1 : (n : \u211d) * (n - 1) / 2 \u2264 n ^ 2 / 2 := by nlinarith [hn0]",
        "+          have hb2 : (1 + 1 / (2 * n : \u211d)) ^ (n - 1) * (n * (n - 1) / 2)",
        "+              \u2264 2 * (n ^ 2 / 2) :=",
        "+            mul_le_mul_of_nonneg hpow hb1 (by positivity) (by positivity)",
        "+          have hb3 : ((1 : \u211d) / \u2016z\u2016) * (1 + 1 / (2 * n)) ^ (n - 1) * (n * (n - 1) / 2)",
        "+              \u2264 (1 / \u2016z\u2016) * (2 * (n ^ 2 / 2)) :=",
        "+            mul_le_mul_of_nonneg_left hb2 (by positivity)",
        "+          exact hb3.trans (by ring_nf)",
        "+  -- Re(f) = n\u00b7z.re/\u2016z\u2016\u00b2 + ((S\u2212n)/z).re",
        "+  have hre1 : ((n : \u2102) * (1 / z)).re = n * z.re / \u2016z\u2016 ^ 2 := by",
        "+    have hinv : ((1 : \u2102) / z).re = z.re / \u2016z\u2016 ^ 2 := by",
        "+      rw [one_div, Complex.inv_re, hnsq]",
        "+    rw [Complex.mul_re]",
        "+    simp [hinv]",
        "+  have hfre : (1 - w ^ n).re = n * z.re / \u2016z\u2016 ^ 2 + ((S - n) / z).re := by",
        "+    rw [hf, hS, Complex.add_re]",
        "+    congr 1",
        "+    rw [mul_comm, div_eq_mul_inv]",
        "+    exact hre1",
        "   rw [hfre]",
        "   have hA : |n * z.re / \u2016z\u2016 ^ 2| \u2264 n / \u2016z\u2016 ^ 2 := by",
        "-    rw [abs_div, abs_mul, abs_of_nonneg hn0.le, abs_pow]",
        "-    apply div_le_div_of_nonneg_right (by positivity)",
        "-    calc |z.re| \u2264 1 := abs_le.mpr \u27e8by linarith, hzre1\u27e9",
        "-      _ \u2264 _ := by linarith",
        "+    rw [abs_div, abs_mul, abs_of_nonneg hn0.le,",
        "+      abs_of_nonneg (sq_nonneg \u2016z\u2016)]",
        "+    apply (div_le_div_right (by positivity)).mpr",
        "+    apply mul_le_mul_of_nonneg_left _ hn0.le",
        "+    exact abs_le.mpr \u27e8hzre0, hzre1\u27e9",
        "   have hB : |((S - n) / z).re| \u2264 n ^ 2 / \u2016z\u2016 ^ 2 := by",
        "     have h1 : |((S - n) / z).re| \u2264 \u2016(S - n) / z\u2016 := Complex.abs_re_le_norm _",
        "     have h2 : \u2016(S - n) / z\u2016 \u2264 (n ^ 2 / \u2016z\u2016) / \u2016z\u2016 := by",
        "       rw [norm_div]",
        "-      apply div_le_div_of_nonneg_right hSn (by positivity)",
        "-    rw [div_pow] at *",
        "-    calc |((S - n) / z).re| \u2264 \u2016(S - n) / z\u2016 := h1",
        "-      _ \u2264 (n ^ 2 / \u2016z\u2016) / \u2016z\u2016 := h2",
        "-      _ = n ^ 2 / \u2016z\u2016 ^ 2 := by ring",
        "+      exact (div_le_div_right hzn).mpr hSn",
        "+    exact h1.trans (h2.trans (by rw [div_div, \u2190 pow_two]))",
        "+  have hnn : (n : \u211d) \u2264 n ^ 2 := by",
        "+    have h : (1 : \u211d) \u2264 n := by exact_mod_cast hn",
        "+    nlinarith",
        "   calc |n * z.re / \u2016z\u2016 ^ 2 + ((S - n) / z).re|",
        "-      \u2264 n / \u2016z\u2016 ^ 2 + n ^ 2 / \u2016z\u2016 ^ 2 := abs_add _ _ |>.trans (add_le_add hA hB)",
        "+      \u2264 n / \u2016z\u2016 ^ 2 + n ^ 2 / \u2016z\u2016 ^ 2 := (abs_add _ _).trans (add_le_add hA hB)",
        "+    _ = (n + n ^ 2) / \u2016z\u2016 ^ 2 := by ring",
        "     _ \u2264 2 * n ^ 2 / \u2016z\u2016 ^ 2 := by",
        "-        have hn1 : (n : \u211d) \u2264 n ^ 2 := by",
        "-          have : (1 : \u211d) \u2264 n := by exact_mod_cast hn",
        "-          nlinarith",
        "-        field_simp at *",
        "-        nlinarith",
        "+        apply (div_le_div_right (by positivity)).mpr",
        "+        linarith",
        " ",
        " end NguyenRH1",
        " "
      ]
    }
  ],
  "originalFile": "/- Session 015 scratch: ii\u2032-1a \u2014 paired liSummand + summability (D-064 \u00a71).\n   When green, merge into LiSkeleton.lean \u00a71 replacing the defective liCoeff. -/\nimport Mathlib\nimport Zeta23.WeilEF.ZeroSummability\nimport Zeta23.Statement.SeamClosed\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-- The paired (symmetrized) Li summand: \u00bd\u00b7mult\u00b7(f \u03c1 + f \u03c1\u0304),\nf \u03c1 = 1 \u2212 (1\u22121/\u03c1)\u207f. Manifestly real-valued (`liSummand_eq`), absolutely\nsummable (`liSummable`) \u2014 the correct Keiper\u2013Li coefficient (D-064 \u00a71:\nthe plain unordered sum of f over the zeros is not absolutely\nsummable; the pairing is the classical Bombieri\u2013Lagarias convention). -/\ndef liSummand (n : \u2115) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) : \u2102 :=\n  (zeroMult \u03c1 : \u2102) / 2 * ((1 - (1 - 1 / (\u03c1 : \u2102)) ^ n) + (1 - (1 - 1 / (star (\u03c1 : \u2102))) ^ n))\n\nlemma summand_star (n : \u2115) (z : \u2102) :\n    (1 : \u2102) - (1 - 1 / (star z)) ^ n = star ((1 - (1 - 1 / z) ^ n)) := by\n  rw [map_sub, map_one, map_pow, map_sub, map_one, map_div\u2080, map_one]\n\nlemma liSummand_eq (n : \u2115) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) :\n    liSummand n \u03c1 = (zeroMult \u03c1 : \u2102) * (((1 - (1 - 1 / (\u03c1 : \u2102)) ^ n).re) : \u2102) := by\n  rw [liSummand, summand_star, Complex.add_conj]\n  ring\n\n/-! ### The geometric-sum estimate: |Re(1 \u2212 w\u207f)| \u2264 2n\u00b2/\u2016z\u2016\u00b2 for w = 1\u22121/z -/\n\n/-- (1 + 1/(2n))\u207f \u2264 2 for n \u2265 1. Via (1+x)\u207f \u2264 exp(nx) and exp(1/2) \u2264 2. -/\nlemma pow_one_add_inv_two_mul_le (n : \u2115) (hn : 1 \u2264 n) :\n    ((1 : \u211d) + 1 / (2 * n)) ^ n \u2264 2 := by\n  have hn0 : (0 : \u211d) < n := by exact_mod_cast hn\n  have h1 : ((1 : \u211d) + 1 / (2 * n)) ^ n \u2264 Real.exp (n * (1 / (2 * n))) := by\n    have h := Real.add_one_le_exp (1 / (2 * (n : \u211d)))\n    calc ((1 : \u211d) + 1 / (2 * n)) ^ n\n        \u2264 (Real.exp (1 / (2 * n))) ^ n := by\n          gcongr\n          \u00b7 linarith\n          \u00b7 exact h\n      _ = Real.exp (n * (1 / (2 * n))) := by\n          rw [\u2190 Real.exp_nat_mul]\n  have h2 : (n : \u211d) * (1 / (2 * n)) = 1 / 2 := by field_simp\n  rw [h2] at h1\n  have h3 : Real.exp (1 / 2 : \u211d) \u2264 2 := by\n    have hsq : (Real.exp (1 / 2 : \u211d)) ^ 2 \u2264 (2 : \u211d) ^ 2 := by\n      rw [\u2190 Real.exp_nat_mul]\n      have : Real.exp (2 * (1 / 2 : \u211d)) = Real.exp 1 := by ring_nf\n      rw [this]\n      have h := Real.exp_one_lt_d9\n      norm_num at h \u22a2; linarith\n    nlinarith [Real.exp_pos (1 / 2 : \u211d)]\n  exact h1.trans h3\n\n/-- Key estimate: for n \u2265 1, z with re z \u2208 [0,1], 2n \u2264 \u2016z\u2016:\n|Re(1 \u2212 (1\u22121/z)\u207f)| \u2264 2n\u00b2/\u2016z\u2016\u00b2. -/\nlemma re_li_summand_le {n : \u2115} (hn : 1 \u2264 n) {z : \u2102}\n    (hzre0 : 0 \u2264 z.re) (hzre1 : z.re \u2264 1) (hz : 2 * n \u2264 \u2016z\u2016) :\n    |(1 - (1 - 1 / z) ^ n).re| \u2264 2 * n ^ 2 / \u2016z\u2016 ^ 2 := by\n  have hn0 : (0 : \u211d) < n := by exact_mod_cast hn\n  have hz0 : z \u2260 0 := by\n    intro h; rw [h, norm_zero] at hz; linarith\n  have hzn : (0 : \u211d) < \u2016z\u2016 := norm_pos_iff.mpr hz0\n  set w : \u2102 := 1 - 1 / z with hw\n  -- kernel identity: 1 \u2212 w\u207f = (1/z)\u00b7S, S = \u03a3_{j<n} w^j\n  have hgeom : w ^ n - 1 = (w - 1) * \u2211 j \u2208 Finset.range n, w ^ j := geom_sum_mul w n\n  have hw1 : w - 1 = -(1 / z) := by rw [hw]; ring\n  have hf : (1 : \u2102) - w ^ n = (1 / z) * \u2211 j \u2208 Finset.range n, w ^ j := by\n    have : (1 : \u2102) - w ^ n = -(w - 1) * \u2211 j \u2208 Finset.range n, w ^ j := by\n      rw [\u2190 hgeom]; ring\n    rw [this, hw1]; ring\n  set S : \u2102 := \u2211 j \u2208 Finset.range n, w ^ j\n  -- S = n + (S \u2212 n); Re(f) = n\u00b7Re(1/z) + Re((S\u2212n)/z)\n  have hS : S = (n : \u2102) + (S - n) := by ring\n  -- |S \u2212 n| \u2264 n\u00b2/\u2016z\u2016\n  have hwnorm : \u2016w\u2016 \u2264 1 + 1 / \u2016z\u2016 := by\n    have : w = 1 - z\u207b\u00b9 := by rw [hw, div_eq_mul_inv, one_mul]\n    rw [this]\n    calc \u2016(1 : \u2102) - z\u207b\u00b9\u2016 \u2264 \u2016(1 : \u2102)\u2016 + \u2016z\u207b\u00b9\u2016 := norm_sub_le _ _\n      _ = 1 + 1 / \u2016z\u2016 := by rw [norm_one, norm_inv]\n  have hwmax : max 1 \u2016w\u2016 \u2264 1 + 1 / (2 * n) := by\n    have h1 : 1 / \u2016z\u2016 \u2264 1 / (2 * n) := by\n      apply one_div_le_one_div_of_le (by positivity : (0:\u211d) < 2 * n)\n      exact_mod_cast hz\n    have h2 : \u2016w\u2016 \u2264 1 + 1 / (2 * n) := hwnorm.trans (by linarith)\n    apply max_le\n    \u00b7 linarith\n    \u00b7 exact h2\n  have hSn : \u2016S - n\u2016 \u2264 n ^ 2 / \u2016z\u2016 := by\n    have hSn1 : S - n = \u2211 j \u2208 Finset.range n, (w ^ j - 1) := by\n      rw [Finset.sum_sub_distrib]\n      simp [Finset.sum_const, Finset.card_range]\n    rw [hSn1]\n    have hbound : \u2200 j \u2208 Finset.range n,\n        \u2016w ^ j - 1\u2016 \u2264 (1 / \u2016z\u2016) * (j * (1 + 1 / (2 * n)) ^ (n - 1)) := by\n      intro j hj\n      have hgeomj : w ^ j - 1 = (w - 1) * \u2211 i \u2208 Finset.range j, w ^ i := geom_sum_mul w j\n      rw [hgeomj, norm_mul, hw1, norm_neg, norm_div, norm_one]\n      have hsum : \u2016\u2211 i \u2208 Finset.range j, w ^ i\u2016 \u2264 j * (1 + 1 / (2 * n)) ^ (n - 1) := by\n        calc \u2016\u2211 i \u2208 Finset.range j, w ^ i\u2016\n            \u2264 \u2211 i \u2208 Finset.range j, \u2016w ^ i\u2016 := norm_sum_le _ _\n          _ \u2264 \u2211 i \u2208 Finset.range j, (1 + 1 / (2 * n)) ^ (n - 1) := by\n              apply Finset.sum_le_sum\n              intro i hi\n              rw [norm_pow]\n              have hjn : i \u2264 n - 1 := by\n                have hij : i < j := Finset.mem_range.mp hi\n                have hjn' : j \u2264 n := le_of_lt (Finset.mem_range.mp hj)\n                omega\n              exact pow_le_pow_left\u2080 (by positivity)\n                (le_trans (le_max_right 1 \u2016w\u2016) hwmax) i (by omega)\n          _ = j * (1 + 1 / (2 * n)) ^ (n - 1) := by\n              rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul]\n      exact mul_le_mul_of_nonneg_left hsum (by positivity)\n    calc \u2016\u2211 j \u2208 Finset.range n, (w ^ j - 1)\u2016\n        \u2264 \u2211 j \u2208 Finset.range n, \u2016w ^ j - 1\u2016 := norm_sum_le _ _\n      _ \u2264 \u2211 j \u2208 Finset.range n, (1 / \u2016z\u2016) * (j * (1 + 1 / (2 * n)) ^ (n - 1)) := by\n          exact Finset.sum_le_sum hbound\n      _ = (1 / \u2016z\u2016) * (1 + 1 / (2 * n)) ^ (n - 1) * \u2211 j \u2208 Finset.range n, j := by\n          rw [Finset.mul_sum]; congr 1; rw [Finset.mul_sum]\n          apply Finset.sum_congr rfl; intro j _; ring\n      _ \u2264 n ^ 2 / \u2016z\u2016 := by\n          have hsumj : (\u2211 j \u2208 Finset.range n, j : \u211d) = n * (n - 1) / 2 := by\n            have h := Finset.sum_range_id_mul_two n\n            have h2 : ((\u2211 j \u2208 Finset.range n, j : \u2115) : \u211d) * 2 = n * (n - 1) := by\n              exact_mod_cast h\n            linarith\n          rw [hsumj]\n          have hpow : (1 + 1 / (2 * n : \u211d)) ^ (n - 1) \u2264 2 := by\n            have h := pow_one_add_inv_two_mul_le n hn\n            calc (1 + 1 / (2 * n : \u211d)) ^ (n - 1)\n                \u2264 (1 + 1 / (2 * n : \u211d)) ^ n := by\n                  apply pow_le_pow_right\u2080 (by positivity)\n                  \u00b7 have : 1 \u2264 (1 : \u211d) + 1 / (2 * n) := by positivity\n                    exact this\n                  \u00b7 omega\n              _ \u2264 2 := h\n          have h3 : (1 / \u2016z\u2016) * (1 + 1 / (2 * n : \u211d)) ^ (n - 1) * (n * (n - 1) / 2)\n              \u2264 (1 / \u2016z\u2016) * 2 * (n ^ 2 / 2) := by\n            apply mul_le_mul_of_nonneg_left _ (by positivity)\n            apply mul_le_mul_of_nonneg_left hpow (by positivity)\n            \u00b7 have : (n : \u211d) * (n - 1) / 2 \u2264 n ^ 2 / 2 := by\n                have hn1 : (1 : \u211d) \u2264 n := by exact_mod_cast hn\n                nlinarith\n              linarith\n          calc (1 / \u2016z\u2016) * (1 + 1 / (2 * n : \u211d)) ^ (n - 1) * (n * (n - 1) / 2)\n              \u2264 (1 / \u2016z\u2016) * 2 * (n ^ 2 / 2) := h3\n            _ = n ^ 2 / \u2016z\u2016 := by ring\n  -- assemble\n  have hfre : (1 - w ^ n).re = n * z.re / \u2016z\u2016 ^ 2 + (((S - n) / z).re) := by\n    rw [hf, hS]\n    have h1 : ((1 / z) * ((n : \u2102) + (S - n))) = (n : \u2102) * (1 / z) + (S - n) / z := by\n      field_simp; ring\n    rw [h1]\n    have hre1 : ((n : \u2102) * (1 / z)).re = n * z.re / \u2016z\u2016 ^ 2 := by\n      have hinv : ((1 : \u2102) / z).re = z.re / \u2016z\u2016 ^ 2 := by\n        rw [one_div, Complex.inv_def]\n        simp [Complex.div_re, Complex.normSq_apply, Complex.norm_eq_abs,\n          Complex.sq_abs]\n      rw [Complex.add_re] at *\n      simp [hinv]\n    rw [Complex.add_re, hre1]\n  rw [hfre]\n  have hA : |n * z.re / \u2016z\u2016 ^ 2| \u2264 n / \u2016z\u2016 ^ 2 := by\n    rw [abs_div, abs_mul, abs_of_nonneg hn0.le, abs_pow]\n    apply div_le_div_of_nonneg_right (by positivity)\n    calc |z.re| \u2264 1 := abs_le.mpr \u27e8by linarith, hzre1\u27e9\n      _ \u2264 _ := by linarith\n  have hB : |((S - n) / z).re| \u2264 n ^ 2 / \u2016z\u2016 ^ 2 := by\n    have h1 : |((S - n) / z).re| \u2264 \u2016(S - n) / z\u2016 := Complex.abs_re_le_norm _\n    have h2 : \u2016(S - n) / z\u2016 \u2264 (n ^ 2 / \u2016z\u2016) / \u2016z\u2016 := by\n      rw [norm_div]\n      apply div_le_div_of_nonneg_right hSn (by positivity)\n    rw [div_pow] at *\n    calc |((S - n) / z).re| \u2264 \u2016(S - n) / z\u2016 := h1\n      _ \u2264 (n ^ 2 / \u2016z\u2016) / \u2016z\u2016 := h2\n      _ = n ^ 2 / \u2016z\u2016 ^ 2 := by ring\n  calc |n * z.re / \u2016z\u2016 ^ 2 + ((S - n) / z).re|\n      \u2264 n / \u2016z\u2016 ^ 2 + n ^ 2 / \u2016z\u2016 ^ 2 := abs_add _ _ |>.trans (add_le_add hA hB)\n    _ \u2264 2 * n ^ 2 / \u2016z\u2016 ^ 2 := by\n        have hn1 : (n : \u211d) \u2264 n ^ 2 := by\n          have : (1 : \u211d) \u2264 n := by exact_mod_cast hn\n          nlinarith\n        field_simp at *\n        nlinarith\n\nend NguyenRH1\n",
  "autoRead": false
}

{
  "command": "cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && for n in \"theorem sq_abs\" \"lemma sq_abs\" \"sum_sub_distrib\" \"sum_range_id_mul_two\" \"theorem pow_le_pow_right\u2080\" \"lemma pow_le_pow_right\u2080\" \"mul_le_mul_of_nonneg \" \"theorem norm_sum_le\" \"lemma norm_sum_le\" \"summable_of_finite_support\" \"theorem pow_le_pow_left\u2080\" \"lemma pow_le_pow_left\u2080\"; do echo \"== $n\"; grep -rn \"$n\" Mathlib/ --include=*.lean -l 2>/dev/null | head -2; done",
  "description": "Verify lemma names exist in vendored Mathlib"
}

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\Scratch.lean",
  "content": "/- Session 015 scratch: ii\u2032-1a \u2014 paired liSummand + summability (D-064 \u00a71).\n   When green, merge into LiSkeleton.lean \u00a71 replacing the defective liCoeff. -/\nimport Mathlib\nimport Zeta23.WeilEF.ZeroSummability\nimport Zeta23.Statement.SeamClosed\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-- The paired (symmetrized) Li summand: \u00bd\u00b7mult\u00b7(f \u03c1 + f \u03c1\u0304),\nf \u03c1 = 1 \u2212 (1\u22121/\u03c1)\u207f. Manifestly real-valued (`liSummand_eq`), absolutely\nsummable (`liSummable`) \u2014 the correct Keiper\u2013Li coefficient (D-064 \u00a71:\nthe plain unordered sum of f over the zeros is not absolutely\nsummable; the pairing is the classical Bombieri\u2013Lagarias convention). -/\ndef liSummand (n : \u2115) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) : \u2102 :=\n  (zeroMult \u03c1 : \u2102) / 2 *\n    ((1 - (1 - 1 / (\u03c1 : \u2102)) ^ n) + (1 - (1 - 1 / (Complex.conj (\u03c1 : \u2102))) ^ n))\n\nlemma summand_conj (n : \u2115) (z : \u2102) :\n    (1 : \u2102) - (1 - 1 / (Complex.conj z)) ^ n = Complex.conj ((1 - (1 - 1 / z) ^ n)) := by\n  rw [map_sub, map_one, map_pow, map_sub, map_one, map_div\u2080, map_one]\n\nlemma liSummand_eq (n : \u2115) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) :\n    liSummand n \u03c1 = (zeroMult \u03c1 : \u2102) * (((1 - (1 - 1 / (\u03c1 : \u2102)) ^ n).re) : \u2102) := by\n  rw [liSummand, summand_conj, Complex.add_conj]\n  push_cast\n  ring\n\n/-! ### The geometric-sum estimate: |Re(1 \u2212 w\u207f)| \u2264 2n\u00b2/\u2016z\u2016\u00b2 for w = 1\u22121/z -/\n\n/-- (1 + 1/(2n))\u207f \u2264 2 for n \u2265 1. Via (1+x)\u207f \u2264 exp(nx) and exp(1/2) \u2264 2. -/\nlemma pow_one_add_inv_two_mul_le (n : \u2115) (hn : 1 \u2264 n) :\n    ((1 : \u211d) + 1 / (2 * n)) ^ n \u2264 2 := by\n  have hn0 : (0 : \u211d) < n := by exact_mod_cast hn\n  have hbase : (1 : \u211d) + 1 / (2 * n) \u2264 Real.exp (1 / (2 * n)) := by\n    have h := Real.add_one_le_exp (1 / (2 * (n : \u211d)))\n    linarith\n  have h1 : ((1 : \u211d) + 1 / (2 * n)) ^ n \u2264 Real.exp (1 / 2) := by\n    calc ((1 : \u211d) + 1 / (2 * n)) ^ n\n        \u2264 (Real.exp (1 / (2 * n))) ^ n := pow_le_pow_left\u2080 (by positivity) hbase n\n      _ = Real.exp (\u2191n * (1 / (2 * n))) := by rw [Real.exp_nat_mul]\n      _ = Real.exp (1 / 2) := by congr 1; field_simp\n  have h2 : Real.exp (1 / 2 : \u211d) \u2264 2 := by\n    have h3 : (1 : \u211d) / 2 \u2264 Real.log 2 := by\n      have h := Real.log_two_gt_d9\n      linarith\n    calc Real.exp (1 / 2 : \u211d) \u2264 Real.exp (Real.log 2) := Real.exp_le_exp.mpr h3\n      _ = 2 := Real.exp_log (by norm_num)\n  exact h1.trans h2\n\n/-- Key estimate: for n \u2265 1, z with re z \u2208 [0,1], 2n \u2264 \u2016z\u2016:\n|Re(1 \u2212 (1\u22121/z)\u207f)| \u2264 2n\u00b2/\u2016z\u2016\u00b2. -/\nlemma re_li_summand_le {n : \u2115} (hn : 1 \u2264 n) {z : \u2102}\n    (hzre0 : 0 \u2264 z.re) (hzre1 : z.re \u2264 1) (hz : 2 * n \u2264 \u2016z\u2016) :\n    |(1 - (1 - 1 / z) ^ n).re| \u2264 2 * n ^ 2 / \u2016z\u2016 ^ 2 := by\n  have hn0 : (0 : \u211d) < n := by exact_mod_cast hn\n  have hz0 : z \u2260 0 := by\n    intro h; rw [h, norm_zero] at hz; linarith\n  have hzn : (0 : \u211d) < \u2016z\u2016 := norm_pos_iff.mpr hz0\n  have hnsq : Complex.normSq z = \u2016z\u2016 ^ 2 := Complex.normSq_eq_norm_sq z\n  set w : \u2102 := 1 - 1 / z with hw\n  -- kernel identity: 1 \u2212 w\u207f = (1/z)\u00b7S, S = \u03a3_{j<n} w^j\n  have hgeom : (w - 1) * \u2211 j \u2208 Finset.range n, w ^ j = w ^ n - 1 := mul_geom_sum w n\n  have hw1 : w - 1 = -(1 / z) := by rw [hw]; ring\n  have hf : (1 : \u2102) - w ^ n = (1 / z) * \u2211 j \u2208 Finset.range n, w ^ j := by\n    have e : (1 : \u2102) - w ^ n = -(w ^ n - 1) := by ring\n    rw [e, \u2190 hgeom, hw1]; ring\n  set S : \u2102 := \u2211 j \u2208 Finset.range n, w ^ j\n  have hS : S = (n : \u2102) + (S - n) := by ring\n  -- \u2016w\u2016 \u2264 1 + 1/\u2016z\u2016\n  have hwnorm : \u2016w\u2016 \u2264 1 + 1 / \u2016z\u2016 := by\n    have e : w = 1 - z\u207b\u00b9 := by rw [hw, div_eq_mul_inv, one_mul]\n    rw [e]\n    calc \u2016(1 : \u2102) - z\u207b\u00b9\u2016 \u2264 \u2016(1 : \u2102)\u2016 + \u2016z\u207b\u00b9\u2016 := norm_sub_le _ _\n      _ = 1 + 1 / \u2016z\u2016 := by rw [norm_one, norm_inv, inv_eq_one_div]\n  have h1z : 1 / \u2016z\u2016 \u2264 1 / (2 * n) :=\n    one_div_le_one_div_of_le (by positivity) (by exact_mod_cast hz)\n  have hwmax : max 1 \u2016w\u2016 \u2264 1 + 1 / (2 * n) := by\n    apply max_le (le_add_of_nonneg_right (by positivity))\n    exact hwnorm.trans (by linarith)\n  -- |S \u2212 n| \u2264 n\u00b2/\u2016z\u2016\n  have hSn : \u2016S - n\u2016 \u2264 n ^ 2 / \u2016z\u2016 := by\n    have hSn1 : S - n = \u2211 j \u2208 Finset.range n, (w ^ j - 1) := by\n      have e : (n : \u2102) = \u2211 j \u2208 Finset.range n, (1 : \u2102) := by\n        simp [Finset.sum_const, Finset.card_range]\n      rw [e, \u2190 Finset.sum_sub_distrib]\n    have hbound : \u2200 j \u2208 Finset.range n,\n        \u2016w ^ j - 1\u2016 \u2264 (1 / \u2016z\u2016) * (j * (1 + 1 / (2 * n)) ^ (n - 1)) := by\n      intro j hj\n      have hgeomj : (w - 1) * \u2211 i \u2208 Finset.range j, w ^ i = w ^ j - 1 := mul_geom_sum w j\n      have hjn : j \u2264 n := le_of_lt (Finset.mem_range.mp hj)\n      have h2 : w ^ j - 1 = -(1 / z) * \u2211 i \u2208 Finset.range j, w ^ i := by\n        have h := hgeomj; rw [hw1] at h\n        rw [\u2190 h]; ring\n      rw [h2, norm_mul, norm_neg, norm_div, norm_one]\n      have hsum : \u2016\u2211 i \u2208 Finset.range j, w ^ i\u2016 \u2264 j * (1 + 1 / (2 * n)) ^ (n - 1) := by\n        calc \u2016\u2211 i \u2208 Finset.range j, w ^ i\u2016\n            \u2264 \u2211 i \u2208 Finset.range j, \u2016w ^ i\u2016 := norm_sum_le _ _\n          _ \u2264 \u2211 i \u2208 Finset.range j, (1 + 1 / (2 * n)) ^ (n - 1) := by\n              apply Finset.sum_le_sum\n              intro i hi\n              rw [norm_pow]\n              have hij : i < j := Finset.mem_range.mp hi\n              have h1 : \u2016w\u2016 ^ i \u2264 (1 + 1 / (2 * n)) ^ i :=\n                pow_le_pow_left\u2080 (norm_nonneg _)\n                  (le_trans (le_max_right 1 \u2016w\u2016) hwmax) i\n              have h2 : (1 + 1 / (2 * n)) ^ i \u2264 (1 + 1 / (2 * n)) ^ (n - 1) := by\n                apply pow_le_pow_right\u2080 (by positivity)\n                omega\n              exact h1.trans h2\n          _ = j * (1 + 1 / (2 * n)) ^ (n - 1) := by\n              rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul]\n      exact mul_le_mul_of_nonneg_left hsum (by positivity)\n    calc \u2016\u2211 j \u2208 Finset.range n, (w ^ j - 1)\u2016\n        \u2264 \u2211 j \u2208 Finset.range n, \u2016w ^ j - 1\u2016 := norm_sum_le _ _\n      _ \u2264 \u2211 j \u2208 Finset.range n, (1 / \u2016z\u2016) * (j * (1 + 1 / (2 * n)) ^ (n - 1)) := by\n          rw [hSn1] at *\n          exact Finset.sum_le_sum hbound\n      _ = ((1 / \u2016z\u2016) * (1 + 1 / (2 * n)) ^ (n - 1)) * \u2211 j \u2208 Finset.range n, j := by\n          rw [\u2190 Finset.mul_sum]\n          apply Finset.sum_congr rfl\n          intro j _; ring\n      _ \u2264 n ^ 2 / \u2016z\u2016 := by\n          have hsumj : (\u2211 j \u2208 Finset.range n, j : \u211d) = n * (n - 1) / 2 := by\n            have h := Finset.sum_range_id_mul_two n\n            have h2 : ((\u2211 j \u2208 Finset.range n, j : \u2115) : \u211d) * 2 = n * (n - 1) := by\n              exact_mod_cast h\n            linarith\n          rw [hsumj]\n          have hpow : (1 + 1 / (2 * n : \u211d)) ^ (n - 1) \u2264 2 :=\n            le_trans (pow_le_pow_right\u2080 (by positivity) (Nat.sub_le n 1))\n              (pow_one_add_inv_two_mul_le n hn)\n          have hb1 : (n : \u211d) * (n - 1) / 2 \u2264 n ^ 2 / 2 := by nlinarith [hn0]\n          have hb2 : (1 + 1 / (2 * n : \u211d)) ^ (n - 1) * (n * (n - 1) / 2)\n              \u2264 2 * (n ^ 2 / 2) :=\n            mul_le_mul_of_nonneg hpow hb1 (by positivity) (by positivity)\n          have hb3 : ((1 : \u211d) / \u2016z\u2016) * (1 + 1 / (2 * n)) ^ (n - 1) * (n * (n - 1) / 2)\n              \u2264 (1 / \u2016z\u2016) * (2 * (n ^ 2 / 2)) :=\n            mul_le_mul_of_nonneg_left hb2 (by positivity)\n          exact hb3.trans (by ring_nf)\n  -- Re(f) = n\u00b7z.re/\u2016z\u2016\u00b2 + ((S\u2212n)/z).re\n  have hre1 : ((n : \u2102) * (1 / z)).re = n * z.re / \u2016z\u2016 ^ 2 := by\n    have hinv : ((1 : \u2102) / z).re = z.re / \u2016z\u2016 ^ 2 := by\n      rw [one_div, Complex.inv_re, hnsq]\n    rw [Complex.mul_re]\n    simp [hinv]\n  have hfre : (1 - w ^ n).re = n * z.re / \u2016z\u2016 ^ 2 + ((S - n) / z).re := by\n    rw [hf, hS, Complex.add_re]\n    congr 1\n    rw [mul_comm, div_eq_mul_inv]\n    exact hre1\n  rw [hfre]\n  have hA : |n * z.re / \u2016z\u2016 ^ 2| \u2264 n / \u2016z\u2016 ^ 2 := by\n    rw [abs_div, abs_mul, abs_of_nonneg hn0.le,\n      abs_of_nonneg (sq_nonneg \u2016z\u2016)]\n    apply (div_le_div_right (by positivity)).mpr\n    apply mul_le_mul_of_nonneg_left _ hn0.le\n    exact abs_le.mpr \u27e8hzre0, hzre1\u27e9\n  have hB : |((S - n) / z).re| \u2264 n ^ 2 / \u2016z\u2016 ^ 2 := by\n    have h1 : |((S - n) / z).re| \u2264 \u2016(S - n) / z\u2016 := Complex.abs_re_le_norm _\n    have h2 : \u2016(S - n) / z\u2016 \u2264 (n ^ 2 / \u2016z\u2016) / \u2016z\u2016 := by\n      rw [norm_div]\n      exact (div_le_div_right hzn).mpr hSn\n    exact h1.trans (h2.trans (by rw [div_div, \u2190 pow_two]))\n  have hnn : (n : \u211d) \u2264 n ^ 2 := by\n    have h : (1 : \u211d) \u2264 n := by exact_mod_cast hn\n    nlinarith\n  calc |n * z.re / \u2016z\u2016 ^ 2 + ((S - n) / z).re|\n      \u2264 n / \u2016z\u2016 ^ 2 + n ^ 2 / \u2016z\u2016 ^ 2 := (abs_add _ _).trans (add_le_add hA hB)\n    _ = (n + n ^ 2) / \u2016z\u2016 ^ 2 := by ring\n    _ \u2264 2 * n ^ 2 / \u2016z\u2016 ^ 2 := by\n        apply (div_le_div_right (by positivity)).mpr\n        linarith\n\nend NguyenRH1\n"
}

{
  "command": "cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"def conj \\|noncomputable def conj\\|abbrev conj \\|notation.*conj\" Mathlib/Data/Complex/Basic.lean Mathlib/Data/Complex/Defs.lean 2>/dev/null | head -5; grep -rn \"conj\" Mathlib/Data/Complex/Defs.lean 2>/dev/null | head -10",
  "description": "Find the conj definition"
}

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\Scratch.lean",
  "content": "/- Session 015 scratch: ii\u2032-1a \u2014 paired liSummand + summability (D-064 \u00a71).\n   When green, merge into LiSkeleton.lean \u00a71 replacing the defective liCoeff. -/\nimport Mathlib\nimport Zeta23.WeilEF.ZeroSummability\nimport Zeta23.Statement.SeamClosed\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-- The paired (symmetrized) Li summand: \u00bd\u00b7mult\u00b7(f \u03c1 + f \u03c1\u0304),\nf \u03c1 = 1 \u2212 (1\u22121/\u03c1)\u207f. Manifestly real-valued (`liSummand_eq`), absolutely\nsummable (`liSummable`) \u2014 the correct Keiper\u2013Li coefficient (D-064 \u00a71:\nthe plain unordered sum of f over the zeros is not absolutely\nsummable; the pairing is the classical Bombieri\u2013Lagarias convention). -/\ndef liSummand (n : \u2115) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) : \u2102 :=\n  (zeroMult \u03c1 : \u2102) / 2 *\n    ((1 - (1 - 1 / (\u03c1 : \u2102)) ^ n) + (1 - (1 - 1 / ((starRingEnd \u2102) (\u03c1 : \u2102))) ^ n))\n\nlemma summand_conj (n : \u2115) (z : \u2102) :\n    (1 : \u2102) - (1 - 1 / ((starRingEnd \u2102) z)) ^ n = (starRingEnd \u2102) ((1 - (1 - 1 / z) ^ n)) := by\n  rw [map_sub, map_one, map_pow, map_sub, map_one, map_div\u2080, map_one]\n\nlemma liSummand_eq (n : \u2115) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) :\n    liSummand n \u03c1 = (zeroMult \u03c1 : \u2102) * (((1 - (1 - 1 / (\u03c1 : \u2102)) ^ n).re) : \u2102) := by\n  rw [liSummand, summand_conj, Complex.add_conj]\n  push_cast\n  ring\n\n/-! ### The geometric-sum estimate: |Re(1 \u2212 w\u207f)| \u2264 2n\u00b2/\u2016z\u2016\u00b2 for w = 1\u22121/z -/\n\n/-- (1 + 1/(2n))\u207f \u2264 2 for n \u2265 1. Via (1+x)\u207f \u2264 exp(nx) and exp(1/2) \u2264 2. -/\nlemma pow_one_add_inv_two_mul_le (n : \u2115) (hn : 1 \u2264 n) :\n    ((1 : \u211d) + 1 / (2 * n)) ^ n \u2264 2 := by\n  have hn0 : (0 : \u211d) < n := by exact_mod_cast hn\n  have hbase : (1 : \u211d) + 1 / (2 * n) \u2264 Real.exp (1 / (2 * n)) := by\n    have h := Real.add_one_le_exp (1 / (2 * (n : \u211d)))\n    linarith\n  have h1 : ((1 : \u211d) + 1 / (2 * n)) ^ n \u2264 Real.exp (1 / 2) := by\n    calc ((1 : \u211d) + 1 / (2 * n)) ^ n\n        \u2264 (Real.exp (1 / (2 * n))) ^ n := pow_le_pow_left\u2080 (by positivity) hbase n\n      _ = Real.exp (\u2191n * (1 / (2 * n))) := by rw [Real.exp_nat_mul]\n      _ = Real.exp (1 / 2) := by congr 1; field_simp\n  have h2 : Real.exp (1 / 2 : \u211d) \u2264 2 := by\n    have h3 : (1 : \u211d) / 2 \u2264 Real.log 2 := by\n      have h := Real.log_two_gt_d9\n      linarith\n    calc Real.exp (1 / 2 : \u211d) \u2264 Real.exp (Real.log 2) := Real.exp_le_exp.mpr h3\n      _ = 2 := Real.exp_log (by norm_num)\n  exact h1.trans h2\n\n/-- Key estimate: for n \u2265 1, z with re z \u2208 [0,1], 2n \u2264 \u2016z\u2016:\n|Re(1 \u2212 (1\u22121/z)\u207f)| \u2264 2n\u00b2/\u2016z\u2016\u00b2. -/\nlemma re_li_summand_le {n : \u2115} (hn : 1 \u2264 n) {z : \u2102}\n    (hzre0 : 0 \u2264 z.re) (hzre1 : z.re \u2264 1) (hz : 2 * n \u2264 \u2016z\u2016) :\n    |(1 - (1 - 1 / z) ^ n).re| \u2264 2 * n ^ 2 / \u2016z\u2016 ^ 2 := by\n  have hn0 : (0 : \u211d) < n := by exact_mod_cast hn\n  have hz0 : z \u2260 0 := by\n    intro h; rw [h, norm_zero] at hz; linarith\n  have hzn : (0 : \u211d) < \u2016z\u2016 := norm_pos_iff.mpr hz0\n  have hnsq : Complex.normSq z = \u2016z\u2016 ^ 2 := Complex.normSq_eq_norm_sq z\n  set M : \u211d := 1 + 1 / (2 * (n : \u211d)) with hM\n  have hM0 : (0 : \u211d) \u2264 1 / (2 * (n : \u211d)) := by positivity\n  have hM1 : (1 : \u211d) \u2264 M := le_add_of_nonneg_right hM0\n  set w : \u2102 := 1 - 1 / z with hw\n  -- kernel identity: 1 \u2212 w\u207f = (1/z)\u00b7S, S = \u03a3_{j<n} w^j\n  have hgeom : (w - 1) * \u2211 j \u2208 Finset.range n, w ^ j = w ^ n - 1 := mul_geom_sum w n\n  have hw1 : w - 1 = -(1 / z) := by rw [hw]; ring\n  have hf : (1 : \u2102) - w ^ n = (1 / z) * \u2211 j \u2208 Finset.range n, w ^ j := by\n    have e : (1 : \u2102) - w ^ n = -(w ^ n - 1) := by ring\n    rw [e, \u2190 hgeom, hw1]; ring\n  set S : \u2102 := \u2211 j \u2208 Finset.range n, w ^ j\n  have hS : S = (n : \u2102) + (S - n) := by ring\n  -- \u2016w\u2016 \u2264 1 + 1/\u2016z\u2016\n  have hwnorm : \u2016w\u2016 \u2264 1 + 1 / \u2016z\u2016 := by\n    have e : w = 1 - z\u207b\u00b9 := by rw [hw, div_eq_mul_inv, one_mul]\n    rw [e]\n    calc \u2016(1 : \u2102) - z\u207b\u00b9\u2016 \u2264 \u2016(1 : \u2102)\u2016 + \u2016z\u207b\u00b9\u2016 := norm_sub_le _ _\n      _ = 1 + 1 / \u2016z\u2016 := by rw [norm_one, norm_inv, inv_eq_one_div]\n  have h1z : 1 / \u2016z\u2016 \u2264 1 / (2 * n) :=\n    one_div_le_one_div_of_le (by positivity) (by exact_mod_cast hz)\n  have hwmax : max 1 \u2016w\u2016 \u2264 M := by\n    apply max_le hM1\n    exact hwnorm.trans (by rw [hM]; linarith)\n  -- |S \u2212 n| \u2264 n\u00b2/\u2016z\u2016\n  have hSn : \u2016S - n\u2016 \u2264 n ^ 2 / \u2016z\u2016 := by\n    have hSn1 : S - n = \u2211 j \u2208 Finset.range n, (w ^ j - 1) := by\n      have e : (n : \u2102) = \u2211 j \u2208 Finset.range n, (1 : \u2102) := by\n        simp [Finset.sum_const, Finset.card_range]\n      rw [e, \u2190 Finset.sum_sub_distrib]\n    rw [hSn1]\n    have hbound : \u2200 j \u2208 Finset.range n,\n        \u2016w ^ j - 1\u2016 \u2264 (1 / \u2016z\u2016) * (j * M ^ (n - 1)) := by\n      intro j hj\n      have hgeomj : (w - 1) * \u2211 i \u2208 Finset.range j, w ^ i = w ^ j - 1 := mul_geom_sum w j\n      have h2 : w ^ j - 1 = -(1 / z) * \u2211 i \u2208 Finset.range j, w ^ i := by\n        rw [\u2190 hgeomj, hw1]\n      rw [h2, norm_mul, norm_neg, norm_div, norm_one]\n      have hsum : \u2016\u2211 i \u2208 Finset.range j, w ^ i\u2016 \u2264 j * M ^ (n - 1) := by\n        calc \u2016\u2211 i \u2208 Finset.range j, w ^ i\u2016\n            \u2264 \u2211 i \u2208 Finset.range j, \u2016w ^ i\u2016 := norm_sum_le _ _\n          _ \u2264 \u2211 i \u2208 Finset.range j, M ^ (n - 1) := by\n              apply Finset.sum_le_sum\n              intro i hi\n              rw [norm_pow]\n              have hij : i < j := Finset.mem_range.mp hi\n              have hjn : j \u2264 n := le_of_lt (Finset.mem_range.mp hj)\n              have h1 : \u2016w\u2016 ^ i \u2264 M ^ i :=\n                pow_le_pow_left\u2080 (norm_nonneg _)\n                  (le_trans (le_max_right 1 \u2016w\u2016) hwmax) i\n              have h2 : M ^ i \u2264 M ^ (n - 1) := pow_le_pow_right\u2080 hM1 (by omega)\n              exact h1.trans h2\n          _ = j * M ^ (n - 1) := by\n              rw [Finset.sum_const, Finset.card_range, nsmul_eq_mul]\n      exact mul_le_mul_of_nonneg_left hsum (by positivity)\n    calc \u2211 j \u2208 Finset.range n, \u2016w ^ j - 1\u2016\n        \u2264 \u2211 j \u2208 Finset.range n, (1 / \u2016z\u2016) * (j * M ^ (n - 1)) :=\n          Finset.sum_le_sum hbound\n      _ = ((1 / \u2016z\u2016) * M ^ (n - 1)) * \u2211 j \u2208 Finset.range n, j := by\n          rw [\u2190 Finset.mul_sum]\n          apply Finset.sum_congr rfl\n          intro j _; ring\n      _ \u2264 n ^ 2 / \u2016z\u2016 := by\n          have hsumj : (\u2211 j \u2208 Finset.range n, j : \u211d) = n * (n - 1) / 2 := by\n            have h := Finset.sum_range_id_mul_two n\n            have h2 : ((\u2211 j \u2208 Finset.range n, j : \u2115) : \u211d) * 2 = n * (n - 1) := by\n              exact_mod_cast h\n            linarith\n          rw [hsumj]\n          have hpow : M ^ (n - 1) \u2264 2 :=\n            le_trans (pow_le_pow_right\u2080 hM1 (Nat.sub_le n 1))\n              (pow_one_add_inv_two_mul_le n hn)\n          have hb1 : (n : \u211d) * (n - 1) / 2 \u2264 n ^ 2 / 2 := by nlinarith [hn0]\n          have hb2 : M ^ (n - 1) * (n * (n - 1) / 2) \u2264 2 * (n ^ 2 / 2) :=\n            mul_le_mul_of_nonneg hpow hb1 (by positivity) (by positivity)\n          have hb3 : ((1 : \u211d) / \u2016z\u2016) * M ^ (n - 1) * (n * (n - 1) / 2)\n              \u2264 (1 / \u2016z\u2016) * (2 * (n ^ 2 / 2)) :=\n            mul_le_mul_of_nonneg_left hb2 (by positivity)\n          exact hb3.trans (by ring_nf)\n  -- Re(f) = n\u00b7z.re/\u2016z\u2016\u00b2 + ((S\u2212n)/z).re\n  have hre1 : ((n : \u2102) * (1 / z)).re = n * z.re / \u2016z\u2016 ^ 2 := by\n    have hinv : ((1 : \u2102) / z).re = z.re / \u2016z\u2016 ^ 2 := by\n      rw [one_div, Complex.inv_re, hnsq]\n    rw [Complex.mul_re]\n    simp [hinv]\n  have hfre : (1 - w ^ n).re = n * z.re / \u2016z\u2016 ^ 2 + ((S - n) / z).re := by\n    rw [hf, hS, Complex.add_re]\n    congr 1\n    rw [mul_comm, div_eq_mul_inv]\n    exact hre1\n  rw [hfre]\n  have hA : |n * z.re / \u2016z\u2016 ^ 2| \u2264 n / \u2016z\u2016 ^ 2 := by\n    rw [abs_div, abs_mul, abs_of_nonneg hn0.le,\n      abs_of_nonneg (sq_nonneg \u2016z\u2016)]\n    apply (div_le_div_right (by positivity)).mpr\n    apply mul_le_mul_of_nonneg_left _ hn0.le\n    exact abs_le.mpr \u27e8hzre0, hzre1\u27e9\n  have hB : |((S - n) / z).re| \u2264 n ^ 2 / \u2016z\u2016 ^ 2 := by\n    have h1 : |((S - n) / z).re| \u2264 \u2016(S - n) / z\u2016 := Complex.abs_re_le_norm _\n    have h2 : \u2016(S - n) / z\u2016 \u2264 (n ^ 2 / \u2016z\u2016) / \u2016z\u2016 := by\n      rw [norm_div]\n      exact (div_le_div_right hzn).mpr hSn\n    exact h1.trans (h2.trans (by rw [div_div, \u2190 pow_two]))\n  have hnn : (n : \u211d) \u2264 n ^ 2 := by\n    have h : (1 : \u211d) \u2264 n := by exact_mod_cast hn\n    nlinarith\n  calc |n * z.re / \u2016z\u2016 ^ 2 + ((S - n) / z).re|\n      \u2264 n / \u2016z\u2016 ^ 2 + n ^ 2 / \u2016z\u2016 ^ 2 := (abs_add _ _).trans (add_le_add hA hB)\n    _ = (n + n ^ 2) / \u2016z\u2016 ^ 2 := by ring\n    _ \u2264 2 * n ^ 2 / \u2016z\u2016 ^ 2 := by\n        apply (div_le_div_right (by positivity)).mpr\n        linarith\n\nend NguyenRH1\n"
}

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\Scratch.lean",
  "old_string": "    calc \u2211 j \u2208 Finset.range n, \u2016w ^ j - 1\u2016\n        \u2264 \u2211 j \u2208 Finset.range n, (1 / \u2016z\u2016) * (j * M ^ (n - 1)) :=\n          Finset.sum_le_sum hbound\n      _ = ((1 / \u2016z\u2016) * M ^ (n - 1)) * \u2211 j \u2208 Finset.range n, j := by\n          rw [\u2190 Finset.mul_sum]\n          apply Finset.sum_congr rfl\n          intro j _; ring\n      _ \u2264 n ^ 2 / \u2016z\u2016 := by\n          have hsumj : (\u2211 j \u2208 Finset.range n, j : \u211d) = n * (n - 1) / 2 := by\n            have h := Finset.sum_range_id_mul_two n\n            have h2 : ((\u2211 j \u2208 Finset.range n, j : \u2115) : \u211d) * 2 = n * (n - 1) := by\n              exact_mod_cast h\n            linarith\n          rw [hsumj]",
  "new_string": "    calc \u2016\u2211 j \u2208 Finset.range n, (w ^ j - 1)\u2016\n        \u2264 \u2211 j \u2208 Finset.range n, \u2016w ^ j - 1\u2016 := norm_sum_le _ _\n      _ \u2264 \u2211 j \u2208 Finset.range n, (1 / \u2016z\u2016) * (j * M ^ (n - 1)) :=\n          Finset.sum_le_sum hbound\n      _ = ((1 / \u2016z\u2016) * M ^ (n - 1)) * \u2211 j \u2208 Finset.range n, j := by\n          symm\n          rw [Finset.mul_sum]\n          apply Finset.sum_congr rfl\n          intro j _; ring\n      _ \u2264 n ^ 2 / \u2016z\u2016 := by\n          have hsumj : (\u2211 j \u2208 Finset.range n, j : \u211d) = n * (n - 1) / 2 := by\n            have h6 := congrArg (Nat.cast : \u2115 \u2192 \u211d) (Finset.sum_range_id_mul_two n)\n            simp only [Nat.cast_mul, Nat.cast_sub hn, Nat.cast_ofNat, Nat.cast_one] at h6\n            have h7 : (\u2211 j \u2208 Finset.range n, (j : \u211d)) =\n                ((\u2211 j \u2208 Finset.range n, j : \u2115) : \u211d) := by simp\n            rw [h7]\n            linarith\n          rw [hsumj]"
}

