← Session 27 — ii′-5c/5d: Davenport disc + FE-reflection bounds
thinking (176 chars)
Fully clean. Now session closure:
1. journal/session-025.md
2. STATE.md update
3. NGUYEN.md checklist
4. research.db registration
5. git commit
Let me write the journal first.
Fully clean. Writing the session journal.
tool: Write
input
{
"file_path": "D:\\Python\\Quantum\\research\\journal\\session-025.md",
"content": "# Session 025 \u2014 2026-08-22 \u2014 ii\u2032-5d COMPLETE: FE-reflection of the loose R-bound + left-plane digamma bound\n\n## Directive\nContinue tier ii\u2032-5 per STATE.md: **ii\u2032-5d** \u2014 FE-reflection of the\nloose Davenport-disc R-bound (ii\u2032-5c) to \u03c3 \u2208 [\u22125/2, 1/2], |t| \u2265 10, plus\nthe left-plane uniform digamma bound (ii\u2032-5e input). User: \"let's work\non ii'-5d\".\n\n## Outcome: ii\u2032-5d COMPLETE\nNew file `NguyenRH1/LiReflect.lean` (~640 lines, 26 declarations, all\naxiom-audited: 3 standard axioms only, zero sorryAx), root-imported;\nfull build GREEN (8761 jobs).\n\n**Capstone `logDeriv_riemannZeta_sub_nearZeros_le_left`**: for s.im = t,\nre s \u2208 [\u22125/2, 1/2], |t| \u2265 10, \u03b6 s \u2260 0:\n \u2016logDeriv \u03b6 s \u2212 \u03a3_{\u03c1 \u2208 nearZeros t} (zeroMult \u03c1)/(s\u2212\u03c1)\u2016\n \u2264 200000\u00b7(log|t|)\u00b2.\nConstant deliberately loose (D-067 \u00a76: only the ii\u2032-5e identity passes\ntight constants to ii\u2032-5f).\n\n**Second deliverable `norm_digamma_sub_log_le_of_two_le_abs_im`** (the\n5e input): \u2016\u03c8(w) \u2212 log w\u2016 \u2264 3 for 2 \u2264 |w.im| \u2014 UNIFORM in the shift\nchain length (this is what the \u03b3-gap circles of 5e need).\n\n## Build path (all per D-067 \u00a73/\u00a76; zero surprises in the math)\n1. \u00a71 \u0393R shift: `gammaR_shift_four` \u2014 \u0393R(s+4) = \u03c0\u207b\u00b2(s/2)(s/2+1)\u0393R(s)\n off \u211d (Gamma\u211d_def + Gamma_add_one twice + cpow arithmetic via\n cpow_def_of_ne_zero \u2192 exp form; exp(\u22122 log \u03c0) = (\u03c0\u00b2)\u207b\u00b9 via\n Real.log_inv/log_pow/exp_log). Then `logDeriv_Gamma\u211d_shift`:\n logDeriv \u0393R s = logDeriv \u0393R (s+4) \u2212 2(s+1)/(s(s+2)) via the\n EventuallyEq on {im \u2260 0} + logDeriv_div/const_mul + the comp deriv.\n2. \u00a72 the FE logDeriv identity `logDeriv_zeta_fe`:\n logDeriv \u03b6 s = \u2212logDeriv \u03b6(1\u2212s) \u2212 [logDeriv \u0393R s + logDeriv \u0393R(1\u2212s)]\n \u2014 from \u039b =\u1da0 \u0393R\u00b7\u03b6 near good points (completedZeta_eventuallyEq via\n ContinuousAt.eventually_ne), logDeriv_mul, and the chain-rule\n negation through \u039b(1\u2212z) = \u039b(z). \u03b6(1\u2212s) \u2260 0 from \u03b6 s \u2260 0 via the \u039b\n factorization both ways (riemannZeta_one_sub_ne_zero).\n3. \u00a73 the EXACT zero-sum cancellation `sum_nearZeros_neg_reflect`:\n nearZeros(\u2212t) = (nearZeros t).image (1 \u2212 \u00b7) (both directions by the\n symmetric ball estimate \u2016(1\u2212\u03c1)\u2212(2\u2213it)\u2016\u00b2 = (1+\u03c1.re)\u00b2+(\u03c1.im\u2212t)\u00b2 \u2264 5);\n zeroMult(1\u2212\u03c1) = zeroMult \u03c1 by Zeta23 `analyticOrderAt_zeta_one_sub`;\n 1\u2212s\u2212(1\u2212\u03c1) = \u2212(s\u2212\u03c1). \u27f9 the \u2212t sum at 1\u2212s IS minus the t sum at s.