← Session 27 — ii′-5c/5d: Davenport disc + FE-reflection bounds
thinking (2394 chars)
Axiom audit PASSES: all 26 new declarations depend on exactly the 3 standard axioms [propext, Classical.choice, Quot.sound], zero sorryAx.
ii′-5d is COMPLETE:
1. ✅ FE logDeriv identity (logDeriv_zeta_fe)
2. ✅ Zero-sum cancellation (sum_nearZeros_neg_reflect)
3. ✅ ΓR-bracket bound (gammaR_bracket_norm_le ≤ 4(log|t|)²)
4. ✅ Left-half R-bound capstone (logDeriv_riemannZeta_sub_nearZeros_le_left ≤ 200000·(log|t|)²)
5. ✅ Left-plane uniform digamma bound (norm_digamma_sub_log_le_of_two_le_abs_im ≤ 3) — the 5e input
6. ✅ Full build green (8761 jobs)
7. ✅ Axiom audit: 26 decls, 3 standard axioms only
The deferred item: zeroMult(−2n) = 1 for all n (5e prep) — STATE.md says it "can slip to its own sub-stage". Let me check the time/energy budget — the session has been productive; the generalization of zeroMult_neg_two is estimated ~0.25 session. Should I do it now or leave it? STATE.md's NEXT ACTION for 5d listed it as "Also in scope for 5e prep (can slip to its own sub-stage)". Let me do it now if it's genuinely quick — actually, let me assess: generalizing zeroMult_neg_two (in LiContour.lean) from −2 to −2n for ALL n. The proof there used the Hfun factorization ζ = (s+2)·Λ/Hfun with Hfun = 4π^{−s/2}Γ(s/2+2)/s. For general −2n: ζ = (s+2n)·Λ/Hfun_n with Hfun_n involving Γ(s/2+n+1)... The LiContour proof was presumably somewhat involved (session 021). Generalizing: either (a) induction on n via the recurrence, or (b) the direct factorization with Γ(s/2+n+1). Given the session is already substantial and this is explicitly allowed to slip (and ii′-5e is its own sub-stage that would consume it), I'll defer it to the start of 5e and note it in STATE. Actually — hmm, let me reconsider: per the pacing directive (sub-stages ending in clean committed states) and the fact that the main 5d deliverables are ALL done and audited, this is a clean boundary. The zeroMult(−2n) generalization belongs with 5e prep. Decision: defer (documented in STATE as the first item of the next session).
Wait — actually, let me reconsider once more. The session protocol says NO RUSH and take the time needed; but also the session has been long and there were MANY compile iterations. The remaining item is small (0.25 sess). Let me look at zeroMult_neg_two quickly to gauge whether the generalization is truly quick. If it's a clean induction/generalistion, do it; if it needs new machinery, defer.
Axiom audit PASSES: all 26 new declarations, 3 standard axioms only, zero sorryAx. Let me check the `zeroMult_neg_two` proof to gauge the 5e-prep generalization (zeroMult(−2n) = 1) before deciding whether to include it now.
tool: Bash
input
{
"command": "grep -n \"zeroMult_neg_two\\|zeroMult_neg\\|theorem zeroMult\" /d/Python/Quantum/research/lean/NguyenRH1/NguyenRH1/LiContour.lean | head -5",
"description": "Locate zeroMult_neg_two in LiContour"
}
raw content_json (database row)
[{"type":"thinking","thinking":"Axiom audit PASSES: all 26 new declarations depend on exactly the 3 standard axioms [propext, Classical.choice, Quot.sound], zero sorryAx.\n\nii′-5d is COMPLETE:\n1. ✅ FE logDeriv identity (logDeriv_zeta_fe)\n2. ✅ Zero-sum cancellation (sum_nearZeros_neg_reflect)\n3. ✅ ΓR-bracket bound (gammaR_bracket_norm_le ≤ 4(log|t|)²)\n4. ✅ Left-half R-bound capstone (logDeriv_riemannZeta_sub_nearZeros_le_left ≤ 200000·(log|t|)²)\n5. ✅ Left-plane uniform digamma bound (norm_digamma_sub_log_le_of_two_le_abs_im ≤ 3) — the 5e input\n6. ✅ Full build green (8761 jobs)\n7. ✅ Axiom audit: 26 decls, 3 standard axioms only\n\nThe deferred item: zeroMult(−2n) = 1 for all n (5e prep) — STATE.md says it \"can slip to its own sub-stage\". Let me check the time/energy budget — the session has been productive; the generalization of zeroMult_neg_two is estimated ~0.25 session. Should I do it now or leave it? STATE.md's NEXT ACTION for 5d listed it as \"Also in scope for 5e prep (can slip to its own sub-stage)\". Let me do it now if it's genuinely quick — actually, let me assess: generalizing zeroMult_neg_two (in LiContour.lean) from −2 to −2n for ALL n. The proof there used the Hfun factorization ζ = (s+2)·Λ/Hfun with Hfun = 4π^{−s/2}Γ(s/2+2)/s. For general −2n: ζ = (s+2n)·Λ/Hfun_n with Hfun_n involving Γ(s/2+n+1)... The LiContour proof was presumably somewhat involved (session 021). Generalizing: either (a) induction on n via the recurrence, or (b) the direct factorization with Γ(s/2+n+1). Given the session is already substantial and this is explicitly allowed to slip (and ii′-5e is its own sub-stage that would consume it), I'll defer it to the start of 5e and note it in STATE. Actually — hmm, let me reconsider: per the pacing directive (sub-stages ending in clean committed states) and the fact that the main 5d deliverables are ALL done and audited, this is a clean boundary. The zeroMult(−2n) generalization belongs with 5e prep. Decision: defer (documented in STATE as the first item of the next session).\n\nWait — actually, let me reconsider once more. The session protocol says NO RUSH and take the time needed; but also the session has been long and there were MANY compile iterations. The remaining item is small (0.25 sess). Let me look at zeroMult_neg_two quickly to gauge whether the generalization is truly quick. If it's a clean induction/generalistion, do it; if it needs new machinery, defer.","signature":""},{"type":"text","text":"Axiom audit PASSES: all 26 new declarations, 3 standard axioms only, zero sorryAx. Let me check the `zeroMult_neg_two` proof to gauge the 5e-prep generalization (zeroMult(−2n) = 1) before deciding whether to include it now."},{"type":"tool_use","id":"Bash_149","name":"Bash","input":{"command":"grep -n \"zeroMult_neg_two\\|zeroMult_neg\\|theorem zeroMult\" /d/Python/Quantum/research/lean/NguyenRH1/NguyenRH1/LiContour.lean | head -5","description":"Locate zeroMult_neg_two in LiContour"},"extra_content":null}]