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\Scratch.lean",
  "old_string": "          have hb3 : ((1 : \u211d) / \u2016z\u2016) * M ^ (n - 1) * (n * (n - 1) / 2)\n              \u2264 (1 / \u2016z\u2016) * (2 * (n ^ 2 / 2)) :=\n            mul_le_mul_of_nonneg_left hb2 (by positivity)\n          exact hb3.trans (by ring_nf)",
  "new_string": "          have hb3 : ((1 : \u211d) / \u2016z\u2016) * (M ^ (n - 1) * (n * (n - 1) / 2))\n              \u2264 (1 / \u2016z\u2016) * (2 * (n ^ 2 / 2)) :=\n            mul_le_mul_of_nonneg_left hb2 (by positivity)\n          exact hb3.trans (by ring_nf)"
}

{
  "command": "cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem div_le_iff \\|lemma div_le_iff \\|theorem div_le_iff\u2080\\|lemma div_le_iff\u2080\" Mathlib/Algebra/Order/Field/Basic.lean Mathlib/Algebra/Order/Unbundled/Basic.lean 2>/dev/null | head -4; grep -rn \"theorem abs_add \\|lemma abs_add \\|abs_add_le\\b\" Mathlib/Algebra/Order/AbsoluteValue.lean Mathlib/Algebra/Order/Unbundled/Abs.lean 2>/dev/null | head -4; grep -rn \"theorem div_le_div_iff_right\\|lemma div_le_div_iff_right\" Mathlib/ --include=*.lean | head -2",
  "description": "Check div_le_iff, abs_add, div_le_div_iff_right names"
}

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\Scratch.lean",
  "old_string": "  have hfre : (1 - w ^ n).re = n * z.re / \u2016z\u2016 ^ 2 + ((S - n) / z).re := by\n    have hd : (1 / z) * ((n : \u2102) + (S - n)) = (n : \u2102) * (1 / z) + (S - n) / z := by\n      field_simp; ring\n    rw [hf, hS, hd, Complex.add_re, hre1]\n  rw [hfre]\n  have hA : |n * z.re / \u2016z\u2016 ^ 2| \u2264 n / \u2016z\u2016 ^ 2 := by\n    rw [abs_div, abs_mul, abs_of_nonneg hn0.le,\n      abs_of_nonneg (sq_nonneg \u2016z\u2016)]\n    apply (div_le_div_right (by positivity)).mpr\n    apply mul_le_mul_of_nonneg_left _ hn0.le\n    exact abs_le.mpr \u27e8hzre0, hzre1\u27e9\n  have hB : |((S - n) / z).re| \u2264 n ^ 2 / \u2016z\u2016 ^ 2 := by\n    have h1 : |((S - n) / z).re| \u2264 \u2016(S - n) / z\u2016 := Complex.abs_re_le_norm _\n    have h2 : \u2016(S - n) / z\u2016 \u2264 (n ^ 2 / \u2016z\u2016) / \u2016z\u2016 := by\n      rw [norm_div]\n      exact (div_le_div_right hzn).mpr hSn\n    exact h1.trans (h2.trans (by rw [div_div, \u2190 pow_two]))\n  have hnn : (n : \u211d) \u2264 n ^ 2 := by\n    have h : (1 : \u211d) \u2264 n := by exact_mod_cast hn\n    nlinarith\n  calc |n * z.re / \u2016z\u2016 ^ 2 + ((S - n) / z).re|\n      \u2264 n / \u2016z\u2016 ^ 2 + n ^ 2 / \u2016z\u2016 ^ 2 := (abs_add _ _).trans (add_le_add hA hB)\n    _ = (n + n ^ 2) / \u2016z\u2016 ^ 2 := by ring\n    _ \u2264 2 * n ^ 2 / \u2016z\u2016 ^ 2 := by\n        apply (div_le_div_right (by positivity)).mpr\n        linarith",
  "new_string": "  have hfre : (1 - w ^ n).re = n * z.re / \u2016z\u2016 ^ 2 + ((S - n) / z).re := by\n    have hd : (1 / z) * ((n : \u2102) + (S - n)) = (n : \u2102) * (1 / z) + (S - n) / z := by\n      field_simp; ring\n    rw [hf]\n    nth_rewrite 1 [show S = (n : \u2102) + (S - n) from by ring]\n    rw [hd, Complex.add_re, hre1]\n  rw [hfre]\n  have hA : |n * z.re / \u2016z\u2016 ^ 2| \u2264 n / \u2016z\u2016 ^ 2 := by\n    have e : |n * z.re / \u2016z\u2016 ^ 2| = n * |z.re| / \u2016z\u2016 ^ 2 := by\n      rw [abs_div, abs_mul, abs_of_nonneg hn0.le,\n        abs_of_nonneg (sq_nonneg \u2016z\u2016)]\n    rw [e, div_le_iff\u2080 (by positivity)]\n    have e2 : (n / \u2016z\u2016 ^ 2) * \u2016z\u2016 ^ 2 = n := by field_simp\n    rw [e2]\n    exact mul_le_mul_of_nonneg_left (abs_le.mpr \u27e8hzre0, hzre1\u27e9) hn0.le\n  have hB : |((S - n) / z).re| \u2264 n ^ 2 / \u2016z\u2016 ^ 2 := by\n    have h1 : |((S - n) / z).re| \u2264 \u2016(S - n) / z\u2016 := Complex.abs_re_le_norm _\n    have h2 : \u2016(S - n) / z\u2016 \u2264 (n ^ 2 / \u2016z\u2016) / \u2016z\u2016 := by\n      rw [norm_div, div_le_iff\u2080 hzn]\n      have e : ((n ^ 2 / \u2016z\u2016) / \u2016z\u2016) * \u2016z\u2016 = n ^ 2 / \u2016z\u2016 := by field_simp\n      rw [e]\n      exact hSn\n    exact h1.trans (h2.trans (by rw [div_div, \u2190 pow_two]))\n  have hnn : (n : \u211d) \u2264 n ^ 2 := by\n    have h : (1 : \u211d) \u2264 n := by exact_mod_cast hn\n    nlinarith\n  calc |n * z.re / \u2016z\u2016 ^ 2 + ((S - n) / z).re|\n      \u2264 n / \u2016z\u2016 ^ 2 + n ^ 2 / \u2016z\u2016 ^ 2 := (abs_add_le _ _).trans (add_le_add hA hB)\n    _ = (n + n ^ 2) / \u2016z\u2016 ^ 2 := by ring\n    _ \u2264 2 * n ^ 2 / \u2016z\u2016 ^ 2 := by\n        rw [div_le_iff\u2080 (by positivity)]\n        have e : (2 * n ^ 2 / \u2016z\u2016 ^ 2) * \u2016z\u2016 ^ 2 = 2 * n ^ 2 := by field_simp\n        rw [e]\n        linarith"
}

{
  "command": "cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem Summable.of_norm_bounded\\|lemma Summable.of_norm_bounded\" Mathlib/ --include=*.lean | head -2; grep -rn \"theorem summable_of_finite_support\\|lemma summable_of_finite_support\" Mathlib/ --include=*.lean | head -2; grep -rn \"theorem Set.Finite.of_finite_image\\|lemma Set.Finite.of_finite_image\" Mathlib/ --include=*.lean | head -2; grep -rn \"norm_natCast\\|norm_ofReal\" Mathlib/Analysis/RCLike/Basic.lean | head -4",
  "description": "Verify summability assembly API names"
}

{
  "command": "cat >> /d/Python/Quantum/research/lean/NguyenRH1/Scratch.lean << 'EOF'\n\n/-! ### The comparison 1 + \u2016\u03b3_\u03c1\u2016\u00b2 \u2264 (9/4)\u2016\u03c1\u2016\u00b2 and the dominator -/\n\n/-- For re z \u2208 [0,1] and 2n \u2264 \u2016z\u2016 (n \u2265 1): 1 + \u2016\u03b3_z\u2016\u00b2 \u2264 (9/4)\u00b7\u2016z\u2016\u00b2. -/\nlemma normSq_gammaOf_le_mul_norm_sq {n : \u2115} (hn : 1 \u2264 n) {z : \u2102}\n    (hzre0 : 0 \u2264 z.re) (hzre1 : z.re \u2264 1) (hz : 2 * n \u2264 \u2016z\u2016) :\n    1 + Complex.normSq (gammaOf z) \u2264 9 / 4 * \u2016z\u2016 ^ 2 := by\n  have hn0 : (0 : \u211d) < n := by exact_mod_cast hn\n  have hz2 : \u2016z\u2016 ^ 2 = z.re ^ 2 + z.im ^ 2 := by\n    rw [\u2190 Complex.normSq_eq_norm_sq, Complex.normSq_apply]\n    ring\n  have hz4 : 4 * n ^ 2 \u2264 \u2016z\u2016 ^ 2 := by\n    have h := pow_le_pow_left\u2080 (by positivity : (0:\u211d) \u2264 2 * n) hz 2\n    nlinarith [h]\n  have hre2 : z.re ^ 2 \u2264 1 := by nlinarith [hzre0, hzre1]\n  have him : 1 \u2264 z.im ^ 2 := by nlinarith [hz2, hz4, hre2]\n  have hgam : Complex.normSq (gammaOf z) = z.im ^ 2 + (1 / 2 - z.re) ^ 2 := by\n    rw [Complex.normSq_apply, WeilEF.gammaOf_re, WeilEF.gammaOf_im]\n    ring\n  rw [hgam, hz2]\n  nlinarith [hzre0, hzre1, him, sq_nonneg (1 / 2 - z.re), sq_nonneg z.re]\n\n/-- Norm of the paired summand: mult\u00b7|Re f(\u03c1)|. -/\nlemma liSummand_norm (n : \u2115) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1}) :\n    \u2016liSummand n \u03c1\u2016 = (zeroMult \u03c1 : \u211d) * |(1 - (1 - 1 / (\u03c1 : \u2102)) ^ n).re| := by\n  rw [liSummand_eq n \u03c1, norm_mul, RCLike.norm_natCast, RCLike.norm_ofReal]\n\n/-- The main bound: for 2n \u2264 \u2016\u03c1\u2016, \u2016liSummand n \u03c1\u2016 \u2264 (9/2)n\u00b2\u00b7mult/(1+\u2016\u03b3_\u03c1\u2016\u00b2). -/\nlemma liSummand_norm_le {n : \u2115} (hn : 1 \u2264 n) (\u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1})\n    (h\u03c1 : (2 : \u211d) * n \u2264 \u2016(\u03c1 : \u2102)\u2016) :\n    \u2016liSummand n \u03c1\u2016 \u2264\n      (9 / 2) * n ^ 2 * (zeroMult \u03c1 : \u211d) / (1 + Complex.normSq (gammaOf (\u03c1 : \u2102))) := by\n  have hz := \u03c1.2\n  have hre0 : 0 \u2264 (\u03c1 : \u2102).re := hz.2.1.le\n  have hre1 : (\u03c1 : \u2102).re \u2264 1 := hz.2.2.le\n  have hcore := re_li_summand_le hn hre0 hre1 h\u03c1\n  have hcomp := normSq_gammaOf_le_mul_norm_sq hn hre0 hre1 h\u03c1\n  have hnpos : (0 : \u211d) < n := by exact_mod_cast hn\n  have hz0 : (0 : \u211d) < \u2016(\u03c1 : \u2102)\u2016 := lt_of_lt_of_le (by positivity) h\u03c1\n  have hg0 : (0 : \u211d) < 1 + Complex.normSq (gammaOf (\u03c1 : \u2102)) := by\n    have h := Complex.normSq_nonneg (gammaOf (\u03c1 : \u2102))\n    linarith\n  have key : (zeroMult \u03c1 : \u211d) * (2 * n ^ 2 / \u2016(\u03c1 : \u2102)\u2016 ^ 2)\n      \u2264 (9 / 2) * n ^ 2 * (zeroMult \u03c1 : \u211d) / (1 + Complex.normSq (gammaOf (\u03c1 : \u2102))) := by\n    rw [le_div_iff\u2080 hg0]\n    have hbound : (zeroMult \u03c1 : \u211d) * (2 * n ^ 2 / \u2016(\u03c1 : \u2102)\u2016 ^ 2) *\n          (1 + Complex.normSq (gammaOf (\u03c1 : \u2102)))\n        \u2264 (zeroMult \u03c1 : \u211d) * (2 * n ^ 2 / \u2016(\u03c1 : \u2102)\u2016 ^ 2) * ((9 / 4) * \u2016(\u03c1 : \u2102)\u2016 ^ 2) :=\n      mul_le_mul_of_nonneg_left hcomp (by positivity)\n    exact hbound.trans (by field_simp; ring)\n  rw [liSummand_norm]\n  exact (mul_le_mul_of_nonneg_left hcore (Nat.cast_nonneg _)).trans key\n\n/-- The paired Li summand is absolutely summable over the nontrivial zeros.\nUses Zeta23's `WeilEF.zero_sum_inv_sq` (\u03a3 mult/(1+\u2016\u03b3_\u03c1\u2016\u00b2) < \u221e, proved) via\nthe tail bound of `liSummand_norm_le`; the ball {\u2016\u03c1\u2016 < 2n} contributes a\nfinite set (Mathlib `IsCompact.inter_riemannZetaZeros_finite`). -/\ntheorem liSummable (n : \u2115) : Summable (liSummand n) := by\n  rcases Nat.eq_zero_or_pos n with h0 | hn\n  \u00b7 subst h0\n    have h : liSummand 0 = fun _ => (0 : \u2102) := by\n      funext \u03c1\n      simp [liSummand]\n    rw [h]\n    exact summable_zero\n  \u00b7 -- the base summable family (Zeta23, proved)\n    have hbase : Summable (fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} =>\n        (9 / 2) * (n : \u211d) ^ 2 * ((zeroMult \u03c1 : \u211d) / (1 + Complex.normSq (gammaOf \u03c1)))) := by\n      have h := (WeilEF.zero_sum_inv_sq zetaSeam).mul_left ((9 / 2) * (n : \u211d) ^ 2)\n      exact h\n    -- the dominator\n    set D : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} \u2192 \u211d := fun \u03c1 =>\n      if (2 : \u211d) * n \u2264 \u2016(\u03c1 : \u2102)\u2016 then\n        (9 / 2) * n ^ 2 * (zeroMult \u03c1 : \u211d) / (1 + Complex.normSq (gammaOf \u03c1))\n      else \u2016liSummand n \u03c1\u2016 with hDdef\n    -- the correction E = D \u2212 base has finite support\n    have hfin : {\u03c1 : {\u03c1 // IsNontrivialZero \u03c1} | \u00ac ((2 : \u211d) * n \u2264 \u2016(\u03c1 : \u2102)\u2016)}.Finite := by\n      have hcomp : IsCompact (Metric.closedBall (0 : \u2102) (2 * n)) :=\n        isCompact_closedBall _ _\n      have hzeros : (Metric.closedBall (0 : \u2102) (2 * n)) \u2229 {\u03c1 : \u2102 | IsNontrivialZero \u03c1}\n          \u2286 (Metric.closedBall (0 : \u2102) (2 * n)) \u2229 riemannZetaZeros := by\n        intro z hz\n        exact \u27e8hz.1, hz.2.1\u27e9\n      have hfin2 := (hcomp.inter_riemannZetaZeros_finite).subset hzeros\n      have himg : (Subtype.val '' {\u03c1 : {\u03c1 // IsNontrivialZero \u03c1} |\n          \u00ac ((2 : \u211d) * n \u2264 \u2016(\u03c1 : \u2102)\u2016)}).Finite := by\n        apply hfin2.subset\n        rintro z \u27e8\u03c1, h\u03c1, rfl\u27e9\n        refine \u27e8?_, \u03c1.2\u27e9\n        simp only [Metric.mem_closedBall, dist_zero_right]\n        exact le_of_lt (lt_of_not_ge h\u03c1)\n      exact Set.Finite.of_finite_image himg (Subtype.val_injective.injOn)\n    have hE : Summable (fun \u03c1 : {\u03c1 // IsNontrivialZero \u03c1} =>\n        D \u03c1 - (9 / 2) * (n : \u211d) ^ 2 * ((zeroMult \u03c1 : \u211d) /\n          (1 + Complex.normSq (gammaOf \u03c1)))) := by\n      apply summable_of_hasFiniteSupport\n      apply hfin.subset\n      intro \u03c1 h\u03c1\n      simp only [Function.mem_support, ne_eq] at h\u03c1\n      by_contra hball\n      apply h\u03c1\n      have hball' : (2 : \u211d) * n \u2264 \u2016(\u03c1 : \u2102)\u2016 := not_not.mp hball\n      simp only [D, if_pos hball', sub_self]\n    have hD : Summable D := by\n      have hsum := hbase.add hE\n      have heq : (fun \u03c1 : {\u03c1 : \u2102 // IsNontrivialZero \u03c1} =>\n          (9 / 2) * (n : \u211d) ^ 2 * ((zeroMult \u03c1 : \u211d) / (1 + Complex.normSq (gammaOf \u03c1))) +\n            (D \u03c1 - (9 / 2) * (n : \u211d) ^ 2 * ((zeroMult \u03c1 : \u211d) /\n              (1 + Complex.normSq (gammaOf \u03c1))))) = D := by\n        funext \u03c1\n        ring\n      rw [heq] at hsum\n      exact hsum\n    refine Summable.of_norm_bounded hD (fun \u03c1 => ?_)\n    by_cases h\u03c1 : (2 : \u211d) * n \u2264 \u2016(\u03c1 : \u2102)\u2016\n    \u00b7 simp only [D, if_pos h\u03c1]\n      exact liSummand_norm_le hn \u03c1 h\u03c1\n    \u00b7 simp only [D, if_neg h\u03c1, le_refl]\nEOF\ncd /d/Python/Quantum/research/lean/NguyenRH1 && export PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean Scratch.lean 2>&1 | head -40",
  "description": "Append summability assembly and check",
  "timeout": 420000
}

{
  "command": "grep -n \"end NguyenRH1\" /d/Python/Quantum/research/lean/NguyenRH1/Scratch.lean",
  "description": "Locate namespace end lines"
}

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiSkeleton.lean",
  "old_string": "import Mathlib\nimport Zeta23.Statement",
  "new_string": "import Mathlib\nimport Zeta23.Statement\nimport Zeta23.WeilEF.ZeroSummability\nimport Zeta23.Statement.SeamClosed"
}

{
  "command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && export PATH=\"$HOME/.elan/bin:$PATH\" && cat > PrintAxioms.lean << 'EOF'\n/- Session 015 axiom audit: ii\u2032-1a new declarations + main theorems. -/\nimport Mathlib\nimport NguyenRH1\n\nopen NguyenRH1\n\n#print axioms liSummable\n#print axioms liSummand_norm_le\n#print axioms re_li_summand_le\n#print axioms normSq_gammaOf_le_mul_norm_sq\n#print axioms liSummand_eq\n#print axioms liSummand_norm\n#print axioms pow_one_add_inv_two_mul_le\n#print axioms lambda_pos_finite_height\n#print axioms lambda_pos_finite_height_verified\n#print axioms sInfty_lower_bound\n#print axioms marginTilde_convex\nEOF\nlake env lean PrintAxioms.lean 2>&1 | head -15",
  "description": "Run axiom audit on new and main declarations",
  "timeout": 300000
}

{
  "todos": [
    {
      "content": "Check zetaZeros carrier def + zeroMult/gammaOf coercion paths",
      "status": "completed",
      "activeForm": "Checking zetaZeros defs"
    },
    {
      "content": "Develop summability proof in Scratch.lean (pair bound + comparison)",
      "status": "completed",
      "activeForm": "Developing summability proof"
    },
    {
      "content": "Replace liCoeff def in LiSkeleton.lean + prove liSummable",
      "status": "completed",
      "activeForm": "Replacing liCoeff def"
    },
    {
      "content": "Full rebuild + axiom audit (LiSkeleton/LiBasePoints unchanged downstream)",
      "status": "completed",
      "activeForm": "Rebuilding + auditing"
    },
    {
      "content": "Session-015 closure (journal, STATE, NGUYEN checkoff, commit)",
      "status": "in_progress",
      "activeForm": "Closing session 015"
    }
  ]
}

{
  "oldTodos": [
    "content='Check zetaZeros carrier def + zeroMult/gammaOf coercion paths' status='in_progress' active_form='Checking zetaZeros defs'",
    "content='Develop summability proof in Scratch.lean (pair bound + comparison)' status='pending' active_form='Developing summability proof'",
    "content='Replace liCoeff def in LiSkeleton.lean + prove liCoeff_summable' status='pending' active_form='Replacing liCoeff def'",
    "content='Full rebuild + axiom audit (LiSkeleton/LiBasePoints unchanged downstream)' status='pending' active_form='Rebuilding + auditing'",
    "content='Session-015 closure (journal, STATE, NGUYEN checkoff, commit)' status='pending' active_form='Closing session 015'"
  ],
  "newTodos": [
    "content='Check zetaZeros carrier def + zeroMult/gammaOf coercion paths' status='completed' active_form='Checking zetaZeros defs'",
    "content='Develop summability proof in Scratch.lean (pair bound + comparison)' status='completed' active_form='Developing summability proof'",
    "content='Replace liCoeff def in LiSkeleton.lean + prove liSummable' status='completed' active_form='Replacing liCoeff def'",
    "content='Full rebuild + axiom audit (LiSkeleton/LiBasePoints unchanged downstream)' status='completed' active_form='Rebuilding + auditing'",
    "content='Session-015 closure (journal, STATE, NGUYEN checkoff, commit)' status='in_progress' active_form='Closing session 015'"
  ],
  "cleared": false
}

{
  "todos": [
    {
      "content": "Verify APIs: iteratedDeriv_mul/pow, vanishesOnRectangle, ResidueTheoremInRectangle",
      "status": "in_progress",
      "activeForm": "Verifying residue APIs"
    },
    {
      "content": "ii\u2032-1b(i): w=1 residue value (Leibniz, ResW1 = \u03a3 C(n,i)\u00b7lcoeff(i\u22121))",
      "status": "pending",
      "activeForm": "Proving w=1 residue value"
    },
    {
      "content": "ii\u2032-1b(ii): principal-part integral \u222e P = ResW1",
      "status": "pending",
      "activeForm": "Proving principal-part integral"
    },
    {
      "content": "ii\u2032-1b(iii): R holomorphic at 1 + rectangle residue theorem for J",
      "status": "pending",
      "activeForm": "Proving rectangle residue theorem"
    },
    {
      "content": "Session-016 closure (journal, STATE, commit)",
      "status": "pending",
      "activeForm": "Closing session 016"
    }
  ]
}

{
  "oldTodos": [],
  "newTodos": [
    "content='Verify APIs: iteratedDeriv_mul/pow, vanishesOnRectangle, ResidueTheoremInRectangle' status='in_progress' active_form='Verifying residue APIs'",
    "content='ii\u2032-1b(i): w=1 residue value (Leibniz, ResW1 = \u03a3 C(n,i)\u00b7lcoeff(i\u22121))' status='pending' active_form='Proving w=1 residue value'",
    "content='ii\u2032-1b(ii): principal-part integral \u222e P = ResW1' status='pending' active_form='Proving principal-part integral'",
    "content='ii\u2032-1b(iii): R holomorphic at 1 + rectangle residue theorem for J' status='pending' active_form='Proving rectangle residue theorem'",
    "content='Session-016 closure (journal, STATE, commit)' status='pending' active_form='Closing session 016'"
  ],
  "cleared": false
}

{
  "command": "cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"logDeriv\" Mathlib/Analysis/Meromorphic/TrailingCoefficient.lean | head -8; grep -rn \"meromorphicTrailingCoeffAt.*logDeriv\\|logDeriv.*meromorphicOrderAt\\|residue.*logDeriv\" Mathlib/Analysis/Meromorphic/*.lean | head -8",
  "description": "Check logDeriv trailing coefficient API"
}

{
  "command": "sed -n '143,200p' /d/Python/Quantum/zeta-23-lean/Zeta23/Analytic/RectangleLogDeriv.lean",
  "description": "Read rectangleIntegral'_mul_logDeriv_of_poles statement"
}

{
  "command": "cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"iteratedDeriv_eq_zero\\|eq_pow_mul\\|exists_pow_mul\\|analyticOrderAt.*ge\\|orderAt_ge\" Mathlib/Analysis/Analytic/Order.lean | head -10; grep -rn \"Taylor\" Mathlib/Analysis/Analytic/*.lean -l | head -5",
  "description": "Find Taylor remainder / analytic order API"
}

{
  "command": "cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"iteratedDeriv_comp\" Mathlib/Analysis/Calculus/IteratedDeriv/*.lean | head -10",
  "description": "Find iteratedDeriv composition lemmas"
}

{
  "command": "cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -n \"completedRiemannZeta\u2080\\|def completedRiemannZeta\" Mathlib/NumberTheory/LSeries/RiemannZeta.lean | head -12; grep -rn \"theorem differentiableAt_Gamma\\b\\|differentiable_completedZeta\u2080\" Mathlib/NumberTheory/LSeries/RiemannZeta.lean Mathlib/Analysis/SpecialFunctions/Gamma/Basic.lean 2>/dev/null | head -5",
  "description": "Check completedRiemannZeta\u2080 definition and Gamma differentiability"
}

{
  "command": "sed -n '95,130p' /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/Complex/RemovableSingularity.lean",
  "description": "Read removable singularity theorem signatures"
}

rewrite -> probe.cgi (temporary)
PATH_INFO     : '/resources/ζ(3) — Status/'
REQUEST_URI   : '/resources/%CE%B6%283%29%20%E2%80%94%20Status/'
SCRIPT_NAME   : '/probe.cgi'
QUERY_STRING  : ''
LANG/LC_ALL   : None / None
fs encoding   : utf-8
PATH_INFO utf8: b'/resources/\xce\xb6(3) \xe2\x80\x94 Status/'

{
  "command": "cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"choose_mul_factorial_mul_factorial\\|factorial_mul_descFactorial\\|descFactorial_eq_div\\|choose_mul_descFactorial\" Mathlib/Data/Nat/Choose/Basic.lean Mathlib/Data/Nat/Factorial/Basic.lean Mathlib/Data/Nat/Factorial/*.lean 2>/dev/null | head -8; grep -rn \"theorem AnalyticAt.contDiffAt\\|lemma AnalyticAt.contDiffAt\" Mathlib/ --include=*.lean | head -3",
  "description": "Check factorial identities and AnalyticAt.contDiffAt"
}

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean",
  "content": "/-\nNguyenRH1/LiResidue.lean \u2014 tier ii\u2032 sub-stage ii\u2032-1b (D-064 \u00a74 R1):\nthe w = 1 residue of the Lagarias-kernel integrand, computed exactly.\n\nContents (Part 1, this file's core):\n* `zetaA w = (w\u22121)\u00b7\u03b6(w)` extended by 1 at w = 1 \u2014 analytic at 1 with\n  value 1 (removable singularity; Mathlib `riemannZeta_residue_one` +\n  `differentiableOn_update_limUnder_of_bddAbove`).\n* `hLi w = 1 \u2212 (w\u22121)\u00b7(zetaA\u2032/zetaA)(w)` \u2014 analytic at 1, hLi 1 = 1;\n  agrees with (w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6)(w) on a punctured neighborhood of 1\n  (`hLi_eq_near_one`).\n* `lcoeff k` \u2014 the Laurent coefficients of \u2212\u03b6\u2032/\u03b6 at 1, DEFINED as\n  iteratedDeriv (k+1) hLi 1 / (k+1)! (operational; no Laurent series\n  needed at this stage).\n* `ResW1 n` \u2014 the residue at w = 1 of J(w) = k_n(w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6(w)),\n  computed by the Leibniz identity (w\u22121)^{n+1}J = [w\u207f\u2212(w\u22121)\u207f]\u00b7hLi\n  (D-064 \u00a74 R1, verified algebraically in session 014):\n  ResW1 n = \u03a3_{j=1}^n C(n,j)\u00b7lcoeff (j\u22121)  =: `ResW1_eq_sum`.\n\nArchive references: D-036, D-037, D-064 (R1), journal-014/015.\n-/\nimport Mathlib\nimport Zeta23.Statement\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. The regularized factor zetaA = (w\u22121)\u03b6(w) at w = 1 -/\n\n/-- `zetaA w = (w\u22121)\u00b7\u03b6(w)` with the removable singularity at w = 1 filled\nby the residue value 1. Analytic at 1 (`zetaA_analyticAt_one`), value\n`zetaA 1 = 1`. -/\ndef zetaA : \u2102 \u2192 \u2102 := Function.update (fun w => (w - 1) * riemannZeta w) 1 1\n\nlemma zetaA_apply {w : \u2102} (hw : w \u2260 1) : zetaA w = (w - 1) * riemannZeta w :=\n  Function.update_of_ne hw _ _\n\nlemma zetaA_one : zetaA 1 = 1 := Function.update_self _ _ _\n\nlemma zetaA_tendsto : Tendsto (fun w => (w - 1) * riemannZeta w) (\ud835\udcdd[\u2260] 1) (\ud835\udcdd 1) :=\n  riemannZeta_residue_one\n\n/-- The regularized factor is analytic at w = 1 (removable singularity). -/\nlemma zetaA_analyticAt_one : AnalyticAt \u2102 zetaA 1 := by\n  have hdiff : DifferentiableOn \u2102 (fun w => (w - 1) * riemannZeta w)\n      (Metric.ball 1 2 \\ {1}) := by\n    intro w hw\n    have hw1 : w \u2260 1 := hw.2\n    exact (differentiableAt_id.sub (differentiableAt_const 1)).mul\n      (differentiableAt_riemannZeta hw1) |>.differentiableWithinAt\n  have hbd : BddAbove (Complex.norm \u2218 (fun w => (w - 1) * riemannZeta w)) ''\n      (Metric.ball 1 2 \\ {1}) := by\n    have hT := zetaA_tendsto\n    rw [Metric.tendsto_nhdsWithin_nhds] at hT\n    obtain \u27e8\u03b4, h\u03b40, h\u03b4\u27e9 := hT 1 one_pos\n    refine \u27e82, ?_\u27e9\n    rintro _ \u27e8z, hz, rfl\u27e9\n    have h1 : dist (z - 1) 0 = \u2016z - 1\u2016 := by\n      rw [dist_eq_norm]; simp [sub_zero]\n    have hz1 : z \u2260 1 := hz.2\n    have hzz : dist z 1 < \u03b4 := by\n      have := hz.1\n      rw [Metric.mem_ball, dist_comm] at this\n      exact this\n    have hlt := h\u03b4 (by rwa [dist_zero_right, h1] at *)\n    have : \u2016(z - 1) * riemannZeta z\u2016 \u2264 \u2016(1 : \u2102)\u2016 + 1 := by\n      have := norm_sub_le_of_le hlt\n      calc \u2016(z - 1) * riemannZeta z\u2016\n          \u2264 \u2016(z - 1) * riemannZeta z - 1\u2016 + \u2016(1 : \u2102)\u2016 := by\n            have := norm_sub_le ((z - 1) * riemannZeta z) 1\n            linarith [this, hlt]\n      _ \u2264 2 := by\n            have h2 := hlt\n            rw [norm_one]\n            linarith [h2]\n    exact this\n  have h := differentiableOn_update_limUnder_of_bddAbove\n    (Metric.ball_mem_nhds 1 (by norm_num : (0 : \u211d) < 2)) hdiff hbd\n  have hlim : limUnder (\ud835\udcdd[\u2260] (1 : \u2102)) (fun w => (w - 1) * riemannZeta w) = 1 :=\n    zetaA_tendsto.limUnder_eq\n  rw [hlim] at h\n  have hopen : IsOpen (Metric.ball (1 : \u2102) 2) := Metric.isOpen_ball\n  have han := h.analyticOnNhd hopen\n  exact han.analyticAt (Metric.ball_mem_nhds 1 (by norm_num : (0 : \u211d) < 2))\n\n/-- zetaA is nonzero on a neighborhood of 1 (value 1 there). -/\nlemma zetaA_ne_zero_near_one : \u2200\u1da0 w in \ud835\udcdd (1 : \u2102), zetaA w \u2260 0 :=\n  (zetaA_analyticAt_one.continuousAt).eventually_ne (by rw [zetaA_one]; norm_num)\n\n/-- zetaA is analytic everywhere except possibly at 1 \u2014 and in fact also\nat 1; we record the off-1 version for quotient computations. -/\nlemma zetaA_analyticAt_of_ne {w : \u2102} (hw : w \u2260 1) : AnalyticAt \u2102 zetaA w := by\n  have h : (fun w => (w - 1) * riemannZeta w) =\u1da0[\ud835\udcdd w] zetaA := by\n    filter_upwards [eventually_ne_nhds hw] with z hz using (zetaA_apply hz).symm\n  have ha : AnalyticAt \u2102 (fun w => (w - 1) * riemannZeta w) w := by\n    exact (analyticAt_id.sub analyticAt_const).mul\n      (differentiableAt_riemannZeta hw).analyticAt\n  exact ha.congr h\n\n/-! ## 2. hLi w = 1 \u2212 (w\u22121)\u00b7(zetaA\u2032/zetaA)(w): analytic at 1, value 1 -/\n\n/-- hLi(w) = 1 \u2212 (w\u22121)\u00b7(zetaA\u2032/zetaA)(w) \u2014 the analytic-at-1 form of\n(w\u22121)(\u2212\u03b6\u2032/\u03b6)(w) (`hLi_eq_near_one`). hLi 1 = 1. -/\ndef hLi (w : \u2102) : \u2102 := 1 - (w - 1) * (deriv zetaA w / zetaA w)\n\nlemma hLi_one : hLi 1 = 1 := by simp [hLi]\n\nlemma hLi_analyticAt_one : AnalyticAt \u2102 hLi 1 := by\n  have ha := zetaA_analyticAt_one\n  have hda : AnalyticAt \u2102 (deriv zetaA) 1 := ha.deriv\n  have hne : zetaA 1 \u2260 0 := by rw [zetaA_one]; norm_num\n  exact (analyticAt_const.sub analyticAt_id).mul\n    (hda.div ha hne) |>.const_sub 1\n\n/-- On a punctured neighborhood of 1, hLi = (w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6). -/\nlemma hLi_eq_of_ne {w : \u2102} (hw1 : w \u2260 1) (hw0 : riemannZeta w \u2260 0) :\n    hLi w = (w - 1) * (-(deriv riemannZeta w) / riemannZeta w) := by\n  have hA : zetaA w = (w - 1) * riemannZeta w := zetaA_apply hw1\n  have hdA : deriv zetaA w = riemannZeta w + (w - 1) * deriv riemannZeta w := by\n    have h1 : deriv (fun z => (z - 1) * riemannZeta z) w =\n        deriv (fun z => z - 1) w * riemannZeta w +\n          (w - 1) * deriv riemannZeta w :=\n      deriv_mul ((differentiableAt_id.sub (differentiableAt_const 1)))\n        (differentiableAt_riemannZeta hw1)\n    have h2 : deriv (fun z => z - 1) w = 1 := by\n      simp [deriv_sub, deriv_id'']\n    have h3 : (fun z => (z - 1) * riemannZeta z) =\u1da0[\ud835\udcdd w] zetaA := by\n      filter_upwards [eventually_ne_nhds hw1] with z hz using (zetaA_apply hz).symm\n    rw [h3.deriv_eq, h1, h2, one_mul]\n  rw [hLi, hdA, hA]\n  have hw1' : w - 1 \u2260 0 := sub_ne_zero.mpr hw1\n  field_simp [hw0, hw1']\n  ring\n\n/-- The punctured-neighborhood form, for later contour work. -/\nlemma hLi_eq_near_one :\n    \u2200\u1da0 w in \ud835\udcdd[\u2260] (1 : \u2102), hLi w = (w - 1) * (-(deriv riemannZeta w) / riemannZeta w) := by\n  have hzero : \u2200\u1da0 w in \ud835\udcdd[\u2260] (1 : \u2102), riemannZeta w \u2260 0 := by\n    have h1 : (1 : \u2102) \u2208 {s : \u2102 | 1 \u2264 s.re} := by simp\n    filter_upwards [eventually_mem_nhdsWithin] with w hw\n    have hwre : 1 \u2264 w.re := by\n      have := hw.2\n      sorry\n    exact riemannZeta_ne_zero_of_one_le_re hwre\n  filter_upwards [hzero, eventually_mem_nhdsWithin] with w hw0 hw1\n  exact hLi_eq_of_ne hw1 hw0\n\n/-! ## 3. The Laurent coefficients of \u2212\u03b6\u2032/\u03b6 at 1 (operational definition) -/\n\n/-- The k-th Laurent coefficient of \u2212\u03b6\u2032/\u03b6 at s = 1:\n\u2212\u03b6\u2032/\u03b6(1+u) = u\u207b\u00b9 + \u03a3_{k\u22650} lcoeff k \u00b7 u^k (identification proved in\ntier ii\u2032-2/3; here lcoeff is DEFINED via the analytic hLi). -/\ndef lcoeff (k : \u2115) : \u2102 := iteratedDeriv (k + 1) hLi 1 / (k + 1) !\n\nlemma lcoeff_eq (k : \u2115) :\n    lcoeff k = iteratedDeriv (k + 1) hLi 1 / (k + 1) ! := rfl\n\n/-- hLi's Taylor coefficient form: iteratedDeriv j hLi 1 / j ! = lcoeff (j\u22121). -/\nlemma hLi_iteratedDeriv_div_factorial {j : \u2115} (hj : 1 \u2264 j) :\n    iteratedDeriv j hLi 1 / j ! = lcoeff (j - 1) := by\n  rw [lcoeff]\n  have : j = (j - 1) + 1 := by omega\n  rw [this]\n\n/-! ## 4. The A-polynomial derivative: A(w) = w\u207f \u2212 (w\u22121)\u207f -/\n\n/-- Iterated derivatives of A(w) = w\u207f \u2212 (w\u22121)\u207f at 1:\nA^{(i)}(1) = n.descFactorial i for i < n, and A^{(n)}(1) = 0. -/\nlemma A_iter (n i : \u2115) (hi : i \u2264 n) :\n    iteratedDeriv i (fun w => w ^ n - (w - 1) ^ n) (1 : \u2102) =\n      if i < n then n.descFactorial i else 0 := by\n  have hf : ContDiffAt \u2102 i (fun w : \u2102 => w ^ n) 1 :=\n    (contDiff_id.pow n).contDiffAt\n  have hg : ContDiffAt \u2102 i (fun w : \u2102 => (w - 1) ^ n) 1 :=\n    ((contDiff_id.sub contDiff_const).pow n).contDiffAt\n  rw [iteratedDeriv_sub hf hg]\n  have h1 : iteratedDeriv i (fun w : \u2102 => w ^ n) 1 = n.descFactorial i := by\n    rw [iteratedDeriv_pow]\n    simp\n  have h2 : iteratedDeriv i (fun w : \u2102 => (w - 1) ^ n) 1 =\n      n.descFactorial i * (0 : \u2102) ^ (n - i) := by\n    have e := iteratedDeriv_comp_sub_const i (fun w : \u2102 => w ^ n) (1 : \u2102)\n    have e2 : iteratedDeriv i (fun w : \u2102 => (w - 1) ^ n) 1 =\n        iteratedDeriv i (fun w : \u2102 => w ^ n) (1 - 1) := by\n      have := congr_fun e 1\n      convert this using 2\n    rw [e2, iteratedDeriv_pow]\n    simp\n  rw [h1, h2]\n  by_cases hin : i < n\n  \u00b7 rw [if_pos hin]\n    have : (0 : \u2102) ^ (n - i) = 0 := by\n      apply zero_pow\n      omega\n    rw [this, mul_zero, sub_zero]\n  \u00b7 rw [if_neg hin]\n    have hin' : i = n := le_antisymm hi (by omega)\n    subst hin'\n    simp [Nat.descFactorial_self]\n\n/-! ## 5. The residue at w = 1: ResW1 n = \u03a3_{j=1}^n C(n,j)\u00b7lcoeff (j\u22121) -/\n\n/-- The residue at w = 1 of the Lagarias-kernel integrand\nJ(w) = k_n(w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6(w)), packaged as (1/n!)\u00b7(d/dw)\u207f[(w\u207f\u2212(w\u22121)\u207f)hLi]\nat w = 1 (D-064 \u00a74 R1: (w\u22121)^{n+1}J = [w\u207f\u2212(w\u22121)\u207f]\u00b7hLi on a punctured\nneighborhood, by `hLi_eq_near_one`). -/\ndef ResW1 (n : \u2115) : \u2102 :=\n  iteratedDeriv n (fun w => (w ^ n - (w - 1) ^ n) * hLi w) (1 : \u2102) / n !\n\n/-- The w = 1 residue value: ResW1 n = \u03a3_{j=1}^n C(n,j)\u00b7lcoeff (j\u22121). -/\ntheorem ResW1_eq_sum (n : \u2115) :\n    ResW1 n = \u2211 j \u2208 Finset.Icc 1 n, (n.choose j : \u2102) * lcoeff (j - 1) := by\n  have hA : ContDiffAt \u2102 n (fun w : \u2102 => w ^ n - (w - 1) ^ n) 1 :=\n    ((contDiff_id.pow n).sub ((contDiff_id.sub contDiff_const).pow n)).contDiffAt\n  have hh : ContDiffAt \u2102 n hLi 1 := hLi_analyticAt_one.contDiffAt\n  have hmul := iteratedDeriv_mul hA hh\n  rw [ResW1, hmul]\n  have hsum : \u2211 i \u2208 Finset.range (n + 1),\n        n.choose i * iteratedDeriv i (fun w : \u2102 => w ^ n - (w - 1) ^ n) 1 *\n          iteratedDeriv (n - i) hLi 1 =\n      \u2211 i \u2208 Finset.range n,\n        n.choose i * (n.descFactorial i : \u2102) * iteratedDeriv (n - i) hLi 1 := by\n    apply Finset.sum_subset (Finset.range_le_iff.mpr (Nat.le_succ n))\n    intro i hi\n    rw [Finset.mem_range] at hi\n    have hi0 : i \u2260 n := by omega\n    rw [A_iter n i (by omega), if_pos (by omega)]\n    \u00b7 rfl\n    \u00b7 rw [Finset.mem_range] at *\n      omega\n  rw [hsum]\n  have hfinal : (\u2211 i \u2208 Finset.range n,\n        n.choose i * (n.descFactorial i : \u2102) * iteratedDeriv (n - i) hLi 1) / n ! =\n      \u2211 j \u2208 Finset.Icc 1 n, (n.choose j : \u2102) * lcoeff (j - 1) := by\n    have hrewrite : \u2200 i \u2208 Finset.range n,\n        n.choose i * (n.descFactorial i : \u2102) * iteratedDeriv (n - i) hLi 1 / n ! =\n          (n.choose (n - i) : \u2102) * lcoeff (n - i - 1) := by\n      intro i hi\n      rw [Finset.mem_range] at hi\n      have hcd : (n.choose i : \u2102) * (n.descFactorial i : \u2102) =\n          n ! * n.choose i / (n - i) ! := by\n        have h1 := Nat.choose_mul_factorial_mul_factorial (show i \u2264 n by omega)\n        have h2 := Nat.factorial_mul_descFactorial (show i \u2264 n by omega)\n        have : (n.choose i : \u2102) * (n.descFactorial i : \u2102) * ((i ! : \u2102) * ((n - i) ! : \u2102)) =\n            (n ! : \u2102) * (n.choose i : \u2102) := by\n          push_cast [h1, h2]\n          ring\n        have hfact : ((i ! : \u2102) * ((n - i) ! : \u2102)) \u2260 0 := by\n          simp [Nat.factorial_ne_zero]\n        have hthis := this\n        field_simp at hthis \u22a2\n        have h3 : (n.choose i : \u2102) * (n.descFactorial i : \u2102) * ((n - i) ! : \u2102) =\n            (n ! : \u2102) * (n.choose i : \u2102) / 1 := by\n          have h4 := hthis\n          rw [mul_comm ((i ! : \u2102)) ((n - i) ! : \u2102)] at h4\n          have h5 : (n.choose i : \u2102) * (n.descFactorial i : \u2102) * ((n - i) ! : \u2102) * (i ! : \u2102) =\n              (n ! : \u2102) * (n.choose i : \u2102) := by\n            rw [mul_comm ((n - i) ! : \u2102) (i ! : \u2102)] at h4\n            calc (n.choose i : \u2102) * (n.descFactorial i : \u2102) * ((n - i) ! : \u2102) * (i ! : \u2102)\n                = (n.choose i : \u2102) * (n.descFactorial i : \u2102) * (((n - i) ! : \u2102) * (i ! : \u2102)) := by ring\n              _ = (n ! : \u2102) * (n.choose i : \u2102) := by\n                  have := h4\n                  ring_nf at this \u22a2\n                  convert this using 1\n                  ring\n          exact (div_eq_one_iff_eq (by simp [Nat.factorial_ne_zero])).mpr h5\n        field_simp at h3 \u22a2\n        calc (n.choose i : \u2102) * (n.descFactorial i : \u2102)\n            = ((n ! : \u2102) * (n.choose i : \u2102) / 1) / ((n - i) ! : \u2102) := by\n              rw [\u2190 h3]\n              field_simp\n              ring\n          _ = (n ! : \u2102) * (n.choose i : \u2102) / ((n - i) ! : \u2102) := by ring\n      rw [hcd]\n      have hni : n - i \u2265 1 := by omega\n      rw [hLi_iteratedDeriv_div_factorial (j := n - i) (by omega)]\n      rw [show n - i - 1 = n - i - 1 from rfl]\n      have hcoe : ((n - i) ! : \u2102) \u2260 0 := by simp [Nat.factorial_ne_zero]\n      field_simp\n      rw [Nat.choose_symm (by omega : i \u2264 n)]\n      ring\n    rw [Finset.sum_div]\n    rw [Finset.sum_congr rfl hrewrite]\n    have hmap : \u2211 i \u2208 Finset.range n, (n.choose (n - i) : \u2102) * lcoeff (n - i - 1) =\n        \u2211 j \u2208 Finset.Icc 1 n, (n.choose j : \u2102) * lcoeff (j - 1) := by\n      rw [Finset.sum_range_reflect]\n      apply Finset.sum_congr rfl\n      intro i hi\n      rw [Finset.mem_range] at hi\n      have : n - 1 - i + 1 = n - i := by omega\n      rw [this]\n      have : n - (n - i) = i := by omega\n      rw [Nat.choose_symm (show n - i \u2264 n by omega)]\n      simp [this]\n      ring\n    exact hmap\n  rw [hfinal]\n\nend NguyenRH1\n"
}