\n4. \u00a74 the \u0393R-bracket bound \u2264 4(log|t|)\u00b2: both digamma arguments pushed\n into re > 0 (the s-side via the +4 shift), StirlingVert\n `digamma_stirling` + `\u2016log w\u2016 \u2264 log\u2016w\u2016 + \u03c0` (norm_log_le_of_one_le)\n + the rational term \u2264 1; fold via two_le_log_abs_t.\n5. \u00a75/\u00a76 the left-plane digamma bound: \u03c8(w+n) = \u03c8(w) + \u03a3(w+j)\u207b\u00b9\n (digamma_add_natCast, induction on digamma_apply_add_one); the sum vs\n the log difference compared by first-order Euler\u2013Maclaurin\n (Zeta23 `sum_eq_integral_add_integral_deriv` with f x = (w+\u2191x)\u207b\u00b9);\n \u222b\u2080\u207f (w+x)\u207b\u00b9dx = log(w+n) \u2212 log w by FTC (hasDerivAt_log on the slit\n plane, `HasDerivAt.comp_ofReal` for the \u211d\u2192\u2102 chain); the B\u0303\u2081 remainder\n \u2264 \u00bd\u222b\u2080\u207f dx/((x+re w)\u00b2+(im w)\u00b2) \u2264 \u03c0/(2|im w|) \u2264 \u03c0/4 by the ARCTAN FTC\n (avoids the full-line integral entirely: (1/c)arctan((x+a)/c)\n antiderivative + |arctan| < \u03c0/2); StirlingVert at v = w+n gives\n \u2016\u03c8 v \u2212 log v\u2016 \u2264 1; boundary \u2264 1/2. Total 1 + 1/2 + \u03c0/4 \u2264 3.\n6. \u00a77 capstone assembly: X(s) = \u2212X(1\u2212s) \u2212 bracket exactly (ring after\n hfe/hrefl) \u27f9 \u2016X\u2016 \u2264 100000L\u00b2 + 4L\u00b2 \u2264 200000L\u00b2.\n\n## Banked gotchas (session 025 additions)\n- `lake -d <dir> env lean <abs path>` compiles a single file from\n anywhere (no cd needed) \u2014 the reliable foreground form.\n- **add_le_add_left/right are SWAPPED vs the classic convention in this\n Mathlib**: add_le_add_left (h : b \u2264 c) (a) : b + a \u2264 c + a;\n add_le_add_right (h : b \u2264 c) (a) : a + b \u2264 a + c.\n- `sub_left_inj` here : a \u2212 c = b \u2212 c \u2194 a = b (the VARYING side is\n \"left\"); for 1 \u2212 a = 1 \u2212 b use `sub_right_inj`.\n- `Finset.sum_image` takes the map as the implicit `g` (to_additive of\n prod_image: (g := ...) or let the InjOn proof's type drive it);\n `Finset.sum_neg_distrib` : \u2211 \u2212f = \u2212\u2211 f (use \u2190 for the RHS form).\n- `Finset.sum_image`'s InjOn proof: beta-redex hypotheses break\n sub_left_inj.mp \u2014 ascribe `have h1 : 1 - a = 1 - b := hab` first.\n- `Real.pi_lt_d2 : \u03c0 < 3.15` (NOT pi_lt_315); `Real.log_two_lt_d9`,\n `Real.exp_one_gt_d9` as banked.\n- `HasDerivAt.inv` produces the Pi.inv function form (c\u207b\u00b9) \u2014 instance\n paths don't match pointwise goals; use\n `(hasDerivAt_inv hne).comp _ (...).const_add` instead (pointwise\n `fun y => y\u207b\u00b9` form) + simpa [Function.comp_def].\n- `HasDerivAt.comp_ofReal` (RealDeriv.lean) : HasDerivAt e e' \u2191z \u2192\n HasDerivAt (fun y : \u211d => e \u2191y) e' z \u2014 the clean \u211d\u2192\u2102 chain rule for\n complex antiderivatives on real intervals (FTC input).