[{"type":"thinking","thinking":"All 8 declarations: 3 standard axioms only, 0 sorryAx. ii′-3c is COMPLETE: the crux identity Sf(n) = Res₁(n) is Lean-proved modulo ZeroCountProp only.\n\nWait — one nuance: sf_eq_resW1 proves `(ResW1 n).re = Sf n`. And lagarias_4_10_a + sf_eq_resW1 take hZC. The crux as designed in D-064 was \"Sf(n) = Res₁(n) EXACT\" — done.\n\nNow archive updates:\n1. Journal session-038 — append the ii′-3c section (the journal already exists with ii′-3b; this session continued — I'll append an ii′-3c part to session-038.md).\n2. STATE.md — update: ii′-3c COMPLETE; next ii′-6.\n3. NGUYEN.md — mark ii′-3 done (ii′-3c complete) in checklist + session log.\n4. research.db — session 038 summary update; D-066 status update.\n5. Commit.\n\nLet me do these. First journal append.","signature":""},{"type":"text","text":"All 8 declarations: 3 standard axioms only, 0 sorryAx. **ii′-3c COMPLETE — the crux Sf(n) = Res₁(n) is Lean-proved.** Updating the archive."},{"type":"tool_use","id":"Edit_150","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\journal\\session-038.md","old_string":"## Next\n\nii′-3c (the (4.10-a) assembly, D-066 §2): τ_j = c_j + (−1)ʲσ_{j+1} −\n(−1)ʲ — needs the FE derivative relation gFun^{(k−1)}(0) =\n(−1)^k·gFun^{(k−1)}(1) (new but small: xiL_one_sub + iteratedDeriv comp\nmachinery) + the ii′-3a pieces (A: iteratedDeriv_logDeriv_Gammaℝ,\nB: iteratedDeriv_hLiExt) + the capstone of 3b → the crux identity\nSf(n) = Res₁(n). Then ii′-6 (L0 assembly incl. D-069 Finding 2's H(T)\nnear-sum repair) → ii′-7 (closure).","new_string":"## ii′-3c COMPLETE (same session, continuation)\n\n**LiCrux.lean NEW (~430 lines, GREEN, root-imported)**: the (4.10-a)\nassembly + the crux identity. Chain (D-066 §2):\n- `gFun_one_sub_fun` — the FE as a function identity gFun(1−s) = −gFun\n  everywhere (xiL entire; xiL_one_sub + deriv_comp chain rule; the\n  `simp at hcomp` beta-normalization + linear_combination -h2).\n- `iteratedDeriv_gFun_zero` — gFun^{(m)}(0) = (−1)^{m+1}·gFun^{(m)}(1)\n  via the UNCONDITIONAL `iteratedDeriv_comp_const_sub` (function-level\n  equality, junk-free!) + iteratedDeriv_neg. No induction needed.\n- `gFun_one_add_punctured` — the shift expansion gFun(1+s) =\n  1/(1+s) + hLiExt s + ΓR′/ΓR(1+s) on 𝓝[≠] 0 (the gFun_eq_momentB\n  logDeriv-algebra pattern at 1+s; the hLiExt identity via zetaA's\n  deriv_mul + mul_div_cancel_left₀).\n- `gFun_one_add_eventuallyEq` — upgraded to full 𝓝 0 (continuity of\n  both sides + eq_of_eventuallyEq_nhd_ne; the Gammaℝ-piece ContDiffAt\n  via ContDiffAt.congr_of_eventuallyEq with the digamma form — NO\n  AnalyticAt-Gammaℝ detour needed).\n- `iteratedDeriv_gFun_one` — coefficient extraction: gFun^{(m)}(1) =\n  (−1)^m·m! − m!·lcoeff m + (−1)^{m+1}·m!·a_{m+1}\n  (iteratedDeriv_comp_add_const + congr + two add splits + the three\n  pieces: iteratedDeriv_one_div_one_add (zpow lemma), iteratedDeriv_hLiExt,\n  iteratedDeriv_logDeriv_Gammaℝ).\n- **`lagarias_4_10_a (hZC)`** — (4.10-a) PROVED: lcoeff (k−1) =\n  (−1)^k·(lagariasA k + sigmaP k) + (−1)^{k−1}, all k ≥ 1. The\n  multiplied-out linear_combination ((−t)·hM2 + t·hFE + t²·hExp +\n  (−F·c + t·F·a + s·F)·ht_sq with F = m!, s = (−1)^m, t = (−1)^{m+1})\n  + mul_left_cancel₀ (m! ≠ 0).