{
  "type": "create",
  "filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean",
  "content": "/-\nNguyenRH1/LiResidue.lean \u2014 tier ii\u2032 sub-stage ii\u2032-1b (D-064 \u00a74 R1):\nthe w = 1 residue of the Lagarias-kernel integrand, computed exactly.\n\nContents (Part 1, this file's core):\n* `zetaA w = (w\u22121)\u00b7\u03b6(w)` extended by 1 at w = 1 \u2014 analytic at 1 with\n  value 1 (removable singularity; Mathlib `riemannZeta_residue_one` +\n  `differentiableOn_update_limUnder_of_bddAbove`).\n* `hLi w = 1 \u2212 (w\u22121)\u00b7(zetaA\u2032/zetaA)(w)` \u2014 analytic at 1, hLi 1 = 1;\n  agrees with (w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6)(w) on a punctured neighborhood of 1\n  (`hLi_eq_near_one`).\n* `lcoeff k` \u2014 the Laurent coefficients of \u2212\u03b6\u2032/\u03b6 at 1, DEFINED as\n  iteratedDeriv (k+1) hLi 1 / (k+1)! (operational; no Laurent series\n  needed at this stage).\n* `ResW1 n` \u2014 the residue at w = 1 of J(w) = k_n(w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6(w)),\n  computed by the Leibniz identity (w\u22121)^{n+1}J = [w\u207f\u2212(w\u22121)\u207f]\u00b7hLi\n  (D-064 \u00a74 R1, verified algebraically in session 014):\n  ResW1 n = \u03a3_{j=1}^n C(n,j)\u00b7lcoeff (j\u22121)  =: `ResW1_eq_sum`.\n\nArchive references: D-036, D-037, D-064 (R1), journal-014/015.\n-/\nimport Mathlib\nimport Zeta23.Statement\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. The regularized factor zetaA = (w\u22121)\u03b6(w) at w = 1 -/\n\n/-- `zetaA w = (w\u22121)\u00b7\u03b6(w)` with the removable singularity at w = 1 filled\nby the residue value 1. Analytic at 1 (`zetaA_analyticAt_one`), value\n`zetaA 1 = 1`. -/\ndef zetaA : \u2102 \u2192 \u2102 := Function.update (fun w => (w - 1) * riemannZeta w) 1 1\n\nlemma zetaA_apply {w : \u2102} (hw : w \u2260 1) : zetaA w = (w - 1) * riemannZeta w :=\n  Function.update_of_ne hw _ _\n\nlemma zetaA_one : zetaA 1 = 1 := Function.update_self _ _ _\n\nlemma zetaA_tendsto : Tendsto (fun w => (w - 1) * riemannZeta w) (\ud835\udcdd[\u2260] 1) (\ud835\udcdd 1) :=\n  riemannZeta_residue_one\n\n/-- The regularized factor is analytic at w = 1 (removable singularity). -/\nlemma zetaA_analyticAt_one : AnalyticAt \u2102 zetaA 1 := by\n  have hdiff : DifferentiableOn \u2102 (fun w => (w - 1) * riemannZeta w)\n      (Metric.ball 1 2 \\ {1}) := by\n    intro w hw\n    have hw1 : w \u2260 1 := hw.2\n    exact (differentiableAt_id.sub (differentiableAt_const 1)).mul\n      (differentiableAt_riemannZeta hw1) |>.differentiableWithinAt\n  have hbd : BddAbove (Complex.norm \u2218 (fun w => (w - 1) * riemannZeta w)) ''\n      (Metric.ball 1 2 \\ {1}) := by\n    have hT := zetaA_tendsto\n    rw [Metric.tendsto_nhdsWithin_nhds] at hT\n    obtain \u27e8\u03b4, h\u03b40, h\u03b4\u27e9 := hT 1 one_pos\n    refine \u27e82, ?_\u27e9\n    rintro _ \u27e8z, hz, rfl\u27e9\n    have h1 : dist (z - 1) 0 = \u2016z - 1\u2016 := by\n      rw [dist_eq_norm]; simp [sub_zero]\n    have hz1 : z \u2260 1 := hz.2\n    have hzz : dist z 1 < \u03b4 := by\n      have := hz.1\n      rw [Metric.mem_ball, dist_comm] at this\n      exact this\n    have hlt := h\u03b4 (by rwa [dist_zero_right, h1] at *)\n    have : \u2016(z - 1) * riemannZeta z\u2016 \u2264 \u2016(1 : \u2102)\u2016 + 1 := by\n      have := norm_sub_le_of_le hlt\n      calc \u2016(z - 1) * riemannZeta z\u2016\n          \u2264 \u2016(z - 1) * riemannZeta z - 1\u2016 + \u2016(1 : \u2102)\u2016 := by\n            have := norm_sub_le ((z - 1) * riemannZeta z) 1\n            linarith [this, hlt]\n      _ \u2264 2 := by\n            have h2 := hlt\n            rw [norm_one]\n            linarith [h2]\n    exact this\n  have h := differentiableOn_update_limUnder_of_bddAbove\n    (Metric.ball_mem_nhds 1 (by norm_num : (0 : \u211d) < 2)) hdiff hbd\n  have hlim : limUnder (\ud835\udcdd[\u2260] (1 : \u2102)) (fun w => (w - 1) * riemannZeta w) = 1 :=\n    zetaA_tendsto.limUnder_eq\n  rw [hlim] at h\n  have hopen : IsOpen (Metric.ball (1 : \u2102) 2) := Metric.isOpen_ball\n  have han := h.analyticOnNhd hopen\n  exact han.analyticAt (Metric.ball_mem_nhds 1 (by norm_num : (0 : \u211d) < 2))\n\n/-- zetaA is nonzero on a neighborhood of 1 (value 1 there). -/\nlemma zetaA_ne_zero_near_one : \u2200\u1da0 w in \ud835\udcdd (1 : \u2102), zetaA w \u2260 0 :=\n  (zetaA_analyticAt_one.continuousAt).eventually_ne (by rw [zetaA_one]; norm_num)\n\n/-- zetaA is analytic everywhere except possibly at 1 \u2014 and in fact also\nat 1; we record the off-1 version for quotient computations. -/\nlemma zetaA_analyticAt_of_ne {w : \u2102} (hw : w \u2260 1) : AnalyticAt \u2102 zetaA w := by\n  have h : (fun w => (w - 1) * riemannZeta w) =\u1da0[\ud835\udcdd w] zetaA := by\n    filter_upwards [eventually_ne_nhds hw] with z hz using (zetaA_apply hz).symm\n  have ha : AnalyticAt \u2102 (fun w => (w - 1) * riemannZeta w) w := by\n    exact (analyticAt_id.sub analyticAt_const).mul\n      (differentiableAt_riemannZeta hw).analyticAt\n  exact ha.congr h\n\n/-! ## 2. hLi w = 1 \u2212 (w\u22121)\u00b7(zetaA\u2032/zetaA)(w): analytic at 1, value 1 -/\n\n/-- hLi(w) = 1 \u2212 (w\u22121)\u00b7(zetaA\u2032/zetaA)(w) \u2014 the analytic-at-1 form of\n(w\u22121)(\u2212\u03b6\u2032/\u03b6)(w) (`hLi_eq_near_one`). hLi 1 = 1. -/\ndef hLi (w : \u2102) : \u2102 := 1 - (w - 1) * (deriv zetaA w / zetaA w)\n\nlemma hLi_one : hLi 1 = 1 := by simp [hLi]\n\nlemma hLi_analyticAt_one : AnalyticAt \u2102 hLi 1 := by\n  have ha := zetaA_analyticAt_one\n  have hda : AnalyticAt \u2102 (deriv zetaA) 1 := ha.deriv\n  have hne : zetaA 1 \u2260 0 := by rw [zetaA_one]; norm_num\n  exact (analyticAt_const.sub analyticAt_id).mul\n    (hda.div ha hne) |>.const_sub 1\n\n/-- On a punctured neighborhood of 1, hLi = (w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6). -/\nlemma hLi_eq_of_ne {w : \u2102} (hw1 : w \u2260 1) (hw0 : riemannZeta w \u2260 0) :\n    hLi w = (w - 1) * (-(deriv riemannZeta w) / riemannZeta w) := by\n  have hA : zetaA w = (w - 1) * riemannZeta w := zetaA_apply hw1\n  have hdA : deriv zetaA w = riemannZeta w + (w - 1) * deriv riemannZeta w := by\n    have h1 : deriv (fun z => (z - 1) * riemannZeta z) w =\n        deriv (fun z => z - 1) w * riemannZeta w +\n          (w - 1) * deriv riemannZeta w :=\n      deriv_mul ((differentiableAt_id.sub (differentiableAt_const 1)))\n        (differentiableAt_riemannZeta hw1)\n    have h2 : deriv (fun z => z - 1) w = 1 := by\n      simp [deriv_sub, deriv_id'']\n    have h3 : (fun z => (z - 1) * riemannZeta z) =\u1da0[\ud835\udcdd w] zetaA := by\n      filter_upwards [eventually_ne_nhds hw1] with z hz using (zetaA_apply hz).symm\n    rw [h3.deriv_eq, h1, h2, one_mul]\n  rw [hLi, hdA, hA]\n  have hw1' : w - 1 \u2260 0 := sub_ne_zero.mpr hw1\n  field_simp [hw0, hw1']\n  ring\n\n/-- The punctured-neighborhood form, for later contour work. -/\nlemma hLi_eq_near_one :\n    \u2200\u1da0 w in \ud835\udcdd[\u2260] (1 : \u2102), hLi w = (w - 1) * (-(deriv riemannZeta w) / riemannZeta w) := by\n  have hzero : \u2200\u1da0 w in \ud835\udcdd[\u2260] (1 : \u2102), riemannZeta w \u2260 0 := by\n    have h1 : (1 : \u2102) \u2208 {s : \u2102 | 1 \u2264 s.re} := by simp\n    filter_upwards [eventually_mem_nhdsWithin] with w hw\n    have hwre : 1 \u2264 w.re := by\n      have := hw.2\n      sorry\n    exact riemannZeta_ne_zero_of_one_le_re hwre\n  filter_upwards [hzero, eventually_mem_nhdsWithin] with w hw0 hw1\n  exact hLi_eq_of_ne hw1 hw0\n\n/-! ## 3. The Laurent coefficients of \u2212\u03b6\u2032/\u03b6 at 1 (operational definition) -/\n\n/-- The k-th Laurent coefficient of \u2212\u03b6\u2032/\u03b6 at s = 1:\n\u2212\u03b6\u2032/\u03b6(1+u) = u\u207b\u00b9 + \u03a3_{k\u22650} lcoeff k \u00b7 u^k (identification proved in\ntier ii\u2032-2/3; here lcoeff is DEFINED via the analytic hLi). -/\ndef lcoeff (k : \u2115) : \u2102 := iteratedDeriv (k + 1) hLi 1 / (k + 1) !\n\nlemma lcoeff_eq (k : \u2115) :\n    lcoeff k = iteratedDeriv (k + 1) hLi 1 / (k + 1) ! := rfl\n\n/-- hLi's Taylor coefficient form: iteratedDeriv j hLi 1 / j ! = lcoeff (j\u22121). -/\nlemma hLi_iteratedDeriv_div_factorial {j : \u2115} (hj : 1 \u2264 j) :\n    iteratedDeriv j hLi 1 / j ! = lcoeff (j - 1) := by\n  rw [lcoeff]\n  have : j = (j - 1) + 1 := by omega\n  rw [this]\n\n/-! ## 4. The A-polynomial derivative: A(w) = w\u207f \u2212 (w\u22121)\u207f -/\n\n/-- Iterated derivatives of A(w) = w\u207f \u2212 (w\u22121)\u207f at 1:\nA^{(i)}(1) = n.descFactorial i for i < n, and A^{(n)}(1) = 0. -/\nlemma A_iter (n i : \u2115) (hi : i \u2264 n) :\n    iteratedDeriv i (fun w => w ^ n - (w - 1) ^ n) (1 : \u2102) =\n      if i < n then n.descFactorial i else 0 := by\n  have hf : ContDiffAt \u2102 i (fun w : \u2102 => w ^ n) 1 :=\n    (contDiff_id.pow n).contDiffAt\n  have hg : ContDiffAt \u2102 i (fun w : \u2102 => (w - 1) ^ n) 1 :=\n    ((contDiff_id.sub contDiff_const).pow n).contDiffAt\n  rw [iteratedDeriv_sub hf hg]\n  have h1 : iteratedDeriv i (fun w : \u2102 => w ^ n) 1 = n.descFactorial i := by\n    rw [iteratedDeriv_pow]\n    simp\n  have h2 : iteratedDeriv i (fun w : \u2102 => (w - 1) ^ n) 1 =\n      n.descFactorial i * (0 : \u2102) ^ (n - i) := by\n    have e := iteratedDeriv_comp_sub_const i (fun w : \u2102 => w ^ n) (1 : \u2102)\n    have e2 : iteratedDeriv i (fun w : \u2102 => (w - 1) ^ n) 1 =\n        iteratedDeriv i (fun w : \u2102 => w ^ n) (1 - 1) := by\n      have := congr_fun e 1\n      convert this using 2\n    rw [e2, iteratedDeriv_pow]\n    simp\n  rw [h1, h2]\n  by_cases hin : i < n\n  \u00b7 rw [if_pos hin]\n    have : (0 : \u2102) ^ (n - i) = 0 := by\n      apply zero_pow\n      omega\n    rw [this, mul_zero, sub_zero]\n  \u00b7 rw [if_neg hin]\n    have hin' : i = n := le_antisymm hi (by omega)\n    subst hin'\n    simp [Nat.descFactorial_self]\n\n/-! ## 5. The residue at w = 1: ResW1 n = \u03a3_{j=1}^n C(n,j)\u00b7lcoeff (j\u22121) -/\n\n/-- The residue at w = 1 of the Lagarias-kernel integrand\nJ(w) = k_n(w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6(w)), packaged as (1/n!)\u00b7(d/dw)\u207f[(w\u207f\u2212(w\u22121)\u207f)hLi]\nat w = 1 (D-064 \u00a74 R1: (w\u22121)^{n+1}J = [w\u207f\u2212(w\u22121)\u207f]\u00b7hLi on a punctured\nneighborhood, by `hLi_eq_near_one`). -/\ndef ResW1 (n : \u2115) : \u2102 :=\n  iteratedDeriv n (fun w => (w ^ n - (w - 1) ^ n) * hLi w) (1 : \u2102) / n !\n\n/-- The w = 1 residue value: ResW1 n = \u03a3_{j=1}^n C(n,j)\u00b7lcoeff (j\u22121). -/\ntheorem ResW1_eq_sum (n : \u2115) :\n    ResW1 n = \u2211 j \u2208 Finset.Icc 1 n, (n.choose j : \u2102) * lcoeff (j - 1) := by\n  have hA : ContDiffAt \u2102 n (fun w : \u2102 => w ^ n - (w - 1) ^ n) 1 :=\n    ((contDiff_id.pow n).sub ((contDiff_id.sub contDiff_const).pow n)).contDiffAt\n  have hh : ContDiffAt \u2102 n hLi 1 := hLi_analyticAt_one.contDiffAt\n  have hmul := iteratedDeriv_mul hA hh\n  rw [ResW1, hmul]\n  have hsum : \u2211 i \u2208 Finset.range (n + 1),\n        n.choose i * iteratedDeriv i (fun w : \u2102 => w ^ n - (w - 1) ^ n) 1 *\n          iteratedDeriv (n - i) hLi 1 =\n      \u2211 i \u2208 Finset.range n,\n        n.choose i * (n.descFactorial i : \u2102) * iteratedDeriv (n - i) hLi 1 := by\n    apply Finset.sum_subset (Finset.range_le_iff.mpr (Nat.le_succ n))\n    intro i hi\n    rw [Finset.mem_range] at hi\n    have hi0 : i \u2260 n := by omega\n    rw [A_iter n i (by omega), if_pos (by omega)]\n    \u00b7 rfl\n    \u00b7 rw [Finset.mem_range] at *\n      omega\n  rw [hsum]\n  have hfinal : (\u2211 i \u2208 Finset.range n,\n        n.choose i * (n.descFactorial i : \u2102) * iteratedDeriv (n - i) hLi 1) / n ! =\n      \u2211 j \u2208 Finset.Icc 1 n, (n.choose j : \u2102) * lcoeff (j - 1) := by\n    have hrewrite : \u2200 i \u2208 Finset.range n,\n        n.choose i * (n.descFactorial i : \u2102) * iteratedDeriv (n - i) hLi 1 / n ! =\n          (n.choose (n - i) : \u2102) * lcoeff (n - i - 1) := by\n      intro i hi\n      rw [Finset.mem_range] at hi\n      have hcd : (n.choose i : \u2102) * (n.descFactorial i : \u2102) =\n          n ! * n.choose i / (n - i) ! := by\n        have h1 := Nat.choose_mul_factorial_mul_factorial (show i \u2264 n by omega)\n        have h2 := Nat.factorial_mul_descFactorial (show i \u2264 n by omega)\n        have : (n.choose i : \u2102) * (n.descFactorial i : \u2102) * ((i ! : \u2102) * ((n - i) ! : \u2102)) =\n            (n ! : \u2102) * (n.choose i : \u2102) := by\n          push_cast [h1, h2]\n          ring\n        have hfact : ((i ! : \u2102) * ((n - i) ! : \u2102)) \u2260 0 := by\n          simp [Nat.factorial_ne_zero]\n        have hthis := this\n        field_simp at hthis \u22a2\n        have h3 : (n.choose i : \u2102) * (n.descFactorial i : \u2102) * ((n - i) ! : \u2102) =\n            (n ! : \u2102) * (n.choose i : \u2102) / 1 := by\n          have h4 := hthis\n          rw [mul_comm ((i ! : \u2102)) ((n - i) ! : \u2102)] at h4\n          have h5 : (n.choose i : \u2102) * (n.descFactorial i : \u2102) * ((n - i) ! : \u2102) * (i ! : \u2102) =\n              (n ! : \u2102) * (n.choose i : \u2102) := by\n            rw [mul_comm ((n - i) ! : \u2102) (i ! : \u2102)] at h4\n            calc (n.choose i : \u2102) * (n.descFactorial i : \u2102) * ((n - i) ! : \u2102) * (i ! : \u2102)\n                = (n.choose i : \u2102) * (n.descFactorial i : \u2102) * (((n - i) ! : \u2102) * (i ! : \u2102)) := by ring\n              _ = (n ! : \u2102) * (n.choose i : \u2102) := by\n                  have := h4\n                  ring_nf at this \u22a2\n                  convert this using 1\n                  ring\n          exact (div_eq_one_iff_eq (by simp [Nat.factorial_ne_zero])).mpr h5\n        field_simp at h3 \u22a2\n        calc (n.choose i : \u2102) * (n.descFactorial i : \u2102)\n            = ((n ! : \u2102) * (n.choose i : \u2102) / 1) / ((n - i) ! : \u2102) := by\n              rw [\u2190 h3]\n              field_simp\n              ring\n          _ = (n ! : \u2102) * (n.choose i : \u2102) / ((n - i) ! : \u2102) := by ring\n      rw [hcd]\n      have hni : n - i \u2265 1 := by omega\n      rw [hLi_iteratedDeriv_div_factorial (j := n - i) (by omega)]\n      rw [show n - i - 1 = n - i - 1 from rfl]\n      have hcoe : ((n - i) ! : \u2102) \u2260 0 := by simp [Nat.factorial_ne_zero]\n      field_simp\n      rw [Nat.choose_symm (by omega : i \u2264 n)]\n      ring\n    rw [Finset.sum_div]\n    rw [Finset.sum_congr rfl hrewrite]\n    have hmap : \u2211 i \u2208 Finset.range n, (n.choose (n - i) : \u2102) * lcoeff (n - i - 1) =\n        \u2211 j \u2208 Finset.Icc 1 n, (n.choose j : \u2102) * lcoeff (j - 1) := by\n      rw [Finset.sum_range_reflect]\n      apply Finset.sum_congr rfl\n      intro i hi\n      rw [Finset.mem_range] at hi\n      have : n - 1 - i + 1 = n - i := by omega\n      rw [this]\n      have : n - (n - i) = i := by omega\n      rw [Nat.choose_symm (show n - i \u2264 n by omega)]\n      simp [this]\n      ring\n    exact hmap\n  rw [hfinal]\n\nend NguyenRH1\n",
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  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean",
  "content": "/-\nNguyenRH1/LiResidue.lean \u2014 tier ii\u2032 sub-stage ii\u2032-1b (D-064 \u00a74 R1):\nthe w = 1 residue of the Lagarias-kernel integrand, computed exactly.\n\nContents (Part 1):\n* `zetaA w = (w\u22121)\u00b7\u03b6(w)` extended by 1 at w = 1 \u2014 analytic at 1 with\n  value 1 (removable singularity; Mathlib `riemannZeta_residue_one` +\n  `differentiableOn_update_limUnder_of_bddAbove`).\n* `hLi w = 1 \u2212 (w\u22121)\u00b7(zetaA\u2032/zetaA)(w)` \u2014 analytic at 1, hLi 1 = 1;\n  agrees with (w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6)(w) on a punctured neighborhood of 1\n  (`hLi_eq_near_one`).\n* `lcoeff k` \u2014 the Laurent coefficients of \u2212\u03b6\u2032/\u03b6 at 1, DEFINED as\n  iteratedDeriv (k+1) hLi 1 / (k+1)!.\n* `ResW1 n` \u2014 the residue at w = 1 of J(w) = k_n(w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6(w)),\n  packaged as (1/n!)\u00b7(d/dw)\u207f[(w\u207f\u2212(w\u22121)\u207f)\u00b7hLi] at w = 1 (D-064 \u00a74 R1:\n  (w\u22121)^{n+1}J = [w\u207f\u2212(w\u22121)\u207f]\u00b7hLi on a punctured neighborhood, by\n  `hLi_eq_near_one`). Value: `ResW1_eq_sum`.\n\nArchive references: D-036, D-037, D-064 (R1), journal-014/015.\n-/\nimport Mathlib\nimport Zeta23.Statement\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. The regularized factor zetaA = (w\u22121)\u03b6(w) at w = 1 -/\n\n/-- `zetaA w = (w\u22121)\u00b7\u03b6(w)` with the removable singularity at w = 1 filled\nby the residue value 1. Analytic at 1 (`zetaA_analyticAt_one`), value\n`zetaA 1 = 1`. -/\ndef zetaA : \u2102 \u2192 \u2102 := Function.update (fun w => (w - 1) * riemannZeta w) 1 1\n\nlemma zetaA_apply {w : \u2102} (hw : w \u2260 1) : zetaA w = (w - 1) * riemannZeta w :=\n  Function.update_of_ne hw _ _\n\nlemma zetaA_one : zetaA 1 = 1 := Function.update_self _ _ _\n\n/-- The regularized factor is analytic at w = 1 (removable singularity). -/\nlemma zetaA_analyticAt_one : AnalyticAt \u2102 zetaA 1 := by\n  have hdiff : DifferentiableOn \u2102 (fun w => (w - 1) * riemannZeta w)\n      (Metric.ball 1 2 \\ {1}) := by\n    intro w hw\n    have hw1 : w \u2260 1 := hw.2\n    exact ((differentiableAt_id.sub (differentiableAt_const 1)).mul\n      (differentiableAt_riemannZeta hw1)).differentiableWithinAt\n  have hbd : BddAbove (Complex.norm \u2218 (fun w => (w - 1) * riemannZeta w)) ''\n      (Metric.ball 1 2 \\ {1}) := by\n    have hT := riemannZeta_residue_one\n    rw [Metric.tendsto_nhdsWithin_nhds] at hT\n    obtain \u27e8\u03b4, h\u03b40, h\u03b4\u27e9 := hT 1 one_pos\n    refine \u27e82, ?_\u27e9\n    rintro _ \u27e8z, hz, rfl\u27e9\n    have hz1 : z \u2260 1 := hz.2\n    have hlt := h\u03b4 (by\n      rw [dist_eq_norm, sub_zero]\n      exact hz.1)\n    have h1 : \u2016(z - 1) * riemannZeta z - 1\u2016 < 1 := by\n      have := hlt\n      rwa [dist_eq_norm] at this\n    have := norm_sub_le ((z - 1) * riemannZeta z) 1\n    simp only [Complex.norm_eq_abs] at *\n    have h2 : \u2016(z - 1) * riemannZeta z\u2016 \u2264 \u2016(z - 1) * riemannZeta z - 1\u2016 + \u2016(1 : \u2102)\u2016 :=\n      norm_sub_le ((z - 1) * riemannZeta z) 1\n    rw [norm_one] at h2\n    linarith [h1]\n  have h := differentiableOn_update_limUnder_of_bddAbove\n    (Metric.ball_mem_nhds 1 (by norm_num : (0 : \u211d) < 2)) hdiff hbd\n  have hlim : limUnder (\ud835\udcdd[\u2260] (1 : \u2102)) (fun w => (w - 1) * riemannZeta w) = 1 :=\n    riemannZeta_residue_one.limUnder_eq\n  rw [hlim] at h\n  have han := h.analyticOnNhd Metric.isOpen_ball\n  exact han.analyticAt (Metric.ball_mem_nhds 1 (by norm_num : (0 : \u211d) < 2))\n\n/-- zetaA is nonzero on a neighborhood of 1 (value 1 there). -/\nlemma zetaA_ne_zero_near_one : \u2200\u1da0 w in \ud835\udcdd (1 : \u2102), zetaA w \u2260 0 :=\n  (zetaA_analyticAt_one.continuousAt).eventually_ne (by rw [zetaA_one]; norm_num)\n\n/-- zetaA is analytic at any w \u2260 1 (agrees with (w\u22121)\u03b6 there). -/\nlemma zetaA_analyticAt_of_ne {w : \u2102} (hw : w \u2260 1) : AnalyticAt \u2102 zetaA w := by\n  have h : (fun w => (w - 1) * riemannZeta w) =\u1da0[\ud835\udcdd w] zetaA := by\n    filter_upwards [eventually_ne_nhds hw] with z hz using (zetaA_apply hz).symm\n  have ha : AnalyticAt \u2102 (fun w => (w - 1) * riemannZeta w) w :=\n    (analyticAt_id.sub analyticAt_const).mul\n      (differentiableAt_riemannZeta hw).analyticAt\n  exact ha.congr h\n\n/-! ## 2. hLi w = 1 \u2212 (w\u22121)\u00b7(zetaA\u2032/zetaA)(w): analytic at 1, value 1 -/\n\n/-- hLi(w) = 1 \u2212 (w\u22121)\u00b7(zetaA\u2032/zetaA)(w) \u2014 the analytic-at-1 form of\n(w\u22121)(\u2212\u03b6\u2032/\u03b6)(w) (`hLi_eq_near_one`). hLi 1 = 1. -/\ndef hLi (w : \u2102) : \u2102 := 1 - (w - 1) * (deriv zetaA w / zetaA w)\n\nlemma hLi_one : hLi 1 = 1 := by simp [hLi]\n\nlemma hLi_analyticAt_one : AnalyticAt \u2102 hLi 1 := by\n  have ha := zetaA_analyticAt_one\n  have hda : AnalyticAt \u2102 (deriv zetaA) 1 := ha.deriv\n  have hne : zetaA 1 \u2260 0 := by rw [zetaA_one]; norm_num\n  exact ((analyticAt_const.sub analyticAt_id).mul\n    (hda.div ha hne)).const_sub 1\n\n/-- On the punctured plane minus zeros of \u03b6, hLi = (w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6). -/\nlemma hLi_eq_of_ne {w : \u2102} (hw1 : w \u2260 1) (hw0 : riemannZeta w \u2260 0) :\n    hLi w = (w - 1) * (-(deriv riemannZeta w) / riemannZeta w) := by\n  have hA : zetaA w = (w - 1) * riemannZeta w := zetaA_apply hw1\n  have hdA : deriv zetaA w = riemannZeta w + (w - 1) * deriv riemannZeta w := by\n    have h1 : deriv (fun z => (z - 1) * riemannZeta z) w =\n        deriv (fun z => z - 1) w * riemannZeta w +\n          (w - 1) * deriv riemannZeta w :=\n      deriv_mul ((differentiableAt_id.sub (differentiableAt_const 1)))\n        (differentiableAt_riemannZeta hw1)\n    have h2 : deriv (fun z => z - 1) w = 1 := by\n      simp [deriv_sub, deriv_id'']\n    have h3 : (fun z => (z - 1) * riemannZeta z) =\u1da0[\ud835\udcdd w] zetaA := by\n      filter_upwards [eventually_ne_nhds hw1] with z hz using (zetaA_apply hz).symm\n    rw [h3.deriv_eq, h1, h2, one_mul]\n  rw [hLi, hdA, hA]\n  have hw1' : w - 1 \u2260 0 := sub_ne_zero.mpr hw1\n  field_simp [hw0, hw1']\n  ring\n\n/-- On a punctured neighborhood of 1, hLi = (w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6). -/\nlemma hLi_eq_near_one :\n    \u2200\u1da0 w in \ud835\udcdd[\u2260] (1 : \u2102), hLi w = (w - 1) * (-(deriv riemannZeta w) / riemannZeta w) := by\n  have hne : \u2200\u1da0 w in \ud835\udcdd[\u2260] (1 : \u2102), (w - 1) * riemannZeta w \u2260 0 :=\n    riemannZeta_residue_one.eventually_ne (show (1 : \u2102) \u2260 0 by norm_num)\n  filter_upwards [hne, eventually_mem_nhdsWithin] with w hw hw1\n  have hz : riemannZeta w \u2260 0 := by\n    have h2 := hw\n    simp only [ne_eq, mul_eq_zero] at h2\n    push_neg at h2\n    exact h2.2\n  exact hLi_eq_of_ne hw1 hz\n\n/-! ## 3. The Laurent coefficients of \u2212\u03b6\u2032/\u03b6 at 1 (operational definition) -/\n\n/-- The k-th Laurent coefficient of \u2212\u03b6\u2032/\u03b6 at s = 1, DEFINED as\niteratedDeriv (k+1) hLi 1 / (k+1)! (the power-series coefficient of the\nanalytic hLi; the identification with \u2212\u03b6\u2032/\u03b6's Laurent expansion is\n`hLi_eq_near_one`). -/\ndef lcoeff (k : \u2115) : \u2102 := iteratedDeriv (k + 1) hLi 1 / (k + 1) !\n\n/-- hLi's Taylor coefficients: iteratedDeriv j hLi 1 / j ! = lcoeff (j\u22121). -/\nlemma hLi_iteratedDeriv_div_factorial {j : \u2115} (hj : 1 \u2264 j) :\n    iteratedDeriv j hLi 1 / j ! = lcoeff (j - 1) := by\n  rw [lcoeff]\n  have : j = (j - 1) + 1 := by omega\n  rw [this]\n\n/-! ## 4. The A-polynomial derivative: A(w) = w\u207f \u2212 (w\u22121)\u207f -/\n\n/-- Iterated derivatives of A(w) = w\u207f \u2212 (w\u22121)\u207f at 1:\nA^{(i)}(1) = n.descFactorial i for i < n, and A^{(n)}(1) = 0. -/\nlemma A_iter (n i : \u2115) (hi : i \u2264 n) :\n    iteratedDeriv i (fun w => w ^ n - (w - 1) ^ n) (1 : \u2102) =\n      if i < n then n.descFactorial i else 0 := by\n  have hf : ContDiffAt \u2102 i (fun w : \u2102 => w ^ n) 1 :=\n    (contDiff_id.pow n).contDiffAt\n  have hg : ContDiffAt \u2102 i (fun w : \u2102 => (w - 1) ^ n) 1 :=\n    ((contDiff_id.sub contDiff_const).pow n).contDiffAt\n  rw [iteratedDeriv_sub hf hg]\n  have h1 : iteratedDeriv i (fun w : \u2102 => w ^ n) 1 = n.descFactorial i := by\n    rw [iteratedDeriv_pow]\n    simp\n  have h2 : iteratedDeriv i (fun w : \u2102 => (w - 1) ^ n) 1 =\n      n.descFactorial i * (0 : \u2102) ^ (n - i) := by\n    have e := congr_fun (iteratedDeriv_comp_sub_const i (fun w : \u2102 => w ^ n) (1 : \u2102)) 1\n    rw [iteratedDeriv_pow] at e\n    simpa using e\n  rw [h1, h2]\n  by_cases hin : i < n\n  \u00b7 rw [if_pos hin]\n    have hz : (0 : \u2102) ^ (n - i) = 0 := zero_pow (by omega)\n    rw [hz, mul_zero, sub_zero]\n  \u00b7 rw [if_neg hin]\n    have hin' : i = n := le_antisymm hi (by omega)\n    subst hin'\n    simp [Nat.descFactorial_self]\n\n/-! ## 5. The residue at w = 1: ResW1 n = \u03a3_{j=1}^n C(n,j)\u00b7lcoeff (j\u22121) -/\n\n/-- The residue at w = 1 of the Lagarias-kernel integrand\nJ(w) = k_n(w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6(w)), packaged as (1/n!)\u00b7(d/dw)\u207f[(w\u207f\u2212(w\u22121)\u207f)hLi]\nat w = 1 (D-064 \u00a74 R1: (w\u22121)^{n+1}J = [w\u207f\u2212(w\u22121)\u207f]\u00b7hLi punctured near 1,\nby `hLi_eq_near_one`). -/\ndef ResW1 (n : \u2115) : \u2102 :=\n  iteratedDeriv n (fun w => (w ^ n - (w - 1) ^ n) * hLi w) (1 : \u2102) / n !\n\n/-- The w = 1 residue value: ResW1 n = \u03a3_{j=1}^n C(n,j)\u00b7lcoeff (j\u22121). -/\ntheorem ResW1_eq_sum (n : \u2115) :\n    ResW1 n = \u2211 j \u2208 Finset.Icc 1 n, (n.choose j : \u2102) * lcoeff (j - 1) := by\n  have hA : ContDiffAt \u2102 n (fun w : \u2102 => w ^ n - (w - 1) ^ n) 1 :=\n    ((contDiff_id.pow n).sub ((contDiff_id.sub contDiff_const).pow n)).contDiffAt\n  have hh : ContDiffAt \u2102 n hLi 1 := hLi_analyticAt_one.contDiffAt\n  have hmul := iteratedDeriv_mul hA hh\n  rw [ResW1, hmul]\n  have hsum : (\u2211 i \u2208 Finset.range (n + 1),\n        n.choose i * iteratedDeriv i (fun w : \u2102 => w ^ n - (w - 1) ^ n) 1 *\n          iteratedDeriv (n - i) hLi 1) =\n      \u2211 i \u2208 Finset.range n,\n        n.choose i * (n.descFactorial i : \u2102) * iteratedDeriv (n - i) hLi 1 := by\n    rw [Finset.sum_range_succ]\n    simp [A_iter n n le_rfl]\n    apply Finset.sum_congr rfl\n    intro i hi\n    rw [Finset.mem_range] at hi\n    rw [A_iter n i (by omega), if_pos hi]\n  rw [hsum]\n  have hterm : \u2200 i \u2208 Finset.range n,\n      n.choose i * (n.descFactorial i : \u2102) * iteratedDeriv (n - i) hLi 1 / n ! =\n        (n.choose i : \u2102) * lcoeff (n - i - 1) := by\n    intro i hi\n    rw [Finset.mem_range] at hi\n    have hdesc : (n.descFactorial i : \u2102) = (n ! : \u2102) / ((n - i) ! : \u2102) := by\n      have h2 := Nat.factorial_mul_descFactorial (show i \u2264 n by omega)\n      have hcast : ((n - i) ! : \u2102) * (n.descFactorial i : \u2102) = (n ! : \u2102) := by\n        exact_mod_cast h2\n      rw [eq_div_iff_mul_eq (by simp [Nat.factorial_ne_zero])]\n      rw [mul_comm]\n      exact hcast\n    rw [hdesc]\n    have hlc : lcoeff (n - i - 1) = iteratedDeriv (n - i) hLi 1 / ((n - i) ! : \u2102) := by\n      rw [hLi_iteratedDeriv_div_factorial (j := n - i) (by omega)]\n    rw [hlc]\n    field_simp\n    ring\n  rw [Finset.sum_div]\n  rw [Finset.sum_congr rfl hterm]\n  have hre : (\u2211 i \u2208 Finset.range n, (n.choose i : \u2102) * lcoeff (n - i - 1)) =\n      \u2211 j \u2208 Finset.Icc 1 n, (n.choose j : \u2102) * lcoeff (j - 1) := by\n    rw [Finset.sum_range_reflect]\n    rw [Finset.sum_Icc_eq_sum_range]\n    apply Finset.sum_congr rfl\n    intro i hi\n    rw [Finset.mem_range] at hi\n    have e1 : n - 1 - i + 1 = n - i := by omega\n    have e2 : n - (n - 1 - i) = i + 1 := by omega\n    have e3 : n.choose (n - 1 - i) = n.choose (i + 1) := by\n      rw [show n - 1 - i = n - (i + 1) by omega]\n      exact Nat.choose_symm (by omega)\n    have e4 : n - (n - 1 - i) - 1 = i := by omega\n    rw [e1, e3]\n    have : n - (i + 1) - 1 = i := by omega\n    rw [this]\n  exact hre\n\nend NguyenRH1\n"
}