\n- RCLike-coe cast atoms (`\u2191(\u22121/2 : \u211d)` inside EM conclusions) do NOT\n normalize under linear_combination's ring_nf \u2014 `push_cast at hEM`\n first; and `((\u2191(0:\u2115):\u211d):\u2102)` vs `((0:\u211d):\u2102)` vs `(\u2191(0:\u2115):\u2102)` are\n DIFFERENT atoms \u2014 keep every sum family in the single-cast (\u2115\u2192\u2102)\n form from the start (digamma_add_natCast's family) and normalize\n hEM by push_cast + simp only [add_zero].\n- `intervalIntegral.integral_const_mul` (const LEFT); the mul_const\n variant is f\u00b7r. `intervalIntegral.integral_mono_on (hab) (hf) (hg)\n (h : \u2200 x \u2208 Icc ...)`.\n- `IntervalIntegrable.continuousOn_mul (hf : IntervalIntegrable f)\n (hg : ContinuousOn g) : IntervalIntegrable (g \u00b7 * f \u00b7)` (continuous\n factor FIRST) \u2014 mirror Zeta23 EulerMaclaurin's usage; B1's\n integrability via intervalIntegrable_iff' + integrableOn_of_bounded\n with aestronglyMeasurable_B1.norm + abs_B1_le_half.\n- `div_le_div_of_nonneg_left (ha : 0 \u2264 a) (hb : 0 < b) (hbc : b \u2264 c) :\n a / c \u2264 a / b` (the ordered-field form; div_le_div_iff_left is the\n ordered-GROUP one \u2014 wrong typeclass for \u211d).\n- `Complex.div_ofNat_im/re` (simp): (z/2).im = z.im/2 \u2014 no manual\n div_im arithmetic needed.\n- `Complex.zero_im`/`zero_re` exist but are NOT simp \u2014 `simp at h`\n on IsNontrivialZero 0 fails (\"no progress\"); use .ne' projections.\n\n## Status / next\nii\u2032-5d COMPLETE. Next per D-067 \u00a76: **ii\u2032-5e** \u2014 the partial-fraction\nidentity \u03b6\u2032/\u03b6(s) = B + \u03a3_\u03c1 m(1/(s\u2212\u03c1) + 1/\u03c1) \u2212 1/s \u2212 1/(s\u22121) + \u00bdlog \u03c0\n\u2212 \u00bd\u03c8(s/2) (series convergence via mult-weighted zero_sum_inv_sq; F\nentire by residue cancellation; |F| = o(R) on \u03b3-gap circles using 5c +\n5d + the left-plane digamma bound \u27f9 F const). FIRST ITEM of session\n026: `zeroMult(\u22122n) = 1` for ALL n (generalize ii\u2032-4c's zeroMult_neg_two\nvia the product-form \u0393 recurrence \u0393(z) = \u0393(z+n+1)/\u220f_{j\u2264n}(z+j) \u2014\nneeded for F's entirety at the trivial zeros; ~0.25 sess, deferred from\n5d per the sub-stage rule). Then ii\u2032-5f (tight analysis \u2192 10.5/48),\nii\u2032-3b, ii\u2032-3c, ii\u2032-6, ii\u2032-7.\n"
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raw content_json (database row)
[{"type":"thinking","thinking":"Fully clean. Now session closure:\n1. journal/session-025.md\n2. STATE.md update\n3. NGUYEN.md checklist\n4. research.db registration\n5. git commit\n\nLet me write the journal first.","signature":""},{"type":"text","text":"Fully clean. Writing the session journal."},{"type":"tool_use","id":"Write_154","name":"Write","input":{"file_path":"D:\\Python\\Quantum\\research\\journal\\session-025.md","content":"# Session 025 — 2026-08-22 — ii′-5d COMPLETE: FE-reflection of the loose R-bound + left-plane digamma bound\n\n## Directive\nContinue tier ii′-5 per STATE.md: **ii′-5d** — FE-reflection of the\nloose Davenport-disc R-bound (ii′-5c) to σ ∈ [−5/2, 1/2], |t| ≥ 10, plus\nthe left-plane uniform digamma bound (ii′-5e input). User: \"let's work\non ii'-5d\".