\n- **`sf_eq_resW1 (hZC)`** — THE CRUX PROVED: (ResW1 n).re = Sf n for\n  n ≥ 1. ResW1_eq_sum + (4.10-a) per-term + the two ii′-2 swaps\n  (sInfty_eq_sum cast by push_cast; liCoeff_eq_sum_sigma with the\n  (−1)^j = −(−1)^{j+1} per-term ring) + the binomial sum\n  Σ_{j=1}^n (−1)^{j−1}C(n,j) = 1 (add_pow (−1) 1 n + insert-0 peel +\n  hsign (−1)^{j−1} = −(−1)^j) + the ℂ identity hid + .re.\n\n**Closure**: LiCrux root-imported; full build GREEN (8771 jobs);\nAuditSession038b.lean + audit-session038b-axioms.txt — 8 declarations,\nALL: [propext, Classical.choice, Quot.sound] only, 0 sorryAx.\n\n**ii′-3 (ALL of (4.10)) → COMPLETE.** Perimeter of both capstones:\nZeroCountProp ONLY.\n\n## Banked gotchas (ii′-3c)\n\n- `iteratedDeriv_comp_const_sub/add_const` are UNCONDITIONAL\n  function-level equalities — no differentiability hypotheses, no\n  induction (the whole FE relation is 4 lines).\n- `ContDiffAt.congr_of_eventuallyEq (hf : ContDiffAt n f x)\n  (h : f₁ =ᶠ[𝓝 x] f)` — the h goes FROM the target TO the proved-smooth\n  function (pass hev, NOT hev.symm).\n- `Finset.sum_add` is RENAMED `Finset.sum_add_distrib` (SummationFilter\n  refactor); `Finset.sum_sub` also gone.\n- `Nat.exists_eq_succ_of_ne_zero` gives m.SUCC form — normalize with\n  `rw [Nat.succ_eq_add_one, Nat.add_sub_cancel]` (Nat.add_sub_cancel\n  doesn't match m.succ − 1).\n- `simp using X.deriv` SILENTLY derailed the tactic block (rw [h1]\n  then can't fire; error reported at the have statement) — use the\n  direct `.deriv` term: `((hasDerivAt_id' x).sub_const 1).deriv :\n  deriv (fun w => w − 1) x = 1` (sub_const's value is ALREADY 1, not\n  1−0 — the ascription (1 − 0) fails).\n- `ContinuousAt.comp`/`Tendsto.comp` numerals: (1:ℂ)+0 is NOT\n  kernel-defeq to 1 — `rw [show (1 : ℂ) = 1 + 0 from (add_zero 1).symm]\n  at hg` BEFORE the comp so the points match syntactically; dot-notation\n  `hg.comp hc` elaborates f := const-ofNat unless pinned.\n- `show` with U+2212 minus (−1) breaks the parser (\"(↥) must have a\n  function type\") — the file's (−1)-forms are ASCII `(-1 : ℂ)`;\n  keep show-statements ASCII.\n- `rw [hS]` vs `rw [← hS]`: when hS : cast = Σ and the goal has the Σ\n  (from a previous sum-split rewrite), the direction is ← hS.\n- `mul_div_cancel_left₀ (b) (ha : a ≠ 0) : a * b / a = b` — kills\n  (s·X)/s goals field_simp leaves behind.\n- Fraction-equality pattern that ALWAYS works: div_add_div to merge\n  the RHS + div_eq_div_iff (both denominators ≠ 0) + ring — avoids\n  field_simp's s⁻¹ leftovers (s·ζ·s⁻¹ is not a ring identity).\n\n## Files (ii′-3c)\n\n- research/lean/NguyenRH1/NguyenRH1/LiCrux.lean (NEW, ~430 lines, GREEN)\n- research/lean/NguyenRH1/NguyenRH1.lean (LiCrux root-imported)\n- research/lean/NguyenRH1/AuditSession038b.lean +\n  audit-session038b-axioms.txt (8 decls, 3 standard axioms only)\n\n## Next\n\nii′-6 (L0 assembly, D-064 R4/R5): the sides identity |sides| bound via\nthe tight R-bound (5f) + the bridge |λ_n(T)| ≤ 2·FAlt(T) (hRH) +\n|Sf n| ≤ Bn FAlt n on [10⁵, N₁]; D-069 Finding 2's H(T) near-sum\nrepair map (honest ≈ 7·Ĉ₀² vs baked 3·Ĉ₀²; margin survives via E-028's\n26× slack) lives here. Then ii′-7 (closure)."},"extra_content":null}]