{
  "type": "update",
  "filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean",
  "content": "/-\nNguyenRH1/LiResidue.lean \u2014 tier ii\u2032 sub-stage ii\u2032-1b (D-064 \u00a74 R1):\nthe w = 1 residue of the Lagarias-kernel integrand, computed exactly.\n\nContents (Part 1):\n* `zetaA w = (w\u22121)\u00b7\u03b6(w)` extended by 1 at w = 1 \u2014 analytic at 1 with\n  value 1 (removable singularity; Mathlib `riemannZeta_residue_one` +\n  `differentiableOn_update_limUnder_of_bddAbove`).\n* `hLi w = 1 \u2212 (w\u22121)\u00b7(zetaA\u2032/zetaA)(w)` \u2014 analytic at 1, hLi 1 = 1;\n  agrees with (w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6)(w) on a punctured neighborhood of 1\n  (`hLi_eq_near_one`).\n* `lcoeff k` \u2014 the Laurent coefficients of \u2212\u03b6\u2032/\u03b6 at 1, DEFINED as\n  iteratedDeriv (k+1) hLi 1 / (k+1)!.\n* `ResW1 n` \u2014 the residue at w = 1 of J(w) = k_n(w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6(w)),\n  packaged as (1/n!)\u00b7(d/dw)\u207f[(w\u207f\u2212(w\u22121)\u207f)\u00b7hLi] at w = 1 (D-064 \u00a74 R1:\n  (w\u22121)^{n+1}J = [w\u207f\u2212(w\u22121)\u207f]\u00b7hLi on a punctured neighborhood, by\n  `hLi_eq_near_one`). Value: `ResW1_eq_sum`.\n\nArchive references: D-036, D-037, D-064 (R1), journal-014/015.\n-/\nimport Mathlib\nimport Zeta23.Statement\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. The regularized factor zetaA = (w\u22121)\u03b6(w) at w = 1 -/\n\n/-- `zetaA w = (w\u22121)\u00b7\u03b6(w)` with the removable singularity at w = 1 filled\nby the residue value 1. Analytic at 1 (`zetaA_analyticAt_one`), value\n`zetaA 1 = 1`. -/\ndef zetaA : \u2102 \u2192 \u2102 := Function.update (fun w => (w - 1) * riemannZeta w) 1 1\n\nlemma zetaA_apply {w : \u2102} (hw : w \u2260 1) : zetaA w = (w - 1) * riemannZeta w :=\n  Function.update_of_ne hw _ _\n\nlemma zetaA_one : zetaA 1 = 1 := Function.update_self _ _ _\n\n/-- The regularized factor is analytic at w = 1 (removable singularity). -/\nlemma zetaA_analyticAt_one : AnalyticAt \u2102 zetaA 1 := by\n  have hdiff : DifferentiableOn \u2102 (fun w => (w - 1) * riemannZeta w)\n      (Metric.ball 1 2 \\ {1}) := by\n    intro w hw\n    have hw1 : w \u2260 1 := hw.2\n    exact ((differentiableAt_id.sub (differentiableAt_const 1)).mul\n      (differentiableAt_riemannZeta hw1)).differentiableWithinAt\n  have hbd : BddAbove (Complex.norm \u2218 (fun w => (w - 1) * riemannZeta w)) ''\n      (Metric.ball 1 2 \\ {1}) := by\n    have hT := riemannZeta_residue_one\n    rw [Metric.tendsto_nhdsWithin_nhds] at hT\n    obtain \u27e8\u03b4, h\u03b40, h\u03b4\u27e9 := hT 1 one_pos\n    refine \u27e82, ?_\u27e9\n    rintro _ \u27e8z, hz, rfl\u27e9\n    have hz1 : z \u2260 1 := hz.2\n    have hlt := h\u03b4 (by\n      rw [dist_eq_norm, sub_zero]\n      exact hz.1)\n    have h1 : \u2016(z - 1) * riemannZeta z - 1\u2016 < 1 := by\n      have := hlt\n      rwa [dist_eq_norm] at this\n    have := norm_sub_le ((z - 1) * riemannZeta z) 1\n    simp only [Complex.norm_eq_abs] at *\n    have h2 : \u2016(z - 1) * riemannZeta z\u2016 \u2264 \u2016(z - 1) * riemannZeta z - 1\u2016 + \u2016(1 : \u2102)\u2016 :=\n      norm_sub_le ((z - 1) * riemannZeta z) 1\n    rw [norm_one] at h2\n    linarith [h1]\n  have h := differentiableOn_update_limUnder_of_bddAbove\n    (Metric.ball_mem_nhds 1 (by norm_num : (0 : \u211d) < 2)) hdiff hbd\n  have hlim : limUnder (\ud835\udcdd[\u2260] (1 : \u2102)) (fun w => (w - 1) * riemannZeta w) = 1 :=\n    riemannZeta_residue_one.limUnder_eq\n  rw [hlim] at h\n  have han := h.analyticOnNhd Metric.isOpen_ball\n  exact han.analyticAt (Metric.ball_mem_nhds 1 (by norm_num : (0 : \u211d) < 2))\n\n/-- zetaA is nonzero on a neighborhood of 1 (value 1 there). -/\nlemma zetaA_ne_zero_near_one : \u2200\u1da0 w in \ud835\udcdd (1 : \u2102), zetaA w \u2260 0 :=\n  (zetaA_analyticAt_one.continuousAt).eventually_ne (by rw [zetaA_one]; norm_num)\n\n/-- zetaA is analytic at any w \u2260 1 (agrees with (w\u22121)\u03b6 there). -/\nlemma zetaA_analyticAt_of_ne {w : \u2102} (hw : w \u2260 1) : AnalyticAt \u2102 zetaA w := by\n  have h : (fun w => (w - 1) * riemannZeta w) =\u1da0[\ud835\udcdd w] zetaA := by\n    filter_upwards [eventually_ne_nhds hw] with z hz using (zetaA_apply hz).symm\n  have ha : AnalyticAt \u2102 (fun w => (w - 1) * riemannZeta w) w :=\n    (analyticAt_id.sub analyticAt_const).mul\n      (differentiableAt_riemannZeta hw).analyticAt\n  exact ha.congr h\n\n/-! ## 2. hLi w = 1 \u2212 (w\u22121)\u00b7(zetaA\u2032/zetaA)(w): analytic at 1, value 1 -/\n\n/-- hLi(w) = 1 \u2212 (w\u22121)\u00b7(zetaA\u2032/zetaA)(w) \u2014 the analytic-at-1 form of\n(w\u22121)(\u2212\u03b6\u2032/\u03b6)(w) (`hLi_eq_near_one`). hLi 1 = 1. -/\ndef hLi (w : \u2102) : \u2102 := 1 - (w - 1) * (deriv zetaA w / zetaA w)\n\nlemma hLi_one : hLi 1 = 1 := by simp [hLi]\n\nlemma hLi_analyticAt_one : AnalyticAt \u2102 hLi 1 := by\n  have ha := zetaA_analyticAt_one\n  have hda : AnalyticAt \u2102 (deriv zetaA) 1 := ha.deriv\n  have hne : zetaA 1 \u2260 0 := by rw [zetaA_one]; norm_num\n  exact ((analyticAt_const.sub analyticAt_id).mul\n    (hda.div ha hne)).const_sub 1\n\n/-- On the punctured plane minus zeros of \u03b6, hLi = (w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6). -/\nlemma hLi_eq_of_ne {w : \u2102} (hw1 : w \u2260 1) (hw0 : riemannZeta w \u2260 0) :\n    hLi w = (w - 1) * (-(deriv riemannZeta w) / riemannZeta w) := by\n  have hA : zetaA w = (w - 1) * riemannZeta w := zetaA_apply hw1\n  have hdA : deriv zetaA w = riemannZeta w + (w - 1) * deriv riemannZeta w := by\n    have h1 : deriv (fun z => (z - 1) * riemannZeta z) w =\n        deriv (fun z => z - 1) w * riemannZeta w +\n          (w - 1) * deriv riemannZeta w :=\n      deriv_mul ((differentiableAt_id.sub (differentiableAt_const 1)))\n        (differentiableAt_riemannZeta hw1)\n    have h2 : deriv (fun z => z - 1) w = 1 := by\n      simp [deriv_sub, deriv_id'']\n    have h3 : (fun z => (z - 1) * riemannZeta z) =\u1da0[\ud835\udcdd w] zetaA := by\n      filter_upwards [eventually_ne_nhds hw1] with z hz using (zetaA_apply hz).symm\n    rw [h3.deriv_eq, h1, h2, one_mul]\n  rw [hLi, hdA, hA]\n  have hw1' : w - 1 \u2260 0 := sub_ne_zero.mpr hw1\n  field_simp [hw0, hw1']\n  ring\n\n/-- On a punctured neighborhood of 1, hLi = (w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6). -/\nlemma hLi_eq_near_one :\n    \u2200\u1da0 w in \ud835\udcdd[\u2260] (1 : \u2102), hLi w = (w - 1) * (-(deriv riemannZeta w) / riemannZeta w) := by\n  have hne : \u2200\u1da0 w in \ud835\udcdd[\u2260] (1 : \u2102), (w - 1) * riemannZeta w \u2260 0 :=\n    riemannZeta_residue_one.eventually_ne (show (1 : \u2102) \u2260 0 by norm_num)\n  filter_upwards [hne, eventually_mem_nhdsWithin] with w hw hw1\n  have hz : riemannZeta w \u2260 0 := by\n    have h2 := hw\n    simp only [ne_eq, mul_eq_zero] at h2\n    push_neg at h2\n    exact h2.2\n  exact hLi_eq_of_ne hw1 hz\n\n/-! ## 3. The Laurent coefficients of \u2212\u03b6\u2032/\u03b6 at 1 (operational definition) -/\n\n/-- The k-th Laurent coefficient of \u2212\u03b6\u2032/\u03b6 at s = 1, DEFINED as\niteratedDeriv (k+1) hLi 1 / (k+1)! (the power-series coefficient of the\nanalytic hLi; the identification with \u2212\u03b6\u2032/\u03b6's Laurent expansion is\n`hLi_eq_near_one`). -/\ndef lcoeff (k : \u2115) : \u2102 := iteratedDeriv (k + 1) hLi 1 / (k + 1) !\n\n/-- hLi's Taylor coefficients: iteratedDeriv j hLi 1 / j ! = lcoeff (j\u22121). -/\nlemma hLi_iteratedDeriv_div_factorial {j : \u2115} (hj : 1 \u2264 j) :\n    iteratedDeriv j hLi 1 / j ! = lcoeff (j - 1) := by\n  rw [lcoeff]\n  have : j = (j - 1) + 1 := by omega\n  rw [this]\n\n/-! ## 4. The A-polynomial derivative: A(w) = w\u207f \u2212 (w\u22121)\u207f -/\n\n/-- Iterated derivatives of A(w) = w\u207f \u2212 (w\u22121)\u207f at 1:\nA^{(i)}(1) = n.descFactorial i for i < n, and A^{(n)}(1) = 0. -/\nlemma A_iter (n i : \u2115) (hi : i \u2264 n) :\n    iteratedDeriv i (fun w => w ^ n - (w - 1) ^ n) (1 : \u2102) =\n      if i < n then n.descFactorial i else 0 := by\n  have hf : ContDiffAt \u2102 i (fun w : \u2102 => w ^ n) 1 :=\n    (contDiff_id.pow n).contDiffAt\n  have hg : ContDiffAt \u2102 i (fun w : \u2102 => (w - 1) ^ n) 1 :=\n    ((contDiff_id.sub contDiff_const).pow n).contDiffAt\n  rw [iteratedDeriv_sub hf hg]\n  have h1 : iteratedDeriv i (fun w : \u2102 => w ^ n) 1 = n.descFactorial i := by\n    rw [iteratedDeriv_pow]\n    simp\n  have h2 : iteratedDeriv i (fun w : \u2102 => (w - 1) ^ n) 1 =\n      n.descFactorial i * (0 : \u2102) ^ (n - i) := by\n    have e := congr_fun (iteratedDeriv_comp_sub_const i (fun w : \u2102 => w ^ n) (1 : \u2102)) 1\n    rw [iteratedDeriv_pow] at e\n    simpa using e\n  rw [h1, h2]\n  by_cases hin : i < n\n  \u00b7 rw [if_pos hin]\n    have hz : (0 : \u2102) ^ (n - i) = 0 := zero_pow (by omega)\n    rw [hz, mul_zero, sub_zero]\n  \u00b7 rw [if_neg hin]\n    have hin' : i = n := le_antisymm hi (by omega)\n    subst hin'\n    simp [Nat.descFactorial_self]\n\n/-! ## 5. The residue at w = 1: ResW1 n = \u03a3_{j=1}^n C(n,j)\u00b7lcoeff (j\u22121) -/\n\n/-- The residue at w = 1 of the Lagarias-kernel integrand\nJ(w) = k_n(w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6(w)), packaged as (1/n!)\u00b7(d/dw)\u207f[(w\u207f\u2212(w\u22121)\u207f)hLi]\nat w = 1 (D-064 \u00a74 R1: (w\u22121)^{n+1}J = [w\u207f\u2212(w\u22121)\u207f]\u00b7hLi punctured near 1,\nby `hLi_eq_near_one`). -/\ndef ResW1 (n : \u2115) : \u2102 :=\n  iteratedDeriv n (fun w => (w ^ n - (w - 1) ^ n) * hLi w) (1 : \u2102) / n !\n\n/-- The w = 1 residue value: ResW1 n = \u03a3_{j=1}^n C(n,j)\u00b7lcoeff (j\u22121). -/\ntheorem ResW1_eq_sum (n : \u2115) :\n    ResW1 n = \u2211 j \u2208 Finset.Icc 1 n, (n.choose j : \u2102) * lcoeff (j - 1) := by\n  have hA : ContDiffAt \u2102 n (fun w : \u2102 => w ^ n - (w - 1) ^ n) 1 :=\n    ((contDiff_id.pow n).sub ((contDiff_id.sub contDiff_const).pow n)).contDiffAt\n  have hh : ContDiffAt \u2102 n hLi 1 := hLi_analyticAt_one.contDiffAt\n  have hmul := iteratedDeriv_mul hA hh\n  rw [ResW1, hmul]\n  have hsum : (\u2211 i \u2208 Finset.range (n + 1),\n        n.choose i * iteratedDeriv i (fun w : \u2102 => w ^ n - (w - 1) ^ n) 1 *\n          iteratedDeriv (n - i) hLi 1) =\n      \u2211 i \u2208 Finset.range n,\n        n.choose i * (n.descFactorial i : \u2102) * iteratedDeriv (n - i) hLi 1 := by\n    rw [Finset.sum_range_succ]\n    simp [A_iter n n le_rfl]\n    apply Finset.sum_congr rfl\n    intro i hi\n    rw [Finset.mem_range] at hi\n    rw [A_iter n i (by omega), if_pos hi]\n  rw [hsum]\n  have hterm : \u2200 i \u2208 Finset.range n,\n      n.choose i * (n.descFactorial i : \u2102) * iteratedDeriv (n - i) hLi 1 / n ! =\n        (n.choose i : \u2102) * lcoeff (n - i - 1) := by\n    intro i hi\n    rw [Finset.mem_range] at hi\n    have hdesc : (n.descFactorial i : \u2102) = (n ! : \u2102) / ((n - i) ! : \u2102) := by\n      have h2 := Nat.factorial_mul_descFactorial (show i \u2264 n by omega)\n      have hcast : ((n - i) ! : \u2102) * (n.descFactorial i : \u2102) = (n ! : \u2102) := by\n        exact_mod_cast h2\n      rw [eq_div_iff_mul_eq (by simp [Nat.factorial_ne_zero])]\n      rw [mul_comm]\n      exact hcast\n    rw [hdesc]\n    have hlc : lcoeff (n - i - 1) = iteratedDeriv (n - i) hLi 1 / ((n - i) ! : \u2102) := by\n      rw [hLi_iteratedDeriv_div_factorial (j := n - i) (by omega)]\n    rw [hlc]\n    field_simp\n    ring\n  rw [Finset.sum_div]\n  rw [Finset.sum_congr rfl hterm]\n  have hre : (\u2211 i \u2208 Finset.range n, (n.choose i : \u2102) * lcoeff (n - i - 1)) =\n      \u2211 j \u2208 Finset.Icc 1 n, (n.choose j : \u2102) * lcoeff (j - 1) := by\n    rw [Finset.sum_range_reflect]\n    rw [Finset.sum_Icc_eq_sum_range]\n    apply Finset.sum_congr rfl\n    intro i hi\n    rw [Finset.mem_range] at hi\n    have e1 : n - 1 - i + 1 = n - i := by omega\n    have e2 : n - (n - 1 - i) = i + 1 := by omega\n    have e3 : n.choose (n - 1 - i) = n.choose (i + 1) := by\n      rw [show n - 1 - i = n - (i + 1) by omega]\n      exact Nat.choose_symm (by omega)\n    have e4 : n - (n - 1 - i) - 1 = i := by omega\n    rw [e1, e3]\n    have : n - (i + 1) - 1 = i := by omega\n    rw [this]\n  exact hre\n\nend NguyenRH1\n",
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      "lines": [
        " NguyenRH1/LiResidue.lean \u2014 tier ii\u2032 sub-stage ii\u2032-1b (D-064 \u00a74 R1):",
        " the w = 1 residue of the Lagarias-kernel integrand, computed exactly.",
        " ",
        "-Contents (Part 1, this file's core):",
        "+Contents (Part 1):",
        " * `zetaA w = (w\u22121)\u00b7\u03b6(w)` extended by 1 at w = 1 \u2014 analytic at 1 with",
        "   value 1 (removable singularity; Mathlib `riemannZeta_residue_one` +",
        "   `differentiableOn_update_limUnder_of_bddAbove`)."
      ]
    },
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        "   agrees with (w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6)(w) on a punctured neighborhood of 1",
        "   (`hLi_eq_near_one`).",
        " * `lcoeff k` \u2014 the Laurent coefficients of \u2212\u03b6\u2032/\u03b6 at 1, DEFINED as",
        "-  iteratedDeriv (k+1) hLi 1 / (k+1)! (operational; no Laurent series",
        "-  needed at this stage).",
        "+  iteratedDeriv (k+1) hLi 1 / (k+1)!.",
        " * `ResW1 n` \u2014 the residue at w = 1 of J(w) = k_n(w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6(w)),",
        "-  computed by the Leibniz identity (w\u22121)^{n+1}J = [w\u207f\u2212(w\u22121)\u207f]\u00b7hLi",
        "-  (D-064 \u00a74 R1, verified algebraically in session 014):",
        "-  ResW1 n = \u03a3_{j=1}^n C(n,j)\u00b7lcoeff (j\u22121)  =: `ResW1_eq_sum`.",
        "+  packaged as (1/n!)\u00b7(d/dw)\u207f[(w\u207f\u2212(w\u22121)\u207f)\u00b7hLi] at w = 1 (D-064 \u00a74 R1:",
        "+  (w\u22121)^{n+1}J = [w\u207f\u2212(w\u22121)\u207f]\u00b7hLi on a punctured neighborhood, by",
        "+  `hLi_eq_near_one`). Value: `ResW1_eq_sum`.",
        " ",
        " Archive references: D-036, D-037, D-064 (R1), journal-014/015.",
        " -/"
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        " ",
        " lemma zetaA_one : zetaA 1 = 1 := Function.update_self _ _ _",
        " ",
        "-lemma zetaA_tendsto : Tendsto (fun w => (w - 1) * riemannZeta w) (\ud835\udcdd[\u2260] 1) (\ud835\udcdd 1) :=",
        "-  riemannZeta_residue_one",
        "-",
        " /-- The regularized factor is analytic at w = 1 (removable singularity). -/",
        " lemma zetaA_analyticAt_one : AnalyticAt \u2102 zetaA 1 := by",
        "   have hdiff : DifferentiableOn \u2102 (fun w => (w - 1) * riemannZeta w)",
        "       (Metric.ball 1 2 \\ {1}) := by",
        "     intro w hw",
        "     have hw1 : w \u2260 1 := hw.2",
        "-    exact (differentiableAt_id.sub (differentiableAt_const 1)).mul",
        "-      (differentiableAt_riemannZeta hw1) |>.differentiableWithinAt",
        "+    exact ((differentiableAt_id.sub (differentiableAt_const 1)).mul",
        "+      (differentiableAt_riemannZeta hw1)).differentiableWithinAt",
        "   have hbd : BddAbove (Complex.norm \u2218 (fun w => (w - 1) * riemannZeta w)) ''",
        "       (Metric.ball 1 2 \\ {1}) := by",
        "-    have hT := zetaA_tendsto",
        "+    have hT := riemannZeta_residue_one",
        "     rw [Metric.tendsto_nhdsWithin_nhds] at hT",
        "     obtain \u27e8\u03b4, h\u03b40, h\u03b4\u27e9 := hT 1 one_pos",
        "     refine \u27e82, ?_\u27e9",
        "     rintro _ \u27e8z, hz, rfl\u27e9",
        "-    have h1 : dist (z - 1) 0 = \u2016z - 1\u2016 := by",
        "-      rw [dist_eq_norm]; simp [sub_zero]",
        "     have hz1 : z \u2260 1 := hz.2",
        "-    have hzz : dist z 1 < \u03b4 := by",
        "-      have := hz.1",
        "-      rw [Metric.mem_ball, dist_comm] at this",
        "-      exact this",
        "-    have hlt := h\u03b4 (by rwa [dist_zero_right, h1] at *)",
        "-    have : \u2016(z - 1) * riemannZeta z\u2016 \u2264 \u2016(1 : \u2102)\u2016 + 1 := by",
        "-      have := norm_sub_le_of_le hlt",
        "-      calc \u2016(z - 1) * riemannZeta z\u2016",
        "-          \u2264 \u2016(z - 1) * riemannZeta z - 1\u2016 + \u2016(1 : \u2102)\u2016 := by",
        "-            have := norm_sub_le ((z - 1) * riemannZeta z) 1",
        "-            linarith [this, hlt]",
        "-      _ \u2264 2 := by",
        "-            have h2 := hlt",
        "-            rw [norm_one]",
        "-            linarith [h2]",
        "-    exact this",
        "+    have hlt := h\u03b4 (by",
        "+      rw [dist_eq_norm, sub_zero]",
        "+      exact hz.1)",
        "+    have h1 : \u2016(z - 1) * riemannZeta z - 1\u2016 < 1 := by",
        "+      have := hlt",
        "+      rwa [dist_eq_norm] at this",
        "+    have := norm_sub_le ((z - 1) * riemannZeta z) 1",
        "+    simp only [Complex.norm_eq_abs] at *",
        "+    have h2 : \u2016(z - 1) * riemannZeta z\u2016 \u2264 \u2016(z - 1) * riemannZeta z - 1\u2016 + \u2016(1 : \u2102)\u2016 :=",
        "+      norm_sub_le ((z - 1) * riemannZeta z) 1",
        "+    rw [norm_one] at h2",
        "+    linarith [h1]",
        "   have h := differentiableOn_update_limUnder_of_bddAbove",
        "     (Metric.ball_mem_nhds 1 (by norm_num : (0 : \u211d) < 2)) hdiff hbd",
        "   have hlim : limUnder (\ud835\udcdd[\u2260] (1 : \u2102)) (fun w => (w - 1) * riemannZeta w) = 1 :=",
        "-    zetaA_tendsto.limUnder_eq",
        "+    riemannZeta_residue_one.limUnder_eq",
        "   rw [hlim] at h",
        "-  have hopen : IsOpen (Metric.ball (1 : \u2102) 2) := Metric.isOpen_ball",
        "-  have han := h.analyticOnNhd hopen",
        "+  have han := h.analyticOnNhd Metric.isOpen_ball",
        "   exact han.analyticAt (Metric.ball_mem_nhds 1 (by norm_num : (0 : \u211d) < 2))",
        " ",
        " /-- zetaA is nonzero on a neighborhood of 1 (value 1 there). -/",
        " lemma zetaA_ne_zero_near_one : \u2200\u1da0 w in \ud835\udcdd (1 : \u2102), zetaA w \u2260 0 :=",
        "   (zetaA_analyticAt_one.continuousAt).eventually_ne (by rw [zetaA_one]; norm_num)",
        " ",
        "-/-- zetaA is analytic everywhere except possibly at 1 \u2014 and in fact also",
        "-at 1; we record the off-1 version for quotient computations. -/",
        "+/-- zetaA is analytic at any w \u2260 1 (agrees with (w\u22121)\u03b6 there). -/",
        " lemma zetaA_analyticAt_of_ne {w : \u2102} (hw : w \u2260 1) : AnalyticAt \u2102 zetaA w := by",
        "   have h : (fun w => (w - 1) * riemannZeta w) =\u1da0[\ud835\udcdd w] zetaA := by",
        "     filter_upwards [eventually_ne_nhds hw] with z hz using (zetaA_apply hz).symm",
        "-  have ha : AnalyticAt \u2102 (fun w => (w - 1) * riemannZeta w) w := by",
        "-    exact (analyticAt_id.sub analyticAt_const).mul",
        "+  have ha : AnalyticAt \u2102 (fun w => (w - 1) * riemannZeta w) w :=",
        "+    (analyticAt_id.sub analyticAt_const).mul",
        "       (differentiableAt_riemannZeta hw).analyticAt",
        "   exact ha.congr h",
        " "
      ]
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      "lines": [
        "   have ha := zetaA_analyticAt_one",
        "   have hda : AnalyticAt \u2102 (deriv zetaA) 1 := ha.deriv",
        "   have hne : zetaA 1 \u2260 0 := by rw [zetaA_one]; norm_num",
        "-  exact (analyticAt_const.sub analyticAt_id).mul",
        "-    (hda.div ha hne) |>.const_sub 1",
        "+  exact ((analyticAt_const.sub analyticAt_id).mul",
        "+    (hda.div ha hne)).const_sub 1",
        " ",
        "-/-- On a punctured neighborhood of 1, hLi = (w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6). -/",
        "+/-- On the punctured plane minus zeros of \u03b6, hLi = (w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6). -/",
        " lemma hLi_eq_of_ne {w : \u2102} (hw1 : w \u2260 1) (hw0 : riemannZeta w \u2260 0) :",
        "     hLi w = (w - 1) * (-(deriv riemannZeta w) / riemannZeta w) := by",
        "   have hA : zetaA w = (w - 1) * riemannZeta w := zetaA_apply hw1"
      ]
    },
    {
      "oldStart": 135,
      "oldLines": 30,
      "newStart": 123,
      "newLines": 28,
      "lines": [
        "   field_simp [hw0, hw1']",
        "   ring",
        " ",
        "-/-- The punctured-neighborhood form, for later contour work. -/",
        "+/-- On a punctured neighborhood of 1, hLi = (w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6). -/",
        " lemma hLi_eq_near_one :",
        "     \u2200\u1da0 w in \ud835\udcdd[\u2260] (1 : \u2102), hLi w = (w - 1) * (-(deriv riemannZeta w) / riemannZeta w) := by",
        "-  have hzero : \u2200\u1da0 w in \ud835\udcdd[\u2260] (1 : \u2102), riemannZeta w \u2260 0 := by",
        "-    have h1 : (1 : \u2102) \u2208 {s : \u2102 | 1 \u2264 s.re} := by simp",
        "-    filter_upwards [eventually_mem_nhdsWithin] with w hw",
        "-    have hwre : 1 \u2264 w.re := by",
        "-      have := hw.2",
        "-      sorry",
        "-    exact riemannZeta_ne_zero_of_one_le_re hwre",
        "-  filter_upwards [hzero, eventually_mem_nhdsWithin] with w hw0 hw1",
        "-  exact hLi_eq_of_ne hw1 hw0",
        "+  have hne : \u2200\u1da0 w in \ud835\udcdd[\u2260] (1 : \u2102), (w - 1) * riemannZeta w \u2260 0 :=",
        "+    riemannZeta_residue_one.eventually_ne (show (1 : \u2102) \u2260 0 by norm_num)",
        "+  filter_upwards [hne, eventually_mem_nhdsWithin] with w hw hw1",
        "+  have hz : riemannZeta w \u2260 0 := by",
        "+    have h2 := hw",
        "+    simp only [ne_eq, mul_eq_zero] at h2",
        "+    push_neg at h2",
        "+    exact h2.2",
        "+  exact hLi_eq_of_ne hw1 hz",
        " ",
        " /-! ## 3. The Laurent coefficients of \u2212\u03b6\u2032/\u03b6 at 1 (operational definition) -/",
        " ",
        "-/-- The k-th Laurent coefficient of \u2212\u03b6\u2032/\u03b6 at s = 1:",
        "-\u2212\u03b6\u2032/\u03b6(1+u) = u\u207b\u00b9 + \u03a3_{k\u22650} lcoeff k \u00b7 u^k (identification proved in",
        "-tier ii\u2032-2/3; here lcoeff is DEFINED via the analytic hLi). -/",
        "+/-- The k-th Laurent coefficient of \u2212\u03b6\u2032/\u03b6 at s = 1, DEFINED as",
        "+iteratedDeriv (k+1) hLi 1 / (k+1)! (the power-series coefficient of the",
        "+analytic hLi; the identification with \u2212\u03b6\u2032/\u03b6's Laurent expansion is",
        "+`hLi_eq_near_one`). -/",
        " def lcoeff (k : \u2115) : \u2102 := iteratedDeriv (k + 1) hLi 1 / (k + 1) !",
        " ",
        "-lemma lcoeff_eq (k : \u2115) :",
        "-    lcoeff k = iteratedDeriv (k + 1) hLi 1 / (k + 1) ! := rfl",
        "-",
        "-/-- hLi's Taylor coefficient form: iteratedDeriv j hLi 1 / j ! = lcoeff (j\u22121). -/",
        "+/-- hLi's Taylor coefficients: iteratedDeriv j hLi 1 / j ! = lcoeff (j\u22121). -/",
        " lemma hLi_iteratedDeriv_div_factorial {j : \u2115} (hj : 1 \u2264 j) :",
        "     iteratedDeriv j hLi 1 / j ! = lcoeff (j - 1) := by",
        "   rw [lcoeff]"
      ]
    },
    {
      "oldStart": 182,
      "oldLines": 20,
      "newStart": 168,
      "newLines": 14,
      "lines": [
        "     simp",
        "   have h2 : iteratedDeriv i (fun w : \u2102 => (w - 1) ^ n) 1 =",
        "       n.descFactorial i * (0 : \u2102) ^ (n - i) := by",
        "-    have e := iteratedDeriv_comp_sub_const i (fun w : \u2102 => w ^ n) (1 : \u2102)",
        "-    have e2 : iteratedDeriv i (fun w : \u2102 => (w - 1) ^ n) 1 =",
        "-        iteratedDeriv i (fun w : \u2102 => w ^ n) (1 - 1) := by",
        "-      have := congr_fun e 1",
        "-      convert this using 2",
        "-    rw [e2, iteratedDeriv_pow]",
        "-    simp",
        "+    have e := congr_fun (iteratedDeriv_comp_sub_const i (fun w : \u2102 => w ^ n) (1 : \u2102)) 1",
        "+    rw [iteratedDeriv_pow] at e",
        "+    simpa using e",
        "   rw [h1, h2]",
        "   by_cases hin : i < n",
        "   \u00b7 rw [if_pos hin]",
        "-    have : (0 : \u2102) ^ (n - i) = 0 := by",
        "-      apply zero_pow",
        "-      omega",
        "-    rw [this, mul_zero, sub_zero]",
        "+    have hz : (0 : \u2102) ^ (n - i) = 0 := zero_pow (by omega)",
        "+    rw [hz, mul_zero, sub_zero]",
        "   \u00b7 rw [if_neg hin]",
        "     have hin' : i = n := le_antisymm hi (by omega)",
        "     subst hin'"
      ]
    },
    {
      "oldStart": 205,
      "oldLines": 8,
      "newStart": 185,
      "newLines": 8,
      "lines": [
        " ",
        " /-- The residue at w = 1 of the Lagarias-kernel integrand",
        " J(w) = k_n(w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6(w)), packaged as (1/n!)\u00b7(d/dw)\u207f[(w\u207f\u2212(w\u22121)\u207f)hLi]",
        "-at w = 1 (D-064 \u00a74 R1: (w\u22121)^{n+1}J = [w\u207f\u2212(w\u22121)\u207f]\u00b7hLi on a punctured",
        "-neighborhood, by `hLi_eq_near_one`). -/",
        "+at w = 1 (D-064 \u00a74 R1: (w\u22121)^{n+1}J = [w\u207f\u2212(w\u22121)\u207f]\u00b7hLi punctured near 1,",
        "+by `hLi_eq_near_one`). -/",
        " def ResW1 (n : \u2115) : \u2102 :=",
        "   iteratedDeriv n (fun w => (w ^ n - (w - 1) ^ n) * hLi w) (1 : \u2102) / n !",
        " "
      ]
    },
    {
      "oldStart": 218,
      "oldLines": 86,
      "newStart": 198,
      "newLines": 55,
      "lines": [
        "   have hh : ContDiffAt \u2102 n hLi 1 := hLi_analyticAt_one.contDiffAt",
        "   have hmul := iteratedDeriv_mul hA hh",
        "   rw [ResW1, hmul]",
        "-  have hsum : \u2211 i \u2208 Finset.range (n + 1),",
        "+  have hsum : (\u2211 i \u2208 Finset.range (n + 1),",
        "         n.choose i * iteratedDeriv i (fun w : \u2102 => w ^ n - (w - 1) ^ n) 1 *",
        "-          iteratedDeriv (n - i) hLi 1 =",
        "+          iteratedDeriv (n - i) hLi 1) =",
        "       \u2211 i \u2208 Finset.range n,",
        "         n.choose i * (n.descFactorial i : \u2102) * iteratedDeriv (n - i) hLi 1 := by",
        "-    apply Finset.sum_subset (Finset.range_le_iff.mpr (Nat.le_succ n))",
        "+    rw [Finset.sum_range_succ]",
        "+    simp [A_iter n n le_rfl]",
        "+    apply Finset.sum_congr rfl",
        "     intro i hi",
        "     rw [Finset.mem_range] at hi",
        "-    have hi0 : i \u2260 n := by omega",
        "-    rw [A_iter n i (by omega), if_pos (by omega)]",
        "-    \u00b7 rfl",
        "-    \u00b7 rw [Finset.mem_range] at *",
        "-      omega",
        "+    rw [A_iter n i (by omega), if_pos hi]",
        "   rw [hsum]",
        "-  have hfinal : (\u2211 i \u2208 Finset.range n,",
        "-        n.choose i * (n.descFactorial i : \u2102) * iteratedDeriv (n - i) hLi 1) / n ! =",
        "+  have hterm : \u2200 i \u2208 Finset.range n,",
        "+      n.choose i * (n.descFactorial i : \u2102) * iteratedDeriv (n - i) hLi 1 / n ! =",
        "+        (n.choose i : \u2102) * lcoeff (n - i - 1) := by",
        "+    intro i hi",
        "+    rw [Finset.mem_range] at hi",
        "+    have hdesc : (n.descFactorial i : \u2102) = (n ! : \u2102) / ((n - i) ! : \u2102) := by",
        "+      have h2 := Nat.factorial_mul_descFactorial (show i \u2264 n by omega)",
        "+      have hcast : ((n - i) ! : \u2102) * (n.descFactorial i : \u2102) = (n ! : \u2102) := by",
        "+        exact_mod_cast h2",
        "+      rw [eq_div_iff_mul_eq (by simp [Nat.factorial_ne_zero])]",
        "+      rw [mul_comm]",
        "+      exact hcast",
        "+    rw [hdesc]",
        "+    have hlc : lcoeff (n - i - 1) = iteratedDeriv (n - i) hLi 1 / ((n - i) ! : \u2102) := by",
        "+      rw [hLi_iteratedDeriv_div_factorial (j := n - i) (by omega)]",
        "+    rw [hlc]",
        "+    field_simp",
        "+    ring",
        "+  rw [Finset.sum_div]",
        "+  rw [Finset.sum_congr rfl hterm]",
        "+  have hre : (\u2211 i \u2208 Finset.range n, (n.choose i : \u2102) * lcoeff (n - i - 1)) =",
        "       \u2211 j \u2208 Finset.Icc 1 n, (n.choose j : \u2102) * lcoeff (j - 1) := by",
        "-    have hrewrite : \u2200 i \u2208 Finset.range n,",
        "-        n.choose i * (n.descFactorial i : \u2102) * iteratedDeriv (n - i) hLi 1 / n ! =",
        "-          (n.choose (n - i) : \u2102) * lcoeff (n - i - 1) := by",
        "-      intro i hi",
        "-      rw [Finset.mem_range] at hi",
        "-      have hcd : (n.choose i : \u2102) * (n.descFactorial i : \u2102) =",
        "-          n ! * n.choose i / (n - i) ! := by",
        "-        have h1 := Nat.choose_mul_factorial_mul_factorial (show i \u2264 n by omega)",
        "-        have h2 := Nat.factorial_mul_descFactorial (show i \u2264 n by omega)",
        "-        have : (n.choose i : \u2102) * (n.descFactorial i : \u2102) * ((i ! : \u2102) * ((n - i) ! : \u2102)) =",
        "-            (n ! : \u2102) * (n.choose i : \u2102) := by",
        "-          push_cast [h1, h2]",
        "-          ring",
        "-        have hfact : ((i ! : \u2102) * ((n - i) ! : \u2102)) \u2260 0 := by",
        "-          simp [Nat.factorial_ne_zero]",
        "-        have hthis := this",
        "-        field_simp at hthis \u22a2",
        "-        have h3 : (n.choose i : \u2102) * (n.descFactorial i : \u2102) * ((n - i) ! : \u2102) =",
        "-            (n ! : \u2102) * (n.choose i : \u2102) / 1 := by",
        "-          have h4 := hthis",
        "-          rw [mul_comm ((i ! : \u2102)) ((n - i) ! : \u2102)] at h4",
        "-          have h5 : (n.choose i : \u2102) * (n.descFactorial i : \u2102) * ((n - i) ! : \u2102) * (i ! : \u2102) =",
        "-              (n ! : \u2102) * (n.choose i : \u2102) := by",
        "-            rw [mul_comm ((n - i) ! : \u2102) (i ! : \u2102)] at h4",
        "-            calc (n.choose i : \u2102) * (n.descFactorial i : \u2102) * ((n - i) ! : \u2102) * (i ! : \u2102)",
        "-                = (n.choose i : \u2102) * (n.descFactorial i : \u2102) * (((n - i) ! : \u2102) * (i ! : \u2102)) := by ring",
        "-              _ = (n ! : \u2102) * (n.choose i : \u2102) := by",
        "-                  have := h4",
        "-                  ring_nf at this \u22a2",
        "-                  convert this using 1",
        "-                  ring",
        "-          exact (div_eq_one_iff_eq (by simp [Nat.factorial_ne_zero])).mpr h5",
        "-        field_simp at h3 \u22a2",
        "-        calc (n.choose i : \u2102) * (n.descFactorial i : \u2102)",
        "-            = ((n ! : \u2102) * (n.choose i : \u2102) / 1) / ((n - i) ! : \u2102) := by",
        "-              rw [\u2190 h3]",
        "-              field_simp",
        "-              ring",
        "-          _ = (n ! : \u2102) * (n.choose i : \u2102) / ((n - i) ! : \u2102) := by ring",
        "-      rw [hcd]",
        "-      have hni : n - i \u2265 1 := by omega",
        "-      rw [hLi_iteratedDeriv_div_factorial (j := n - i) (by omega)]",
        "-      rw [show n - i - 1 = n - i - 1 from rfl]",
        "-      have hcoe : ((n - i) ! : \u2102) \u2260 0 := by simp [Nat.factorial_ne_zero]",
        "-      field_simp",
        "-      rw [Nat.choose_symm (by omega : i \u2264 n)]",
        "-      ring",
        "-    rw [Finset.sum_div]",
        "-    rw [Finset.sum_congr rfl hrewrite]",
        "-    have hmap : \u2211 i \u2208 Finset.range n, (n.choose (n - i) : \u2102) * lcoeff (n - i - 1) =",
        "-        \u2211 j \u2208 Finset.Icc 1 n, (n.choose j : \u2102) * lcoeff (j - 1) := by",
        "-      rw [Finset.sum_range_reflect]",
        "-      apply Finset.sum_congr rfl",
        "-      intro i hi",
        "-      rw [Finset.mem_range] at hi",
        "-      have : n - 1 - i + 1 = n - i := by omega",
        "-      rw [this]",
        "-      have : n - (n - i) = i := by omega",
        "-      rw [Nat.choose_symm (show n - i \u2264 n by omega)]",
        "-      simp [this]",
        "-      ring",
        "-    exact hmap",
        "-  rw [hfinal]",
        "+    rw [Finset.sum_range_reflect]",
        "+    rw [Finset.sum_Icc_eq_sum_range]",
        "+    apply Finset.sum_congr rfl",
        "+    intro i hi",
        "+    rw [Finset.mem_range] at hi",
        "+    have e1 : n - 1 - i + 1 = n - i := by omega",
        "+    have e2 : n - (n - 1 - i) = i + 1 := by omega",
        "+    have e3 : n.choose (n - 1 - i) = n.choose (i + 1) := by",
        "+      rw [show n - 1 - i = n - (i + 1) by omega]",
        "+      exact Nat.choose_symm (by omega)",
        "+    have e4 : n - (n - 1 - i) - 1 = i := by omega",
        "+    rw [e1, e3]",
        "+    have : n - (i + 1) - 1 = i := by omega",
        "+    rw [this]",
        "+  exact hre",
        " ",
        " end NguyenRH1",
        " "
      ]
    }
  ],
  "originalFile": "/-\nNguyenRH1/LiResidue.lean \u2014 tier ii\u2032 sub-stage ii\u2032-1b (D-064 \u00a74 R1):\nthe w = 1 residue of the Lagarias-kernel integrand, computed exactly.\n\nContents (Part 1, this file's core):\n* `zetaA w = (w\u22121)\u00b7\u03b6(w)` extended by 1 at w = 1 \u2014 analytic at 1 with\n  value 1 (removable singularity; Mathlib `riemannZeta_residue_one` +\n  `differentiableOn_update_limUnder_of_bddAbove`).\n* `hLi w = 1 \u2212 (w\u22121)\u00b7(zetaA\u2032/zetaA)(w)` \u2014 analytic at 1, hLi 1 = 1;\n  agrees with (w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6)(w) on a punctured neighborhood of 1\n  (`hLi_eq_near_one`).\n* `lcoeff k` \u2014 the Laurent coefficients of \u2212\u03b6\u2032/\u03b6 at 1, DEFINED as\n  iteratedDeriv (k+1) hLi 1 / (k+1)! (operational; no Laurent series\n  needed at this stage).\n* `ResW1 n` \u2014 the residue at w = 1 of J(w) = k_n(w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6(w)),\n  computed by the Leibniz identity (w\u22121)^{n+1}J = [w\u207f\u2212(w\u22121)\u207f]\u00b7hLi\n  (D-064 \u00a74 R1, verified algebraically in session 014):\n  ResW1 n = \u03a3_{j=1}^n C(n,j)\u00b7lcoeff (j\u22121)  =: `ResW1_eq_sum`.\n\nArchive references: D-036, D-037, D-064 (R1), journal-014/015.\n-/\nimport Mathlib\nimport Zeta23.Statement\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. The regularized factor zetaA = (w\u22121)\u03b6(w) at w = 1 -/\n\n/-- `zetaA w = (w\u22121)\u00b7\u03b6(w)` with the removable singularity at w = 1 filled\nby the residue value 1. Analytic at 1 (`zetaA_analyticAt_one`), value\n`zetaA 1 = 1`. -/\ndef zetaA : \u2102 \u2192 \u2102 := Function.update (fun w => (w - 1) * riemannZeta w) 1 1\n\nlemma zetaA_apply {w : \u2102} (hw : w \u2260 1) : zetaA w = (w - 1) * riemannZeta w :=\n  Function.update_of_ne hw _ _\n\nlemma zetaA_one : zetaA 1 = 1 := Function.update_self _ _ _\n\nlemma zetaA_tendsto : Tendsto (fun w => (w - 1) * riemannZeta w) (\ud835\udcdd[\u2260] 1) (\ud835\udcdd 1) :=\n  riemannZeta_residue_one\n\n/-- The regularized factor is analytic at w = 1 (removable singularity). -/\nlemma zetaA_analyticAt_one : AnalyticAt \u2102 zetaA 1 := by\n  have hdiff : DifferentiableOn \u2102 (fun w => (w - 1) * riemannZeta w)\n      (Metric.ball 1 2 \\ {1}) := by\n    intro w hw\n    have hw1 : w \u2260 1 := hw.2\n    exact (differentiableAt_id.sub (differentiableAt_const 1)).mul\n      (differentiableAt_riemannZeta hw1) |>.differentiableWithinAt\n  have hbd : BddAbove (Complex.norm \u2218 (fun w => (w - 1) * riemannZeta w)) ''\n      (Metric.ball 1 2 \\ {1}) := by\n    have hT := zetaA_tendsto\n    rw [Metric.tendsto_nhdsWithin_nhds] at hT\n    obtain \u27e8\u03b4, h\u03b40, h\u03b4\u27e9 := hT 1 one_pos\n    refine \u27e82, ?_\u27e9\n    rintro _ \u27e8z, hz, rfl\u27e9\n    have h1 : dist (z - 1) 0 = \u2016z - 1\u2016 := by\n      rw [dist_eq_norm]; simp [sub_zero]\n    have hz1 : z \u2260 1 := hz.2\n    have hzz : dist z 1 < \u03b4 := by\n      have := hz.1\n      rw [Metric.mem_ball, dist_comm] at this\n      exact this\n    have hlt := h\u03b4 (by rwa [dist_zero_right, h1] at *)\n    have : \u2016(z - 1) * riemannZeta z\u2016 \u2264 \u2016(1 : \u2102)\u2016 + 1 := by\n      have := norm_sub_le_of_le hlt\n      calc \u2016(z - 1) * riemannZeta z\u2016\n          \u2264 \u2016(z - 1) * riemannZeta z - 1\u2016 + \u2016(1 : \u2102)\u2016 := by\n            have := norm_sub_le ((z - 1) * riemannZeta z) 1\n            linarith [this, hlt]\n      _ \u2264 2 := by\n            have h2 := hlt\n            rw [norm_one]\n            linarith [h2]\n    exact this\n  have h := differentiableOn_update_limUnder_of_bddAbove\n    (Metric.ball_mem_nhds 1 (by norm_num : (0 : \u211d) < 2)) hdiff hbd\n  have hlim : limUnder (\ud835\udcdd[\u2260] (1 : \u2102)) (fun w => (w - 1) * riemannZeta w) = 1 :=\n    zetaA_tendsto.limUnder_eq\n  rw [hlim] at h\n  have hopen : IsOpen (Metric.ball (1 : \u2102) 2) := Metric.isOpen_ball\n  have han := h.analyticOnNhd hopen\n  exact han.analyticAt (Metric.ball_mem_nhds 1 (by norm_num : (0 : \u211d) < 2))\n\n/-- zetaA is nonzero on a neighborhood of 1 (value 1 there). -/\nlemma zetaA_ne_zero_near_one : \u2200\u1da0 w in \ud835\udcdd (1 : \u2102), zetaA w \u2260 0 :=\n  (zetaA_analyticAt_one.continuousAt).eventually_ne (by rw [zetaA_one]; norm_num)\n\n/-- zetaA is analytic everywhere except possibly at 1 \u2014 and in fact also\nat 1; we record the off-1 version for quotient computations. -/\nlemma zetaA_analyticAt_of_ne {w : \u2102} (hw : w \u2260 1) : AnalyticAt \u2102 zetaA w := by\n  have h : (fun w => (w - 1) * riemannZeta w) =\u1da0[\ud835\udcdd w] zetaA := by\n    filter_upwards [eventually_ne_nhds hw] with z hz using (zetaA_apply hz).symm\n  have ha : AnalyticAt \u2102 (fun w => (w - 1) * riemannZeta w) w := by\n    exact (analyticAt_id.sub analyticAt_const).mul\n      (differentiableAt_riemannZeta hw).analyticAt\n  exact ha.congr h\n\n/-! ## 2. hLi w = 1 \u2212 (w\u22121)\u00b7(zetaA\u2032/zetaA)(w): analytic at 1, value 1 -/\n\n/-- hLi(w) = 1 \u2212 (w\u22121)\u00b7(zetaA\u2032/zetaA)(w) \u2014 the analytic-at-1 form of\n(w\u22121)(\u2212\u03b6\u2032/\u03b6)(w) (`hLi_eq_near_one`). hLi 1 = 1. -/\ndef hLi (w : \u2102) : \u2102 := 1 - (w - 1) * (deriv zetaA w / zetaA w)\n\nlemma hLi_one : hLi 1 = 1 := by simp [hLi]\n\nlemma hLi_analyticAt_one : AnalyticAt \u2102 hLi 1 := by\n  have ha := zetaA_analyticAt_one\n  have hda : AnalyticAt \u2102 (deriv zetaA) 1 := ha.deriv\n  have hne : zetaA 1 \u2260 0 := by rw [zetaA_one]; norm_num\n  exact (analyticAt_const.sub analyticAt_id).mul\n    (hda.div ha hne) |>.const_sub 1\n\n/-- On a punctured neighborhood of 1, hLi = (w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6). -/\nlemma hLi_eq_of_ne {w : \u2102} (hw1 : w \u2260 1) (hw0 : riemannZeta w \u2260 0) :\n    hLi w = (w - 1) * (-(deriv riemannZeta w) / riemannZeta w) := by\n  have hA : zetaA w = (w - 1) * riemannZeta w := zetaA_apply hw1\n  have hdA : deriv zetaA w = riemannZeta w + (w - 1) * deriv riemannZeta w := by\n    have h1 : deriv (fun z => (z - 1) * riemannZeta z) w =\n        deriv (fun z => z - 1) w * riemannZeta w +\n          (w - 1) * deriv riemannZeta w :=\n      deriv_mul ((differentiableAt_id.sub (differentiableAt_const 1)))\n        (differentiableAt_riemannZeta hw1)\n    have h2 : deriv (fun z => z - 1) w = 1 := by\n      simp [deriv_sub, deriv_id'']\n    have h3 : (fun z => (z - 1) * riemannZeta z) =\u1da0[\ud835\udcdd w] zetaA := by\n      filter_upwards [eventually_ne_nhds hw1] with z hz using (zetaA_apply hz).symm\n    rw [h3.deriv_eq, h1, h2, one_mul]\n  rw [hLi, hdA, hA]\n  have hw1' : w - 1 \u2260 0 := sub_ne_zero.mpr hw1\n  field_simp [hw0, hw1']\n  ring\n\n/-- The punctured-neighborhood form, for later contour work. -/\nlemma hLi_eq_near_one :\n    \u2200\u1da0 w in \ud835\udcdd[\u2260] (1 : \u2102), hLi w = (w - 1) * (-(deriv riemannZeta w) / riemannZeta w) := by\n  have hzero : \u2200\u1da0 w in \ud835\udcdd[\u2260] (1 : \u2102), riemannZeta w \u2260 0 := by\n    have h1 : (1 : \u2102) \u2208 {s : \u2102 | 1 \u2264 s.re} := by simp\n    filter_upwards [eventually_mem_nhdsWithin] with w hw\n    have hwre : 1 \u2264 w.re := by\n      have := hw.2\n      sorry\n    exact riemannZeta_ne_zero_of_one_le_re hwre\n  filter_upwards [hzero, eventually_mem_nhdsWithin] with w hw0 hw1\n  exact hLi_eq_of_ne hw1 hw0\n\n/-! ## 3. The Laurent coefficients of \u2212\u03b6\u2032/\u03b6 at 1 (operational definition) -/\n\n/-- The k-th Laurent coefficient of \u2212\u03b6\u2032/\u03b6 at s = 1:\n\u2212\u03b6\u2032/\u03b6(1+u) = u\u207b\u00b9 + \u03a3_{k\u22650} lcoeff k \u00b7 u^k (identification proved in\ntier ii\u2032-2/3; here lcoeff is DEFINED via the analytic hLi). -/\ndef lcoeff (k : \u2115) : \u2102 := iteratedDeriv (k + 1) hLi 1 / (k + 1) !\n\nlemma lcoeff_eq (k : \u2115) :\n    lcoeff k = iteratedDeriv (k + 1) hLi 1 / (k + 1) ! := rfl\n\n/-- hLi's Taylor coefficient form: iteratedDeriv j hLi 1 / j ! = lcoeff (j\u22121). -/\nlemma hLi_iteratedDeriv_div_factorial {j : \u2115} (hj : 1 \u2264 j) :\n    iteratedDeriv j hLi 1 / j ! = lcoeff (j - 1) := by\n  rw [lcoeff]\n  have : j = (j - 1) + 1 := by omega\n  rw [this]\n\n/-! ## 4. The A-polynomial derivative: A(w) = w\u207f \u2212 (w\u22121)\u207f -/\n\n/-- Iterated derivatives of A(w) = w\u207f \u2212 (w\u22121)\u207f at 1:\nA^{(i)}(1) = n.descFactorial i for i < n, and A^{(n)}(1) = 0. -/\nlemma A_iter (n i : \u2115) (hi : i \u2264 n) :\n    iteratedDeriv i (fun w => w ^ n - (w - 1) ^ n) (1 : \u2102) =\n      if i < n then n.descFactorial i else 0 := by\n  have hf : ContDiffAt \u2102 i (fun w : \u2102 => w ^ n) 1 :=\n    (contDiff_id.pow n).contDiffAt\n  have hg : ContDiffAt \u2102 i (fun w : \u2102 => (w - 1) ^ n) 1 :=\n    ((contDiff_id.sub contDiff_const).pow n).contDiffAt\n  rw [iteratedDeriv_sub hf hg]\n  have h1 : iteratedDeriv i (fun w : \u2102 => w ^ n) 1 = n.descFactorial i := by\n    rw [iteratedDeriv_pow]\n    simp\n  have h2 : iteratedDeriv i (fun w : \u2102 => (w - 1) ^ n) 1 =\n      n.descFactorial i * (0 : \u2102) ^ (n - i) := by\n    have e := iteratedDeriv_comp_sub_const i (fun w : \u2102 => w ^ n) (1 : \u2102)\n    have e2 : iteratedDeriv i (fun w : \u2102 => (w - 1) ^ n) 1 =\n        iteratedDeriv i (fun w : \u2102 => w ^ n) (1 - 1) := by\n      have := congr_fun e 1\n      convert this using 2\n    rw [e2, iteratedDeriv_pow]\n    simp\n  rw [h1, h2]\n  by_cases hin : i < n\n  \u00b7 rw [if_pos hin]\n    have : (0 : \u2102) ^ (n - i) = 0 := by\n      apply zero_pow\n      omega\n    rw [this, mul_zero, sub_zero]\n  \u00b7 rw [if_neg hin]\n    have hin' : i = n := le_antisymm hi (by omega)\n    subst hin'\n    simp [Nat.descFactorial_self]\n\n/-! ## 5. The residue at w = 1: ResW1 n = \u03a3_{j=1}^n C(n,j)\u00b7lcoeff (j\u22121) -/\n\n/-- The residue at w = 1 of the Lagarias-kernel integrand\nJ(w) = k_n(w\u22121)\u00b7(\u2212\u03b6\u2032/\u03b6(w)), packaged as (1/n!)\u00b7(d/dw)\u207f[(w\u207f\u2212(w\u22121)\u207f)hLi]\nat w = 1 (D-064 \u00a74 R1: (w\u22121)^{n+1}J = [w\u207f\u2212(w\u22121)\u207f]\u00b7hLi on a punctured\nneighborhood, by `hLi_eq_near_one`). -/\ndef ResW1 (n : \u2115) : \u2102 :=\n  iteratedDeriv n (fun w => (w ^ n - (w - 1) ^ n) * hLi w) (1 : \u2102) / n !\n\n/-- The w = 1 residue value: ResW1 n = \u03a3_{j=1}^n C(n,j)\u00b7lcoeff (j\u22121). -/\ntheorem ResW1_eq_sum (n : \u2115) :\n    ResW1 n = \u2211 j \u2208 Finset.Icc 1 n, (n.choose j : \u2102) * lcoeff (j - 1) := by\n  have hA : ContDiffAt \u2102 n (fun w : \u2102 => w ^ n - (w - 1) ^ n) 1 :=\n    ((contDiff_id.pow n).sub ((contDiff_id.sub contDiff_const).pow n)).contDiffAt\n  have hh : ContDiffAt \u2102 n hLi 1 := hLi_analyticAt_one.contDiffAt\n  have hmul := iteratedDeriv_mul hA hh\n  rw [ResW1, hmul]\n  have hsum : \u2211 i \u2208 Finset.range (n + 1),\n        n.choose i * iteratedDeriv i (fun w : \u2102 => w ^ n - (w - 1) ^ n) 1 *\n          iteratedDeriv (n - i) hLi 1 =\n      \u2211 i \u2208 Finset.range n,\n        n.choose i * (n.descFactorial i : \u2102) * iteratedDeriv (n - i) hLi 1 := by\n    apply Finset.sum_subset (Finset.range_le_iff.mpr (Nat.le_succ n))\n    intro i hi\n    rw [Finset.mem_range] at hi\n    have hi0 : i \u2260 n := by omega\n    rw [A_iter n i (by omega), if_pos (by omega)]\n    \u00b7 rfl\n    \u00b7 rw [Finset.mem_range] at *\n      omega\n  rw [hsum]\n  have hfinal : (\u2211 i \u2208 Finset.range n,\n        n.choose i * (n.descFactorial i : \u2102) * iteratedDeriv (n - i) hLi 1) / n ! =\n      \u2211 j \u2208 Finset.Icc 1 n, (n.choose j : \u2102) * lcoeff (j - 1) := by\n    have hrewrite : \u2200 i \u2208 Finset.range n,\n        n.choose i * (n.descFactorial i : \u2102) * iteratedDeriv (n - i) hLi 1 / n ! =\n          (n.choose (n - i) : \u2102) * lcoeff (n - i - 1) := by\n      intro i hi\n      rw [Finset.mem_range] at hi\n      have hcd : (n.choose i : \u2102) * (n.descFactorial i : \u2102) =\n          n ! * n.choose i / (n - i) ! := by\n        have h1 := Nat.choose_mul_factorial_mul_factorial (show i \u2264 n by omega)\n        have h2 := Nat.factorial_mul_descFactorial (show i \u2264 n by omega)\n        have : (n.choose i : \u2102) * (n.descFactorial i : \u2102) * ((i ! : \u2102) * ((n - i) ! : \u2102)) =\n            (n ! : \u2102) * (n.choose i : \u2102) := by\n          push_cast [h1, h2]\n          ring\n        have hfact : ((i ! : \u2102) * ((n - i) ! : \u2102)) \u2260 0 := by\n          simp [Nat.factorial_ne_zero]\n        have hthis := this\n        field_simp at hthis \u22a2\n        have h3 : (n.choose i : \u2102) * (n.descFactorial i : \u2102) * ((n - i) ! : \u2102) =\n            (n ! : \u2102) * (n.choose i : \u2102) / 1 := by\n          have h4 := hthis\n          rw [mul_comm ((i ! : \u2102)) ((n - i) ! : \u2102)] at h4\n          have h5 : (n.choose i : \u2102) * (n.descFactorial i : \u2102) * ((n - i) ! : \u2102) * (i ! : \u2102) =\n              (n ! : \u2102) * (n.choose i : \u2102) := by\n            rw [mul_comm ((n - i) ! : \u2102) (i ! : \u2102)] at h4\n            calc (n.choose i : \u2102) * (n.descFactorial i : \u2102) * ((n - i) ! : \u2102) * (i ! : \u2102)\n                = (n.choose i : \u2102) * (n.descFactorial i : \u2102) * (((n - i) ! : \u2102) * (i ! : \u2102)) := by ring\n              _ = (n ! : \u2102) * (n.choose i : \u2102) := by\n                  have := h4\n                  ring_nf at this \u22a2\n                  convert this using 1\n                  ring\n          exact (div_eq_one_iff_eq (by simp [Nat.factorial_ne_zero])).mpr h5\n        field_simp at h3 \u22a2\n        calc (n.choose i : \u2102) * (n.descFactorial i : \u2102)\n            = ((n ! : \u2102) * (n.choose i : \u2102) / 1) / ((n - i) ! : \u2102) := by\n              rw [\u2190 h3]\n              field_simp\n              ring\n          _ = (n ! : \u2102) * (n.choose i : \u2102) / ((n - i) ! : \u2102) := by ring\n      rw [hcd]\n      have hni : n - i \u2265 1 := by omega\n      rw [hLi_iteratedDeriv_div_factorial (j := n - i) (by omega)]\n      rw [show n - i - 1 = n - i - 1 from rfl]\n      have hcoe : ((n - i) ! : \u2102) \u2260 0 := by simp [Nat.factorial_ne_zero]\n      field_simp\n      rw [Nat.choose_symm (by omega : i \u2264 n)]\n      ring\n    rw [Finset.sum_div]\n    rw [Finset.sum_congr rfl hrewrite]\n    have hmap : \u2211 i \u2208 Finset.range n, (n.choose (n - i) : \u2102) * lcoeff (n - i - 1) =\n        \u2211 j \u2208 Finset.Icc 1 n, (n.choose j : \u2102) * lcoeff (j - 1) := by\n      rw [Finset.sum_range_reflect]\n      apply Finset.sum_congr rfl\n      intro i hi\n      rw [Finset.mem_range] at hi\n      have : n - 1 - i + 1 = n - i := by omega\n      rw [this]\n      have : n - (n - i) = i := by omega\n      rw [Nat.choose_symm (show n - i \u2264 n by omega)]\n      simp [this]\n      ring\n    exact hmap\n  rw [hfinal]\n\nend NguyenRH1\n",
  "autoRead": false
}

{
  "command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiResidue.lean'\ns = open(p, encoding='utf-8').read()\n\n# 1. Complex.norm -> norm\ns = s.replace(\"BddAbove (Complex.norm \u2218 (fun w => (w - 1) * riemannZeta w)) ''\",\n              \"BddAbove (norm \u2218 (fun w => (w - 1) * riemannZeta w)) ''\")\n\n# 2. fix the boundedness block (remove Complex.norm_eq_abs line)\ns = s.replace(\"\"\"    have h1 : \u2016(z - 1) * riemannZeta z - 1\u2016 < 1 := by\n      have := hlt\n      rwa [dist_eq_norm] at this\n    have := norm_sub_le ((z - 1) * riemannZeta z) 1\n    simp only [Complex.norm_eq_abs] at *\n    have h2 : \u2016(z - 1) * riemannZeta z\u2016 \u2264 \u2016(z - 1) * riemannZeta z - 1\u2016 + \u2016(1 : \u2102)\u2016 :=\n      norm_sub_le ((z - 1) * riemannZeta z) 1\n    rw [norm_one] at h2\n    linarith [h1]\"\"\",\n\"\"\"    have h1 : \u2016(z - 1) * riemannZeta z - 1\u2016 < 1 := by\n      have := hlt\n      rwa [dist_eq_norm] at this\n    have h2 : \u2016(z - 1) * riemannZeta z\u2016 \u2264 \u2016(z - 1) * riemannZeta z - 1\u2016 + \u2016(1 : \u2102)\u2016 :=\n      norm_sub_le ((z - 1) * riemannZeta z) 1\n    rw [norm_one] at h2\n    linarith [h1]\"\"\")\n\n# 3. analyticAt from analyticOnNhd membership\ns = s.replace(\"\"\"  have han := h.analyticOnNhd Metric.isOpen_ball\n  exact han.analyticAt (Metric.ball_mem_nhds 1 (by norm_num : (0 : \u211d) < 2))\"\"\",\n\"\"\"  have han := h.analyticOnNhd Metric.isOpen_ball\n  exact han 1 (Metric.mem_ball_self (by norm_num : (0 : \u211d) < 2))\"\"\")\n\n# 4. DifferentiableAt.analyticAt explicit\ns = s.replace(\"(differentiableAt_riemannZeta hw).analyticAt\",\n              \"DifferentiableAt.analyticAt (differentiableAt_riemannZeta hw)\")\n\n# 5. hLi_analyticAt_one restructure\ns = s.replace(\"\"\"  exact ((analyticAt_const.sub analyticAt_id).mul\n    (hda.div ha hne)).const_sub 1\"\"\",\n\"\"\"  have hprod : AnalyticAt \u2102 (fun w => (w - 1) * (deriv zetaA w / zetaA w)) 1 :=\n    (analyticAt_id.sub analyticAt_const).mul (hda.div ha hne)\n  exact analyticAt_const.sub hprod\"\"\")\n\n# 6. deriv_eq direction + deriv (z - 1)\ns = s.replace(\"\"\"    have h2 : deriv (fun z => z - 1) w = 1 := by\n      simp [deriv_sub, deriv_id'']\n    have h3 : (fun z => (z - 1) * riemannZeta z) =\u1da0[\ud835\udcdd w] zetaA := by\n      filter_upwards [eventually_ne_nhds hw1] with z hz using (zetaA_apply hz).symm\n    rw [h3.deriv_eq, h1, h2, one_mul]\"\"\",\n\"\"\"    have h2 : deriv (fun z => z - 1) w = 1 := by\n      simp [deriv_sub_const, deriv_id'']\n    have h3 : (fun z => (z - 1) * riemannZeta z) =\u1da0[\ud835\udcdd w] zetaA := by\n      filter_upwards [eventually_ne_nhds hw1] with z hz using (zetaA_apply hz).symm\n    rw [\u2190 h3.deriv_eq, h1, h2, one_mul]\"\"\")\n\n# 7. push_neg deprecation -> push Not\ns = s.replace(\"push_neg at h2\", \"push Not at h2\")\n\n# 8. factorial spacing\ns = s.replace(\"iteratedDeriv (k + 1) hLi 1 / (k + 1) !\", \"iteratedDeriv (k + 1) hLi 1 / (k + 1)!\")\ns = s.replace(\"iteratedDeriv n (fun w => (w ^ n - (w - 1) ^ n) * hLi w) (1 : \u2102) / n !\",\n              \"iteratedDeriv n (fun w => (w ^ n - (w - 1) ^ n) * hLi w) (1 : \u2102) / n !\")\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiResidue.lean 2>&1 | head -50",
  "description": "Apply fixes and recompile LiResidue.lean",
  "timeout": 420000
}

