[{"type":"thinking","thinking":"The zeroMult_neg_two proof is ~100 lines with the Hfun machinery (Hfun = 4π^{−s/2}Γ(s/2+2)/s presumably defined just above line 440, with Hfun_analyticAt_neg_two, Hfun_neg_two_ne_zero, Hfun_eq helpers). Generalizing to all −2n:\n\nThe general factorization: near s = −2n (n ≥ 1): ζ(s) = Λ(s)/ΓR(s), ΓR(s) = π^{−s/2}Γ(s/2). Γ(s/2) has a simple pole at s = −2n (Γ has simple poles at 0,−1,−2,...; s/2 = −n). Write Γ(s/2) = Γ(s/2 + n + 1)/((s/2)(s/2+1)⋯(s/2+n)) via n+1 applications of the recurrence. The denominator product Q_n(s) = ∏_{j=0}^{n} (s/2 + j) vanishes to order exactly 1 at s = −2n (the j = n factor), and Γ(s/2+n+1) is analytic nonzero at −2n (value Γ(1) = 1). So:\n\nζ(s) = (s + 2n) · G_n(s) with G_n(s) = Λ(s) · 2·(−1)... let me compute: ΓR(s) = π^{−s/2} Γ(s/2+n+1)/P_n(s) where P_n(s) = ∏_{j=0}^{n}(s/2+j). Near −2n, P_n(s) = (s/2+n)·Ũ_n(s) with Ũ_n(s) = ∏_{j=0}^{n−1}(s/2+j) nonzero at −2n (values −n+j ≠ 0 for j < n ✓; at s = −2n: s/2+j = −n+j ≤ −1 ≠ 0 ✓). So ζ = Λ · P_n/(π^{−s/2}Γ(s/2+n+1)) = (s/2+n) · [Λ Ũ_n/(π^{−s/2}Γ(s/2+n+1))] and (s/2+n) = (s+2n)/2: ζ(s) = (s+2n)·G_n(s), G_n(s) = Λ(s)·Ũ_n(s)/(2·π^{−s/2}·Γ(s/2+n+1)).\n\nAt s = −2n: G_n(−2n) = Λ(−2n)·Ũ_n(−2n)/(2 π^{n} Γ(1)) = Λ(1+2n)·∏_{j=0}^{n−1}(−n+j)/(2πⁿ·1) ≠ 0: Λ(1+2n) = ΓR(1+2n)ζ(1+2n) ≠ 0 (re > 1: ζ ≠ 0 by riemannZeta_ne_zero_of_one_lt_re; ΓR ≠ 0 by re > 0) ✓; ∏(−n+j) ≠ 0 ✓; 2πⁿ ≠ 0 ✓.\n\nIn Lean, this generalization needs:\n1. The product form of the Γ recurrence iterated: Γ(z) = Γ(z+n+1)/∏_{j=0}^{n}(z+j) — a finite-product identity (induction with Gamma_add_one). New but standard: `Complex.Gamma_eq_div_prod` or similar — needs to be built (~30–50 lines with Finset.prod).\n2. Hfun_n analyticity/nonvanishing — similar to the n=1 case but with a product; AnalyticAt of products exists (Finset.prod analyticAt).\n3. The assembly mirrors zeroMult_neg_two.\n\nThis is genuinely ~0.25–0.5 session of careful work (product form of the recurrence, AnalyticAt of the cofactor, etc.). Given the session length so far (14 compile iterations on the main file) and that this is 5e-prep that \"can slip to its own sub-stage\" per STATE.md, the right call per the pacing directive (clean sub-stage boundaries) is: commit ii′-5d now as the session-025 deliverable, and put zeroMult(−2n) = 1 as the first item of session 026 (bundled with ii′-5e design). Actually — STATE.md said \"Also in scope for 5e prep (can slip to its own sub-stage): zeroMult(−2n) = 1 for ALL n ... ~0.25 sess\". \n\nHmm, let me reconsider — the fork rule and pacing: the main 5d deliverables are done, green, audited. The extra item is small but NOT trivial (product-form recurrence). Time-wise I don't have a hard signal, but the session has already produced a full sub-stage. The user directive (8.5) says break into sub-stages each ending in clean committed state — this IS the boundary. I'll defer zeroMult(−2n) to next session (it's explicitly 5e-prep, and 5e is its own sub-stage).\n\nLet me now do the session closure:\n1. journal/session-025.md\n2. Update STATE.md (NEXT ACTION → ii′-5e design + zeroMult(−2n))\n3. Update NGUYEN.md checklist (ii′-5d [x])\n4. Register session in research.db\n5. git commit\n\nLet me also fix the section-ordering cosmetic issue in LiReflect.lean: §5.5/§6 (digamma) come BEFORE §5 capstone in the file. Let me check the actual order... The file order: §0, §1, §2, §3, §4, §5.5 (digamma_add_natCast), §6 (norm_digamma_sub_log_le...), §5 capstone (logDeriv_riemannZeta_sub_nearZeros_le_left). Wait — that's what happened because I inserted §6 before the §5 header. Cosmetic but confusing. Options: (a) reorder sections in the file (move capstone before 5.5), or (b) renumber the capstone section to §7. Since the capstone USES §3/§4 results but not §6's, and §6 is independent, the cleanest cosmetic fix: renumber sections so the file order is 0,1,2,3,4,5(digamma recurrence),6(left-plane digamma),7(capstone). Let me just relabel: \"## 5.5\" → \"## 5\", \"## 6\" stays \"## 6\", and the capstone \"## 5\" → \"## 7\". Wait — currently: \"§5.5 The digamma recurrence\" then \"§6 left-plane\" then \"§5 capstone\". Renumber to: §5 digamma recurrence, §6 left-plane, §7 capstone. Two header edits + capstone header edit. Also update the file's top comment if it references section numbers (it doesn't appear to).\n\nActually let me check what the current headers are:\n- \"/-! ## 5.5. The digamma recurrence over a natural shift -/\"\n- \"/-! ## 6. The left-plane uniform digamma bound (ii′-5e input) -/\"\n- \"/-! ## 5. The left-half loose R-bound (ii′-5d capstone) -/\" (with the /-- **The loose R-bound... -/ doc)\n\nFix: 5.5 → 5, 6 stays, capstone 5 → 7. Do it.","signature":""},{"type":"text","text":"The `zeroMult(−2n)` generalization needs a product-form Γ recurrence — genuinely a separate 0.25-session build; per STATE.md it can slip to 5e-prep. ii′-5d's main deliverables are all done. Fixing the section-ordering cosmetic issue, then closing the session."},{"type":"tool_use","id":"Edit_151","name":"Edit","input":{"file_path":"D:\\Python\\Quantum\\research\\lean\\NguyenRH1\\NguyenRH1\\LiReflect.lean","old_string":"/-! ## 5.5. The digamma recurrence over a natural shift -/","new_string":"/-! ## 5. The digamma recurrence over a natural shift -/"},"extra_content":null}]