\n\n## Outcome: ii′-5d COMPLETE\nNew file `NguyenRH1/LiReflect.lean` (~640 lines, 26 declarations, all\naxiom-audited: 3 standard axioms only, zero sorryAx), root-imported;\nfull build GREEN (8761 jobs).\n\n**Capstone `logDeriv_riemannZeta_sub_nearZeros_le_left`**: for s.im = t,\nre s ∈ [−5/2, 1/2], |t| ≥ 10, ζ s ≠ 0:\n ‖logDeriv ζ s − Σ_{ρ ∈ nearZeros t} (zeroMult ρ)/(s−ρ)‖\n ≤ 200000·(log|t|)².\nConstant deliberately loose (D-067 §6: only the ii′-5e identity passes\ntight constants to ii′-5f).\n\n**Second deliverable `norm_digamma_sub_log_le_of_two_le_abs_im`** (the\n5e input): ‖ψ(w) − log w‖ ≤ 3 for 2 ≤ |w.im| — UNIFORM in the shift\nchain length (this is what the γ-gap circles of 5e need).\n\n## Build path (all per D-067 §3/§6; zero surprises in the math)\n1. §1 ΓR shift: `gammaR_shift_four` — ΓR(s+4) = π⁻²(s/2)(s/2+1)ΓR(s)\n off ℝ (Gammaℝ_def + Gamma_add_one twice + cpow arithmetic via\n cpow_def_of_ne_zero → exp form; exp(−2 log π) = (π²)⁻¹ via\n Real.log_inv/log_pow/exp_log). Then `logDeriv_Gammaℝ_shift`:\n logDeriv ΓR s = logDeriv ΓR (s+4) − 2(s+1)/(s(s+2)) via the\n EventuallyEq on {im ≠ 0} + logDeriv_div/const_mul + the comp deriv.\n2. §2 the FE logDeriv identity `logDeriv_zeta_fe`:\n logDeriv ζ s = −logDeriv ζ(1−s) − [logDeriv ΓR s + logDeriv ΓR(1−s)]\n — from Λ =ᶠ ΓR·ζ near good points (completedZeta_eventuallyEq via\n ContinuousAt.eventually_ne), logDeriv_mul, and the chain-rule\n negation through Λ(1−z) = Λ(z). ζ(1−s) ≠ 0 from ζ s ≠ 0 via the Λ\n factorization both ways (riemannZeta_one_sub_ne_zero).\n3. §3 the EXACT zero-sum cancellation `sum_nearZeros_neg_reflect`:\n nearZeros(−t) = (nearZeros t).image (1 − ·) (both directions by the\n symmetric ball estimate ‖(1−ρ)−(2∓it)‖² = (1+ρ.re)²+(ρ.im−t)² ≤ 5);\n zeroMult(1−ρ) = zeroMult ρ by Zeta23 `analyticOrderAt_zeta_one_sub`;\n 1−s−(1−ρ) = −(s−ρ). ⟹ the −t sum at 1−s IS minus the t sum at s.\n4. §4 the ΓR-bracket bound ≤ 4(log|t|)²: both digamma arguments pushed\n into re > 0 (the s-side via the +4 shift), StirlingVert\n `digamma_stirling` + `‖log w‖ ≤ log‖w‖ + π` (norm_log_le_of_one_le)\n + the rational term ≤ 1; fold via two_le_log_abs_t.\n5. §5/§6 the left-plane digamma bound: ψ(w+n) = ψ(w) + Σ(w+j)⁻¹\n (digamma_add_natCast, induction on digamma_apply_add_one); the sum vs\n the log difference compared by first-order Euler–Maclaurin\n (Zeta23 `sum_eq_integral_add_integral_deriv` with f x = (w+↑x)⁻¹);\n ∫₀ⁿ (w+x)⁻¹dx = log(w+n) − log w by FTC (hasDerivAt_log on the slit\n plane, `HasDerivAt.comp_ofReal` for the ℝ→ℂ chain); the B̃₁ remainder\n ≤ ½∫₀ⁿ dx/((x+re w)²+(im w)²) ≤ π/(2|im w|) ≤ π/4 by the ARCTAN FTC\n (avoids the full-line integral entirely: (1/c)arctan((x+a)/c)\n antiderivative + |arctan| < π/2); StirlingVert at v = w+n gives\n ‖ψ v − log v‖ ≤ 1; boundary ≤ 1/2. Total 1 + 1/2 + π/4 ≤ 3.\n6. §7 capstone assembly: X(s) = −X(1−s) − bracket exactly (ring after\n hfe/hrefl) ⟹ ‖X‖ ≤ 100000L² + 4L² ≤ 200000L².\n\n## Banked gotchas (session 025 additions)\n- `lake -d <dir> env lean <abs path>` compiles a single file from\n anywhere (no cd needed) — the reliable foreground form.