{
  "command": "cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"DifferentiableAt.analyticAt\\|HasDerivAt.analyticAt\" Mathlib/ --include=*.lean | head -4; grep -rn \"theorem analyticAt_of_differentiableAt\" Mathlib/ --include=*.lean | head -3; grep -rn \"notation \\\"!\\\"\\|scoped.*\\\"!\\\"\" Mathlib/Data/Nat/Factorial/Basic.lean Mathlib/Data/Nat/Factorial/Defs.lean 2>/dev/null | head -4",
  "description": "Find analyticAt-of-differentiableAt and factorial notation"
}

{
  "command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiResidue.lean'\ns = open(p, encoding='utf-8').read()\n\n# import + scope\ns = s.replace(\"\"\"import Mathlib\nimport Zeta23.Statement\n\nopen Zeta23 Complex Filter Topology\"\"\",\n\"\"\"import Mathlib\nimport Zeta23.Statement\nimport Zeta23.FromPNTPlus.ZetaBounds\n\nopen Zeta23 Complex Filter Topology\nopen scoped Nat\"\"\")\n\n# BddAbove image: explicit lambda\ns = s.replace(\"BddAbove (norm \u2218 (fun w => (w - 1) * riemannZeta w)) ''\",\n              \"BddAbove ((fun z => \u2016(z - 1) * riemannZeta z\u2016) ''\")\n\n# analyticAt_riemannZeta from ZetaBounds\ns = s.replace(\"\"\"  have ha : AnalyticAt \u2102 (fun w => (w - 1) * riemannZeta w) w :=\n    (analyticAt_id.sub analyticAt_const).mul\n      DifferentiableAt.analyticAt (differentiableAt_riemannZeta hw)\n  exact ha.congr h\"\"\",\n\"\"\"  have ha : AnalyticAt \u2102 (fun w => (w - 1) * riemannZeta w) w :=\n    (analyticAt_id.sub analyticAt_const).mul (analyticAt_riemannZeta hw)\n  exact ha.congr h\"\"\")\n\n# A_iter: function-sub form\ns = s.replace(\"\"\"  rw [iteratedDeriv_sub hf hg]\n  have h1 : iteratedDeriv i (fun w : \u2102 => w ^ n) 1 = n.descFactorial i := by\n    rw [iteratedDeriv_pow]\n    simp\"\"\",\n\"\"\"  have hsub : (fun w : \u2102 => w ^ n - (w - 1) ^ n) =\n      (fun w : \u2102 => w ^ n) - (fun w : \u2102 => (w - 1) ^ n) := by\n    funext w\n    simp [Pi.sub_apply]\n  rw [hsub, iteratedDeriv_sub hf hg]\n  have h1 : iteratedDeriv i (fun w : \u2102 => w ^ n) 1 = n.descFactorial i := by\n    rw [iteratedDeriv_pow]\n    simp\"\"\")\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiResidue.lean 2>&1 | head -50",
  "description": "Apply second round of fixes and recompile",
  "timeout": 420000
}

{
  "command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && sed -n '50,56p' NguyenRH1/LiResidue.lean",
  "description": "View broken BddAbove line"
}

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiResidue.lean",
  "offset": 148,
  "limit": 75
}

{
  "type": "text",
  "file": {
    "filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiRBound.lean",
    "content": "\n/-- The unit-interval IBP: over each integer interval [k, k+1] (k \u2265 1),\n\u222b B\u0303\u2081\u00b7x^{\u2212s\u22121} = (s+1)\u00b7\u222b B\u0303\u2082\u00b7x^{\u2212s\u22122} (boundary terms vanish: B\u0303\u2082 = 0 at\nthe integers). -/\ntheorem intervalIntegral_b1_cpow (k : \u2115) (hk : 1 \u2264 k) (s : \u2102) (hs : s \u2260 -1) :\n    \u222b x in (k : \u211d)..(k + 1 : \u211d), (B1 x : \u2102) * (x : \u2102) ^ (-s - 1)\n      = (s + 1) * \u222b x in (k : \u211d)..(k + 1 : \u211d), (b2Fun x : \u2102) * (x : \u2102) ^ (-s - 2) := by\n  have hk0 : (0:\u211d) < (k : \u211d) := by exact_mod_cast hk\n  have hu : ContinuousOn (fun x : \u211d => (b2Fun x : \u2102)) (Set.Icc (k : \u211d) (k + 1 : \u211d)) :=\n    (continuous_ofReal.comp continuous_b2Fun).continuousOn\n  have hv : ContinuousOn (fun x : \u211d => (x : \u2102) ^ (-s - 1)) (Set.Icc (k : \u211d) (k + 1 : \u211d)) := by\n    intro x hx\n    have hx0 : x \u2260 0 := by\n      have : (k : \u211d) \u2264 x := hx.1\n      linarith",
    "numLines": 15,
    "startLine": 98,
    "totalLines": 146
  }
}

{
  "command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiResidue.lean'\ns = open(p, encoding='utf-8').read()\n\n# 1. annotate lambda in A_iter statement\ns = s.replace(\"iteratedDeriv i (fun w => w ^ n - (w - 1) ^ n) (1 : \u2102) =\",\n              \"iteratedDeriv i (fun w : \u2102 => w ^ n - (w - 1) ^ n) (1 : \u2102) =\")\n\n# 2. annotate lambda in ResW1 def\ns = s.replace(\"iteratedDeriv n (fun w => (w ^ n - (w - 1) ^ n) * hLi w) (1 : \u2102) / n !\",\n              \"iteratedDeriv n (fun w : \u2102 => (w ^ n - (w - 1) ^ n) * hLi w) (1 : \u2102) / n !\")\n\n# 3. rewrite hbd with correct radius handling\nold_hbd = s[s.index(\"  have hbd : BddAbove ((fun z => \u2016(z - 1) * riemannZeta z\u2016) ''\"):s.index(\"  have h := differentiableOn_update_limUnder_of_bddAbove\")]\nnew_hbd = \"\"\"  have hT := riemannZeta_residue_one\n  rw [Metric.tendsto_nhdsWithin_nhds] at hT\n  obtain \u27e8\u03b4, h\u03b40, h\u03b4\u27e9 := hT 1 one_pos\n  have hdiff : DifferentiableOn \u2102 (fun w => (w - 1) * riemannZeta w)\n      (Metric.ball 1 (\u03b4 / 2) \\\\ {1}) := by\n    intro w hw\n    have hw1 : w \u2260 1 := hw.2\n    exact ((differentiableAt_id.sub (differentiableAt_const 1)).mul\n      (differentiableAt_riemannZeta hw1)).differentiableWithinAt\n  have hbd : BddAbove ((fun z => \u2016(z - 1) * riemannZeta z\u2016) ''\n      (Metric.ball 1 (\u03b4 / 2) \\\\ {1})) := by\n    refine \u27e82, ?_\u27e9\n    rintro _ \u27e8z, hz, rfl\u27e9\n    have hz1 : z \u2260 1 := hz.2\n    have hdist : dist z 1 < \u03b4 := by\n      have := hz.1\n      rw [Metric.mem_ball] at this\n      linarith [this, h\u03b40]\n    have h1 := h\u03b4 (Set.mem_compl_singleton_iff.mpr hz1) hdist\n    rw [dist_eq_norm] at h1\n    have h2 : \u2016(z - 1) * riemannZeta z\u2016 \u2264 \u2016(z - 1) * riemannZeta z - 1\u2016 + \u2016(1 : \u2102)\u2016 :=\n      norm_sub_le ((z - 1) * riemannZeta z) 1\n    rw [norm_one] at h2\n    linarith [h1]\n\"\"\"\ns = s.replace(old_hbd, new_hbd)\n\n# 4. remove the old hdiff (now duplicated)\nold_hdiff_first = \"\"\"  have hdiff : DifferentiableOn \u2102 (fun w => (w - 1) * riemannZeta w)\n      (Metric.ball 1 2 \\\\ {1}) := by\n    intro w hw\n    have hw1 : w \u2260 1 := hw.2\n    exact ((differentiableAt_id.sub (differentiableAt_const 1)).mul\n      (differentiableAt_riemannZeta hw1)).differentiableWithinAt\n\"\"\"\n# remove only the FIRST occurrence (before hT)\nidx_hT = s.index(\"  have hT := riemannZeta_residue_one\")\nidx_first = s.index(old_hdiff_first)\nif idx_first < idx_hT:\n    s = s[:idx_first] + s[idx_first:].replace(old_hdiff_first, \"\", 1)\n\n# 5. fix the ball radius in the final analytic call\ns = s.replace(\"(Metric.ball_mem_nhds 1 (by norm_num : (0 : \u211d) < 2)) hdiff hbd\",\n              \"(Metric.ball_mem_nhds 1 (by positivity : (0 : \u211d) < \u03b4 / 2)) hdiff hbd\")\ns = s.replace(\"exact han 1 (Metric.mem_ball_self (by norm_num : (0 : \u211d) < 2))\",\n              \"exact han 1 (Metric.mem_ball_self (by positivity : (0 : \u211d) < \u03b4 / 2))\")\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiResidue.lean 2>&1 | head -40",
  "description": "Apply fourth round of fixes and recompile",
  "timeout": 420000
}

{
  "command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiResidue.lean'\ns = open(p, encoding='utf-8').read()\n\n# 1. h2 norm calc fix\ns = s.replace(\"\"\"    have h2 : \u2016(z - 1) * riemannZeta z\u2016 \u2264 \u2016(z - 1) * riemannZeta z - 1\u2016 + \u2016(1 : \u2102)\u2016 := by\n      have h3 : (z - 1) * riemannZeta z = ((z - 1) * riemannZeta z - 1) + 1 := by ring\n      rw [h3]\n      exact norm_add_le _ _\"\"\",\n\"\"\"    have h2 : \u2016(z - 1) * riemannZeta z\u2016 \u2264 \u2016(z - 1) * riemannZeta z - 1\u2016 + \u2016(1 : \u2102)\u2016 := by\n      have h3 : (z - 1) * riemannZeta z = ((z - 1) * riemannZeta z - 1) + 1 := by ring\n      calc \u2016(z - 1) * riemannZeta z\u2016 = \u2016((z - 1) * riemannZeta z - 1) + 1\u2016 := by rw [h3]\n        _ \u2264 \u2016(z - 1) * riemannZeta z - 1\u2016 + \u2016(1 : \u2102)\u2016 := norm_add_le _ _\"\"\")\n\n# 2. hterm field_simp with ne facts\ns = s.replace(\"\"\"    rw [hlc]\n    field_simp\n    ring\"\"\",\n\"\"\"    rw [hlc]\n    have hf1 : (n ! : \u2102) \u2260 0 := by simp [Nat.factorial_ne_zero]\n    have hf2 : ((n - i) ! : \u2102) \u2260 0 := by simp [Nat.factorial_ne_zero]\n    field_simp [hf1, hf2]\n    ring\"\"\")\n\n# 3. hre via sum_bij\nold_hre = s[s.index(\"  have hre : (\u2211 i \u2208 Finset.range n, (n.choose i : \u2102) * lcoeff (n - i - 1)) =\"):]\nold_hre = old_hre[:old_hre.index(\"  exact hre\") + len(\"  exact hre\")]\nnew_hre = \"\"\"  have hre : (\u2211 i \u2208 Finset.range n, (n.choose i : \u2102) * lcoeff (n - i - 1)) =\n      \u2211 j \u2208 Finset.Icc 1 n, (n.choose j : \u2102) * lcoeff (j - 1) := by\n    apply Finset.sum_bij (fun i _ => n - i)\n    \u00b7 intro i hi\n      rw [Finset.mem_range] at hi\n      rw [Finset.mem_Icc]\n      omega\n    \u00b7 intro i hi\n      rw [Finset.mem_range] at hi\n      have e2 : n.choose (n - i) = n.choose i := Nat.choose_symm (by omega)\n      rw [e2]\n    \u00b7 intro a b ha hb hab\n      rw [Finset.mem_range] at ha hb\n      omega\n    \u00b7 intro j hj\n      rw [Finset.mem_Icc] at hj\n      exact \u27e8n - j, by rw [Finset.mem_range]; omega, by omega\u27e9\n  exact hre\"\"\"\ns = s.replace(old_hre, new_hre)\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiResidue.lean 2>&1 | head -40",
  "description": "Apply sixth round of fixes and recompile",
  "timeout": 420000
}

{
  "command": "cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem Finset.sum_bij \\|lemma Finset.sum_bij \" Mathlib/Algebra/BigOperators/Basic.lean | head -3; grep -A3 \"theorem Finset.sum_bij \" Mathlib/Algebra/BigOperators/Basic.lean | head -6",
  "description": "Check Finset.sum_bij signature"
}

{
  "command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && export PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiResidue.lean 2>&1 | head -10; echo \"=== exit: $?\" && cat > PrintAxioms.lean << 'EOF'\n/- Session 016 axiom audit: ii\u2032-1b(i) new declarations. -/\nimport Mathlib\nimport NguyenRH1.LiResidue\n\nopen NguyenRH1\n\n#print axioms ResW1_eq_sum\n#print axioms zetaA_analyticAt_one\n#print axioms hLi_analyticAt_one\n#print axioms hLi_eq_near_one\n#print axioms A_iter\nEOF\nlake env lean PrintAxioms.lean 2>&1 | head -8",
  "description": "Verify clean compile and axiom audit",
  "timeout": 420000
}

{
  "command": "cd /d/Python/Quantum && git add research/lean/NguyenRH1/NguyenRH1/LiResidue.lean research/lean/NguyenRH1/PrintAxioms.lean && git -c user.name=\"NGUYEN-RH-1\" -c user.email=\"research@local\" commit -m \"$(cat <<'EOF'\nSession 016 (WIP): ii'-1b(i) \u2014 the w=1 residue value theorem\n\nLiResidue.lean Part 1, PROVED (axiom audit: 3 standard axioms only):\n- zetaA = (w-1)*zeta(w) extended by 1 at 1: analytic (removable\n  singularity via riemannZeta_residue_one + Mathlib RemovableSingularity)\n- hLi = 1 - (w-1)*(zetaA'/zetaA): analytic at 1, hLi 1 = 1, agrees with\n  (w-1)(-zeta'/zeta) on a punctured nhd of 1 (hLi_eq_near_one)\n- lcoeff k := iteratedDeriv (k+1) hLi 1 / (k+1)! (operational Laurent\n  coefficients of -zeta'/zeta at 1)\n- A_iter: derivatives of A(w) = w^n - (w-1)^n at 1\n- ResW1_eq_sum: the w=1 residue of the Lagarias-kernel integrand equals\n  sum_{j=1}^n C(n,j)*lcoeff(j-1) \u2014 via iteratedDeriv_mul (Leibniz),\n  exactly as designed in D-064 R1 / verified numerically session 014.\nEOF\n)\" && git log --oneline -1",
  "description": "WIP commit of ii\u2032-1b(i)"
}

{
  "command": "sed -n '30,80p' /d/Python/Quantum/zeta-23-lean/Zeta23/FromPNTPlus/ResidueCalcOnRectangles.lean",
  "description": "Read HIntegral/VIntegral/RectangleIntegral definitions"
}

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean",
  "content": "/-\nNguyenRH1/LiPrincipal.lean \u2014 tier ii\u2032 sub-stage ii\u2032-1b(ii) (D-064 \u00a74 R1):\nthe principal-part integral. For the explicit principal part\n  P(w) = \u03a3_{i=1}^{n+1} A_i/(w\u22121)^i,   A_i = F^{(n+1\u2212i)}(1)/(n+1\u2212i)!,\nwith F(w) = (w\u207f\u2212(w\u22121)\u207f)\u00b7hLi(w), on any rectangle with 1 in its interior:\n  RectangleIntegral' P z w = A_1 = ResW1 n.\n\nMethod: P = A_1/(w\u22121) + (\u03a3_{i\u22652} A_i/(w\u22121)^i). The first term is handled\nby the vendored `ResidueTheoremInRectangle`; the tail has the explicit\nglobal primitive G(w) = \u2212\u03a3_{i\u22652} A_i/((i\u22121)(w\u22121)^{i\u22121}), differentiable\non the border (1 is interior, hence off the border), so its rectangle\nintegral vanishes by per-side FTC (`rectInt_deriv_eq_zero`).\n\nArchive references: D-036, D-064 (R1), journal-016.\n-/\nimport Mathlib\nimport Zeta23.FromPNTPlus.ResidueCalcOnRectangles\nimport NguyenRH1.LiResidue\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. Linearity of rectangle integrals -/\n\nlemma HIntegral_add (f g : \u2102 \u2192 \u2102) (x\u2081 x\u2082 y : \u211d)\n    (hf : IntervalIntegrable (fun x => f (x + y * I)) volume x\u2081 x\u2082)\n    (hg : IntervalIntegrable (fun x => g (x + y * I)) volume x\u2081 x\u2082) :\n    HIntegral (f + g) x\u2081 x\u2082 y = HIntegral f x\u2081 x\u2082 y + HIntegral g x\u2081 x\u2082 y := by\n  rw [HIntegral, HIntegral, HIntegral]\n  have e : (fun x => (f + g) (x + y * I)) = (fun x => f (x + y * I) + g (x + y * I)) := by\n    funext x\n    simp [Pi.add_apply]\n  rw [e, intervalIntegral.integral_add hf hg]\n\nlemma VIntegral_add (f g : \u2102 \u2192 \u2102) (x y\u2081 y\u2082 : \u211d)\n    (hf : IntervalIntegrable (fun y => f (x + y * I)) volume y\u2081 y\u2082)\n    (hg : IntervalIntegrable (fun y => g (x + y * I)) volume y\u2081 y\u2082) :\n    VIntegral (f + g) x y\u2081 y\u2082 = VIntegral f x y\u2081 y\u2082 + VIntegral g x y\u2081 y\u2082 := by\n  rw [VIntegral, VIntegral, VIntegral]\n  have e : (fun y => (f + g) (x + y * I)) = (fun y => f (x + y * I) + g (x + y * I)) := by\n    funext y\n    simp [Pi.add_apply]\n  rw [e, intervalIntegral.integral_add hf hg, smul_add]\n\nlemma RectangleIntegral_add (f g : \u2102 \u2192 \u2102) (z w : \u2102)\n    (hf : RectangleBorderIntegrable f z w) (hg : RectangleBorderIntegrable g z w) :\n    RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n  rw [RectangleIntegral, RectangleIntegral, RectangleIntegral]\n  obtain \u27e8\u27e8hf1, hf2\u27e9, hf3, hf4\u27e9 := hf\n  obtain \u27e8\u27e8hg1, hg2\u27e9, hg3, hg4\u27e9 := hg\n  rw [HIntegral_add f g _ _ _ hf1 hg1, HIntegral_add f g _ _ _ hf2 hg2,\n    VIntegral_add f g _ _ _ hf3 hg3, VIntegral_add f g _ _ _ hf4 hg4]\n  ring\n\nlemma RectangleIntegral'_add (f g : \u2102 \u2192 \u2102) (z w : \u2102)\n    (hf : RectangleBorderIntegrable f z w) (hg : RectangleBorderIntegrable g z w) :\n    RectangleIntegral' (f + g) z w = RectangleIntegral' f z w + RectangleIntegral' g z w := by\n  rw [RectangleIntegral', RectangleIntegral_add f g z w hf hg, RectangleIntegral', RectangleIntegral',\n    smul_add]\n\n/-! ## 2. The per-side FTC vanishing lemma -/\n\n/-- If `G : \u2102 \u2192 \u2102` is differentiable at every point of the rectangle\nborder, then the rectangle integral of `deriv G` vanishes (each side\ntelescopes by FTC). -/\nlemma rectInt_deriv_eq_zero {G : \u2102 \u2192 \u2102} {z w : \u2102}\n    (hG : \u2200 s \u2208 RectangleBorder z w, DifferentiableAt \u2102 G s)\n    (hGc : \u2200 s \u2208 RectangleBorder z w, ContinuousAt (deriv G) s) :\n    RectangleIntegral (fun s => deriv G s) z w = 0 := by\n  rw [RectangleIntegral]\n  have hsideH : \u2200 y : \u211d, (\u2200 x : \u211d, x \u2208 Set.uIcc z.re w.re \u2192 (x + y * I) \u2208 RectangleBorder z w) \u2192\n      \u222b x in z.re..w.re, deriv G (x + y * I) = G (w.re + y * I) - G (z.re + y * I) := by\n    intro y hy\n    have hd : \u2200 x \u2208 Set.uIcc z.re w.re,\n        HasDerivAt (fun x => G (x + y * I)) (deriv G (x + y * I)) x := by\n      intro x hx\n      have h1 : HasDerivAt G (deriv G (x + y * I)) (x + y * I) :=\n        (hG _ (hy x hx)).hasDerivAt\n      have h2 : HasDerivAt (fun x => x + y * I) 1 x := by\n        have hid := hasDerivAt_id x\n        have hconst : HasDerivAt (fun x => y * I) 0 x := hasDerivAt_const x _\n        convert hid.add hconst using 1\n        simp\n      have := h1.comp x h2\n      convert this using 1\n      simp\n    have hint : IntervalIntegrable (fun x => deriv G (x + y * I)) volume z.re w.re := by\n      apply ContinuousOn.intervalIntegrable_of_Icc (a := z.re) (b := w.re)\n      intro x hx\n      have hmem : (x + y * I) \u2208 RectangleBorder z w := hy x (Set.uIcc_of_le ?_ hx |>.elim (fun h => h.1) )\n      \u00b7 exact (hGc _ hmem).continuousWithinAt\n      \u00b7 sorry\n    have := intervalIntegral.integral_deriv_eq_sub hd hint\n    convert this using 1\n  sorry\n\nend NguyenRH1\n"
}

{
  "type": "create",
  "filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean",
  "content": "/-\nNguyenRH1/LiPrincipal.lean \u2014 tier ii\u2032 sub-stage ii\u2032-1b(ii) (D-064 \u00a74 R1):\nthe principal-part integral. For the explicit principal part\n  P(w) = \u03a3_{i=1}^{n+1} A_i/(w\u22121)^i,   A_i = F^{(n+1\u2212i)}(1)/(n+1\u2212i)!,\nwith F(w) = (w\u207f\u2212(w\u22121)\u207f)\u00b7hLi(w), on any rectangle with 1 in its interior:\n  RectangleIntegral' P z w = A_1 = ResW1 n.\n\nMethod: P = A_1/(w\u22121) + (\u03a3_{i\u22652} A_i/(w\u22121)^i). The first term is handled\nby the vendored `ResidueTheoremInRectangle`; the tail has the explicit\nglobal primitive G(w) = \u2212\u03a3_{i\u22652} A_i/((i\u22121)(w\u22121)^{i\u22121}), differentiable\non the border (1 is interior, hence off the border), so its rectangle\nintegral vanishes by per-side FTC (`rectInt_deriv_eq_zero`).\n\nArchive references: D-036, D-064 (R1), journal-016.\n-/\nimport Mathlib\nimport Zeta23.FromPNTPlus.ResidueCalcOnRectangles\nimport NguyenRH1.LiResidue\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. Linearity of rectangle integrals -/\n\nlemma HIntegral_add (f g : \u2102 \u2192 \u2102) (x\u2081 x\u2082 y : \u211d)\n    (hf : IntervalIntegrable (fun x => f (x + y * I)) volume x\u2081 x\u2082)\n    (hg : IntervalIntegrable (fun x => g (x + y * I)) volume x\u2081 x\u2082) :\n    HIntegral (f + g) x\u2081 x\u2082 y = HIntegral f x\u2081 x\u2082 y + HIntegral g x\u2081 x\u2082 y := by\n  rw [HIntegral, HIntegral, HIntegral]\n  have e : (fun x => (f + g) (x + y * I)) = (fun x => f (x + y * I) + g (x + y * I)) := by\n    funext x\n    simp [Pi.add_apply]\n  rw [e, intervalIntegral.integral_add hf hg]\n\nlemma VIntegral_add (f g : \u2102 \u2192 \u2102) (x y\u2081 y\u2082 : \u211d)\n    (hf : IntervalIntegrable (fun y => f (x + y * I)) volume y\u2081 y\u2082)\n    (hg : IntervalIntegrable (fun y => g (x + y * I)) volume y\u2081 y\u2082) :\n    VIntegral (f + g) x y\u2081 y\u2082 = VIntegral f x y\u2081 y\u2082 + VIntegral g x y\u2081 y\u2082 := by\n  rw [VIntegral, VIntegral, VIntegral]\n  have e : (fun y => (f + g) (x + y * I)) = (fun y => f (x + y * I) + g (x + y * I)) := by\n    funext y\n    simp [Pi.add_apply]\n  rw [e, intervalIntegral.integral_add hf hg, smul_add]\n\nlemma RectangleIntegral_add (f g : \u2102 \u2192 \u2102) (z w : \u2102)\n    (hf : RectangleBorderIntegrable f z w) (hg : RectangleBorderIntegrable g z w) :\n    RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n  rw [RectangleIntegral, RectangleIntegral, RectangleIntegral]\n  obtain \u27e8\u27e8hf1, hf2\u27e9, hf3, hf4\u27e9 := hf\n  obtain \u27e8\u27e8hg1, hg2\u27e9, hg3, hg4\u27e9 := hg\n  rw [HIntegral_add f g _ _ _ hf1 hg1, HIntegral_add f g _ _ _ hf2 hg2,\n    VIntegral_add f g _ _ _ hf3 hg3, VIntegral_add f g _ _ _ hf4 hg4]\n  ring\n\nlemma RectangleIntegral'_add (f g : \u2102 \u2192 \u2102) (z w : \u2102)\n    (hf : RectangleBorderIntegrable f z w) (hg : RectangleBorderIntegrable g z w) :\n    RectangleIntegral' (f + g) z w = RectangleIntegral' f z w + RectangleIntegral' g z w := by\n  rw [RectangleIntegral', RectangleIntegral_add f g z w hf hg, RectangleIntegral', RectangleIntegral',\n    smul_add]\n\n/-! ## 2. The per-side FTC vanishing lemma -/\n\n/-- If `G : \u2102 \u2192 \u2102` is differentiable at every point of the rectangle\nborder, then the rectangle integral of `deriv G` vanishes (each side\ntelescopes by FTC). -/\nlemma rectInt_deriv_eq_zero {G : \u2102 \u2192 \u2102} {z w : \u2102}\n    (hG : \u2200 s \u2208 RectangleBorder z w, DifferentiableAt \u2102 G s)\n    (hGc : \u2200 s \u2208 RectangleBorder z w, ContinuousAt (deriv G) s) :\n    RectangleIntegral (fun s => deriv G s) z w = 0 := by\n  rw [RectangleIntegral]\n  have hsideH : \u2200 y : \u211d, (\u2200 x : \u211d, x \u2208 Set.uIcc z.re w.re \u2192 (x + y * I) \u2208 RectangleBorder z w) \u2192\n      \u222b x in z.re..w.re, deriv G (x + y * I) = G (w.re + y * I) - G (z.re + y * I) := by\n    intro y hy\n    have hd : \u2200 x \u2208 Set.uIcc z.re w.re,\n        HasDerivAt (fun x => G (x + y * I)) (deriv G (x + y * I)) x := by\n      intro x hx\n      have h1 : HasDerivAt G (deriv G (x + y * I)) (x + y * I) :=\n        (hG _ (hy x hx)).hasDerivAt\n      have h2 : HasDerivAt (fun x => x + y * I) 1 x := by\n        have hid := hasDerivAt_id x\n        have hconst : HasDerivAt (fun x => y * I) 0 x := hasDerivAt_const x _\n        convert hid.add hconst using 1\n        simp\n      have := h1.comp x h2\n      convert this using 1\n      simp\n    have hint : IntervalIntegrable (fun x => deriv G (x + y * I)) volume z.re w.re := by\n      apply ContinuousOn.intervalIntegrable_of_Icc (a := z.re) (b := w.re)\n      intro x hx\n      have hmem : (x + y * I) \u2208 RectangleBorder z w := hy x (Set.uIcc_of_le ?_ hx |>.elim (fun h => h.1) )\n      \u00b7 exact (hGc _ hmem).continuousWithinAt\n      \u00b7 sorry\n    have := intervalIntegral.integral_deriv_eq_sub hd hint\n    convert this using 1\n  sorry\n\nend NguyenRH1\n",
  "structuredPatch": [],
  "originalFile": null
}

{
  "command": "grep -n \"def RectangleBorder\\|RectangleBorder \" /d/Python/Quantum/zeta-23-lean/Zeta23/FromPNTPlus/Rectangle.lean | head -8; sed -n \"$(grep -n 'def RectangleBorder' /d/Python/Quantum/zeta-23-lean/Zeta23/FromPNTPlus/Rectangle.lean | head -1 | cut -d: -f1),+15p\" /d/Python/Quantum/zeta-23-lean/Zeta23/FromPNTPlus/Rectangle.lean",
  "description": "Read RectangleBorder definition"
}

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean",
  "content": "/-\nNguyenRH1/LiPrincipal.lean \u2014 tier ii\u2032 sub-stage ii\u2032-1b(ii) (D-064 \u00a74 R1):\nthe principal-part integral. For the explicit principal part\n  P(w) = \u03a3_{i=1}^{n+1} A_i/(w\u22121)^i,   A_i = F^{(n+1\u2212i)}(1)/(n+1\u2212i)!,\nwith F(w) = (w\u207f\u2212(w\u22121)\u207f)\u00b7hLi(w), on any rectangle with 1 in its interior:\n  RectangleIntegral' P z w = A_1 = ResW1 n.\n\nMethod: P = A_1/(w\u22121) + (\u03a3_{i\u22652} A_i/(w\u22121)^i). The first term is handled\nby the vendored `ResidueTheoremInRectangle`; the tail has the explicit\nglobal primitive G(w) = \u2212\u03a3_{i\u22652} A_i/((i\u22121)(w\u22121)^{i\u22121}), differentiable\non the border (1 is interior, hence off the border), so its rectangle\nintegral vanishes by per-side FTC (`rectInt_deriv_eq_zero`).\n\nArchive references: D-036, D-064 (R1), journal-016.\n-/\nimport Mathlib\nimport Zeta23.FromPNTPlus.ResidueCalcOnRectangles\nimport NguyenRH1.LiResidue\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. Linearity of rectangle integrals -/\n\nlemma HIntegral_add (f g : \u2102 \u2192 \u2102) (x\u2081 x\u2082 y : \u211d)\n    (hf : IntervalIntegrable (fun x => f (x + y * I)) volume x\u2081 x\u2082)\n    (hg : IntervalIntegrable (fun x => g (x + y * I)) volume x\u2081 x\u2082) :\n    HIntegral (f + g) x\u2081 x\u2082 y = HIntegral f x\u2081 x\u2082 y + HIntegral g x\u2081 x\u2082 y := by\n  rw [HIntegral, HIntegral, HIntegral]\n  have e : (fun x => (f + g) (x + y * I)) = (fun x => f (x + y * I) + g (x + y * I)) := by\n    funext x\n    simp [Pi.add_apply]\n  rw [e, intervalIntegral.integral_add hf hg]\n\nlemma VIntegral_add (f g : \u2102 \u2192 \u2102) (x y\u2081 y\u2082 : \u211d)\n    (hf : IntervalIntegrable (fun y => f (x + y * I)) volume y\u2081 y\u2082)\n    (hg : IntervalIntegrable (fun y => g (x + y * I)) volume y\u2081 y\u2082) :\n    VIntegral (f + g) x y\u2081 y\u2082 = VIntegral f x y\u2081 y\u2082 + VIntegral g x y\u2081 y\u2082 := by\n  rw [VIntegral, VIntegral, VIntegral]\n  have e : (fun y => (f + g) (x + y * I)) = (fun y => f (x + y * I) + g (x + y * I)) := by\n    funext y\n    simp [Pi.add_apply]\n  rw [e, intervalIntegral.integral_add hf hg, smul_add]\n\nlemma RectangleIntegral_add (f g : \u2102 \u2192 \u2102) (z w : \u2102)\n    (hf : RectangleBorderIntegrable f z w) (hg : RectangleBorderIntegrable g z w) :\n    RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n  rw [RectangleIntegral, RectangleIntegral, RectangleIntegral]\n  obtain \u27e8\u27e8hf1, hf2\u27e9, hf3, hf4\u27e9 := hf\n  obtain \u27e8\u27e8hg1, hg2\u27e9, hg3, hg4\u27e9 := hg\n  rw [HIntegral_add f g _ _ _ hf1 hg1, HIntegral_add f g _ _ _ hf2 hg2,\n    VIntegral_add f g _ _ _ hf3 hg3, VIntegral_add f g _ _ _ hf4 hg4]\n  ring\n\nlemma RectangleIntegral'_add (f g : \u2102 \u2192 \u2102) (z w : \u2102)\n    (hf : RectangleBorderIntegrable f z w) (hg : RectangleBorderIntegrable g z w) :\n    RectangleIntegral' (f + g) z w = RectangleIntegral' f z w + RectangleIntegral' g z w := by\n  rw [RectangleIntegral', RectangleIntegral_add f g z w hf hg, RectangleIntegral', RectangleIntegral',\n    smul_add]\n\n/-! ## 2. Parametrized side derivatives and integrability -/\n\nlemma hasDerivAt_add_const_I (y : \u211d) (x : \u211d) :\n    HasDerivAt (fun t => t + y * I) 1 x := by\n  have hid := hasDerivAt_id x\n  have hconst : HasDerivAt (fun _ => y * I) 0 x := hasDerivAt_const x _\n  convert hid.add hconst using 1\n  simp\n\nlemma hasDerivAt_const_add_I (x y : \u211d) :\n    HasDerivAt (fun t => x + t * I) I y := by\n  have h1 : HasDerivAt (fun t => t * I) I y := by\n    have := (hasDerivAt_id y).mul_const I\n    simpa using this\n  have hconst : HasDerivAt (fun _ => x) 0 y := hasDerivAt_const y x\n  convert hconst.add h1 using 1\n  simp\n\n/-- FTC on a horizontal side: \u222b_{x\u2081}^{x\u2082} (deriv G)(x + yI) dx\n= G(x\u2082 + yI) \u2212 G(x\u2081 + yI). -/\nlemma HIntegral_deriv {G : \u2102 \u2192 \u2102} {z w : \u2102} (y : \u211d)\n    (hG : \u2200 s \u2208 RectangleBorder z w, DifferentiableAt \u2102 G s)\n    (hGc : \u2200 s \u2208 RectangleBorder z w, ContinuousAt (deriv G) s)\n    (hmap : MapsTo (fun x : \u211d => \u2191x + y * I) [[z.re, w.re]] (RectangleBorder z w)) :\n    \u222b x in z.re..w.re, deriv G (x + y * I) = G (w.re + y * I) - G (z.re + y * I) := by\n  have hd : \u2200 x \u2208 Set.uIcc z.re w.re,\n      HasDerivAt (fun x => G (x + y * I)) (deriv G (x + y * I)) x := by\n    intro x hx\n    have h1 : HasDerivAt G (deriv G (x + y * I)) (x + y * I) :=\n      (hG _ (hmap (Set.uIcc_of_le (by simp [Set.uIcc]) hx |>.elim (fun h => h.1) |>.1))).hasDerivAt\n    have h2 : HasDerivAt (fun x => x + y * I) 1 x := hasDerivAt_add_const_I y x\n    have hcomp := h1.comp x h2\n    convert hcomp using 1\n    simp\n  have hint : IntervalIntegrable (fun x => deriv G (x + y * I)) volume z.re w.re := by\n    apply ContinuousOn.intervalIntegrable\n    intro x hx\n    have hmem : (x + y * I) \u2208 RectangleBorder z w := by\n      apply hmap\n      rcases Set.uIcc_of_le (by simp [Set.uIcc]) hx with h | h\n      \u00b7 exact h.1\n      \u00b7 exact h.1\n    exact (hGc _ hmem).continuousWithinAt\n  have hftc := intervalIntegral.integral_deriv_eq_sub hd hint\n  convert hftc using 1\n  \u00b7 simp\n  \u00b7 simp\n\n/-- FTC on a vertical side: \u222b_{y\u2081}^{y\u2082} (deriv G)(x + yI) dy\n= \u2212I\u00b7(G(x + y\u2082I) \u2212 G(x + y\u2081I)). -/\nlemma VIntegral_deriv {G : \u2102 \u2192 \u2102} {z w : \u2102} (x : \u211d)\n    (hG : \u2200 s \u2208 RectangleBorder z w, DifferentiableAt \u2102 G s)\n    (hGc : \u2200 s \u2208 RectangleBorder z w, ContinuousAt (deriv G) s)\n    (hmap : MapsTo (fun y : \u211d => \u2191x + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w)) :\n    \u222b y in z.im..w.im, deriv G (x + y * I) =\n      -I * (G (x + w.im * I) - G (x + z.im * I)) := by\n  have hd : \u2200 y \u2208 Set.uIcc z.im w.im,\n      HasDerivAt (fun y => G (x + y * I)) (I * deriv G (x + y * I)) y := by\n    intro y hy\n    have h1 : HasDerivAt G (deriv G (x + y * I)) (x + y * I) :=\n      (hG _ (hmap (Set.uIcc_of_le (by simp [Set.uIcc]) hy |>.elim (fun h => h.1) |>.1))).hasDerivAt\n    have h2 : HasDerivAt (fun y => x + y * I) I y := hasDerivAt_const_add_I x y\n    have hcomp := h1.comp y h2\n    convert hcomp using 1\n    ring\n  have hint : IntervalIntegrable (fun y => I * deriv G (x + y * I)) volume z.im w.im := by\n    apply ContinuousOn.intervalIntegrable\n    intro y hy\n    apply ContinuousWithinAt.mul continuousWithinAt_const\n    have hmem : (x + y * I) \u2208 RectangleBorder z w := by\n      apply hmap\n      rcases Set.uIcc_of_le (by simp [Set.uIcc]) hy with h | h\n      \u00b7 exact h.1\n      \u00b7 exact h.1\n    exact (hGc _ hmem).continuousWithinAt\n  have hftc := intervalIntegral.integral_deriv_eq_sub hd hint\n  rw [\u2190 hftc]\n  rw [integral_smul]\n  field_simp [Complex.I_ne_zero]\n\n/-- If `G : \u2102 \u2192 \u2102` is differentiable at every point of the rectangle\nborder, then the rectangle integral of `deriv G` vanishes (each side\ntelescopes by FTC). -/\nlemma rectInt_deriv_eq_zero {G : \u2102 \u2192 \u2102} {z w : \u2102}\n    (hG : \u2200 s \u2208 RectangleBorder z w, DifferentiableAt \u2102 G s)\n    (hGc : \u2200 s \u2208 RectangleBorder z w, ContinuousAt (deriv G) s) :\n    RectangleIntegral (fun s => deriv G s) z w = 0 := by\n  rw [RectangleIntegral]\n  rw [HIntegral_deriv z.im hG hGc (mapsTo_rectangleBorder_left_im z w),\n    HIntegral_deriv w.im hG hGc (mapsTo_rectangleBorder_right_im z w)]\n  have hVr := VIntegral_deriv w.re hG hGc (mapsTo_rectangleBorder_right_re z w)\n  have hVl := VIntegral_deriv z.re hG hGc (mapsTo_rectangleBorder_left_re z w)\n  rw [VIntegral, VIntegral, hVr, hVl]\n  simp [Complex.I_mul_I]\n  ring\n\nend NguyenRH1\n"
}

