← Session 35 — ii′-6-R: Lagarias 6.1 right-tail re-proof (E-032/D-073)
thinking (1017 chars)
Starting DL-0. Per STATE.md's work order:
1. §2 coordinate audit (the audit table)
2. §3 exact derivation
3. §4 candidate rational constants
4. §5 E-034 (script + validation + fold)
5. §7 A–L report, then STOP
Let me set up the todos, then start with the coordinate audit. I need to inspect the actual Lean definitions to build the audit table. Key things to verify:
- Jlag definition (the integrand)
- k_n / kFun definition
- the rectangle convention (rectInt'_Jlag: z = −5/2 − iT, w = 2√n + iT)
- orientations (RectangleIntegral' convention — which direction do the sides go?)
- X_R = 2√n ✓
- T_cut = √n+ε (the pigeonhole T ∈ [√n, √n+1])
- existing conjugation lemmas
- right-half-plane ζ′/ζ bounds (dirichletC2, the LSeries vonMangoldt form)
- I_I bound (IIMaj)
- where "3.3" enters Bn (the horizontal share 2(e⁴+1)(7Ĥ+3.3))
- the residue-theorem-side identity (sf_abs_le_rectInt_add_zeroSum from LiAssembly)
Let me look at the Lean files for these. Start with Jlag/kFun definitions and the rectangle machinery.
Starting DL-0. First the coordinate/convention audit (directive §2) — inspecting the actual Lean definitions.
tool: TodoWrite
input
{
"todos": [
{
"content": "DL-0.1: \u00a72 coordinate audit (Jlag/k_n/rectangle/orientations/bounds/where-3.3-enters) \u2192 D-074 \u00a72 table",
"status": "in_progress",
"activeForm": "Auditing contour conventions"
},
{
"content": "DL-0.2: \u00a73 exact derivation (deformation identity + analyticity + piece bounds + \u00a710 dependency statement) \u2192 D-074 \u00a73",
"status": "pending",
"activeForm": "Writing the exact derivation"
},
{
"content": "DL-0.3: \u00a74 candidate rational constants (C_zeta3, e^{5/2}+1, e^{4/9}<2, q<1 certified) \u2192 D-074 \u00a74",
"status": "pending",
"activeForm": "Certifying rational constants"
},
{
"content": "DL-0.4: \u00a75 E-034 script (identity check + piece bounds + slack) + \u00a76 temporary fold vs N\u2081, run + db",
"status": "pending",
"activeForm": "Building and running E-034"
},
{
"content": "DL-0.5: \u00a77 A\u2013L report \u2192 D-074 \u00a77 + commit + STOP for user gate",
"status": "pending",
"activeForm": "Writing the A\u2013L report"
}
]
}
tool: Grep ok
input
{
"pattern": "def Jlag|def kFun|RectangleIntegral'|rectInt'",
"path": "D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1",
"output_mode": "content",
"-n": true,
"head_limit": 40
}
result 2026-08-26 04:13:37Z → 2026-08-26 04:13:37Z
mode='content' num_files=0 filenames=[] content="research\\lean\\NguyenRH1\\NguyenRH1\\LiAssembly.lean:6:`rectInt'_Jlag` (ii′-1b) on the D-036 variant-B rectangle\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiAssembly.lean:74:|Sf n| ≤ ‖RectangleIntegral' (Jlag n) z w‖ + ‖Σ_{Sz} mult·(1−(ρ/(ρ−1))ⁿ)‖,\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiAssembly.lean:80: |Sf n| ≤ ‖RectangleIntegral' (Jlag n) (rectZ T) (rectW n T)‖ +\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiAssembly.lean:289: have hid := rectInt'_Jlag n hre him h1int hborder Sz hSz\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiAssembly.lean:293: have e : ResW1 n = RectangleIntegral' (Jlag n) z w -\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean:6: RectangleIntegral' P z w = A_1 = ResW1 n.\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean:28:lemma RectangleIntegral'_add (f g : ℂ → ℂ) (z w : ℂ)\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean:30: RectangleIntegral' (f + g) z w = RectangleIntegral' f z w + RectangleIntegral' g z w := by\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean:31: rw [RectangleIntegral', RectangleBorderIntegrable.add hf hg, RectangleIntegral', RectangleIntegral',\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean:266:RectangleIntegral' (ppart n) z w = ResW1 n. -/\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean:267:theorem rectInt'_ppart {n : ℕ} {z w : ℂ}\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean:269: RectangleIntegral' (ppart n) z w = ResW1 n := by\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean:276: rw [RectangleIntegral'_congr hdecomp]\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean:295: rw [RectangleIntegral'_add _ _ _ _ hAint hGint]\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiPrincipal.lean:309: rw [RectangleIntegral', hzero]\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean:23:plus session 016's `rectInt'_ppart` give `rectInt'_Jlag`:\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean:26: RectangleIntegral' (Jlag n) z w\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean:55:def kFun (n : ℕ) (s : ℂ) : ℂ := (s / (s - 1)) ^ n - 1\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean:58:def Jlag (n : ℕ) (s : ℂ) : ℂ := kFun n s * (-(deriv riemannZeta s / riemannZeta s))\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean:434: RectangleIntegral' (Jlag n) z w\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean:437:theorem rectInt'_Jlag (n : ℕ) {z w : ℂ} (hre : z.re ≤ w.re) (him : z.im ≤ w.im)\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean:441: RectangleIntegral' (Jlag n) z w =\nresearch\\lean\\NguyenRH1\\NguyenRH1\\LiSides.lean:518: rw [heq, RectangleIntegral'_add _ _ _ _ hRint hPint, key, rectInt'_ppart hre him h1]" num_lines=23 num_matches=None applied_limit=None applied_offset=None
raw content_json (database row)
[{"type":"thinking","thinking":"Starting DL-0. Per STATE.md's work order:\n1. §2 coordinate audit (the audit table)\n2. §3 exact derivation\n3. §4 candidate rational constants\n4. §5 E-034 (script + validation + fold)\n5. §7 A–L report, then STOP\n\nLet me set up the todos, then start with the coordinate audit. I need to inspect the actual Lean definitions to build the audit table. Key things to verify:\n- Jlag definition (the integrand)\n- k_n / kFun definition\n- the rectangle convention (rectInt'_Jlag: z = −5/2 − iT, w = 2√n + iT)\n- orientations (RectangleIntegral' convention — which direction do the sides go?)\n- X_R = 2√n ✓\n- T_cut = √n+ε (the pigeonhole T ∈ [√n, √n+1])\n- existing conjugation lemmas\n- right-half-plane ζ′/ζ bounds (dirichletC2, the LSeries vonMangoldt form)\n- I_I bound (IIMaj)\n- where \"3.3\" enters Bn (the horizontal share 2(e⁴+1)(7Ĥ+3.3))\n- the residue-theorem-side identity (sf_abs_le_rectInt_add_zeroSum from LiAssembly)\n\nLet me look at the Lean files for these. Start with Jlag/kFun definitions and the rectangle machinery.","signature":""},{"type":"text","text":"Starting DL-0. First the coordinate/convention audit (directive §2) — inspecting the actual Lean definitions."},{"type":"tool_use","id":"TodoWrite_110","name":"TodoWrite","input":{"todos":[{"content":"DL-0.1: §2 coordinate audit (Jlag/k_n/rectangle/orientations/bounds/where-3.3-enters) → D-074 §2 table","status":"in_progress","activeForm":"Auditing contour conventions"},{"content":"DL-0.2: §3 exact derivation (deformation identity + analyticity + piece bounds + §10 dependency statement) → D-074 §3","status":"pending","activeForm":"Writing the exact derivation"},{"content":"DL-0.3: §4 candidate rational constants (C_zeta3, e^{5/2}+1, e^{4/9}<2, q<1 certified) → D-074 §4","status":"pending","activeForm":"Certifying rational constants"},{"content":"DL-0.4: §5 E-034 script (identity check + piece bounds + slack) + §6 temporary fold vs N₁, run + db","status":"pending","activeForm":"Building and running E-034"},{"content":"DL-0.5: §7 A–L report → D-074 §7 + commit + STOP for user gate","status":"pending","activeForm":"Writing the A–L report"}]},"extra_content":null},{"type":"tool_use","id":"Grep_111","name":"Grep","input":{"pattern":"def Jlag|def kFun|RectangleIntegral'|rectInt'","path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1","output_mode":"content","-n":true,"head_limit":40},"extra_content":null}]