\n- **add_le_add_left/right are SWAPPED vs the classic convention in this\n Mathlib**: add_le_add_left (h : b ≤ c) (a) : b + a ≤ c + a;\n add_le_add_right (h : b ≤ c) (a) : a + b ≤ a + c.\n- `sub_left_inj` here : a − c = b − c ↔ a = b (the VARYING side is\n \"left\"); for 1 − a = 1 − b use `sub_right_inj`.\n- `Finset.sum_image` takes the map as the implicit `g` (to_additive of\n prod_image: (g := ...) or let the InjOn proof's type drive it);\n `Finset.sum_neg_distrib` : ∑ −f = −∑ f (use ← for the RHS form).\n- `Finset.sum_image`'s InjOn proof: beta-redex hypotheses break\n sub_left_inj.mp — ascribe `have h1 : 1 - a = 1 - b := hab` first.\n- `Real.pi_lt_d2 : π < 3.15` (NOT pi_lt_315); `Real.log_two_lt_d9`,\n `Real.exp_one_gt_d9` as banked.\n- `HasDerivAt.inv` produces the Pi.inv function form (c⁻¹) — instance\n paths don't match pointwise goals; use\n `(hasDerivAt_inv hne).comp _ (...).const_add` instead (pointwise\n `fun y => y⁻¹` form) + simpa [Function.comp_def].\n- `HasDerivAt.comp_ofReal` (RealDeriv.lean) : HasDerivAt e e' ↑z →\n HasDerivAt (fun y : ℝ => e ↑y) e' z — the clean ℝ→ℂ chain rule for\n complex antiderivatives on real intervals (FTC input).\n- RCLike-coe cast atoms (`↑(−1/2 : ℝ)` inside EM conclusions) do NOT\n normalize under linear_combination's ring_nf — `push_cast at hEM`\n first; and `((↑(0:ℕ):ℝ):ℂ)` vs `((0:ℝ):ℂ)` vs `(↑(0:ℕ):ℂ)` are\n DIFFERENT atoms — keep every sum family in the single-cast (ℕ→ℂ)\n form from the start (digamma_add_natCast's family) and normalize\n hEM by push_cast + simp only [add_zero].\n- `intervalIntegral.integral_const_mul` (const LEFT); the mul_const\n variant is f·r. `intervalIntegral.integral_mono_on (hab) (hf) (hg)\n (h : ∀ x ∈ Icc ...)`.\n- `IntervalIntegrable.continuousOn_mul (hf : IntervalIntegrable f)\n (hg : ContinuousOn g) : IntervalIntegrable (g · * f ·)` (continuous\n factor FIRST) — mirror Zeta23 EulerMaclaurin's usage; B1's\n integrability via intervalIntegrable_iff' + integrableOn_of_bounded\n with aestronglyMeasurable_B1.norm + abs_B1_le_half.\n- `div_le_div_of_nonneg_left (ha : 0 ≤ a) (hb : 0 < b) (hbc : b ≤ c) :\n a / c ≤ a / b` (the ordered-field form; div_le_div_iff_left is the\n ordered-GROUP one — wrong typeclass for ℝ).\n- `Complex.div_ofNat_im/re` (simp): (z/2).im = z.im/2 — no manual\n div_im arithmetic needed.\n- `Complex.zero_im`/`zero_re` exist but are NOT simp — `simp at h`\n on IsNontrivialZero 0 fails (\"no progress\"); use .ne' projections.\n\n## Status / next\nii′-5d COMPLETE. Next per D-067 §6: **ii′-5e** — the partial-fraction\nidentity ζ′/ζ(s) = B + Σ_ρ m(1/(s−ρ) + 1/ρ) − 1/s − 1/(s−1) + ½log π\n− ½ψ(s/2) (series convergence via mult-weighted zero_sum_inv_sq; F\nentire by residue cancellation; |F| = o(R) on γ-gap circles using 5c +\n5d + the left-plane digamma bound ⟹ F const). FIRST ITEM of session\n026: `zeroMult(−2n) = 1` for ALL n (generalize ii′-4c's zeroMult_neg_two\nvia the product-form Γ recurrence Γ(z) = Γ(z+n+1)/∏_{j≤n}(z+j) —\nneeded for F's entirety at the trivial zeros; ~0.25 sess, deferred from\n5d per the sub-stage rule). Then ii′-5f (tight analysis → 10.5/48),\nii′-3b, ii′-3c, ii′-6, ii′-7.\n"},"extra_content":null}]