{
  "type": "update",
  "filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean",
  "content": "/-\nNguyenRH1/LiPrincipal.lean \u2014 tier ii\u2032 sub-stage ii\u2032-1b(ii) (D-064 \u00a74 R1):\nthe principal-part integral. For the explicit principal part\n  P(w) = \u03a3_{i=1}^{n+1} A_i/(w\u22121)^i,   A_i = F^{(n+1\u2212i)}(1)/(n+1\u2212i)!,\nwith F(w) = (w\u207f\u2212(w\u22121)\u207f)\u00b7hLi(w), on any rectangle with 1 in its interior:\n  RectangleIntegral' P z w = A_1 = ResW1 n.\n\nMethod: P = A_1/(w\u22121) + (\u03a3_{i\u22652} A_i/(w\u22121)^i). The first term is handled\nby the vendored `ResidueTheoremInRectangle`; the tail has the explicit\nglobal primitive G(w) = \u2212\u03a3_{i\u22652} A_i/((i\u22121)(w\u22121)^{i\u22121}), differentiable\non the border (1 is interior, hence off the border), so its rectangle\nintegral vanishes by per-side FTC (`rectInt_deriv_eq_zero`).\n\nArchive references: D-036, D-064 (R1), journal-016.\n-/\nimport Mathlib\nimport Zeta23.FromPNTPlus.ResidueCalcOnRectangles\nimport NguyenRH1.LiResidue\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. Linearity of rectangle integrals -/\n\nlemma HIntegral_add (f g : \u2102 \u2192 \u2102) (x\u2081 x\u2082 y : \u211d)\n    (hf : IntervalIntegrable (fun x => f (x + y * I)) volume x\u2081 x\u2082)\n    (hg : IntervalIntegrable (fun x => g (x + y * I)) volume x\u2081 x\u2082) :\n    HIntegral (f + g) x\u2081 x\u2082 y = HIntegral f x\u2081 x\u2082 y + HIntegral g x\u2081 x\u2082 y := by\n  rw [HIntegral, HIntegral, HIntegral]\n  have e : (fun x => (f + g) (x + y * I)) = (fun x => f (x + y * I) + g (x + y * I)) := by\n    funext x\n    simp [Pi.add_apply]\n  rw [e, intervalIntegral.integral_add hf hg]\n\nlemma VIntegral_add (f g : \u2102 \u2192 \u2102) (x y\u2081 y\u2082 : \u211d)\n    (hf : IntervalIntegrable (fun y => f (x + y * I)) volume y\u2081 y\u2082)\n    (hg : IntervalIntegrable (fun y => g (x + y * I)) volume y\u2081 y\u2082) :\n    VIntegral (f + g) x y\u2081 y\u2082 = VIntegral f x y\u2081 y\u2082 + VIntegral g x y\u2081 y\u2082 := by\n  rw [VIntegral, VIntegral, VIntegral]\n  have e : (fun y => (f + g) (x + y * I)) = (fun y => f (x + y * I) + g (x + y * I)) := by\n    funext y\n    simp [Pi.add_apply]\n  rw [e, intervalIntegral.integral_add hf hg, smul_add]\n\nlemma RectangleIntegral_add (f g : \u2102 \u2192 \u2102) (z w : \u2102)\n    (hf : RectangleBorderIntegrable f z w) (hg : RectangleBorderIntegrable g z w) :\n    RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n  rw [RectangleIntegral, RectangleIntegral, RectangleIntegral]\n  obtain \u27e8\u27e8hf1, hf2\u27e9, hf3, hf4\u27e9 := hf\n  obtain \u27e8\u27e8hg1, hg2\u27e9, hg3, hg4\u27e9 := hg\n  rw [HIntegral_add f g _ _ _ hf1 hg1, HIntegral_add f g _ _ _ hf2 hg2,\n    VIntegral_add f g _ _ _ hf3 hg3, VIntegral_add f g _ _ _ hf4 hg4]\n  ring\n\nlemma RectangleIntegral'_add (f g : \u2102 \u2192 \u2102) (z w : \u2102)\n    (hf : RectangleBorderIntegrable f z w) (hg : RectangleBorderIntegrable g z w) :\n    RectangleIntegral' (f + g) z w = RectangleIntegral' f z w + RectangleIntegral' g z w := by\n  rw [RectangleIntegral', RectangleIntegral_add f g z w hf hg, RectangleIntegral', RectangleIntegral',\n    smul_add]\n\n/-! ## 2. Parametrized side derivatives and integrability -/\n\nlemma hasDerivAt_add_const_I (y : \u211d) (x : \u211d) :\n    HasDerivAt (fun t => t + y * I) 1 x := by\n  have hid := hasDerivAt_id x\n  have hconst : HasDerivAt (fun _ => y * I) 0 x := hasDerivAt_const x _\n  convert hid.add hconst using 1\n  simp\n\nlemma hasDerivAt_const_add_I (x y : \u211d) :\n    HasDerivAt (fun t => x + t * I) I y := by\n  have h1 : HasDerivAt (fun t => t * I) I y := by\n    have := (hasDerivAt_id y).mul_const I\n    simpa using this\n  have hconst : HasDerivAt (fun _ => x) 0 y := hasDerivAt_const y x\n  convert hconst.add h1 using 1\n  simp\n\n/-- FTC on a horizontal side: \u222b_{x\u2081}^{x\u2082} (deriv G)(x + yI) dx\n= G(x\u2082 + yI) \u2212 G(x\u2081 + yI). -/\nlemma HIntegral_deriv {G : \u2102 \u2192 \u2102} {z w : \u2102} (y : \u211d)\n    (hG : \u2200 s \u2208 RectangleBorder z w, DifferentiableAt \u2102 G s)\n    (hGc : \u2200 s \u2208 RectangleBorder z w, ContinuousAt (deriv G) s)\n    (hmap : MapsTo (fun x : \u211d => \u2191x + y * I) [[z.re, w.re]] (RectangleBorder z w)) :\n    \u222b x in z.re..w.re, deriv G (x + y * I) = G (w.re + y * I) - G (z.re + y * I) := by\n  have hd : \u2200 x \u2208 Set.uIcc z.re w.re,\n      HasDerivAt (fun x => G (x + y * I)) (deriv G (x + y * I)) x := by\n    intro x hx\n    have h1 : HasDerivAt G (deriv G (x + y * I)) (x + y * I) :=\n      (hG _ (hmap (Set.uIcc_of_le (by simp [Set.uIcc]) hx |>.elim (fun h => h.1) |>.1))).hasDerivAt\n    have h2 : HasDerivAt (fun x => x + y * I) 1 x := hasDerivAt_add_const_I y x\n    have hcomp := h1.comp x h2\n    convert hcomp using 1\n    simp\n  have hint : IntervalIntegrable (fun x => deriv G (x + y * I)) volume z.re w.re := by\n    apply ContinuousOn.intervalIntegrable\n    intro x hx\n    have hmem : (x + y * I) \u2208 RectangleBorder z w := by\n      apply hmap\n      rcases Set.uIcc_of_le (by simp [Set.uIcc]) hx with h | h\n      \u00b7 exact h.1\n      \u00b7 exact h.1\n    exact (hGc _ hmem).continuousWithinAt\n  have hftc := intervalIntegral.integral_deriv_eq_sub hd hint\n  convert hftc using 1\n  \u00b7 simp\n  \u00b7 simp\n\n/-- FTC on a vertical side: \u222b_{y\u2081}^{y\u2082} (deriv G)(x + yI) dy\n= \u2212I\u00b7(G(x + y\u2082I) \u2212 G(x + y\u2081I)). -/\nlemma VIntegral_deriv {G : \u2102 \u2192 \u2102} {z w : \u2102} (x : \u211d)\n    (hG : \u2200 s \u2208 RectangleBorder z w, DifferentiableAt \u2102 G s)\n    (hGc : \u2200 s \u2208 RectangleBorder z w, ContinuousAt (deriv G) s)\n    (hmap : MapsTo (fun y : \u211d => \u2191x + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w)) :\n    \u222b y in z.im..w.im, deriv G (x + y * I) =\n      -I * (G (x + w.im * I) - G (x + z.im * I)) := by\n  have hd : \u2200 y \u2208 Set.uIcc z.im w.im,\n      HasDerivAt (fun y => G (x + y * I)) (I * deriv G (x + y * I)) y := by\n    intro y hy\n    have h1 : HasDerivAt G (deriv G (x + y * I)) (x + y * I) :=\n      (hG _ (hmap (Set.uIcc_of_le (by simp [Set.uIcc]) hy |>.elim (fun h => h.1) |>.1))).hasDerivAt\n    have h2 : HasDerivAt (fun y => x + y * I) I y := hasDerivAt_const_add_I x y\n    have hcomp := h1.comp y h2\n    convert hcomp using 1\n    ring\n  have hint : IntervalIntegrable (fun y => I * deriv G (x + y * I)) volume z.im w.im := by\n    apply ContinuousOn.intervalIntegrable\n    intro y hy\n    apply ContinuousWithinAt.mul continuousWithinAt_const\n    have hmem : (x + y * I) \u2208 RectangleBorder z w := by\n      apply hmap\n      rcases Set.uIcc_of_le (by simp [Set.uIcc]) hy with h | h\n      \u00b7 exact h.1\n      \u00b7 exact h.1\n    exact (hGc _ hmem).continuousWithinAt\n  have hftc := intervalIntegral.integral_deriv_eq_sub hd hint\n  rw [\u2190 hftc]\n  rw [integral_smul]\n  field_simp [Complex.I_ne_zero]\n\n/-- If `G : \u2102 \u2192 \u2102` is differentiable at every point of the rectangle\nborder, then the rectangle integral of `deriv G` vanishes (each side\ntelescopes by FTC). -/\nlemma rectInt_deriv_eq_zero {G : \u2102 \u2192 \u2102} {z w : \u2102}\n    (hG : \u2200 s \u2208 RectangleBorder z w, DifferentiableAt \u2102 G s)\n    (hGc : \u2200 s \u2208 RectangleBorder z w, ContinuousAt (deriv G) s) :\n    RectangleIntegral (fun s => deriv G s) z w = 0 := by\n  rw [RectangleIntegral]\n  rw [HIntegral_deriv z.im hG hGc (mapsTo_rectangleBorder_left_im z w),\n    HIntegral_deriv w.im hG hGc (mapsTo_rectangleBorder_right_im z w)]\n  have hVr := VIntegral_deriv w.re hG hGc (mapsTo_rectangleBorder_right_re z w)\n  have hVl := VIntegral_deriv z.re hG hGc (mapsTo_rectangleBorder_left_re z w)\n  rw [VIntegral, VIntegral, hVr, hVl]\n  simp [Complex.I_mul_I]\n  ring\n\nend NguyenRH1\n",
  "structuredPatch": [
    {
      "oldStart": 61,
      "oldLines": 7,
      "newStart": 61,
      "newLines": 85,
      "lines": [
        "   rw [RectangleIntegral', RectangleIntegral_add f g z w hf hg, RectangleIntegral', RectangleIntegral',",
        "     smul_add]",
        " ",
        "-/-! ## 2. The per-side FTC vanishing lemma -/",
        "+/-! ## 2. Parametrized side derivatives and integrability -/",
        "+",
        "+lemma hasDerivAt_add_const_I (y : \u211d) (x : \u211d) :",
        "+    HasDerivAt (fun t => t + y * I) 1 x := by",
        "+  have hid := hasDerivAt_id x",
        "+  have hconst : HasDerivAt (fun _ => y * I) 0 x := hasDerivAt_const x _",
        "+  convert hid.add hconst using 1",
        "+  simp",
        "+",
        "+lemma hasDerivAt_const_add_I (x y : \u211d) :",
        "+    HasDerivAt (fun t => x + t * I) I y := by",
        "+  have h1 : HasDerivAt (fun t => t * I) I y := by",
        "+    have := (hasDerivAt_id y).mul_const I",
        "+    simpa using this",
        "+  have hconst : HasDerivAt (fun _ => x) 0 y := hasDerivAt_const y x",
        "+  convert hconst.add h1 using 1",
        "+  simp",
        "+",
        "+/-- FTC on a horizontal side: \u222b_{x\u2081}^{x\u2082} (deriv G)(x + yI) dx",
        "+= G(x\u2082 + yI) \u2212 G(x\u2081 + yI). -/",
        "+lemma HIntegral_deriv {G : \u2102 \u2192 \u2102} {z w : \u2102} (y : \u211d)",
        "+    (hG : \u2200 s \u2208 RectangleBorder z w, DifferentiableAt \u2102 G s)",
        "+    (hGc : \u2200 s \u2208 RectangleBorder z w, ContinuousAt (deriv G) s)",
        "+    (hmap : MapsTo (fun x : \u211d => \u2191x + y * I) [[z.re, w.re]] (RectangleBorder z w)) :",
        "+    \u222b x in z.re..w.re, deriv G (x + y * I) = G (w.re + y * I) - G (z.re + y * I) := by",
        "+  have hd : \u2200 x \u2208 Set.uIcc z.re w.re,",
        "+      HasDerivAt (fun x => G (x + y * I)) (deriv G (x + y * I)) x := by",
        "+    intro x hx",
        "+    have h1 : HasDerivAt G (deriv G (x + y * I)) (x + y * I) :=",
        "+      (hG _ (hmap (Set.uIcc_of_le (by simp [Set.uIcc]) hx |>.elim (fun h => h.1) |>.1))).hasDerivAt",
        "+    have h2 : HasDerivAt (fun x => x + y * I) 1 x := hasDerivAt_add_const_I y x",
        "+    have hcomp := h1.comp x h2",
        "+    convert hcomp using 1",
        "+    simp",
        "+  have hint : IntervalIntegrable (fun x => deriv G (x + y * I)) volume z.re w.re := by",
        "+    apply ContinuousOn.intervalIntegrable",
        "+    intro x hx",
        "+    have hmem : (x + y * I) \u2208 RectangleBorder z w := by",
        "+      apply hmap",
        "+      rcases Set.uIcc_of_le (by simp [Set.uIcc]) hx with h | h",
        "+      \u00b7 exact h.1",
        "+      \u00b7 exact h.1",
        "+    exact (hGc _ hmem).continuousWithinAt",
        "+  have hftc := intervalIntegral.integral_deriv_eq_sub hd hint",
        "+  convert hftc using 1",
        "+  \u00b7 simp",
        "+  \u00b7 simp",
        "+",
        "+/-- FTC on a vertical side: \u222b_{y\u2081}^{y\u2082} (deriv G)(x + yI) dy",
        "+= \u2212I\u00b7(G(x + y\u2082I) \u2212 G(x + y\u2081I)). -/",
        "+lemma VIntegral_deriv {G : \u2102 \u2192 \u2102} {z w : \u2102} (x : \u211d)",
        "+    (hG : \u2200 s \u2208 RectangleBorder z w, DifferentiableAt \u2102 G s)",
        "+    (hGc : \u2200 s \u2208 RectangleBorder z w, ContinuousAt (deriv G) s)",
        "+    (hmap : MapsTo (fun y : \u211d => \u2191x + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w)) :",
        "+    \u222b y in z.im..w.im, deriv G (x + y * I) =",
        "+      -I * (G (x + w.im * I) - G (x + z.im * I)) := by",
        "+  have hd : \u2200 y \u2208 Set.uIcc z.im w.im,",
        "+      HasDerivAt (fun y => G (x + y * I)) (I * deriv G (x + y * I)) y := by",
        "+    intro y hy",
        "+    have h1 : HasDerivAt G (deriv G (x + y * I)) (x + y * I) :=",
        "+      (hG _ (hmap (Set.uIcc_of_le (by simp [Set.uIcc]) hy |>.elim (fun h => h.1) |>.1))).hasDerivAt",
        "+    have h2 : HasDerivAt (fun y => x + y * I) I y := hasDerivAt_const_add_I x y",
        "+    have hcomp := h1.comp y h2",
        "+    convert hcomp using 1",
        "+    ring",
        "+  have hint : IntervalIntegrable (fun y => I * deriv G (x + y * I)) volume z.im w.im := by",
        "+    apply ContinuousOn.intervalIntegrable",
        "+    intro y hy",
        "+    apply ContinuousWithinAt.mul continuousWithinAt_const",
        "+    have hmem : (x + y * I) \u2208 RectangleBorder z w := by",
        "+      apply hmap",
        "+      rcases Set.uIcc_of_le (by simp [Set.uIcc]) hy with h | h",
        "+      \u00b7 exact h.1",
        "+      \u00b7 exact h.1",
        "+    exact (hGc _ hmem).continuousWithinAt",
        "+  have hftc := intervalIntegral.integral_deriv_eq_sub hd hint",
        "+  rw [\u2190 hftc]",
        "+  rw [integral_smul]",
        "+  field_simp [Complex.I_ne_zero]",
        " ",
        " /-- If `G : \u2102 \u2192 \u2102` is differentiable at every point of the rectangle",
        " border, then the rectangle integral of `deriv G` vanishes (each side"
      ]
    },
    {
      "oldStart": 71,
      "oldLines": 31,
      "newStart": 149,
      "newLines": 13,
      "lines": [
        "     (hGc : \u2200 s \u2208 RectangleBorder z w, ContinuousAt (deriv G) s) :",
        "     RectangleIntegral (fun s => deriv G s) z w = 0 := by",
        "   rw [RectangleIntegral]",
        "-  have hsideH : \u2200 y : \u211d, (\u2200 x : \u211d, x \u2208 Set.uIcc z.re w.re \u2192 (x + y * I) \u2208 RectangleBorder z w) \u2192",
        "-      \u222b x in z.re..w.re, deriv G (x + y * I) = G (w.re + y * I) - G (z.re + y * I) := by",
        "-    intro y hy",
        "-    have hd : \u2200 x \u2208 Set.uIcc z.re w.re,",
        "-        HasDerivAt (fun x => G (x + y * I)) (deriv G (x + y * I)) x := by",
        "-      intro x hx",
        "-      have h1 : HasDerivAt G (deriv G (x + y * I)) (x + y * I) :=",
        "-        (hG _ (hy x hx)).hasDerivAt",
        "-      have h2 : HasDerivAt (fun x => x + y * I) 1 x := by",
        "-        have hid := hasDerivAt_id x",
        "-        have hconst : HasDerivAt (fun x => y * I) 0 x := hasDerivAt_const x _",
        "-        convert hid.add hconst using 1",
        "-        simp",
        "-      have := h1.comp x h2",
        "-      convert this using 1",
        "-      simp",
        "-    have hint : IntervalIntegrable (fun x => deriv G (x + y * I)) volume z.re w.re := by",
        "-      apply ContinuousOn.intervalIntegrable_of_Icc (a := z.re) (b := w.re)",
        "-      intro x hx",
        "-      have hmem : (x + y * I) \u2208 RectangleBorder z w := hy x (Set.uIcc_of_le ?_ hx |>.elim (fun h => h.1) )",
        "-      \u00b7 exact (hGc _ hmem).continuousWithinAt",
        "-      \u00b7 sorry",
        "-    have := intervalIntegral.integral_deriv_eq_sub hd hint",
        "-    convert this using 1",
        "-  sorry",
        "+  rw [HIntegral_deriv z.im hG hGc (mapsTo_rectangleBorder_left_im z w),",
        "+    HIntegral_deriv w.im hG hGc (mapsTo_rectangleBorder_right_im z w)]",
        "+  have hVr := VIntegral_deriv w.re hG hGc (mapsTo_rectangleBorder_right_re z w)",
        "+  have hVl := VIntegral_deriv z.re hG hGc (mapsTo_rectangleBorder_left_re z w)",
        "+  rw [VIntegral, VIntegral, hVr, hVl]",
        "+  simp [Complex.I_mul_I]",
        "+  ring",
        " ",
        " end NguyenRH1",
        " "
      ]
    }
  ],
  "originalFile": "/-\nNguyenRH1/LiPrincipal.lean \u2014 tier ii\u2032 sub-stage ii\u2032-1b(ii) (D-064 \u00a74 R1):\nthe principal-part integral. For the explicit principal part\n  P(w) = \u03a3_{i=1}^{n+1} A_i/(w\u22121)^i,   A_i = F^{(n+1\u2212i)}(1)/(n+1\u2212i)!,\nwith F(w) = (w\u207f\u2212(w\u22121)\u207f)\u00b7hLi(w), on any rectangle with 1 in its interior:\n  RectangleIntegral' P z w = A_1 = ResW1 n.\n\nMethod: P = A_1/(w\u22121) + (\u03a3_{i\u22652} A_i/(w\u22121)^i). The first term is handled\nby the vendored `ResidueTheoremInRectangle`; the tail has the explicit\nglobal primitive G(w) = \u2212\u03a3_{i\u22652} A_i/((i\u22121)(w\u22121)^{i\u22121}), differentiable\non the border (1 is interior, hence off the border), so its rectangle\nintegral vanishes by per-side FTC (`rectInt_deriv_eq_zero`).\n\nArchive references: D-036, D-064 (R1), journal-016.\n-/\nimport Mathlib\nimport Zeta23.FromPNTPlus.ResidueCalcOnRectangles\nimport NguyenRH1.LiResidue\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. Linearity of rectangle integrals -/\n\nlemma HIntegral_add (f g : \u2102 \u2192 \u2102) (x\u2081 x\u2082 y : \u211d)\n    (hf : IntervalIntegrable (fun x => f (x + y * I)) volume x\u2081 x\u2082)\n    (hg : IntervalIntegrable (fun x => g (x + y * I)) volume x\u2081 x\u2082) :\n    HIntegral (f + g) x\u2081 x\u2082 y = HIntegral f x\u2081 x\u2082 y + HIntegral g x\u2081 x\u2082 y := by\n  rw [HIntegral, HIntegral, HIntegral]\n  have e : (fun x => (f + g) (x + y * I)) = (fun x => f (x + y * I) + g (x + y * I)) := by\n    funext x\n    simp [Pi.add_apply]\n  rw [e, intervalIntegral.integral_add hf hg]\n\nlemma VIntegral_add (f g : \u2102 \u2192 \u2102) (x y\u2081 y\u2082 : \u211d)\n    (hf : IntervalIntegrable (fun y => f (x + y * I)) volume y\u2081 y\u2082)\n    (hg : IntervalIntegrable (fun y => g (x + y * I)) volume y\u2081 y\u2082) :\n    VIntegral (f + g) x y\u2081 y\u2082 = VIntegral f x y\u2081 y\u2082 + VIntegral g x y\u2081 y\u2082 := by\n  rw [VIntegral, VIntegral, VIntegral]\n  have e : (fun y => (f + g) (x + y * I)) = (fun y => f (x + y * I) + g (x + y * I)) := by\n    funext y\n    simp [Pi.add_apply]\n  rw [e, intervalIntegral.integral_add hf hg, smul_add]\n\nlemma RectangleIntegral_add (f g : \u2102 \u2192 \u2102) (z w : \u2102)\n    (hf : RectangleBorderIntegrable f z w) (hg : RectangleBorderIntegrable g z w) :\n    RectangleIntegral (f + g) z w = RectangleIntegral f z w + RectangleIntegral g z w := by\n  rw [RectangleIntegral, RectangleIntegral, RectangleIntegral]\n  obtain \u27e8\u27e8hf1, hf2\u27e9, hf3, hf4\u27e9 := hf\n  obtain \u27e8\u27e8hg1, hg2\u27e9, hg3, hg4\u27e9 := hg\n  rw [HIntegral_add f g _ _ _ hf1 hg1, HIntegral_add f g _ _ _ hf2 hg2,\n    VIntegral_add f g _ _ _ hf3 hg3, VIntegral_add f g _ _ _ hf4 hg4]\n  ring\n\nlemma RectangleIntegral'_add (f g : \u2102 \u2192 \u2102) (z w : \u2102)\n    (hf : RectangleBorderIntegrable f z w) (hg : RectangleBorderIntegrable g z w) :\n    RectangleIntegral' (f + g) z w = RectangleIntegral' f z w + RectangleIntegral' g z w := by\n  rw [RectangleIntegral', RectangleIntegral_add f g z w hf hg, RectangleIntegral', RectangleIntegral',\n    smul_add]\n\n/-! ## 2. The per-side FTC vanishing lemma -/\n\n/-- If `G : \u2102 \u2192 \u2102` is differentiable at every point of the rectangle\nborder, then the rectangle integral of `deriv G` vanishes (each side\ntelescopes by FTC). -/\nlemma rectInt_deriv_eq_zero {G : \u2102 \u2192 \u2102} {z w : \u2102}\n    (hG : \u2200 s \u2208 RectangleBorder z w, DifferentiableAt \u2102 G s)\n    (hGc : \u2200 s \u2208 RectangleBorder z w, ContinuousAt (deriv G) s) :\n    RectangleIntegral (fun s => deriv G s) z w = 0 := by\n  rw [RectangleIntegral]\n  have hsideH : \u2200 y : \u211d, (\u2200 x : \u211d, x \u2208 Set.uIcc z.re w.re \u2192 (x + y * I) \u2208 RectangleBorder z w) \u2192\n      \u222b x in z.re..w.re, deriv G (x + y * I) = G (w.re + y * I) - G (z.re + y * I) := by\n    intro y hy\n    have hd : \u2200 x \u2208 Set.uIcc z.re w.re,\n        HasDerivAt (fun x => G (x + y * I)) (deriv G (x + y * I)) x := by\n      intro x hx\n      have h1 : HasDerivAt G (deriv G (x + y * I)) (x + y * I) :=\n        (hG _ (hy x hx)).hasDerivAt\n      have h2 : HasDerivAt (fun x => x + y * I) 1 x := by\n        have hid := hasDerivAt_id x\n        have hconst : HasDerivAt (fun x => y * I) 0 x := hasDerivAt_const x _\n        convert hid.add hconst using 1\n        simp\n      have := h1.comp x h2\n      convert this using 1\n      simp\n    have hint : IntervalIntegrable (fun x => deriv G (x + y * I)) volume z.re w.re := by\n      apply ContinuousOn.intervalIntegrable_of_Icc (a := z.re) (b := w.re)\n      intro x hx\n      have hmem : (x + y * I) \u2208 RectangleBorder z w := hy x (Set.uIcc_of_le ?_ hx |>.elim (fun h => h.1) )\n      \u00b7 exact (hGc _ hmem).continuousWithinAt\n      \u00b7 sorry\n    have := intervalIntegral.integral_deriv_eq_sub hd hint\n    convert this using 1\n  sorry\n\nend NguyenRH1\n",
  "autoRead": false
}

{
  "command": "grep -n \"def RectangleBorderIntegrable\" -A 8 /d/Python/Quantum/zeta-23-lean/Zeta23/FromPNTPlus/ResidueCalcOnRectangles.lean | head -12",
  "description": "Read RectangleBorderIntegrable definition"
}

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean",
  "content": "/-\nNguyenRH1/LiPrincipal.lean \u2014 tier ii\u2032 sub-stage ii\u2032-1b(ii) (D-064 \u00a74 R1):\nthe principal-part integral. For the explicit principal part\n  P(w) = \u03a3_{i=1}^{n+1} A_i/(w\u22121)^i,   A_i = F^{(n+1\u2212i)}(1)/(n+1\u2212i)!,\nwith F(w) = (w\u207f\u2212(w\u22121)\u207f)\u00b7hLi(w), on any rectangle with 1 in its interior:\n  RectangleIntegral' P z w = A_1 = ResW1 n.\n\nMethod: P = A_1/(w\u22121) + (\u03a3_{i\u22652} A_i/(w\u22121)^i). The first term is handled\nby the vendored `ResidueTheoremInRectangle`; the tail has the explicit\nglobal primitive G(w) = \u2212\u03a3_{i\u22652} A_i/((i\u22121)(w\u22121)^{i\u22121}), differentiable\non the border (1 is interior, hence off the border), so its rectangle\nintegral vanishes by per-side FTC (`rectInt_deriv_eq_zero`).\n\nArchive references: D-036, D-064 (R1), journal-016.\n-/\nimport Mathlib\nimport Zeta23.FromPNTPlus.ResidueCalcOnRectangles\nimport NguyenRH1.LiResidue\n\nopen Zeta23 Complex Filter Topology\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. Additivity of the primed rectangle integral -/\n\nlemma RectangleIntegral'_add (f g : \u2102 \u2192 \u2102) (z w : \u2102)\n    (hf : RectangleBorderIntegrable f z w) (hg : RectangleBorderIntegrable g z w) :\n    RectangleIntegral' (f + g) z w = RectangleIntegral' f z w + RectangleIntegral' g z w := by\n  rw [RectangleIntegral', RectangleBorderIntegrable.add hf hg, RectangleIntegral', RectangleIntegral',\n    smul_add]\n\n/-! ## 2. Parametrized side derivatives and integrability -/\n\nlemma hasDerivAt_add_const_I (y : \u211d) (x : \u211d) :\n    HasDerivAt (fun t => t + y * I) 1 x := by\n  have hid := hasDerivAt_id x\n  have hconst : HasDerivAt (fun _ => y * I) 0 x := hasDerivAt_const x _\n  convert hid.add hconst using 1\n  simp\n\nlemma hasDerivAt_const_add_I (x y : \u211d) :\n    HasDerivAt (fun t => x + t * I) I y := by\n  have h1 : HasDerivAt (fun t => t * I) I y := by\n    have := (hasDerivAt_id y).mul_const I\n    simpa using this\n  have hconst : HasDerivAt (fun _ => x) 0 y := hasDerivAt_const y x\n  convert hconst.add h1 using 1\n  simp\n\n/-- FTC on a horizontal side: \u222b_{x\u2081}^{x\u2082} (deriv G)(x + yI) dx\n= G(x\u2082 + yI) \u2212 G(x\u2081 + yI). -/\nlemma HIntegral_deriv {G : \u2102 \u2192 \u2102} {z w : \u2102} (y : \u211d)\n    (hG : \u2200 s \u2208 RectangleBorder z w, DifferentiableAt \u2102 G s)\n    (hGc : \u2200 s \u2208 RectangleBorder z w, ContinuousAt (deriv G) s)\n    (hmap : MapsTo (fun x : \u211d => \u2191x + y * I) [[z.re, w.re]] (RectangleBorder z w)) :\n    \u222b x in z.re..w.re, deriv G (x + y * I) = G (w.re + y * I) - G (z.re + y * I) := by\n  have hd : \u2200 x \u2208 Set.uIcc z.re w.re,\n      HasDerivAt (fun x => G (x + y * I)) (deriv G (x + y * I)) x := by\n    intro x hx\n    have h1 : HasDerivAt G (deriv G (x + y * I)) (x + y * I) :=\n      (hG _ (hmap hx)).hasDerivAt\n    have h2 : HasDerivAt (fun x => x + y * I) 1 x := hasDerivAt_add_const_I y x\n    have hcomp := h1.comp x h2\n    convert hcomp using 1\n    simp\n  have hint : IntervalIntegrable (fun x => deriv G (x + y * I)) volume z.re w.re := by\n    apply ContinuousOn.intervalIntegrable\n    intro x hx\n    exact (hGc _ (hmap hx)).continuousWithinAt\n  have hftc := intervalIntegral.integral_deriv_eq_sub hd hint\n  convert hftc using 1\n  \u00b7 simp\n  \u00b7 simp\n\n/-- FTC on a vertical side: \u222b_{y\u2081}^{y\u2082} (deriv G)(x + yI) dy\n= \u2212I\u00b7(G(x + y\u2082I) \u2212 G(x + y\u2081I)). -/\nlemma VIntegral_deriv {G : \u2102 \u2192 \u2102} {z w : \u2102} (x : \u211d)\n    (hG : \u2200 s \u2208 RectangleBorder z w, DifferentiableAt \u2102 G s)\n    (hGc : \u2200 s \u2208 RectangleBorder z w, ContinuousAt (deriv G) s)\n    (hmap : MapsTo (fun y : \u211d => \u2191x + \u2191y * I) [[z.im, w.im]] (RectangleBorder z w)) :\n    \u222b y in z.im..w.im, deriv G (x + y * I) =\n      -I * (G (x + w.im * I) - G (x + z.im * I)) := by\n  have hd : \u2200 y \u2208 Set.uIcc z.im w.im,\n      HasDerivAt (fun y => G (x + y * I)) (I * deriv G (x + y * I)) y := by\n    intro y hy\n    have h1 : HasDerivAt G (deriv G (x + y * I)) (x + y * I) :=\n      (hG _ (hmap hy)).hasDerivAt\n    have h2 : HasDerivAt (fun y => x + y * I) I y := hasDerivAt_const_add_I x y\n    have hcomp := h1.comp y h2\n    convert hcomp using 1\n    ring\n  have hint : IntervalIntegrable (fun y => I * deriv G (x + y * I)) volume z.im w.im := by\n    apply ContinuousOn.intervalIntegrable\n    intro y hy\n    exact ContinuousWithinAt.mul continuousWithinAt_const\n      ((hGc _ (hmap hy)).continuousWithinAt)\n  have hftc := intervalIntegral.integral_deriv_eq_sub hd hint\n  rw [\u2190 hftc, integral_smul]\n  field_simp [Complex.I_ne_zero]\n\n/-- If `G : \u2102 \u2192 \u2102` is differentiable at every point of the rectangle\nborder, then the rectangle integral of `deriv G` vanishes (each side\ntelescopes by FTC). -/\nlemma rectInt_deriv_eq_zero {G : \u2102 \u2192 \u2102} {z w : \u2102}\n    (hG : \u2200 s \u2208 RectangleBorder z w, DifferentiableAt \u2102 G s)\n    (hGc : \u2200 s \u2208 RectangleBorder z w, ContinuousAt (deriv G) s) :\n    RectangleIntegral (fun s => deriv G s) z w = 0 := by\n  rw [RectangleIntegral]\n  rw [HIntegral_deriv z.im hG hGc (mapsTo_rectangleBorder_left_im z w),\n    HIntegral_deriv w.im hG hGc (mapsTo_rectangleBorder_right_im z w)]\n  have hVr := VIntegral_deriv w.re hG hGc (mapsTo_rectangleBorder_right_re z w)\n  have hVl := VIntegral_deriv z.re hG hGc (mapsTo_rectangleBorder_left_re z w)\n  rw [VIntegral, VIntegral, hVr, hVl]\n  simp [Complex.I_mul_I]\n  ring\n\nend NguyenRH1\n"
}

{
  "type": "update",
  "filePath": "D:\\Python\\quantum-deploy\\docroot\\app.cgi",
  "content": "#!/opt/alt/python311/bin/python3.11\n\"\"\"WSGI->CGI adapter \u2014 runs the RH-WEBSITE Django app on Apache shared hosting.\n\nApache rewrites every non-file request to /app.cgi/<path> (see .htaccess).\nwsgiref's CGIHandler bridges the CGI environment onto Django's WSGI app.\n\"\"\"\nimport os\nimport sys\n\nDOCROOT = os.path.dirname(os.path.abspath(__file__))\nAPP = os.path.join(DOCROOT, \"app\")\n\n# Vendored pure-Python deps (Django 5.2, markdown) + the project package.\nsys.path.insert(0, os.path.join(APP, \"vendor\"))\nsys.path.insert(0, os.path.join(APP, \"web\"))\n\nos.environ.setdefault(\"DJANGO_SETTINGS_MODULE\", \"rhweb.settings\")\n# Apache mounts us at /app.cgi/<path>; present clean root URLs to Django.\nos.environ[\"SCRIPT_NAME\"] = \"\"\n\nfrom wsgiref.handlers import CGIHandler\n\nfrom rhweb.wsgi import application\n\n\ndef _script_url_fix(environ, start_response):\n    \"\"\"PEP 3333 fix for non-ASCII URLs (e.g. resources files with Greek\n    letters/dashes in their names).\n\n    wsgiref's read_environ() already latin-1-mangles PATH_INFO (and any\n    REDIRECT_* var that is itself transcodable) so Django can re-decode\n    it \u2014 do NOT touch PATH_INFO here (a second mangle breaks routing,\n    404s). But SCRIPT_URL and REDIRECT_URL pass through read_environ\n    UNCHANGED as proper unicode, and Django's get_script_name() then\n    crashes encoding them as iso-8859-1 (UnicodeEncodeError -> 500).\n    Mangle exactly those two.\"\"\"\n    for key in (\"SCRIPT_URL\", \"REDIRECT_URL\"):\n        v = environ.get(key)\n        if v:\n            environ[key] = v.encode(\"utf-8\", \"surrogateescape\") \\\n                             .decode(\"latin-1\")\n    return application(environ, start_response)\n\n\nCGIHandler().run(_script_url_fix)\n",
  "structuredPatch": [
    {
      "oldStart": 9,
      "oldLines": 17,
      "newStart": 9,
      "newLines": 6,
      "lines": [
        " ",
        " DOCROOT = os.path.dirname(os.path.abspath(__file__))",
        " APP = os.path.join(DOCROOT, \"app\")",
        "-",
        "-",
        "-def _dbg(msg):",
        "-    if \"resources\" in (os.environ.get(\"PATH_INFO\") or \"\"):",
        "-        with open(os.path.join(APP, \"debug-environ.txt\"), \"a\",",
        "-                  encoding=\"utf-8\") as fh:",
        "-            fh.write(msg + \"\\n\")",
        "-",
        "-",
        "-_dbg(\"TOP fsenc=\" + sys.getfilesystemencoding()",
        "-     + \" PATH_INFO=\" + repr(os.environ.get(\"PATH_INFO\")))",
        " ",
        " # Vendored pure-Python deps (Django 5.2, markdown) + the project package.",
        " sys.path.insert(0, os.path.join(APP, \"vendor\"))"
      ]
    },
    {
      "oldStart": 33,
      "oldLines": 56,
      "newStart": 22,
      "newLines": 25,
      "lines": [
        " ",
        " from rhweb.wsgi import application",
        " ",
        "-_dbg(\"AFTER-IMPORT PATH_INFO=\" + repr(os.environ.get(\"PATH_INFO\")))",
        " ",
        "-from urllib.parse import unquote_to_bytes",
        "+def _script_url_fix(environ, start_response):",
        "+    \"\"\"PEP 3333 fix for non-ASCII URLs (e.g. resources files with Greek",
        "+    letters/dashes in their names).",
        " ",
        "-",
        "-def _pep3333_paths(environ, start_response):",
        "-    \"\"\"Normalize Apache's path vars to the PEP 3333 form Django needs.",
        "-",
        "-    Apache hands these vars either percent-encoded (mod_rewrite path)",
        "-    or UTF-8-decoded (direct CGI path). Django (a) crashes with",
        "-    UnicodeEncodeError on any non-latin-1 char (500) and (b) needs the",
        "-    decoded path for URL resolution (else the resolver sees %XX and 404s).",
        "-    So: bytes -> unquote -> re-mangle as latin-1. SCRIPT_URL/REDIRECT_URL",
        "-    (set by mod_rewrite) are consulted by get_script_name BEFORE",
        "-    PATH_INFO \u2014 all three must be normalized.\"\"\"",
        "-    if \"resources\" in (environ.get(\"PATH_INFO\") or \"\"):",
        "-        with open(os.path.join(APP, \"debug-environ.txt\"), \"a\",",
        "-                  encoding=\"utf-8\") as fh:",
        "-            fh.write(\"ENTRY PATH_INFO=\" + repr(environ.get(\"PATH_INFO\"))",
        "-                     + \" SCRIPT_URL=\" + repr(environ.get(\"SCRIPT_URL\")) + \"\\n\")",
        "-    for key in (\"PATH_INFO\", \"SCRIPT_URL\", \"REDIRECT_URL\"):",
        "+    wsgiref's read_environ() already latin-1-mangles PATH_INFO (and any",
        "+    REDIRECT_* var that is itself transcodable) so Django can re-decode",
        "+    it \u2014 do NOT touch PATH_INFO here (a second mangle breaks routing,",
        "+    404s). But SCRIPT_URL and REDIRECT_URL pass through read_environ",
        "+    UNCHANGED as proper unicode, and Django's get_script_name() then",
        "+    crashes encoding them as iso-8859-1 (UnicodeEncodeError -> 500).",
        "+    Mangle exactly those two.\"\"\"",
        "+    for key in (\"SCRIPT_URL\", \"REDIRECT_URL\"):",
        "         v = environ.get(key)",
        "         if v:",
        "-            raw = unquote_to_bytes(v)  # %XX-decodes; utf-8-encodes str first",
        "-            environ[key] = raw.decode(\"latin-1\")",
        "-    if \"resources\" in (environ.get(\"PATH_INFO\") or \"\"):",
        "-        with open(os.path.join(APP, \"debug-environ.txt\"), \"a\",",
        "-                  encoding=\"utf-8\") as fh:",
        "-            fh.write(repr({k: environ.get(k) for k in",
        "-                           (\"PATH_INFO\", \"SCRIPT_NAME\", \"SCRIPT_URL\",",
        "-                            \"REDIRECT_URL\", \"REQUEST_METHOD\", \"HTTP_HOST\")})",
        "-                     + \"\\n\")",
        "-            try:",
        "-                from django.core.handlers.wsgi import WSGIRequest",
        "-                req = WSGIRequest(dict(environ))",
        "-                fh.write(\"path_info=\" + repr(req.path_info) + \"\\n\")",
        "-                from django.urls import resolve",
        "-                m = resolve(req.path_info)",
        "-                fh.write(\"match=\" + repr(m.kwargs) + \"\\n\")",
        "-            except Exception as e:  # noqa: BLE001",
        "-                fh.write(\"ERR \" + repr(e) + \"\\n\")",
        "-            try:",
        "-                raw = open(\"/proc/self/environ\", \"rb\").read()",
        "-                for entry in raw.split(b\"\\x00\"):",
        "-                    if entry.startswith((b\"PATH_INFO=\", b\"SCRIPT_URL=\")):",
        "-                        fh.write(\"ENVBYTES \" + repr(entry) + \"\\n\")",
        "-            except Exception as e:  # noqa: BLE001",
        "-                fh.write(\"ENVBYTES ERR \" + repr(e) + \"\\n\")",
        "+            environ[key] = v.encode(\"utf-8\", \"surrogateescape\") \\",
        "+                             .decode(\"latin-1\")",
        "     return application(environ, start_response)",
        " ",
        " ",
        "-CGIHandler().run(_pep3333_paths)",
        "+CGIHandler().run(_script_url_fix)",
        " "
      ]
    }
  ],
  "originalFile": "#!/opt/alt/python311/bin/python3.11\n\"\"\"WSGI->CGI adapter \u2014 runs the RH-WEBSITE Django app on Apache shared hosting.\n\nApache rewrites every non-file request to /app.cgi/<path> (see .htaccess).\nwsgiref's CGIHandler bridges the CGI environment onto Django's WSGI app.\n\"\"\"\nimport os\nimport sys\n\nDOCROOT = os.path.dirname(os.path.abspath(__file__))\nAPP = os.path.join(DOCROOT, \"app\")\n\n\ndef _dbg(msg):\n    if \"resources\" in (os.environ.get(\"PATH_INFO\") or \"\"):\n        with open(os.path.join(APP, \"debug-environ.txt\"), \"a\",\n                  encoding=\"utf-8\") as fh:\n            fh.write(msg + \"\\n\")\n\n\n_dbg(\"TOP fsenc=\" + sys.getfilesystemencoding()\n     + \" PATH_INFO=\" + repr(os.environ.get(\"PATH_INFO\")))\n\n# Vendored pure-Python deps (Django 5.2, markdown) + the project package.\nsys.path.insert(0, os.path.join(APP, \"vendor\"))\nsys.path.insert(0, os.path.join(APP, \"web\"))\n\nos.environ.setdefault(\"DJANGO_SETTINGS_MODULE\", \"rhweb.settings\")\n# Apache mounts us at /app.cgi/<path>; present clean root URLs to Django.\nos.environ[\"SCRIPT_NAME\"] = \"\"\n\nfrom wsgiref.handlers import CGIHandler\n\nfrom rhweb.wsgi import application\n\n_dbg(\"AFTER-IMPORT PATH_INFO=\" + repr(os.environ.get(\"PATH_INFO\")))\n\nfrom urllib.parse import unquote_to_bytes\n\n\ndef _pep3333_paths(environ, start_response):\n    \"\"\"Normalize Apache's path vars to the PEP 3333 form Django needs.\n\n    Apache hands these vars either percent-encoded (mod_rewrite path)\n    or UTF-8-decoded (direct CGI path). Django (a) crashes with\n    UnicodeEncodeError on any non-latin-1 char (500) and (b) needs the\n    decoded path for URL resolution (else the resolver sees %XX and 404s).\n    So: bytes -> unquote -> re-mangle as latin-1. SCRIPT_URL/REDIRECT_URL\n    (set by mod_rewrite) are consulted by get_script_name BEFORE\n    PATH_INFO \u2014 all three must be normalized.\"\"\"\n    if \"resources\" in (environ.get(\"PATH_INFO\") or \"\"):\n        with open(os.path.join(APP, \"debug-environ.txt\"), \"a\",\n                  encoding=\"utf-8\") as fh:\n            fh.write(\"ENTRY PATH_INFO=\" + repr(environ.get(\"PATH_INFO\"))\n                     + \" SCRIPT_URL=\" + repr(environ.get(\"SCRIPT_URL\")) + \"\\n\")\n    for key in (\"PATH_INFO\", \"SCRIPT_URL\", \"REDIRECT_URL\"):\n        v = environ.get(key)\n        if v:\n            raw = unquote_to_bytes(v)  # %XX-decodes; utf-8-encodes str first\n            environ[key] = raw.decode(\"latin-1\")\n    if \"resources\" in (environ.get(\"PATH_INFO\") or \"\"):\n        with open(os.path.join(APP, \"debug-environ.txt\"), \"a\",\n                  encoding=\"utf-8\") as fh:\n            fh.write(repr({k: environ.get(k) for k in\n                           (\"PATH_INFO\", \"SCRIPT_NAME\", \"SCRIPT_URL\",\n                            \"REDIRECT_URL\", \"REQUEST_METHOD\", \"HTTP_HOST\")})\n                     + \"\\n\")\n            try:\n                from django.core.handlers.wsgi import WSGIRequest\n                req = WSGIRequest(dict(environ))\n                fh.write(\"path_info=\" + repr(req.path_info) + \"\\n\")\n                from django.urls import resolve\n                m = resolve(req.path_info)\n                fh.write(\"match=\" + repr(m.kwargs) + \"\\n\")\n            except Exception as e:  # noqa: BLE001\n                fh.write(\"ERR \" + repr(e) + \"\\n\")\n            try:\n                raw = open(\"/proc/self/environ\", \"rb\").read()\n                for entry in raw.split(b\"\\x00\"):\n                    if entry.startswith((b\"PATH_INFO=\", b\"SCRIPT_URL=\")):\n                        fh.write(\"ENVBYTES \" + repr(entry) + \"\\n\")\n            except Exception as e:  # noqa: BLE001\n                fh.write(\"ENVBYTES ERR \" + repr(e) + \"\\n\")\n    return application(environ, start_response)\n\n\nCGIHandler().run(_pep3333_paths)\n",
  "autoRead": false
}

{
  "command": "cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem HasDerivAt.restrict_scalars\\|lemma HasDerivAt.restrict_scalars\\|restrict_scalars.*HasDerivAt\" Mathlib/ --include=*.lean | head -4; grep -rn \"hasDerivAt_ofReal\\|ofRealCLM\" Mathlib/Analysis/Complex/Basic.lean | head -5; grep -rn \"theorem HasDerivAt.comp\\b\" Mathlib/Analysis/Calculus/Deriv/Comp.lean | head -3",
  "description": "Find restrict_scalars and composition lemmas for mixed fields"
}

{
  "command": "sed -n '1,60p' /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib/Mathlib/Analysis/Calculus/Deriv/Comp.lean | grep -n \"variable\\|section\" | head -15",
  "description": "Check Comp.lean variable declarations"
}

{
  "command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiPrincipal.lean'\ns = open(p, encoding='utf-8').read()\n\ns = s.replace(\"open Zeta23 Complex Filter Topology\", \"open Zeta23 Complex Filter Topology Set\")\n\ns = s.replace(\"\"\"lemma hasDerivAt_add_const_I (y : \u211d) (x : \u211d) :\n    HasDerivAt (fun t => t + y * I) 1 x := by\n  have hid := hasDerivAt_id x\n  have hconst : HasDerivAt (fun _ => y * I) 0 x := hasDerivAt_const x _\n  convert hid.add hconst using 1\n  simp\"\"\",\n\"\"\"lemma hasDerivAt_add_const_I (y : \u211d) (x : \u211d) :\n    HasDerivAt (fun t : \u211d => t + y * I) 1 x := by\n  have h := HasDerivAt.add (hasDerivAt_id x) (hasDerivAt_const x (y * I))\n  simpa using h\"\"\")\n\ns = s.replace(\"\"\"lemma hasDerivAt_const_add_I (x y : \u211d) :\n    HasDerivAt (fun t => x + t * I) I y := by\n  have h1 : HasDerivAt (fun t => t * I) I y := by\n    have := (hasDerivAt_id y).mul_const I\n    simpa using this\n  have hconst : HasDerivAt (fun _ => x) 0 y := hasDerivAt_const y x\n  convert hconst.add h1 using 1\n  simp\"\"\",\n\"\"\"lemma hasDerivAt_const_add_I (x y : \u211d) :\n    HasDerivAt (fun t : \u211d => x + t * I) I y := by\n  have hof : HasDerivAt (fun t : \u211d => (t : \u2102)) 1 y := by\n    have h := Complex.ofRealCLM.hasDerivAt (x := y)\n    simpa using h\n  have h1 : HasDerivAt (fun t : \u211d => (t : \u2102) * I) I y := by\n    have := hof.mul_const I\n    simpa using this\n  have h := HasDerivAt.add (hasDerivAt_const y x) h1\n  simpa using h\"\"\")\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiPrincipal.lean 2>&1 | head -40",
  "description": "Fix path derivative lemmas and recompile",
  "timeout": 420000
}

{
  "command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiPrincipal.lean'\ns = open(p, encoding='utf-8').read()\n\ns = s.replace(\"\"\"lemma hasDerivAt_add_const_I (y : \u211d) (x : \u211d) :\n    HasDerivAt (fun t : \u211d => t + y * I) 1 x := by\n  simpa using (hasDerivAt_id x).add_const (y * I)\"\"\",\n\"\"\"lemma hasDerivAt_add_const_I (y : \u211d) (x : \u211d) :\n    HasDerivAt (fun t : \u211d => t + y * I) 1 x := by\n  have hof : HasDerivAt (fun t : \u211d => (t : \u2102)) 1 x := by\n    have h := Complex.ofRealCLM.hasDerivAt (x := x)\n    rw [Complex.ofRealCLM_apply] at h\n    exact h\n  have h := hof.add_const (y * I)\n  simpa using h\"\"\")\n\ns = s.replace(\"\"\"  have h1 : HasDerivAt (fun t : \u211d => (t : \u2102) * I) I y := by\n    have := hof.mul_const I\n    simpa using this\n  have h := h1.const_add x\n  simpa using h\"\"\",\n\"\"\"  have h1 : HasDerivAt (fun t : \u211d => (t : \u2102) * I) I y := by\n    have := hof.mul_const I\n    simpa using this\n  have h := HasDerivAt.const_add x h1\n  simpa using h\"\"\")\n\n# HIntegral_deriv: annotate h2's lambda domain and fix hint\ns = s.replace(\"\"\"    have h2 : HasDerivAt (fun x => x + y * I) 1 x := hasDerivAt_add_const_I y x\n    have hcomp := h1.comp x h2\n    convert hcomp using 1\n    simp\n  have hint : IntervalIntegrable (fun x => deriv G (x + y * I)) volume z.re w.re := by\n    apply ContinuousOn.intervalIntegrable\n    intro x hx\n    exact (hGc _ (hmap hx)).continuousWithinAt\"\"\",\n\"\"\"    have h2 : HasDerivAt (fun x : \u211d => x + y * I) 1 x := hasDerivAt_add_const_I y x\n    have hcomp := h1.comp x h2\n    convert hcomp using 1\n    simp\n  have hint : IntervalIntegrable (fun x => deriv G (x + y * I)) MeasureTheory.volume z.re w.re := by\n    apply ContinuousOn.intervalIntegrable\n    have hcon : ContinuousOn (deriv G) (RectangleBorder z w) := by\n      intro s hs\n      exact (hGc s hs).continuousWithinAt\n    have hpath : ContinuousOn (fun x : \u211d => x + y * I) (Set.uIcc z.re w.re) := by\n      apply Continuous.continuousOn\n      have hof : Continuous (fun t : \u211d => (t : \u2102)) := Complex.continuous_ofReal\n      exact hof.add_const (y * I)\n    exact hcon.comp hpath (by intro x hx; exact hmap hx)\"\"\")\n\n# VIntegral_deriv: fix hint similarly\ns = s.replace(\"\"\"  have hint : IntervalIntegrable (fun y => I * deriv G (x + y * I)) MeasureTheory.volume z.im w.im := by\n    apply ContinuousOn.intervalIntegrable\n    intro y hy\n    exact ContinuousWithinAt.mul continuousWithinAt_const\n      ((hGc _ (hmap hy)).continuousWithinAt)\"\"\",\n\"\"\"  have hint : IntervalIntegrable (fun y => I * deriv G (x + y * I)) MeasureTheory.volume z.im w.im := by\n    apply ContinuousOn.intervalIntegrable\n    have hcon : ContinuousOn (deriv G) (RectangleBorder z w) := by\n      intro s hs\n      exact (hGc s hs).continuousWithinAt\n    have hpath : ContinuousOn (fun y : \u211d => x + y * I) (Set.uIcc z.im w.im) := by\n      apply Continuous.continuousOn\n      have hof : Continuous (fun t : \u211d => (t : \u2102)) := Complex.continuous_ofReal\n      have hmul : Continuous (fun t : \u211d => (t : \u2102) * I) := hof.mul_const I\n      exact hmul.const_add x\n    have hcomp : ContinuousOn (fun y : \u211d => I * deriv G (x + y * I)) (Set.uIcc z.im w.im) := by\n      apply ContinuousOn.mul continuousOn_const\n      exact hcon.comp hpath (by intro y hy; exact hmap hy)\n    exact hcomp\"\"\")\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiPrincipal.lean 2>&1 | head -40",
  "description": "Apply fixes round 3 and recompile",
  "timeout": 420000
}

R-004-finite-height-li-theorem.md
# R-004 — Explicit finite-height Li theorem: λ_n > 0 for all n ≤ (T₀−1)² = 8.999…×10²⁴
Date: 2026-08-19 | Session: 008 | Track: C (task 3.8)
Tag: **[PROVED-CONDITIONAL(computational RH-to-height-T₀; computational
finite verification)]** — R-bound FULLY TRACED 2026-08-19 (D-056,
task 4.5a: reference-point method, A = 10.5, B₀ = 48 with the honest
C₀ incl. log log term; digamma remainder 1.2/|w|² proved; E-028
verified, 26× slack); N₀ = 80 000 (6.25× overlap with
Misguich–Voros). M₀ remains grid-verified (E-023 S1; additive ~53).
Margin convexity: PROVED (D-057, task 4.5b; E-029 ALL CHECKS PASS).
History: tag downgraded then restored same day (D-054 erratum: broken
D-039/D-036-§1 proofs; repairs D-056–D-059).

## Statement
Let λ_n = Σ′_ρ [1−(1−1/ρ)ⁿ] be the Li coefficients of the Riemann zeta
function. Assume the Riemann hypothesis verified to height
T₀ = 3×10¹² (Platt–Trudgian 2021, interval arithmetic). Then

  **λ_n > 0 for every 1 ≤ n ≤ N₁ := (T₀−1)²
  = 8 999 999 999 994 000 000 000 001 ≈ 9.0×10²⁴.**

Conservative variant (using Lagarias's printed bridge window (1.16)
instead of the traced proof): N₁* = T₀²/(4 log²T₀) = 2.726×10²¹.

This is the first assembled EXPLICIT finite-height Li bound: the chain
"RH to height T₀ + fully traced constants ⟹ λ_n > 0 to n ~ T₀²" existed

{
  "command": "cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem intervalIntegral.integral_deriv_eq_sub\\|theorem intervalIntegral.integral_eq_sub_of_hasDerivAt\" Mathlib/ --include=*.lean | head -4; grep -A4 \"theorem intervalIntegral.integral_deriv_eq_sub \" Mathlib/MeasureTheory/Integral/FundThmCalculus.lean 2>/dev/null | head -8; grep -rn \"theorem intervalIntegral.integral_deriv_eq_sub \" Mathlib/ --include=*.lean -A3 | head -8",
  "description": "Check integral_deriv_eq_sub signature"
}

{
  "command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiPrincipal.lean'\ns = open(p, encoding='utf-8').read()\n\ns = s.replace(\"\"\"  have hftc := intervalIntegral.integral_deriv_eq_sub hd hint\n  convert hftc using 1\n  \u00b7 simp\n  \u00b7 simp\"\"\",\n\"\"\"  have hftc := intervalIntegral.integral_eq_sub_of_hasDerivAt\n    (f := fun x => G (x + y * I)) (f' := fun x => deriv G (x + y * I)) hd hint\n  convert hftc using 1\n  \u00b7 simp\n  \u00b7 simp\"\"\")\n\ns = s.replace(\"\"\"  have hftc := intervalIntegral.integral_deriv_eq_sub hd hint\n  rw [\u2190 hftc, integral_smul]\n  field_simp [Complex.I_ne_zero]\"\"\",\n\"\"\"  have hftc := intervalIntegral.integral_eq_sub_of_hasDerivAt\n    (f := fun y => G (x + y * I)) (f' := fun y => I * deriv G (x + y * I)) hd hint\n  rw [\u2190 hftc, integral_smul]\n  field_simp [Complex.I_ne_zero]\"\"\")\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiPrincipal.lean 2>&1 | head -40",
  "description": "Use correct FTC lemma with explicit args",
  "timeout": 420000
}

done
NguyenRH1/LiTightR.lean:3593:36: error: unexpected token '/--'; expected ':='
NguyenRH1/LiTightR.lean:3592:2: error: invalid 'calc' step, relation expected
  ?m.236
NguyenRH1/LiTightR.lean:3570:46: error: unsolved goals
case neg
t : ℝ
ht : 10 ≤ t
k : ℕ
hk : ¬⌊t⌋₊ + 1 ≤ k
⊢ farGRightMaj t k ≤ farGRightMajCombo k
NguyenRH1/LiTightR.lean:3674:35: error: unsolved goals
t : ℝ
ht : 10 ≤ t
hsplit : ∑' (k : ℕ), farGRightMaj t k = ∑ k ∈ Finset.range 11, farGRightMaj t k + ∑' (k : ℕ), farGRightMaj t (k + 11)
k : ℕ
hk : k < 11
⊢ 10 ≤ t
NguyenRH1/LiTightR.lean:3678:5: error: (deterministic) timeout at `whnf`, maximum number of heartbeats (200000) has been reached

Note: Use `set_option maxHeartbeats <num>` to set the limit.

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean",
  "offset": 42,
  "limit": 80
}

{
  "type": "text",
  "file": {
    "filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiContour.lean",
    "content": "\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. The unit ordinate window and the perimeter count hypothesis -/\n\n/-- The nontrivial zeros with ordinate in the unit window [a, a+1]. -/\ndef zeroWindow (a : \u211d) : Set \u2102 :=\n  {\u03c1 | IsNontrivialZero \u03c1 \u2227 a \u2264 \u03c1.im \u2227 \u03c1.im \u2264 a + 1}\n\n/-- Finiteness of the unit ordinate window (via Zeta23 `zerosIn_finite`). -/\ntheorem zeroWindow_finite (a : \u211d) : (zeroWindow a).Finite := by\n  apply Set.Finite.subset (zerosIn_finite (a - 1) (a + 1))\n  intro \u03c1 h\u03c1\n  refine \u27e8h\u03c1.1, ?_, h\u03c1.2.2\u27e9\n  linarith [h\u03c1.2.1]\n\n/-- A nontrivial zero has multiplicity \u2265 1 (zeroMult is the analytic\norder). Bridges the mult-weighted perimeter count to the plain window\ncardinality (card \u2264 \u03a3 mult). -/\ntheorem one_le_zeroMult {\u03c1 : \u2102} (h\u03c1 : IsNontrivialZero \u03c1) : 1 \u2264 zeroMult \u03c1 := by\n  have h\u03c11 : \u03c1 \u2260 1 := by\n    rintro rfl\n    have h := h\u03c1.2.2\n    rw [Complex.one_re] at h\n    exact (lt_irrefl (1 : \u211d)) h\n  have htop := zeta_order_ne_top_of_ne h\u03c11\n  have hne : analyticOrderAt riemannZeta \u03c1 \u2260 0 :=\n    (analyticAt_riemannZeta h\u03c11).analyticOrderAt_ne_zero.mpr h\u03c1.1\n  exact ENat.toNat_pos hne htop\n\n/-- [PERIMETER \u2014 Trudgian 2014 N(T)/S(T), D-062 \u00a74] The unit-window\nzero-count bound: the multiplicity-weighted count of nontrivial zeros\nwith ordinates in [t, t+1] is at most \u0108\u2080(t+1) (t \u2265 2). Declared\ncounting input for the L3 pigeonhole and the tight R-bound; \u0108\u2080 =\n`C0hat` is D-056 \u00a72's honest majorant (LiSkeleton).\n[Erratum 2026-08-23 (session 035): strengthened from the unweighted\nset-card form `(Nat.card (zeroWindow t) : \u211d) \u2264 \u0108\u2080(t+1)` to the\nmult-weighted form \u2014 Trudgian's N(T) counts zeros with multiplicity,\nand the mult-weighted form is what the 5f/ii\u2032-6 near sums\n(zeroMult-weighted, from `seriesFun`) consume; the set-card encoding\nunder-captured the declared source (a second ii\u2032-1a-class encoding\ndefect, found at the point of consumption). The set-card bound follows\nfrom this form via `one_le_zeroMult` (card \u2264 \u03a3 mult); the two\nconsumers (exists_zero_avoiding_T, exists_gapCircle_radius) adapted.] -/\ndef ZeroCountProp : Prop :=\n  \u2200 t : \u211d, 2 \u2264 t \u2192\n    (((zeroWindow_finite t).toFinset.sum fun \u03c1 => zeroMult \u03c1) : \u211d) \u2264 C0hat (t + 1)\n",
    "numLines": 50,
    "startLine": 40,
    "totalLines": 647
  }
}

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean",
  "old_string": "  have h := HasDerivAt.const_add x h1\n  simpa using h",
  "new_string": "  have h := HasDerivAt.const_add (\u2191x : \u2102) h1\n  simpa using h"
}

{
  "command": "cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem intervalIntegral.integral_smul\\b\" Mathlib/ --include=*.lean | head -2; grep -rn \"theorem Complex.inv_I\\|lemma Complex.inv_I\" Mathlib/Data/Complex/Basic.lean | head -2; grep -rn \"theorem integral_smul\\b\" Mathlib/MeasureTheory/Integral/Bochner.lean Mathlib/MeasureTheory/Integral/Bochner/*.lean 2>/dev/null | head -2",
  "description": "Verify integral_smul and Complex.inv_I names"
}

{
  "command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiPrincipal.lean'\ns = open(p, encoding='utf-8').read()\n\n# 1. H-side hd: simpa\ns = s.replace(\"\"\"    have hcomp := h1.comp x h2\n    convert hcomp using 1\n    simp\n  have hint : IntervalIntegrable (fun x => deriv G (x + y * I)) MeasureTheory.volume z.re w.re := by\"\"\",\n\"\"\"    have hcomp := h1.comp x h2\n    simpa [Function.comp_apply] using hcomp\n  have hint : IntervalIntegrable (fun x => deriv G (x + y * I)) MeasureTheory.volume z.re w.re := by\"\"\")\n\n# 2. H-side hftc: simpa\ns = s.replace(\"\"\"  have hftc := intervalIntegral.integral_eq_sub_of_hasDerivAt\n    (f := fun x : \u211d => G (x + y * I)) (f' := fun x : \u211d => deriv G (x + y * I)) hd hint\n  convert hftc using 1\n  \u00b7 simp\n  \u00b7 simp\"\"\",\n\"\"\"  have hftc := intervalIntegral.integral_eq_sub_of_hasDerivAt\n    (f := fun x : \u211d => G (x + y * I)) (f' := fun x : \u211d => deriv G (x + y * I)) hd hint\n  simpa using hftc\"\"\")\n\n# 3. V-side hd: simpa\ns = s.replace(\"\"\"    have hcomp := h1.comp y h2\n    convert hcomp using 1\n    ring\"\"\",\n\"\"\"    have hcomp := h1.comp y h2\n    simpa [Function.comp_apply] using hcomp\"\"\")\n\n# 4. V-side final computation\ns = s.replace(\"\"\"  have hftc := intervalIntegral.integral_eq_sub_of_hasDerivAt\n    (f := fun y : \u211d => G (x + y * I)) (f' := fun y : \u211d => I * deriv G (x + y * I)) hd hint\n  rw [\u2190 hftc, integral_smul]\n  field_simp [Complex.I_ne_zero]\"\"\",\n\"\"\"  have hftc := intervalIntegral.integral_eq_sub_of_hasDerivAt\n    (f := fun y : \u211d => G (x + y * I)) (f' := fun y : \u211d => I * deriv G (x + y * I)) hd hint\n  have h6 : I * \u222b y in z.im..w.im, deriv G (x + y * I) =\n      G (x + w.im * I) - G (x + z.im * I) := by\n    rw [\u2190 intervalIntegral.integral_smul, smul_eq_mul]\n    exact hftc\n  have h7 : \u222b y in z.im..w.im, deriv G (x + y * I) =\n      (G (x + w.im * I) - G (x + z.im * I)) / I :=\n    eq_div_of_mul_eq Complex.I_ne_zero (by rw [mul_comm]; exact h6)\n  rw [h7, div_eq_mul_inv, Complex.inv_I]\n  ring\"\"\")\n\n# 5. rectInt_deriv_eq_zero: unfold first\ns = s.replace(\"\"\"  rw [RectangleIntegral]\n  rw [HIntegral_deriv z.im hG hGc (mapsTo_rectangleBorder_left_im z w),\n    HIntegral_deriv w.im hG hGc (mapsTo_rectangleBorder_right_im z w)]\n  have hVr := VIntegral_deriv w.re hG hGc (mapsTo_rectangleBorder_right_re z w)\n  have hVl := VIntegral_deriv z.re hG hGc (mapsTo_rectangleBorder_left_re z w)\n  rw [VIntegral, VIntegral, hVr, hVl]\n  simp [Complex.I_mul_I]\n  ring\"\"\",\n\"\"\"  rw [RectangleIntegral, HIntegral, HIntegral, VIntegral, VIntegral]\n  rw [HIntegral_deriv z.im hG hGc (mapsTo_rectangleBorder_left_im z w),\n    HIntegral_deriv w.im hG hGc (mapsTo_rectangleBorder_right_im z w)]\n  have hVr := VIntegral_deriv w.re hG hGc (mapsTo_rectangleBorder_right_re z w)\n  have hVl := VIntegral_deriv z.re hG hGc (mapsTo_rectangleBorder_left_re z w)\n  rw [hVr, hVl]\n  simp [smul_eq_mul, Complex.I_mul_I]\n  ring\"\"\")\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiPrincipal.lean 2>&1 | head -40",
  "description": "Apply final fixes and recompile LiPrincipal",
  "timeout": 420000
}

NguyenRH1/LiDogLeg.lean:430:4: error: No goals to be solved

{
  "command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiPrincipal.lean'\ns = open(p, encoding='utf-8').read()\n\ns = s.replace(\"\"\"    have hcomp := h1.comp x h2\n    exact hcomp\n  have hint : IntervalIntegrable (fun x => deriv G (x + y * I)) MeasureTheory.volume z.re w.re := by\"\"\",\n\"\"\"    have hcomp := h1.comp x h2\n    rw [mul_one] at hcomp\n    exact hcomp\n  have hint : IntervalIntegrable (fun x => deriv G (x + y * I)) MeasureTheory.volume z.re w.re := by\"\"\")\n\ns = s.replace(\"\"\"    have hcomp := h1.comp y h2\n    exact hcomp\n  have hint : IntervalIntegrable (fun y : \u211d => I * deriv G (x + y * I)) MeasureTheory.volume z.im w.im := by\"\"\",\n\"\"\"    have hcomp := h1.comp y h2\n    rw [mul_comm] at hcomp\n    exact hcomp\n  have hint : IntervalIntegrable (fun y : \u211d => I * deriv G (x + y * I)) MeasureTheory.volume z.im w.im := by\"\"\")\n\ns = s.replace(\"\"\"    rw [\u2190 smul_eq_mul, \u2190 intervalIntegral.integral_smul, smul_eq_mul]\n    exact hftc\"\"\",\n\"\"\"    have e := intervalIntegral.integral_smul (\u03bc := MeasureTheory.volume) (c := I)\n      (f := fun y : \u211d => deriv G (x + y * I)) (a := z.im) (b := w.im)\n    simp only [Pi.smul_apply, smul_eq_mul] at e\n    rw [\u2190 e]\n    exact hftc\"\"\")\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiPrincipal.lean 2>&1 | head -30",
  "description": "Apply comp mul and smul fixes, recompile",
  "timeout": 420000
}

{
  "command": "grep -n \"rectangleBorderNoPIntegrable\\|rectangleBorderIntegrable\" /d/Python/Quantum/zeta-23-lean/Zeta23/FromPNTPlus/ResidueCalcOnRectangles.lean | head -8; sed -n \"$(grep -n 'theorem ContinuousOn.rectangleBorderNoPIntegrable' /d/Python/Quantum/zeta-23-lean/Zeta23/FromPNTPlus/ResidueCalcOnRectangles.lean | head -1 | cut -d: -f1),+8p\" /d/Python/Quantum/zeta-23-lean/Zeta23/FromPNTPlus/ResidueCalcOnRectangles.lean",
  "description": "Check punctured border integrability helper"
}

{
  "command": "cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem deriv_zpow\\|lemma deriv_zpow\" Mathlib/ --include=*.lean | head -3; grep -rn \"HasDerivAt.zpow\\|hasDerivAt_zpow\" Mathlib/Analysis/Calculus/Deriv/*.lean 2>/dev/null | head -4",
  "description": "Check zpow derivative lemmas"
}

{
  "command": "cat >> /d/Python/Quantum/research/lean/NguyenRH1/NguyenRH1/LiPrincipal.lean << 'EOF'\n\n/-! ## 3. The principal part and its integral -/\n\n/-- The principal-part coefficients: A_i = F^{(n+1\u2212i)}(1)/(n+1\u2212i)!,\nF(w) = (w\u207f\u2212(w\u22121)\u207f)\u00b7hLi(w). -/\ndef Acoeff (n i : \u2115) : \u2102 :=\n  iteratedDeriv (n + 1 - i) (fun w : \u2102 => (w ^ n - (w - 1) ^ n) * hLi w) 1 / (n + 1 - i)!\n\nlemma Acoeff_one (n : \u2115) : Acoeff n 1 = ResW1 n := by\n  rw [Acoeff, ResW1, show n + 1 - 1 = n from by omega]\n\n/-- The principal part of J at w = 1: P(w) = \u03a3_{i=1}^{n+1} A_i/(w\u22121)^i. -/\ndef ppart (n : \u2115) (w : \u2102) : \u2102 :=\n  \u2211 i \u2208 Finset.Icc 1 (n + 1), Acoeff n i / (w - 1) ^ i\n\n/-- The tail primitive G(w) = \u2212\u03a3_{i=2}^{n+1} (A_i/(i\u22121))\u00b7(w\u22121)^{\u2212(i\u22121)}. -/\ndef Gprim (n : \u2115) (w : \u2102) : \u2102 :=\n  \u2211 i \u2208 Finset.Icc 2 (n + 1), -(Acoeff n i / ((i : \u2102) - 1)) * (w - 1) ^ (-((i : \u2124) - 1))\n\nlemma Gprim_differentiableAt {n : \u2115} {w : \u2102} (hw : w \u2260 1) :\n    DifferentiableAt \u2102 (Gprim n) w := by\n  rw [Gprim]\n  apply DifferentiableAt.sum\n  intro i hi\n  apply DifferentiableAt.mul (differentiableAt_const _)\n  have hw1 : w - 1 \u2260 0 := sub_ne_zero.mpr hw\n  exact (hasDerivAt_zpow _ (w - 1) (Or.inl hw1)).differentiableAt.comp w\n    ((hasDerivAt_id w).sub_const 1)\n\nlemma deriv_Gprim {n : \u2115} {w : \u2102} (hw : w \u2260 1) :\n    deriv (Gprim n) w = \u2211 i \u2208 Finset.Icc 2 (n + 1), Acoeff n i / (w - 1) ^ i := by\n  rw [Gprim, deriv_sum (fun i hi => ?_)]\n  \u00b7 apply Finset.sum_congr rfl\n    intro i hi\n    rw [Finset.mem_Icc] at hi\n    have hw1 : w - 1 \u2260 0 := sub_ne_zero.mpr hw\n    have hi1 : ((i : \u2102) - 1) \u2260 0 := by\n      have : 1 < (i : \u211d) := by exact_mod_cast (by omega : 1 < i)\n      intro heq\n      rw [sub_eq_zero] at heq\n      have : (i : \u211d) = 1 := by exact_mod_cast heq\n      linarith\n    have hd : deriv (fun w => (w - 1) ^ (-((i : \u2124) - 1))) w\n        = ((-((i : \u2124) - 1)) : \u2102) * (w - 1) ^ (-((i : \u2124) - 1) - 1) := by\n      have h2 : HasDerivAt (fun x : \u2102 => x ^ (-((i : \u2124) - 1)))\n          ((-((i : \u2124) - 1)) * (w - 1) ^ (-((i : \u2124) - 1) - 1)) (w - 1) :=\n        hasDerivAt_zpow _ _ (Or.inl hw1)\n      have h3 : HasDerivAt (fun w : \u2102 => w - 1) 1 w := by\n        simpa using (hasDerivAt_id w).sub_const 1\n      have hcomp := h2.comp w h3\n      rw [Function.comp_apply, mul_one] at hcomp\n      exact hcomp.deriv\n    rw [deriv_const_mul _ (by\n      have hw1' : w - 1 \u2260 0 := sub_ne_zero.mpr hw\n      exact (hasDerivAt_zpow _ (w - 1) (Or.inl hw1')).differentiableAt.comp w\n        ((hasDerivAt_id w).sub_const 1)), hd]\n    have hm : (-((i : \u2124) - 1) - 1 : \u2124) = -(i : \u2124) := by omega\n    rw [hm]\n    have hzi : ((-(i : \u2124)) : \u2102) = -((i : \u2102)) := by simp\n    rw [hzi]\n    field_simp [hw1, hi1]\n    rw [zpow_neg, zpow_natCast]\n    field_simp [hw1]\n    ring\n  \u00b7 intro i hi\n    apply DifferentiableAt.mul (differentiableAt_const _)\n    have hw1 : w - 1 \u2260 0 := sub_ne_zero.mpr hw\n    exact (hasDerivAt_zpow _ (w - 1) (Or.inl hw1)).differentiableAt.comp w\n      ((hasDerivAt_id w).sub_const 1)\n\nlemma tail_continuousAt {n : \u2115} {w : \u2102} (hw : w \u2260 1) :\n    ContinuousAt (\u2211 i \u2208 Finset.Icc 2 (n + 1), Acoeff n i / (w - 1) ^ i) w := by\n  apply ContinuousAt.sum\n  intro i hi\n  rw [Finset.mem_Icc] at hi\n  have hw1 : w - 1 \u2260 0 := sub_ne_zero.mpr hw\n  apply ContinuousAt.div (continuousAt_const)\n  exact (continuousAt_id.sub continuousAt_const).pow hw1.ne' |> fun h => h.continuousAt\n  \u00b7 exact pow_ne_zero _ hw1\n\nlemma ppart_eq_deriv_add {n : \u2115} {w : \u2102} (hw : w \u2260 1) :\n    ppart n w = Acoeff n 1 / (w - 1) + deriv (Gprim n) w := by\n  rw [ppart, deriv_Gprim hw]\n  have hsplit : Finset.Icc 1 (n + 1) = {1} \u222a Finset.Icc 2 (n + 1) := by\n    ext j\n    simp [Finset.mem_Icc]\n    omega\n  rw [hsplit, Finset.sum_union (by simp)]\n  simp [pow_one]\n\n/-- The principal-part integral: on any rectangle with 1 in its interior,\nRectangleIntegral' (ppart n) z w = ResW1 n. -/\ntheorem rectInt'_ppart {n : \u2115} {z w : \u2102}\n    (hre : z.re \u2264 w.re) (him : z.im \u2264 w.im) (h1 : Rectangle z w \u2208 \ud835\udcdd (1 : \u2102)) :\n    RectangleIntegral' (ppart n) z w = ResW1 n := by\n  have h1not : (1 : \u2102) \u2209 RectangleBorder z w := not_mem_rectangleBorder_of_rectangle_mem_nhds h1\n  have hdecomp : Set.EqOn (ppart n) (fun w => Acoeff n 1 / (w - 1) + deriv (Gprim n) w)\n      (RectangleBorder z w) := by\n    intro s hs\n    have hs1 : s \u2260 1 := fun heq => h1not (heq \u25b8 hs)\n    exact ppart_eq_deriv_add hs1\n  rw [RectangleIntegral'_congr hdecomp]\n  have hAint : RectangleBorderIntegrable (fun w => Acoeff n 1 / (w - 1)) z w := by\n    apply HolomorphicOn.rectangleBorderIntegrable' _ h1\n    intro s hs\n    have hs1 : s \u2260 1 := hs.2\n    exact (differentiableAt_const.div ((differentiableAt_id.sub (differentiableAt_const 1)))\n      (sub_ne_zero.mpr hs1)).differentiableWithinAt\n  have hGint : RectangleBorderIntegrable (deriv (Gprim n)) z w := by\n    apply HolomorphicOn.rectangleBorderIntegrable' _ h1\n    intro s hs\n    have hs1 : s \u2260 1 := hs.2\n    rw [deriv_Gprim hs1]\n    exact (tail_continuousAt hs1).differentiableAt.continuousWithinAt\n  rw [RectangleIntegral'_add _ _ hAint hGint]\n  rw [ResidueTheoremInRectangle hre him h1]\n  have hzero : RectangleIntegral (deriv (Gprim n)) z w = 0 := by\n    apply rectInt_deriv_eq_zero\n    \u00b7 intro s hs\n      have hs1 : s \u2260 1 := fun heq => h1not (heq \u25b8 hs)\n      exact Gprim_differentiableAt hs1\n    \u00b7 intro s hs\n      have hs1 : s \u2260 1 := fun heq => h1not (heq \u25b8 hs)\n      rw [deriv_Gprim hs1]\n      exact tail_continuousAt hs1\n  rw [RectangleIntegral', hzero]\n  rw [Acoeff_one]\n  simp\nEOF\ncd /d/Python/Quantum/research/lean/NguyenRH1 && export PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiPrincipal.lean 2>&1 | head -40",
  "description": "Append assembly section and compile",
  "timeout": 420000
}

{
  "command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiPrincipal.lean'\ns = open(p, encoding='utf-8').read()\n\n# 1. namespace: remove end before section 3, add at file end\ns = s.replace(\"end NguyenRH1\\n\\n/-! ## 3. The principal part and its integral -/\",\n              \"/-! ## 3. The principal part and its integral -/\")\ns = s.rstrip() + \"\\n\\nend NguyenRH1\\n\"\n\n# 2. ContinuousAt.sum -> use explicit induction-free form\ns = s.replace(\"\"\"  apply ContinuousAt.sum\n  intro i hi\n  rw [Finset.mem_Icc] at hi\n  have hw1 : w - 1 \u2260 0 := sub_ne_zero.mpr hw\n  apply ContinuousAt.div (continuousAt_const)\n  exact (continuousAt_id.sub continuousAt_const).pow hw1.ne' |> fun h => h.continuousAt\n  \u00b7 exact pow_ne_zero _ hw1\"\"\",\n\"\"\"  apply ContinuousAt.sum\n  intro i hi\n  rw [Finset.mem_Icc] at hi\n  have hw1 : w - 1 \u2260 0 := sub_ne_zero.mpr hw\n  apply ContinuousAt.div (continuousAt_const)\n  \u00b7 exact (continuousAt_id.sub continuousAt_const).pow _\n  \u00b7 exact pow_ne_zero _ hw1\"\"\")\n\n# 3. differentiableAt_const.div fix\ns = s.replace(\"\"\"    exact (differentiableAt_const.div ((differentiableAt_id.sub (differentiableAt_const 1)))\n      (sub_ne_zero.mpr hs1)).differentiableWithinAt\"\"\",\n\"\"\"    exact (((differentiableAt_const (Acoeff n 1)).div\n      ((differentiableAt_id.sub (differentiableAt_const 1)))\n      (sub_ne_zero.mpr hs1))).differentiableWithinAt\"\"\")\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiPrincipal.lean 2>&1 | head -40",
  "description": "Fix namespace and remaining issues, recompile",
  "timeout": 420000
}

{
  "command": "cd /d/Python/Quantum/zeta-23-lean/.lake/packages/mathlib && grep -rn \"theorem ContinuousAt.sum\\|lemma ContinuousAt.sum\" Mathlib/ --include=*.lean | head -3; grep -rn \"theorem DifferentiableOn.deriv\\b\\|lemma DifferentiableOn.deriv\\b\" Mathlib/ --include=*.lean | head -3; grep -rn \"theorem isOpen_compl_singleton\" Mathlib/ --include=*.lean | head -2",
  "description": "Check ContinuousAt.sum, DifferentiableOn.deriv, isOpen_compl_singleton"
}

{
  "command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiPrincipal.lean'\ns = open(p, encoding='utf-8').read()\n\n# 1. Acoeff: explicit Nat.factorial\ns = s.replace(\"\"\"def Acoeff (n i : \u2115) : \u2102 :=\n  iteratedDeriv (n + 1 - i) (fun w : \u2102 => (w ^ n - (w - 1) ^ n) * hLi w) 1 / (n + 1 - i)!\"\"\",\n\"\"\"def Acoeff (n i : \u2115) : \u2102 :=\n  iteratedDeriv (n + 1 - i) (fun w : \u2102 => (w ^ n - (w - 1) ^ n) * hLi w) 1 /\n    Nat.factorial (n + 1 - i)\"\"\")\n\n# 2. Acoeff_one\ns = s.replace(\"\"\"lemma Acoeff_one (n : \u2115) : Acoeff n 1 = ResW1 n := by\n  rw [Acoeff, ResW1, show n + 1 - 1 = n from by omega]\"\"\",\n\"\"\"lemma Acoeff_one (n : \u2115) : Acoeff n 1 = ResW1 n := by\n  simp only [Acoeff, ResW1, show n + 1 - 1 = n from by omega]\"\"\")\n\n# 3. Gprim unfolds\ns = s.replace(\"\"\"  rw [Gprim]\n  apply DifferentiableAt.sum\"\"\",\n\"\"\"  simp only [Gprim]\n  apply DifferentiableAt.sum\"\"\")\ns = s.replace(\"\"\"  rw [Gprim, deriv_sum (fun i hi => ?_)]\"\"\",\n\"\"\"  simp only [Gprim]\n  rw [deriv_sum (fun i hi => ?_)]\"\"\")\n\n# 4. tail_continuousAt via induction\ns = s.replace(\"\"\"lemma tail_continuousAt {n : \u2115} {w : \u2102} (hw : w \u2260 1) :\n    ContinuousAt (\u2211 i \u2208 Finset.Icc 2 (n + 1), Acoeff n i / (w - 1) ^ i) w := by\n  apply ContinuousAt.sum\n  intro i hi\n  rw [Finset.mem_Icc] at hi\n  have hw1 : w - 1 \u2260 0 := sub_ne_zero.mpr hw\n  apply ContinuousAt.div (continuousAt_const)\n  \u00b7 exact (continuousAt_id.sub continuousAt_const).pow _\n  \u00b7 exact pow_ne_zero _ hw1\"\"\",\n\"\"\"lemma tail_continuousAt {n : \u2115} {w : \u2102} (hw : w \u2260 1) :\n    ContinuousAt (\u2211 i \u2208 Finset.Icc 2 (n + 1), Acoeff n i / (w - 1) ^ i) w := by\n  have hterm : \u2200 i : \u2115, ContinuousAt (fun w => Acoeff n i / (w - 1) ^ i) w := by\n    intro i\n    have hw1 : w - 1 \u2260 0 := sub_ne_zero.mpr hw\n    apply ContinuousAt.div (continuousAt_const)\n    \u00b7 exact (continuousAt_id.sub continuousAt_const).pow _\n    \u00b7 exact pow_ne_zero _ hw1\n  have : \u2200 s : Finset \u2115, ContinuousAt (\u2211 i \u2208 s, Acoeff n i / (w - 1) ^ i) w := by\n    intro s\n    induction s using Finset.induction with\n    | empty => simp\n    | insert has hs =>\n      rename_i a s\n      rw [Finset.sum_insert has]\n      exact ContinuousAt.add hs (hterm a)\n  exact this _\"\"\")\n\n# 5. hGint restructure\ns = s.replace(\"\"\"  have hGint : RectangleBorderIntegrable (deriv (Gprim n)) z w := by\n    apply HolomorphicOn.rectangleBorderIntegrable' _ h1\n    intro s hs\n    have hs1 : s \u2260 1 := hs.2\n    rw [deriv_Gprim hs1]\n    exact (tail_continuousAt hs1).differentiableAt.continuousWithinAt\"\"\",\n\"\"\"  have hGint : RectangleBorderIntegrable (deriv (Gprim n)) z w := by\n    apply HolomorphicOn.rectangleBorderIntegrable' _ h1\n    have hopen : DifferentiableOn \u2102 (deriv (Gprim n)) {1}\u1d9c := by\n      have hd : DifferentiableOn \u2102 (Gprim n) {1}\u1d9c := by\n        intro s hs\n        exact (Gprim_differentiableAt (Set.mem_compl_singleton_iff.mp hs)).differentiableWithinAt\n      exact hd.deriv isOpen_compl_singleton\n    exact hopen.mono (Set.diff_subset_diff_left (Set.subset_univ _))\"\"\")\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiPrincipal.lean 2>&1 | head -40",
  "description": "Apply assembly fixes and recompile",
  "timeout": 420000
}

{
  "command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiPrincipal.lean'\ns = open(p, encoding='utf-8').read()\n\n# 1. Gprim_differentiableAt: show family form\ns = s.replace(\"\"\"  unfold Gprim\n  apply DifferentiableAt.sum\n  intro i hi\n  apply DifferentiableAt.mul (differentiableAt_const _)\"\"\",\n\"\"\"  unfold Gprim\n  show DifferentiableAt \u2102 (\u2211 i \u2208 Finset.Icc 2 (n + 1),\n    fun w : \u2102 => -(Acoeff n i / (\u2191i - 1)) * (w - 1) ^ (-(\u2191i - 1))) w\n  apply DifferentiableAt.sum\n  intro i hi\n  apply DifferentiableAt.mul (differentiableAt_const _)\"\"\")\n\n# 2. deriv_Gprim: show family form\ns = s.replace(\"\"\"  unfold Gprim\n  rw [deriv_sum (fun i hi => ?_)]\"\"\",\n\"\"\"  show deriv (\u2211 i \u2208 Finset.Icc 2 (n + 1),\n    fun w : \u2102 => -(Acoeff n i / (\u2191i - 1)) * (w - 1) ^ (-(\u2191i - 1))) w = _\n  rw [deriv_sum (fun i hi => ?_)]\"\"\")\n\n# 3. tail_continuousAt: explicit fun w =>\ns = s.replace(\"\"\"lemma tail_continuousAt {n : \u2115} {w : \u2102} (hw : w \u2260 1) :\n    ContinuousAt (\u2211 i \u2208 Finset.Icc 2 (n + 1), Acoeff n i / (w - 1) ^ i) w := by\"\"\",\n\"\"\"lemma tail_continuousAt {n : \u2115} {w : \u2102} (hw : w \u2260 1) :\n    ContinuousAt (fun w => \u2211 i \u2208 Finset.Icc 2 (n + 1), Acoeff n i / (w - 1) ^ i) w := by\"\"\")\ns = s.replace(\"\"\"  have : \u2200 s : Finset \u2115, ContinuousAt (\u2211 i \u2208 s, Acoeff n i / (w - 1) ^ i) w := by\"\"\",\n\"\"\"  have : \u2200 s : Finset \u2115, ContinuousAt (fun w => \u2211 i \u2208 s, Acoeff n i / (w - 1) ^ i) w := by\"\"\")\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiPrincipal.lean 2>&1 | head -40",
  "description": "Fix lambda abstraction issues, recompile",
  "timeout": 420000
}

{
  "command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && python - << 'PYEOF'\np = 'NguyenRH1/LiPrincipal.lean'\ns = open(p, encoding='utf-8').read()\n\n# 1. Gprim_differentiableAt: proper function-form conversion\ns = s.replace(\"\"\"  unfold Gprim\n  show DifferentiableAt \u2102 (\u2211 i \u2208 Finset.Icc 2 (n + 1),\n    fun w : \u2102 => -(Acoeff n i / (\u2191i - 1)) * (w - 1) ^ (-(\u2191i - 1))) w\n  apply DifferentiableAt.sum\n  intro i hi\n  apply DifferentiableAt.mul (differentiableAt_const _)\"\"\",\n\"\"\"  unfold Gprim\n  rw [show (fun w : \u2102 => \u2211 i \u2208 Finset.Icc 2 (n + 1),\n      -(Acoeff n i / (\u2191i - 1)) * (w - 1) ^ (-((i : \u2124) - 1))) =\n    \u2211 i \u2208 Finset.Icc 2 (n + 1),\n      (fun w : \u2102 => -(Acoeff n i / (\u2191i - 1)) * (w - 1) ^ (-((i : \u2124) - 1))) from by\n    funext w\n    simp [Finset.sum_apply]]\n  apply DifferentiableAt.sum\n  intro i hi\n  apply DifferentiableAt.mul (differentiableAt_const _)\"\"\")\n\n# 2. deriv_Gprim: same conversion\ns = s.replace(\"\"\"  show deriv (\u2211 i \u2208 Finset.Icc 2 (n + 1),\n    fun w : \u2102 => -(Acoeff n i / (\u2191i - 1)) * (w - 1) ^ (-(\u2191i - 1))) w = _\n  rw [deriv_sum (fun i hi => ?_)]\"\"\",\n\"\"\"  unfold Gprim\n  rw [show (fun w : \u2102 => \u2211 i \u2208 Finset.Icc 2 (n + 1),\n      -(Acoeff n i / (\u2191i - 1)) * (w - 1) ^ (-((i : \u2124) - 1))) =\n    \u2211 i \u2208 Finset.Icc 2 (n + 1),\n      (fun w : \u2102 => -(Acoeff n i / (\u2191i - 1)) * (w - 1) ^ (-((i : \u2124) - 1))) from by\n    funext w\n    simp [Finset.sum_apply]]\n  rw [deriv_sum (fun i hi => ?_)]\"\"\")\n\n# 3. tail induction fixes\ns = s.replace(\"\"\"    induction s using Finset.induction with\n    | empty => simp\n    | insert has hs =>\n      rename_i a s\n      rw [Finset.sum_insert has]\n      exact ContinuousAt.add hs (hterm a)\n  exact this _\"\"\",\n\"\"\"    induction s using Finset.induction with\n    | empty => exact continuousAt_const\n    | insert a s has hs =>\n      rw [Finset.sum_insert has]\n      exact ContinuousAt.add hs (hterm a)\n  exact this _\"\"\")\n\n# 4. subset by hand\ns = s.replace(\"exact hopen.mono (Set.sdiff_subset_sdiff_left (Set.subset_univ _))\",\n              \"exact hopen.mono (by intro s hs; exact hs.2)\")\n\nopen(p, 'w', encoding='utf-8').write(s)\nprint(\"patched\")\nPYEOF\nexport PATH=\"$HOME/.elan/bin:$PATH\" && lake env lean NguyenRH1/LiPrincipal.lean 2>&1 | head -40",
  "description": "Apply structural fixes and recompile",
  "timeout": 420000
}

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean",
  "old_string": "  apply DifferentiableAt.sum\n  intro i hi\n  apply DifferentiableAt.mul (differentiableAt_const _)\n  have hw1 : w - 1 \u2260 0 := sub_ne_zero.mpr hw\n  exact (hasDerivAt_zpow _ (w - 1) (Or.inl hw1)).differentiableAt.comp w\n    ((hasDerivAt_id w).sub_const 1)\n\nlemma deriv_Gprim {n : \u2115} {w : \u2102} (hw : w \u2260 1) :",
  "new_string": "  apply DifferentiableAt.sum\n  intro i hi\n  apply DifferentiableAt.mul (differentiableAt_const _)\n  have hw1 : w - 1 \u2260 0 := sub_ne_zero.mpr hw\n  exact (((hasDerivAt_zpow (-((i : \u2124) - 1)) (w - 1) (Or.inl hw1)).differentiableAt).comp w\n    (((hasDerivAt_id w).sub_const 1).differentiableAt))\n\nlemma deriv_Gprim {n : \u2115} {w : \u2102} (hw : w \u2260 1) :"
}

{
  "file_path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean",
  "old_string": "    have hd : deriv (fun w => (w - 1) ^ (-((i : \u2124) - 1))) w\n        = ((-((i : \u2124) - 1)) : \u2102) * (w - 1) ^ (-((i : \u2124) - 1) - 1) := by\n      have h2 : HasDerivAt (fun x : \u2102 => x ^ (-((i : \u2124) - 1)))\n          ((-((i : \u2124) - 1)) * (w - 1) ^ (-((i : \u2124) - 1) - 1)) (w - 1) :=\n        hasDerivAt_zpow (-((i : \u2124) - 1)) (w - 1) (Or.inl hw1)\n      have h3 : HasDerivAt (fun w : \u2102 => w - 1) 1 w := by\n        simpa using (hasDerivAt_id w).sub_const 1\n      have hcomp := h2.comp w h3\n      rw [Function.comp_apply, mul_one] at hcomp\n      exact hcomp.deriv\n    rw [deriv_const_mul _ (by\n      have hw1' : w - 1 \u2260 0 := sub_ne_zero.mpr hw\n      exact (((hasDerivAt_zpow (-((i : \u2124) - 1)) (w - 1) (Or.inl hw1')).differentiableAt).comp w\n        (((hasDerivAt_id w).sub_const 1).differentiableAt))), hd]\n    have hm : (-((i : \u2124) - 1) - 1 : \u2124) = -(i : \u2124) := by omega\n    rw [hm]\n    have hc : ((-((i : \u2124) - 1)) : \u2102) = 1 - (i : \u2102) := by\n      rw [show (-((i : \u2124) - 1) : \u2124) = 1 - (i : \u2124) from by omega]\n      simp [Int.cast_sub, Int.cast_one]\n    rw [hc]\n    field_simp [hw1, hi1]\n    rw [zpow_neg, zpow_natCast]\n    field_simp [hw1]\n    ring",
  "new_string": "    have hd : deriv (fun w => (w - 1) ^ (-((i : \u2124) - 1))) w\n        = ((-((i : \u2124) - 1) : \u2124) : \u2102) * (w - 1) ^ (-((i : \u2124) - 1) - 1) := by\n      have h2 : HasDerivAt (fun x : \u2102 => x ^ (-((i : \u2124) - 1)))\n          (((-((i : \u2124) - 1) : \u2124) : \u2102) * (w - 1) ^ (-((i : \u2124) - 1) - 1)) (w - 1) :=\n        hasDerivAt_zpow (-((i : \u2124) - 1)) (w - 1) (Or.inl hw1)\n      have h3 : HasDerivAt (fun w : \u2102 => w - 1) 1 w := by\n        simpa using (hasDerivAt_id w).sub_const 1\n      have hcomp := h2.comp w h3\n      rw [mul_one] at hcomp\n      exact hcomp.deriv\n    rw [deriv_const_mul _ (by\n      have hw1' : w - 1 \u2260 0 := sub_ne_zero.mpr hw\n      exact (((hasDerivAt_zpow (-((i : \u2124) - 1)) (w - 1) (Or.inl hw1')).differentiableAt).comp w\n        (((hasDerivAt_id w).sub_const 1).differentiableAt))), hd]\n    have hm : (-((i : \u2124) - 1) - 1 : \u2124) = -(i : \u2124) := by omega\n    rw [hm]\n    push_cast\n    field_simp [hw1, hi1]\n    rw [zpow_neg, zpow_natCast]\n    field_simp [hw1]\n    ring"
}

{
  "filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiDogLeg.lean",
  "oldString": "  rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hF hint]\n  have h1 : (0 : \u211d) \u2264 Real.exp (-c * XR) / c := by positivity\n  have e : Real.exp (-c * 3) = Real.exp (-3 * c) := by\n    congr 1\n    ring\n  rw [e]\n  linarith [h1]",
  "newString": "  rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hF hint, e, sub_le_iff_le_add]\n  have h1 : (0 : \u211d) \u2264 Real.exp (-c * XR) := (Real.exp_pos _).le\n  have h2 : -Real.exp (-c * XR) / c \u2264 0 := by\n    apply div_nonpos_of_nonpos_of_nonneg _ hc.le\n    linarith [h1]\n  linarith [h2]",
  "originalFile": "/-\nNguyenRH1/LiDogLeg.lean \u2014 ii\u2032-6-DL-1 (D-074, session 040): the zero-free\ndog-leg deformation of the Lagarias horizontal right tail.\n\nTHE CONTENT (D-074 \u00a73.1\u20133.2): on the dog-leg rectangle [3, X_R]\u00d7[T, U]\n(T > 0), the Lagarias integrand Jlag n s = (s/(s\u22121))\u207f\u22121 \u00b7 (\u2212\u03b6\u2032/\u03b6(s)) is\nholomorphic UNCONDITIONALLY (s \u2260 1 since im > 0; \u03b6 s \u2260 0 for re s \u2265 3 > 1\nby Mathlib's Euler-product nonvanishing `riemannZeta_ne_zero_of_one_lt_re`\n\u2014 NO perimeter hypothesis, no zero-avoidance at U). Cauchy's theorem\n(`HolomorphicOn.vanishesOnRectangle`) then gives the EXACT deformation\nidentity\n  H(3\u2192X_R, T) = H(3\u2192X_R, U) \u2212 V(X_R, T\u2192U) + V(3, T\u2192U)\n(the upper right tail = the high horizontal \u2212 the right drop + the left\nlift), plus the Jlag conjugation identity for the lower-side transfer.\n\nThis is the repair of Finding F-D (D-072): the broken low-path envelope\nis ABANDONED (not bounded) \u2014 the magnitude bound comes later from the\nthree dog-leg pieces, all in the unconditionally zero-free region.\n-/\nimport Mathlib\nimport NguyenRH1.LiSides\nimport NguyenRH1.LiTightR\nimport NguyenRH1.LiBasePoints\n\nopen Zeta23 Complex Filter Topology\nopen scoped Nat\n\nnoncomputable section\n\nnamespace NguyenRH1\n\n/-! ## 1. Dog-leg analyticity (D-074 \u00a73.1; no perimeter) -/\n\n/-- The dog-leg corners (same real-part/imaginary-part form as rectZ/rectW). -/\ndef dogZ (T : \u211d) : \u2102 := ((3 : \u211d) : \u2102) + (T : \u211d) * Complex.I\ndef dogW (XR U : \u211d) : \u2102 := ((XR : \u211d) : \u2102) + (U : \u211d) * Complex.I\n\nlemma dogZ_re (T : \u211d) : (dogZ T).re = 3 := by simp [dogZ]\nlemma dogZ_im (T : \u211d) : (dogZ T).im = T := by simp [dogZ]\nlemma dogW_re (XR U : \u211d) : (dogW XR U).re = XR := by simp [dogW]\nlemma dogW_im (XR U : \u211d) : (dogW XR U).im = U := by simp [dogW]\n\n/-- Jlag is differentiable at every point with 3 \u2264 re, 0 < im:\n\u03b6 is zero-free for 1 < re (Mathlib `riemannZeta_ne_zero_of_one_lt_re`,\nUNCONDITIONAL \u2014 the Euler-product region) and s \u2260 1 (im \u2260 0).\nNO perimeter hypothesis. -/\ntheorem Jlag_differentiableAt_dogleg {n : \u2115} {s : \u2102}\n    (hre : 3 \u2264 s.re) (him : 0 < s.im) :\n    DifferentiableAt \u2102 (Jlag n) s := by\n  apply Jlag_differentiableAt\n  \u00b7 rintro rfl\n    simp at him\n  \u00b7 exact riemannZeta_ne_zero_of_one_lt_re (by linarith)\n\n/-- Jlag is holomorphic on the whole dog-leg rectangle\n[3, X_R]\u00d7[T, U] (T > 0). -/\ntheorem holomorphicOn_Jlag_dogleg {n : \u2115} {T U XR : \u211d}\n    (hT : 0 < T) (hTU : T \u2264 U) (hXR : 3 \u2264 XR) :\n    HolomorphicOn (Jlag n) (Rectangle (dogZ T) (dogW XR U)) := by\n  intro s hs\n  rw [Rectangle, Complex.mem_reProdIm, dogZ_re, dogW_re, dogZ_im, dogW_im] at hs\n  rw [Set.uIcc_of_le hXR, Set.uIcc_of_le hTU] at hs\n  exact (Jlag_differentiableAt_dogleg hs.1.1 (by linarith [hs.2.1])).differentiableWithinAt\n\n/-- **The dog-leg rectangle integral vanishes** (Cauchy; the region is\nunconditionally zero-free \u2014 the residue sum is EMPTY, no perimeter). -/\ntheorem rectangleIntegral_Jlag_dogleg_eq_zero {n : \u2115} {T U XR : \u211d}\n    (hT : 0 < T) (hTU : T \u2264 U) (hXR : 3 \u2264 XR) :\n    RectangleIntegral (Jlag n) (dogZ T) (dogW XR U) = 0 :=\n  (holomorphicOn_Jlag_dogleg hT hTU hXR).vanishesOnRectangle subset_rfl\n\n/-- The (2\u03c0i)\u207b\u00b9-normalized form (API symmetry with rectInt'_Jlag). -/\ntheorem rectangleIntegral'_Jlag_dogleg_eq_zero {n : \u2115} {T U XR : \u211d}\n    (hT : 0 < T) (hTU : T \u2264 U) (hXR : 3 \u2264 XR) :\n    RectangleIntegral' (Jlag n) (dogZ T) (dogW XR U) = 0 := by\n  rw [RectangleIntegral', rectangleIntegral_Jlag_dogleg_eq_zero hT hTU hXR,\n    smul_zero]\n\n/-- **THE EXACT DOG-LEG DEFORMATION IDENTITY** (D-074 \u00a73.2 (\u2605), the\ncapstone for ii\u2032-6-c): the low right-horizontal equals the high\nhorizontal minus the right drop plus the left lift. Orientations: all\nH-integrals left-to-right, all V-integrals upward (the main contour's\ntop side is their reversal, handled at the fold site). -/\ntheorem upper_right_horizontal_eq_dogleg {n : \u2115} {T U XR : \u211d}\n    (hT : 0 < T) (hTU : T \u2264 U) (hXR : 3 \u2264 XR) :\n    HIntegral (Jlag n) 3 XR T =\n      HIntegral (Jlag n) 3 XR U - VIntegral (Jlag n) XR T U +\n        VIntegral (Jlag n) 3 T U := by\n  have h0 := rectangleIntegral_Jlag_dogleg_eq_zero (n := n) hT hTU hXR\n  unfold RectangleIntegral at h0\n  rw [dogZ_re, dogZ_im, dogW_re, dogW_im] at h0\n  linear_combination h0\n\n/-! ## 2. The conjugation identity (for the lower-side transfer, DL-5) -/\n\n/-- kFun is conjugation-symmetric (real-coefficient rational function). -/\ntheorem kFun_conj (n : \u2115) (s : \u2102) :\n    kFun n ((starRingEnd \u2102) s) = (starRingEnd \u2102) (kFun n s) := by\n  unfold kFun\n  simp [map_sub, map_pow, map_one]\n\n/-- **Jlag is conjugation-symmetric** (from `riemannZeta_conj` +\n`iteratedDeriv_riemannZeta_conj`): Jlag n (s\u0304) = conj (Jlag n s).\nNeeded for the lower-horizontal transfer (DL-5). -/\ntheorem Jlag_conj (n : \u2115) {s : \u2102} (hs : s \u2260 1) :\n    Jlag n ((starRingEnd \u2102) s) = (starRingEnd \u2102) (Jlag n s) := by\n  have h1 : deriv riemannZeta ((starRingEnd \u2102) s) =\n      (starRingEnd \u2102) (deriv riemannZeta s) := by\n    have h := iteratedDeriv_riemannZeta_conj 1 hs\n    rwa [iteratedDeriv_one] at h\n  unfold Jlag\n  rw [kFun_conj, _root_.riemannZeta_conj, h1]\n  simp only [map_mul, map_neg, map_div\u2080]\n\n/-! ## 3. The left-lift bound (DL-2, D-074 \u00a73.3 V_left) -/\n\n/-- **Kernel on the lift**: \u2016kFun n (3+it)\u2016 \u2264 e^{5/2}+1 for n \u2264 t\u00b2\n(D-074 \u00a76: |(3+it)/(2+it)|\u00b2 = (9+t\u00b2)/(4+t\u00b2) = 1 + 5/(4+t\u00b2), whose n-th\npower is \u2264 e^{5n/(2(4+t\u00b2))} \u2264 e^{5/2}). -/\ntheorem kFun_three_add_it_norm_le_exp_five_halves_add_one {n : \u2115} {t : \u211d}\n    (hT : (n : \u211d) \u2264 t ^ 2) :\n    \u2016kFun n ((3 : \u211d) + t * Complex.I)\u2016 \u2264 Real.exp (5 / 2) + 1 := by\n  rcases Nat.eq_zero_or_pos n with hn | hn\n  \u00b7 subst hn\n    simp only [kFun, pow_zero, sub_self, norm_zero]\n    positivity\n  \u00b7 have hnR : (1 : \u211d) \u2264 (n : \u211d) := by exact_mod_cast hn\n    have ht2 : (0 : \u211d) < 4 + t ^ 2 := by positivity\n    have hn4t2 : (n : \u211d) \u2264 4 + t ^ 2 := by linarith [hT]\n    have hQpos : (0 : \u211d) \u2264 (9 + t ^ 2) / (4 + t ^ 2) := by positivity\n    have hQle : (9 + t ^ 2) / (4 + t ^ 2) \u2264 Real.exp (5 / (n : \u211d)) := by\n      have h1 : (9 + t ^ 2) / (4 + t ^ 2) = 1 + 5 / (4 + t ^ 2) := by\n        field_simp [ht2.ne']\n        ring\n      rw [h1]\n      have h2 : 5 / (4 + t ^ 2) \u2264 5 / (n : \u211d) :=\n        (div_le_div_iff_of_pos_left (by norm_num) ht2 (by positivity)).mpr hn4t2\n      have h3 := Real.add_one_le_exp (5 / (n : \u211d))\n      linarith\n    have hQn : ((9 + t ^ 2) / (4 + t ^ 2)) ^ n \u2264 Real.exp 5 := by\n      calc ((9 + t ^ 2) / (4 + t ^ 2)) ^ n \u2264 (Real.exp (5 / (n : \u211d))) ^ n :=\n            pow_le_pow_left\u2080 hQpos hQle n\n        _ = Real.exp 5 := by\n            rw [\u2190 Real.exp_nat_mul]\n            congr 1\n            have hn0 : (n : \u211d) \u2260 0 := by exact_mod_cast hn.ne'\n            field_simp [hn0]\n    have hns3 : Complex.normSq ((3 : \u211d) + t * Complex.I) = 9 + t ^ 2 := by\n      simp [Complex.normSq_apply]\n      ring\n    have hns2 : Complex.normSq ((2 : \u211d) + t * Complex.I) = 4 + t ^ 2 := by\n      simp [Complex.normSq_apply]\n      ring\n    have hnormA : \u2016((3 : \u211d) + t * Complex.I)\u2016 = Real.sqrt (9 + t ^ 2) := by\n      have h1 : \u2016((3 : \u211d) + t * Complex.I)\u2016 =\n          Real.sqrt (Complex.normSq ((3 : \u211d) + t * Complex.I)) := by\n        rw [Complex.normSq_eq_norm_sq, Real.sqrt_sq (norm_nonneg _)]\n      rw [h1, hns3]\n    have hnormB : \u2016((2 : \u211d) + t * Complex.I)\u2016 = Real.sqrt (4 + t ^ 2) := by\n      have h1 : \u2016((2 : \u211d) + t * Complex.I)\u2016 =\n          Real.sqrt (Complex.normSq ((2 : \u211d) + t * Complex.I)) := by\n        rw [Complex.normSq_eq_norm_sq, Real.sqrt_sq (norm_nonneg _)]\n      rw [h1, hns2]\n    have hnorm : \u2016((3 : \u211d) + t * Complex.I) / ((2 : \u211d) + t * Complex.I)\u2016 =\n        Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2)) := by\n      rw [norm_div, hnormA, hnormB,\n        \u2190 Real.sqrt_div (by positivity : (0 : \u211d) \u2264 9 + t ^ 2)]\n    have hs1 : ((3 : \u211d) + t * Complex.I) - 1 = (2 : \u211d) + t * Complex.I := by\n      push_cast\n      ring\n    unfold kFun\n    rw [hs1]\n    calc \u2016(((3 : \u211d) + t * Complex.I) / ((2 : \u211d) + t * Complex.I)) ^ n - 1\u2016\n        \u2264 \u2016(((3 : \u211d) + t * Complex.I) / ((2 : \u211d) + t * Complex.I)) ^ n\u2016 + \u2016(1 : \u2102)\u2016 :=\n          norm_sub_le _ _\n      _ = (Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n + 1 := by\n          rw [norm_pow, hnorm, norm_one]\n      _ \u2264 Real.exp (5 / 2) + 1 := by\n          have h4 : (Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n \u2264 Real.exp (5 / 2) := by\n            have hx0 : (0 : \u211d) \u2264 Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2)) :=\n              Real.sqrt_nonneg _\n            have hx : (0 : \u211d) \u2264 (Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n :=\n              pow_nonneg hx0 n\n            have h5 : ((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2 \u2264 Real.exp 5 := by\n              have e : ((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2 =\n                  ((9 + t ^ 2) / (4 + t ^ 2)) ^ n := by\n                rw [\u2190 pow_mul, mul_comm n 2, pow_mul, Real.sq_sqrt hQpos]\n              rw [e]\n              exact hQn\n            have h8 : (Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n =\n                Real.sqrt (((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2) := by\n              rw [Real.sqrt_sq hx]\n            rw [h8]\n            calc Real.sqrt (((Real.sqrt ((9 + t ^ 2) / (4 + t ^ 2))) ^ n) ^ 2)\n                \u2264 Real.sqrt (Real.exp 5) := Real.sqrt_le_sqrt h5\n              _ = Real.exp (5 / 2) := (Real.exp_half 5).symm\n          linarith [h4]\n\n/-- The Dirichlet majorant at s = 3 (summable since 3 > 1). -/\ndef dirichletC3 : \u211d :=\n  \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016\n\n/-- Termwise 3-vs-2 comparison for the \u039b L-series: \u039b(n)/n\u00b3 \u2264 (1/2)\u00b7\u039b(n)/n\u00b2. -/\nlemma norm_term_vonMangoldt_three_le_half (n : \u2115) :\n    \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016 \u2264\n      (1 / 2) * \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016 := by\n  have h3re : ((3 : \u2102).re) = 3 := rfl\n  have h2re : ((2 : \u2102).re) = 2 := rfl\n  simp only [LSeries.norm_term_eq, h3re, h2re]\n  split\n  \u00b7 norm_num\n  \u00b7 next hn =>\n    rcases eq_or_ne n 1 with rfl | hn1\n    \u00b7 simp [ArithmeticFunction.vonMangoldt_apply_one]\n    \u00b7 have hn2 : 2 \u2264 n := by omega\n      have h\u039b : \u2016(ArithmeticFunction.vonMangoldt n : \u2102)\u2016 =\n          ArithmeticFunction.vonMangoldt n :=\n        Complex.norm_of_nonneg ArithmeticFunction.vonMangoldt_nonneg\n      rw [h\u039b]\n      have e : ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (3 : \u211d) =\n          (1 / (n : \u211d)) * (ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (2 : \u211d)) := by\n        rw [show (3 : \u211d) = 2 + 1 by norm_num,\n          Real.rpow_add (by positivity : (0 : \u211d) < (n : \u211d)), Real.rpow_one]\n        field_simp [show (n : \u211d) \u2260 0 by positivity]\n      have h12 : (1 : \u211d) / (n : \u211d) \u2264 1 / 2 :=\n        one_div_le_one_div_of_le (by norm_num : (0 : \u211d) < 2)\n          (by exact_mod_cast hn2 : (2 : \u211d) \u2264 (n : \u211d))\n      have hpos : (0 : \u211d) \u2264 ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (2 : \u211d) :=\n        div_nonneg ArithmeticFunction.vonMangoldt_nonneg\n          (Real.rpow_nonneg (by positivity : (0 : \u211d) \u2264 (n : \u211d)) _)\n      calc ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (3 : \u211d)\n          = (1 / (n : \u211d)) * (ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (2 : \u211d)) := e\n        _ \u2264 (1 / 2) * (ArithmeticFunction.vonMangoldt n / (n : \u211d) ^ (2 : \u211d)) :=\n            mul_le_mul_of_nonneg_right h12 hpos\n\n/-- dirichletC3 \u2264 1/2 (the termwise comparison + dirichletC2 \u2264 1) \u2014\nthe certified C_zeta3 = 0.5 of D-074 \u00a74. (A tighter 0.17 build is\npossible via a 6-prime partial sum + antitone-integral tail; the fold\nis indifferent to the constant, so the cheap certified 0.5 is charged.) -/\nlemma dirichletC3_le_half : dirichletC3 \u2264 1 / 2 := by\n  have hsum3 : Summable (fun n : \u2115 =>\n      \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016) := by\n    rw [summable_norm_iff]\n    exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n  have hsum2 : Summable (fun n : \u2115 =>\n      \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016) := by\n    rw [summable_norm_iff]\n    exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n  calc dirichletC3\n      = \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016 := rfl\n    _ \u2264 \u2211' n : \u2115, (1 / 2) *\n          \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016 :=\n        Summable.tsum_le_tsum (fun n => norm_term_vonMangoldt_three_le_half n) hsum3\n          (hsum2.mul_left (1 / 2))\n    _ = (1 / 2) * dirichletC2 := by rw [dirichletC2, tsum_mul_left]\n    _ \u2264 1 / 2 := by linarith [dirichletC2_le_one]\n\n/-- \u2016\u03b6\u2032/\u03b6 s\u2016 \u2264 dirichletC3 for re s \u2265 3 (the s = 3 clone of the\nDirichlet-series majorant). -/\nlemma norm_logDeriv_zeta_le_dirichletC3 {s : \u2102} (hs : 3 \u2264 s.re) :\n    \u2016logDeriv riemannZeta s\u2016 \u2264 dirichletC3 := by\n  have hs1 : (1 : \u211d) < s.re := by linarith\n  have h1 : -logDeriv riemannZeta s\n      = LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s := by\n    have h2 := ArithmeticFunction.LSeries_vonMangoldt_eq_deriv_riemannZeta_div hs1\n    rw [logDeriv, Pi.div_apply, h2]\n    ring\n  have hsum3 : Summable (fun n : \u2115 =>\n      \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016) := by\n    rw [summable_norm_iff]\n    exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n  have hsumS : Summable (fun n : \u2115 =>\n      \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016) := by\n    rw [summable_norm_iff]\n    exact ArithmeticFunction.LSeriesSummable_vonMangoldt hs1\n  have hle : \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s\u2016\n      \u2264 dirichletC3 := by\n    show \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016 \u2264 _\n    calc \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016\n        \u2264 \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s n\u2016 :=\n          norm_tsum_le_tsum_norm hsumS\n      _ \u2264 \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (3 : \u2102) n\u2016 :=\n          Summable.tsum_le_tsum (fun n => LSeries.norm_term_le_of_re_le_re _ (by\n            simp\n            exact hs) n) hsumS hsum3\n      _ = dirichletC3 := rfl\n  calc \u2016logDeriv riemannZeta s\u2016\n      = \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) s\u2016 := by\n        rw [\u2190 h1, norm_neg]\n    _ \u2264 dirichletC3 := hle\n\n/-- \u2016\u03b6\u2032/\u03b6 s\u2016 \u2264 1/2 for re s \u2265 3 (the charged C_zeta3 = 0.5). -/\nlemma norm_logDeriv_zeta_le_half {s : \u2102} (hs : 3 \u2264 s.re) :\n    \u2016logDeriv riemannZeta s\u2016 \u2264 1 / 2 :=\n  (norm_logDeriv_zeta_le_dirichletC3 hs).trans dirichletC3_le_half\n\n/-- **The left-lift bound** (D-074 \u00a73.3 V_left): for 0 < T \u2264 U and\nn \u2264 T\u00b2, \u2016VIntegral (Jlag n) 3 T U\u2016 \u2264 (U \u2212 T)\u00b7(e^{5/2}+1)/2. The kernel\n\u2264 e^{5/2}+1 (the 1+5/(4+t\u00b2) computation), the zeta factor \u2264 1/2\n(certified, no perimeter). -/\ntheorem vIntegral_Jlag_three_norm_le {n : \u2115} {T U : \u211d}\n    (hT0 : (0 : \u211d) < T) (hTU : T \u2264 U) (hTn : (n : \u211d) \u2264 T ^ 2) :\n    \u2016VIntegral (Jlag n) 3 T U\u2016 \u2264 (U - T) * (Real.exp (5 / 2) + 1) / 2 := by\n  have hnorm : \u2016VIntegral (Jlag n) 3 T U\u2016 =\n      \u2016\u222b t in T..U, Jlag n ((3 : \u211d) + t * Complex.I)\u2016 := by\n    unfold VIntegral\n    rw [norm_smul, Complex.norm_I, one_mul]\n  rw [hnorm]\n  have hbound : \u2200 t \u2208 Set.Ioc T U,\n      \u2016Jlag n ((3 : \u211d) + t * Complex.I)\u2016 \u2264 (Real.exp (5 / 2) + 1) * (1 / 2) := by\n    intro t ht\n    have htT : T < t := ht.1\n    have ht2 : (n : \u211d) \u2264 t ^ 2 := by nlinarith [hTn, hT0]\n    have hre : (3 : \u211d) \u2264 ((3 : \u211d) + t * Complex.I).re := by simp\n    have h1 := kFun_three_add_it_norm_le_exp_five_halves_add_one ht2\n    have h2 := norm_logDeriv_zeta_le_half (s := (3 : \u211d) + t * Complex.I) hre\n    have hnorm3 : \u2016Jlag n ((3 : \u211d) + t * Complex.I)\u2016 =\n        \u2016kFun n ((3 : \u211d) + t * Complex.I)\u2016 *\n          \u2016logDeriv riemannZeta ((3 : \u211d) + t * Complex.I)\u2016 := by\n      show \u2016kFun n ((3 : \u211d) + t * Complex.I) *\n          (-logDeriv riemannZeta ((3 : \u211d) + t * Complex.I))\u2016 = _\n      rw [norm_mul, norm_neg]\n    rw [hnorm3]\n    exact mul_le_mul h1 h2 (norm_nonneg _) (by positivity)\n  have hint : \u2016\u222b t in T..U, Jlag n ((3 : \u211d) + t * Complex.I)\u2016 \u2264\n      ((Real.exp (5 / 2) + 1) * (1 / 2)) * |U - T| := by\n    apply intervalIntegral.norm_integral_le_of_norm_le_const\n    intro t ht\n    rw [Set.uIoc_of_le hTU] at ht\n    exact hbound t ht\n  have habs : |U - T| = U - T := abs_of_nonneg (by linarith)\n  rw [habs] at hint\n  calc \u2016\u222b t in T..U, Jlag n ((3 : \u211d) + t * Complex.I)\u2016\n      \u2264 ((Real.exp (5 / 2) + 1) * (1 / 2)) * (U - T) := hint\n    _ = (U - T) * (Real.exp (5 / 2) + 1) / 2 := by ring\n\n/-! ## 4. The high-horizontal stepping (DL-3, D-074 \u00a77 + D-073 \u00a74) -/\n\n/-- Termwise \u03c3-vs-2 comparison for the \u039b L-series (the (6.10) halving):\n\u039b(m)/m^\u03c3 \u2264 2^{\u2212(\u03c3\u22122)}\u00b7\u039b(m)/m\u00b2 for \u03c3 \u2265 2. -/\nlemma norm_term_vonMangoldt_le_two_pow {\u03c3 t : \u211d} (h\u03c3 : 2 \u2264 \u03c3) (m : \u2115) :\n    \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n        ((\u03c3 : \u211d) + t * Complex.I) m\u2016 \u2264\n      (2 : \u211d) ^ (-(\u03c3 - 2)) * \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) m\u2016 := by\n  have h\u03c3re : ((\u03c3 : \u211d) + t * Complex.I).re = \u03c3 := by simp\n  have h2re : ((2 : \u2102).re) = 2 := rfl\n  simp only [LSeries.norm_term_eq, h\u03c3re, h2re]\n  split\n  \u00b7 norm_num\n  \u00b7 next hm =>\n    have h\u039b : \u2016(ArithmeticFunction.vonMangoldt m : \u2102)\u2016 =\n        ArithmeticFunction.vonMangoldt m :=\n      Complex.norm_of_nonneg ArithmeticFunction.vonMangoldt_nonneg\n    rw [h\u039b]\n    rcases eq_or_ne m 1 with rfl | hm1\n    \u00b7 simp [ArithmeticFunction.vonMangoldt_apply_one]\n    \u00b7 have hm2 : 2 \u2264 m := by omega\n      have e : ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ \u03c3 =\n          (1 / (m : \u211d) ^ (\u03c3 - 2)) * (ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ (2 : \u211d)) := by\n        rw [show (\u03c3 : \u211d) = (\u03c3 - 2) + 2 by ring,\n          Real.rpow_add (by positivity : (0 : \u211d) < (m : \u211d))]\n        field_simp [show (m : \u211d) \u2260 0 by positivity]\n        ring_nf\n      rw [e]\n      have hle : (1 : \u211d) / (m : \u211d) ^ (\u03c3 - 2) \u2264 (2 : \u211d) ^ (-(\u03c3 - 2)) := by\n        rw [Real.rpow_neg (by norm_num : (0 : \u211d) \u2264 2), \u2190 one_div]\n        apply one_div_le_one_div_of_le (Real.rpow_pos_of_pos (by norm_num : (0 : \u211d) < 2) _)\n        exact Real.rpow_le_rpow (by norm_num : (0 : \u211d) \u2264 2)\n          (by exact_mod_cast hm2 : (2 : \u211d) \u2264 (m : \u211d)) (by linarith : (0 : \u211d) \u2264 \u03c3 - 2)\n      have hpos : (0 : \u211d) \u2264 ArithmeticFunction.vonMangoldt m / (m : \u211d) ^ (2 : \u211d) :=\n        div_nonneg ArithmeticFunction.vonMangoldt_nonneg\n          (Real.rpow_nonneg (by positivity : (0 : \u211d) \u2264 (m : \u211d)) _)\n      exact mul_le_mul_of_nonneg_right hle hpos\n\n/-- \u2016\u03b6\u2032/\u03b6(\u03c3+it)\u2016 \u2264 2^{\u2212(\u03c3\u22122)}\u00b7dirichletC2 for \u03c3 \u2265 2 (the (6.10)\nhalving in LSeries form; dirichletC2 \u2264 1 makes the tail free). -/\nlemma norm_logDeriv_zeta_le_two_pow {\u03c3 t : \u211d} (h\u03c3 : 2 \u2264 \u03c3) :\n    \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264\n      (2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2 := by\n  have hs1 : (1 : \u211d) < ((\u03c3 : \u211d) + t * Complex.I).re := by simp; linarith\n  have h1 : -logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I)\n      = LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) ((\u03c3 : \u211d) + t * Complex.I) := by\n    have h2 := ArithmeticFunction.LSeries_vonMangoldt_eq_deriv_riemannZeta_div hs1\n    rw [logDeriv, Pi.div_apply, h2]\n    ring\n  have hsum2 : Summable (fun n : \u2115 =>\n      \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016) := by\n    rw [summable_norm_iff]\n    exact ArithmeticFunction.LSeriesSummable_vonMangoldt (by norm_num)\n  have hsumS : Summable (fun n : \u2115 =>\n      \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n        ((\u03c3 : \u211d) + t * Complex.I) n\u2016) := by\n    rw [summable_norm_iff]\n    exact ArithmeticFunction.LSeriesSummable_vonMangoldt hs1\n  have hle : \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n      ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264 (2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2 := by\n    show \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n      ((\u03c3 : \u211d) + t * Complex.I) n\u2016 \u2264 _\n    calc \u2016\u2211' n : \u2115, LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n          ((\u03c3 : \u211d) + t * Complex.I) n\u2016\n        \u2264 \u2211' n : \u2115, \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n            ((\u03c3 : \u211d) + t * Complex.I) n\u2016 :=\n          norm_tsum_le_tsum_norm hsumS\n      _ \u2264 \u2211' n : \u2115, (2 : \u211d) ^ (-(\u03c3 - 2)) *\n            \u2016LSeries.term (fun n => (ArithmeticFunction.vonMangoldt n : \u2102)) (2 : \u2102) n\u2016 :=\n          Summable.tsum_le_tsum (fun n => norm_term_vonMangoldt_le_two_pow h\u03c3 n) hsumS\n            (hsum2.mul_left _)\n      _ = (2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2 := by rw [dirichletC2, tsum_mul_left]\n  calc \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I)\u2016\n      = \u2016LSeries (fun n => (ArithmeticFunction.vonMangoldt n : \u2102))\n          ((\u03c3 : \u211d) + t * Complex.I)\u2016 := by\n        rw [\u2190 h1, norm_neg]\n    _ \u2264 (2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2 := hle\n\n/-- **Kernel envelope** (D-074 \u00a77, the single-exponential form that\nreplaces the unit-step geometric series): \u2016(1+1/((\u03c3\u22121)+iU))\u207f\u2016 \u2264\ne^{(4/9)\u03c3} for \u03c3 \u2265 3 and n \u2264 (4/9)U\u00b2. From the (6.11) core\n\u20161+1/s\u2016 \u2264 1+(re s+1)/normSq s \u2264 1+\u03c3/U\u00b2. -/\ntheorem one_add_div_pow_norm_le_exp_four_ninths {n : \u2115} {\u03c3 U : \u211d}\n    (h\u03c3 : 3 \u2264 \u03c3) (hU : 0 < U) (hnU : (n : \u211d) \u2264 (4 / 9) * U ^ 2) :\n    \u2016(1 + 1/(((\u03c3 : \u211d) - 1) + U * Complex.I))\u2016 ^ n \u2264 Real.exp ((4 / 9) * \u03c3) := by\n  have hs0 : (((\u03c3 : \u211d) - 1) + U * Complex.I) \u2260 0 := by\n    intro h\n    have him : (((\u03c3 : \u211d) - 1) + U * Complex.I).im = 0 := by rw [h]; simp\n    simp at him\n    linarith\n  have hle := norm_one_add_one_div_le hs0\n  have hre : (((\u03c3 : \u211d) - 1) + U * Complex.I).re + 1 = \u03c3 := by\n    simp [Complex.add_re, Complex.ofReal_re, Complex.mul_re, Complex.I_re, Complex.I_im,\n      Complex.sub_re, Complex.one_re]\n  have hns : Complex.normSq (((\u03c3 : \u211d) - 1) + U * Complex.I) = (\u03c3 - 1) ^ 2 + U ^ 2 := by\n    simp [Complex.normSq_apply]\n    ring\n  have hstep : ((((\u03c3 : \u211d) - 1) + U * Complex.I).re + 1) /\n      Complex.normSq (((\u03c3 : \u211d) - 1) + U * Complex.I) \u2264 \u03c3 / U ^ 2 := by\n    rw [hre, hns]\n    have hU2 : (0 : \u211d) < U ^ 2 := by positivity\n    have h1 : (0 : \u211d) \u2264 (\u03c3 - 1) ^ 2 := sq_nonneg _\n    exact (div_le_div_iff_of_pos_left (by linarith : (0 : \u211d) < \u03c3)\n      (by positivity) hU2).mpr (by linarith [h1])\n  have hstep1 : 1 + ((((\u03c3 : \u211d) - 1) + U * Complex.I).re + 1) /\n      Complex.normSq (((\u03c3 : \u211d) - 1) + U * Complex.I) \u2264 1 + \u03c3 / U ^ 2 := by\n    linarith [hstep]\n  have hpow1 : \u2016(1 + 1/(((\u03c3 : \u211d) - 1) + U * Complex.I))\u2016 ^ n \u2264 (1 + \u03c3 / U ^ 2) ^ n :=\n    pow_le_pow_left\u2080 (norm_nonneg _) (hle.trans hstep1) n\n  have hpow2 : (1 + \u03c3 / U ^ 2) ^ n \u2264 Real.exp ((n : \u211d) * (\u03c3 / U ^ 2)) := by\n    calc (1 + \u03c3 / U ^ 2) ^ n \u2264 (Real.exp (\u03c3 / U ^ 2)) ^ n := by\n          have h := Real.add_one_le_exp (\u03c3 / U ^ 2)\n          exact pow_le_pow_left\u2080 (by positivity) (by linarith) n\n      _ = Real.exp ((n : \u211d) * (\u03c3 / U ^ 2)) := (Real.exp_nat_mul (\u03c3 / U ^ 2) n).symm\n  have hexp : (n : \u211d) * (\u03c3 / U ^ 2) \u2264 (4 / 9) * \u03c3 := by\n    have hU2 : (0 : \u211d) < U ^ 2 := by positivity\n    have h\u03c30 : (0 : \u211d) < \u03c3 := by linarith\n    have h1 : (n : \u211d) / U ^ 2 \u2264 4 / 9 := by\n      have h2 := div_le_div_of_nonneg_right hnU hU2.le\n      rwa [mul_div_cancel_right\u2080 _ hU2.ne'] at h2\n    rw [show (n : \u211d) * (\u03c3 / U ^ 2) = ((n : \u211d) / U ^ 2) * \u03c3 by ring]\n    exact mul_le_mul_of_nonneg_right h1 h\u03c30.le\n  calc \u2016(1 + 1/(((\u03c3 : \u211d) - 1) + U * Complex.I))\u2016 ^ n\n      \u2264 Real.exp ((n : \u211d) * (\u03c3 / U ^ 2)) := hpow1.trans hpow2\n    _ \u2264 Real.exp ((4 / 9) * \u03c3) := Real.exp_le_exp.mpr hexp\n\n/-! ## 5. The product envelope and the integral assembly (DL-3c/d) -/\n\n/-- **The product envelope** (D-074 \u00a77): for \u03c3 \u2265 3, 0 < t, n \u2264 (4/9)t\u00b2,\n\u2016Jlag n (\u03c3+it)\u2016 \u2264 (4\u00b7e^{\u2212c\u03c3} + 4\u00b72^{\u2212\u03c3})\u00b7dirichletC2, c = log 2 \u2212 4/9.\nThe kernel envelope (e^{(4/9)\u03c3}) times the zeta halving (2^{\u2212(\u03c3\u22122)})\ncollapses to the single exponential 4\u00b7e^{\u2212c\u03c3}: no geometric series. -/\ntheorem Jlag_dogleg_norm_le_envelope {n : \u2115} {\u03c3 t : \u211d}\n    (h\u03c3 : 3 \u2264 \u03c3) (ht : 0 < t) (hnU : (n : \u211d) \u2264 (4 / 9) * t ^ 2) :\n    \u2016Jlag n ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264\n      (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) * dirichletC2 := by\n  have hw1 : ((\u03c3 : \u211d) + t * Complex.I) - 1 = ((\u03c3 : \u211d) - 1) + t * Complex.I := by\n    push_cast\n    ring\n  have hs1ne : (((\u03c3 : \u211d) - 1) + t * Complex.I) \u2260 0 := by\n    intro h\n    have him : (((\u03c3 : \u211d) - 1) + t * Complex.I).im = 0 := by rw [h]; simp\n    simp at him\n    linarith\n  have hwid : ((\u03c3 : \u211d) + t * Complex.I) / (((\u03c3 : \u211d) + t * Complex.I) - 1) =\n      1 + 1 / (((\u03c3 : \u211d) - 1) + t * Complex.I) := by\n    rw [hw1]\n    field_simp [hs1ne]\n    ring\n  have hker := one_add_div_pow_norm_le_exp_four_ninths h\u03c3 ht hnU\n  have hzeta := norm_logDeriv_zeta_le_two_pow (show (2 : \u211d) \u2264 \u03c3 by linarith) (t := t)\n  have hk : \u2016kFun n ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264 Real.exp ((4 / 9) * \u03c3) + 1 := by\n    unfold kFun\n    rw [hwid]\n    calc \u2016(1 + 1 / (((\u03c3 : \u211d) - 1) + t * Complex.I)) ^ n - 1\u2016\n        \u2264 \u2016(1 + 1 / (((\u03c3 : \u211d) - 1) + t * Complex.I)) ^ n\u2016 + \u2016(1 : \u2102)\u2016 := norm_sub_le _ _\n      _ = \u2016(1 + 1 / (((\u03c3 : \u211d) - 1) + t * Complex.I))\u2016 ^ n + 1 := by rw [norm_pow, norm_one]\n      _ \u2264 Real.exp ((4 / 9) * \u03c3) + 1 := by linarith [hker]\n  have hprod : \u2016Jlag n ((\u03c3 : \u211d) + t * Complex.I)\u2016 \u2264\n      (Real.exp ((4 / 9) * \u03c3) + 1) * ((2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2) := by\n    have hn3 : \u2016Jlag n ((\u03c3 : \u211d) + t * Complex.I)\u2016 =\n        \u2016kFun n ((\u03c3 : \u211d) + t * Complex.I)\u2016 *\n          \u2016logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I)\u2016 := by\n      show \u2016kFun n ((\u03c3 : \u211d) + t * Complex.I) *\n          (-logDeriv riemannZeta ((\u03c3 : \u211d) + t * Complex.I))\u2016 = _\n      rw [norm_mul, norm_neg]\n    rw [hn3]\n    exact mul_le_mul hk hzeta (norm_nonneg _)\n      (add_nonneg (Real.exp_nonneg _) (by norm_num))\n  have h2b : (2 : \u211d) ^ (-\u03c3) = Real.exp (-(Real.log 2) * \u03c3) := by\n    rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 2)]\n    congr 1\n    ring\n  have h22 : (2 : \u211d) ^ (2 : \u211d) = 4 := by\n    rw [Real.rpow_two]\n    norm_num\n  have h2a : (2 : \u211d) ^ (-(\u03c3 - 2)) = 4 * (2 : \u211d) ^ (-\u03c3) := by\n    rw [show (-(\u03c3 - 2)) = -\u03c3 + 2 by ring, Real.rpow_add (by norm_num : (0 : \u211d) < 2), h2b, h22]\n    ring\n  have hexp_alg : Real.exp ((4 / 9) * \u03c3) * Real.exp (-(Real.log 2) * \u03c3) =\n      Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) := by\n    rw [\u2190 Real.exp_add]\n    congr 1\n    ring\n  have halg : (Real.exp ((4 / 9) * \u03c3) + 1) * ((2 : \u211d) ^ (-(\u03c3 - 2))) =\n      4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3) := by\n    rw [h2a, h2b]\n    calc (Real.exp ((4 / 9) * \u03c3) + 1) * (4 * Real.exp (-(Real.log 2) * \u03c3))\n        = 4 * (Real.exp ((4 / 9) * \u03c3) * Real.exp (-(Real.log 2) * \u03c3)) +\n          4 * Real.exp (-(Real.log 2) * \u03c3) := by ring\n      _ = 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * Real.exp (-(Real.log 2) * \u03c3) := by\n        rw [hexp_alg]\n  calc \u2016Jlag n ((\u03c3 : \u211d) + t * Complex.I)\u2016\n      \u2264 (Real.exp ((4 / 9) * \u03c3) + 1) * ((2 : \u211d) ^ (-(\u03c3 - 2)) * dirichletC2) := hprod\n    _ = dirichletC2 * ((Real.exp ((4 / 9) * \u03c3) + 1) * (2 : \u211d) ^ (-(\u03c3 - 2))) := by ring\n    _ = dirichletC2 * (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) := by\n        rw [halg]\n\n/-- The FTC-bound \u222b\u2083^XR e^{\u2212c\u03c3}d\u03c3 \u2264 e^{\u22123c}/c for c > 0. -/\nlemma integral_exp_neg_le {c XR : \u211d} (hc : 0 < c) (_hXR : 3 \u2264 XR) :\n    \u222b \u03c3 in (3 : \u211d)..XR, Real.exp (-c * \u03c3) \u2264 Real.exp (-3 * c) / c := by\n  have hF : \u2200 x \u2208 Set.uIcc (3 : \u211d) XR,\n      HasDerivAt (fun \u03c3 => -Real.exp (-c * \u03c3) / c) (Real.exp (-c * x)) x := by\n    intro x hx\n    have h2 : HasDerivAt (fun \u03c3 => Real.exp (-c * \u03c3)) (Real.exp (-c * x) * (-c)) x := by\n      have h := ((hasDerivAt_id' x).const_mul (-c)).exp\n      rwa [mul_one] at h\n    have h3 := h2.const_mul (-1 / c)\n    rw [show (-1 / c) * (Real.exp (-c * x) * (-c)) = Real.exp (-c * x) by\n      field_simp [hc.ne']] at h3\n    have h5 : (fun \u03c3 => (-1 / c) * Real.exp (-c * \u03c3)) =\n        (fun \u03c3 => -Real.exp (-c * \u03c3) / c) := by\n      funext \u03c3\n      field_simp [hc.ne']\n    rw [h5] at h3\n    exact h3\n  have hint : IntervalIntegrable (fun \u03c3 => Real.exp (-c * \u03c3)) MeasureTheory.volume 3 XR := by\n    apply ContinuousOn.intervalIntegrable\n    exact (Real.continuous_exp.comp (continuous_const.mul continuous_id)).continuousOn\n  rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hF hint]\n  have h1 : (0 : \u211d) \u2264 Real.exp (-c * XR) / c := by positivity\n  have e : Real.exp (-c * 3) = Real.exp (-3 * c) := by\n    congr 1\n    ring\n  rw [e]\n  linarith [h1]\n\n/-- The certified exponent-decay constant c = log 2 \u2212 4/9 \u2265 0.2487. -/\nlemma cdog_gt : (0.2487 : \u211d) \u2264 Real.log 2 - 4 / 9 := by\n  have h := Real.log_two_gt_d9\n  norm_num at h\n  linarith\n\n/-- The certified e^{4/3} \u2264 3.8073 (the (1\u2212x/16)\u207b\u00b9\u2076 route + d9 e). -/\nlemma exp_four_thirds_le : Real.exp (4 / 3) \u2264 (3.8073 : \u211d) := by\n  have he13 : Real.exp (1 / 3) \u2264 (1.4006 : \u211d) :=\n    le_trans (exp_le_inv_sub_sixteen (by norm_num) (by norm_num)) (by norm_num)\n  rw [show (4 / 3 : \u211d) = 1 + 1 / 3 by norm_num, Real.exp_add]\n  calc Real.exp 1 * Real.exp (1 / 3) \u2264 2.7182818286 * 1.4006 :=\n        mul_le_mul Real.exp_one_lt_d9.le he13 (Real.exp_pos _).le (by norm_num)\n    _ \u2264 3.8073 := by norm_num\n\n/-- **THE HIGH-HORIZONTAL BOUND** (D-074 \u00a77 capstone; D-073 Cor 4.2's\nR(1.5) in single-exponential-envelope form): for 0 < U, n \u2264 (4/9)U\u00b2\n(i.e. U \u2265 (3/2)\u221an) and 3 \u2264 XR, \u2016HIntegral (Jlag n) 3 XR U\u2016 \u2264 8.38\n(charged; the true value \u2264 1.2 at n = 10\u2075, E-034 V5). Charged with\ndirichletC2 \u2264 1 \u2014 the tight R(1.5) = 5.94 would need a dirichletC2 \u2264\n0.57 build (documented as the tightening path; the fold is\nindifferent: \u0394 \u2248 +2.4 per side). -/\ntheorem hIntegral_Jlag_dogleg_norm_le {n : \u2115} {U XR : \u211d}\n    (hU : 0 < U) (hnU : (n : \u211d) \u2264 (4 / 9) * U ^ 2) (hXR : 3 \u2264 XR) :\n    \u2016HIntegral (Jlag n) 3 XR U\u2016 \u2264 (8.38 : \u211d) := by\n  have hc : (0 : \u211d) < Real.log 2 - 4 / 9 := by\n    have h := cdog_gt\n    linarith\n  have hpt : \u2200 \u03c3 \u2208 Set.Ioc 3 XR,\n      \u2016Jlag n ((\u03c3 : \u211d) + U * Complex.I)\u2016 \u2264\n        (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) * dirichletC2 :=\n    fun \u03c3 h\u03c3 => Jlag_dogleg_norm_le_envelope (by linarith [h\u03c3.1]) hU hnU\n  have hcontA : Continuous (fun \u03c3 : \u211d => Real.exp (-(Real.log 2 - 4 / 9) * \u03c3)) :=\n    Real.continuous_exp.comp (continuous_const.mul continuous_id)\n  have hcontB : Continuous (fun \u03c3 : \u211d => (2 : \u211d) ^ (-\u03c3)) := by\n    have h1 : Continuous (fun \u03c3 : \u211d => Real.exp (-(Real.log 2) * \u03c3)) :=\n      Real.continuous_exp.comp (continuous_const.mul continuous_id)\n    have h2 : (fun \u03c3 : \u211d => (2 : \u211d) ^ (-\u03c3)) =\n        (fun \u03c3 : \u211d => Real.exp (-(Real.log 2) * \u03c3)) := by\n      funext \u03c3\n      rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 2)]\n      congr 1\n      ring\n    rw [h2]\n    exact h1\n  have hIntA : IntervalIntegrable (fun \u03c3 : \u211d => 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3))\n      MeasureTheory.volume 3 XR := by\n    apply ContinuousOn.intervalIntegrable\n    exact (hcontA.const_mul 4).continuousOn\n  have hIntB : IntervalIntegrable (fun \u03c3 : \u211d => 4 * (2 : \u211d) ^ (-\u03c3)) MeasureTheory.volume 3 XR := by\n    apply ContinuousOn.intervalIntegrable\n    exact (hcontB.const_mul 4).continuousOn\n  have hIntG : IntervalIntegrable (fun \u03c3 : \u211d =>\n      (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) * dirichletC2)\n      MeasureTheory.volume 3 XR := by\n    have h := (hIntA.add hIntB).const_mul dirichletC2\n    convert h using 1\n    ext \u03c3\n    ring\n  -- step 1: \u2016H\u2016 \u2264 \u222b envelope\n  have h1 : \u2016HIntegral (Jlag n) 3 XR U\u2016 \u2264\n      \u222b \u03c3 in (3 : \u211d)..XR,\n        (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) * dirichletC2 := by\n    unfold HIntegral\n    apply intervalIntegral.norm_integral_le_of_norm_le hXR _ hIntG\n    exact Filter.Eventually.of_forall (fun \u03c3 h\u03c3 => hpt \u03c3 h\u03c3)\n  -- step 2: \u222b envelope = dirichletC2\u00b7(4\u222be^{\u2212c\u03c3} + 4\u222b2^{\u2212\u03c3})\n  have h2 : \u222b \u03c3 in (3 : \u211d)..XR,\n        (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) * dirichletC2\n      = dirichletC2 * (\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) +\n          \u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3)) := by\n    rw [intervalIntegral.integral_const_mul, intervalIntegral.integral_add hIntA hIntB]\n  -- step 3: the two integral bounds\n  have h3a : \u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) \u2264\n      Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) := by\n    rw [intervalIntegral.integral_const_mul]\n    have h := integral_exp_neg_le hc hXR\n    have he3c : Real.exp (-3 * (Real.log 2 - 4 / 9)) = Real.exp (4 / 3) / 8 := by\n      have h3l2 : Real.exp (-(3 * Real.log 2)) = 1 / 8 := by\n        have e1 : -(3 * Real.log 2) = Real.log (1 / 8) := by\n          rw [show ((1 : \u211d) / 8) = (2 ^ 3 : \u211d)\u207b\u00b9 by norm_num, Real.log_inv, Real.log_pow]\n          push_cast\n          ring\n        rw [e1, Real.exp_log (by norm_num : (0 : \u211d) < 1 / 8)]\n      calc Real.exp (-3 * (Real.log 2 - 4 / 9))\n          = Real.exp ((4 / 3) + -(3 * Real.log 2)) := by congr 1; ring\n        _ = Real.exp (4 / 3) * Real.exp (-(3 * Real.log 2)) := by rw [Real.exp_add]\n        _ = Real.exp (4 / 3) / 8 := by rw [h3l2]; ring\n    rw [he3c] at h\n    calc 4 * (Real.exp (-3 * (Real.log 2 - 4 / 9)) / (Real.log 2 - 4 / 9))\n        = 4 * ((Real.exp (4 / 3) / 8) / (Real.log 2 - 4 / 9)) := by rw [he3c]\n      _ = Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) := by\n        field_simp [hc.ne']\n        ring\n  have h3b : \u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3) \u2264 1 / (2 * Real.log 2) := by\n    have h2e : (fun \u03c3 : \u211d => (2 : \u211d) ^ (-\u03c3)) =\n        (fun \u03c3 : \u211d => Real.exp (-(Real.log 2) * \u03c3)) := by\n      funext \u03c3\n      rw [Real.rpow_def_of_pos (by norm_num : (0 : \u211d) < 2)]\n      congr 1\n      ring\n    rw [h2e, intervalIntegral.integral_const_mul]\n    have h := integral_exp_neg_le (Real.log_pos (by norm_num : (1 : \u211d) < 2)) hXR\n    have h3l2 : Real.exp (-3 * Real.log 2) = 1 / 8 := by\n      have e1 : (-3 * Real.log 2) = Real.log (1 / 8) := by\n        rw [show ((1 : \u211d) / 8) = (2 ^ 3 : \u211d)\u207b\u00b9 by norm_num, Real.log_inv, Real.log_pow]\n        push_cast\n        ring\n      rw [e1, Real.exp_log (by norm_num : (0 : \u211d) < 1 / 8)]\n    rw [h3l2] at h\n    calc 4 * \u222b \u03c3 in (3 : \u211d)..XR, Real.exp (-(Real.log 2) * \u03c3)\n        \u2264 4 * ((1 / 8) / Real.log 2) := mul_le_mul_of_nonneg_left h (by norm_num)\n      _ = 1 / (2 * Real.log 2) := by\n        field_simp [Real.log_pos (by norm_num : (1 : \u211d) < 2)].ne'\n        ring\n  -- step 4: the certified assembly\n  have hlog2 : (0.6931471803 : \u211d) \u2264 Real.log 2 := Real.log_two_gt_d9\n  have h48 : Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) \u2264 (7.6545 : \u211d) := by\n    have hcge := cdog_gt\n    have h1 : Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) \u2264 3.8073 / (2 * 0.2487) :=\n      div_le_div exp_four_thirds_le (by linarith) (by norm_num) (by positivity)\n    calc Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9))\n        \u2264 3.8073 / (2 * 0.2487) := h1\n      _ \u2264 7.6545 := by norm_num\n  have h72 : 1 / (2 * Real.log 2) \u2264 (0.7214 : \u211d) := by\n    have h1 : 1 / (2 * Real.log 2) \u2264 1 / (2 * 0.6931471803) :=\n      one_div_le_one_div_of_le (by positivity) (by linarith)\n    calc 1 / (2 * Real.log 2) \u2264 1 / (2 * 0.6931471803) := h1\n      _ \u2264 0.7214 := by norm_num\n  have hC2 : dirichletC2 \u2264 1 := dirichletC2_le_one\n  have hsum : dirichletC2 * (\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) +\n      \u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3)) \u2264 (8.38 : \u211d) := by\n    have hA0 : (0 : \u211d) \u2264 Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) := by positivity\n    have hB0 : (0 : \u211d) \u2264 1 / (2 * Real.log 2) := by positivity\n    calc dirichletC2 * (\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) +\n          \u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3))\n        \u2264 1 * (Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) + 1 / (2 * Real.log 2)) := by\n          apply mul_le_mul hC2 (add_le_add h3a h3b) (by positivity) (by norm_num)\n      _ = Real.exp (4 / 3) / (2 * (Real.log 2 - 4 / 9)) + 1 / (2 * Real.log 2) := by ring\n      _ \u2264 7.6545 + 0.7214 := add_le_add h48 h72\n      _ \u2264 8.38 := by norm_num\n  calc \u2016HIntegral (Jlag n) 3 XR U\u2016\n      \u2264 \u222b \u03c3 in (3 : \u211d)..XR,\n          (4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) + 4 * (2 : \u211d) ^ (-\u03c3)) * dirichletC2 := h1\n    _ = dirichletC2 * (\u222b \u03c3 in (3 : \u211d)..XR, 4 * Real.exp (-(Real.log 2 - 4 / 9) * \u03c3) +\n          \u222b \u03c3 in (3 : \u211d)..XR, 4 * (2 : \u211d) ^ (-\u03c3)) := h2\n    _ \u2264 8.38 := hsum\n\nend NguyenRH1\n\nend\n",
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    {
      "oldStart": 553,
      "oldLines": 13,
      "newStart": 553,
      "newLines": 12,
      "lines": [
        "   have hint : IntervalIntegrable (fun \u03c3 => Real.exp (-c * \u03c3)) MeasureTheory.volume 3 XR := by",
        "     apply ContinuousOn.intervalIntegrable",
        "     exact (Real.continuous_exp.comp (continuous_const.mul continuous_id)).continuousOn",
        "-  rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hF hint]",
        "-  have h1 : (0 : \u211d) \u2264 Real.exp (-c * XR) / c := by positivity",
        "-  have e : Real.exp (-c * 3) = Real.exp (-3 * c) := by",
        "-    congr 1",
        "-    ring",
        "-  rw [e]",
        "-  linarith [h1]",
        "+  rw [intervalIntegral.integral_eq_sub_of_hasDerivAt hF hint, e, sub_le_iff_le_add]",
        "+  have h1 : (0 : \u211d) \u2264 Real.exp (-c * XR) := (Real.exp_pos _).le",
        "+  have h2 : -Real.exp (-c * XR) / c \u2264 0 := by",
        "+    apply div_nonpos_of_nonpos_of_nonneg _ hc.le",
        "+    linarith [h1]",
        "+  linarith [h2]",
        " ",
        " /-- The certified exponent-decay constant c = log 2 \u2212 4/9 \u2265 0.2487. -/",
        " lemma cdog_gt : (0.2487 : \u211d) \u2264 Real.log 2 - 4 / 9 := by"
      ]
    }
  ],
  "userModified": false,
  "replaceAll": false
}

{
  "command": "cd /d/Python/Quantum/research/lean/NguyenRH1 && export PATH=\"$HOME/.elan/bin:$PATH\" && lake build NguyenRH1.LiPrincipal 2>&1 | tail -3 && cat > PrintAxioms.lean << 'EOF'\n/- Session 016 axiom audit: ii\u2032-1b(ii) new declarations. -/\nimport Mathlib\nimport NguyenRH1.LiPrincipal\n\nopen NguyenRH1\n\n#print axioms rectInt'_ppart\n#print axioms rectInt_deriv_eq_zero\n#print axioms deriv_Gprim\n#print axioms Acoeff_one\nEOF\nlake env lean PrintAxioms.lean 2>&1 | head -6",
  "description": "Build LiPrincipal and audit axioms",
  "timeout": 600000
}

{
  "file_path": "D:\\Python\\Quantum\\research\\journal\\session-016.md",
  "content": "# Session 016 \u2014 2026-08-21 \u2014 Tier ii\u2032 sub-stage ii\u2032-1b (i)+(ii) COMPLETE\n\n## Directive\nContinue tier ii\u2032 per D-064: sub-stage ii\u2032-1b (R1 residue machinery).\n\n## Work done\n**ii\u2032-1b(i) COMPLETE (commit 1c35d31)** \u2014 LiResidue.lean:\n- `zetaA = (w\u22121)\u00b7\u03b6(w)` extended by 1 at 1: analytic (removable\n  singularity via Mathlib `riemannZeta_residue_one` +\n  RemovableSingularity.lean's `differentiableOn_update_limUnder_of_bddAbove`).\n- `hLi = 1 \u2212 (w\u22121)\u00b7(zetaA\u2032/zetaA)`: analytic at 1, value 1, agrees with\n  (w\u22121)(\u2212\u03b6\u2032/\u03b6) on a punctured neighborhood of 1 (`hLi_eq_near_one`,\n  via riemannZeta_residue_one.eventually_ne).\n- `lcoeff k := iteratedDeriv (k+1) hLi 1 / (k+1)!` \u2014 operational\n  Laurent coefficients of \u2212\u03b6\u2032/\u03b6 at 1.\n- `A_iter`: iterated derivatives of A(w) = w\u207f\u2212(w\u22121)\u207f at 1\n  (descFactorial for i < n, 0 for i = n; via iteratedDeriv_pow +\n  iteratedDeriv_comp_sub_const).\n- **`ResW1_eq_sum`**: the w=1 residue of the Lagarias-kernel integrand\n  equals \u03a3_{j=1}^n C(n,j)\u00b7lcoeff(j\u22121) \u2014 via iteratedDeriv_mul\n  (Leibniz), as designed (D-064 R1) and verified numerically (session 014).\n\n**ii\u2032-1b(ii) COMPLETE (commit 973fe66)** \u2014 LiPrincipal.lean:\n- `rectInt_deriv_eq_zero`: RectangleIntegral (deriv G) = 0 when G is\n  differentiable on the border \u2014 per-side FTC with the mixed-field chain\n  rule (HasDerivAt.comp in the \u211d\u2192\u2102\u2192\u2102 algebra tower), border MapsTo from\n  vendored Rectangle.lean.\n- `RectangleIntegral'_add` linearity.\n- `Acoeff/ppart/Gprim`; `deriv_Gprim` (termwise: hasDerivAt_zpow + comp);\n  `tail_continuousAt` (Finset induction); `ppart_eq_deriv_add`.\n- **`rectInt'_ppart`**: RectangleIntegral' (ppart n) z w = ResW1 n on\n  any rectangle with 1 interior \u2014 A_1 term by the vendored\n  `ResidueTheoremInRectangle`, i \u2265 2 tail vanishes by the primitive FTC.\n\nBoth axiom-audited: 3 standard axioms only.\n\n## Lean notes (hard-won this session, additions to the handoff list)\n- Factorial `n !` is SCOPED notation \u2014 needs `open scoped Nat`.\n- `conj` is scope-locked (ComplexConjugate) \u2014 use `starRingEnd \u2102`.\n- Parametrized lambdas over \u211d into \u2102 need explicit `(x : \u211d)` annotations\n  (else x elaborates at \u2102 and every downstream unify fails).\n- `HasDerivAt.comp` handles mixed fields via NormedAlgebra/IsScalarTower.\n- `ContinuousLinearMap.hasDerivAt : HasDerivAt e (e 1) x` (ofRealCLM).\n- `intervalIntegral.integral_eq_sub_of_hasDerivAt` (FTC-2) with explicit\n  (f := ...) (f' := ...) args \u2014 unification is brittle otherwise.\n- `ContinuousAt` of expressions containing the point variable: write the\n  `fun w =>` lambda EXPLICITLY (Lean's eta-unification misfires).\n- Finset.sum_apply / funext for sum-of-functions \u2194 lambda-of-sums.\n- Finset.sum_bij arg order: (i, hi, i_inj, i_surj, h).\n- differentiableOn_update_limUnder_of_bddAbove (removable singularity);\n  Tendsto.eventually_ne; DifferentiableOn.deriv (CauchyIntegral.lean).\n\n## State at stop\nii\u2032-1b(i)+(ii) complete. Remaining for ii\u2032-1b: (iii) the full rectangle\nresidue theorem for J \u2014 R = J \u2212 P holomorphic off {zeros \u222a {\u22122}}\n(order \u2265 0 at 1 via Taylor-remainder), simple poles at zeros (residue\nmult\u00b7(1\u2212(\u03c1/(\u03c1\u22121))\u207f)) and at \u22122 (residue 1\u2212(2/3)\u207f), via vendored\n`residueTheorem_finset` (Zeta23.Analytic.RectangleLogDeriv), then\nassemble sides = ResW1 + \u03a3 + triv.\nNext session: ii\u2032-1b(iii).\n"
}

{
  "type": "update",
  "filePath": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\PrintAxioms.lean",
  "content": "/- Session 020 axiom audit: ii\u2032-3a piece (A) part 1 new declarations. -/\nimport Mathlib\nimport NguyenRH1.LiArchimedean\n\nopen NguyenRH1\n\n#print axioms logDeriv_Gamma\u211d_one\n#print axioms logDeriv_Gamma\u211d_one_add\n#print axioms pi_cpow_neg_half_eq_exp\n#print axioms log_four_pi\n",
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      "lines": [
        "-/- Session 019 axiom audit: ii\u2032-3a piece (B) new declarations. -/",
        "+/- Session 020 axiom audit: ii\u2032-3a piece (A) part 1 new declarations. -/",
        " import Mathlib",
        "-import NguyenRH1.LiZetaCoeff",
        "+import NguyenRH1.LiArchimedean",
        " ",
        " open NguyenRH1",
        " ",
        "-#print axioms iteratedDeriv_hLiExt",
        "-#print axioms hLiExt_analyticAt_zero",
        "-#print axioms hLiExt_mul_id",
        "-#print axioms hLiExt_tendsto",
        "-#print axioms iteratedDeriv_hLi_one_add",
        "-#print axioms hLiExt_eq_near_zero",
        "+#print axioms logDeriv_Gamma\u211d_one",
        "+#print axioms logDeriv_Gamma\u211d_one_add",
        "+#print axioms pi_cpow_neg_half_eq_exp",
        "+#print axioms log_four_pi",
        " "
      ]
    }
  ],
  "originalFile": "/- Session 019 axiom audit: ii\u2032-3a piece (B) new declarations. -/\nimport Mathlib\nimport NguyenRH1.LiZetaCoeff\n\nopen NguyenRH1\n\n#print axioms iteratedDeriv_hLiExt\n#print axioms hLiExt_analyticAt_zero\n#print axioms hLiExt_mul_id\n#print axioms hLiExt_tendsto\n#print axioms iteratedDeriv_hLi_one_add\n#print axioms hLiExt_eq_near_zero\n",
  "autoRead": false
}