← Session 25 — ii′-5 R-bound: design + level-2 EM disc bound
thinking (557 chars)
The user says "Update me here" — they want a status update on where things stand. The session just completed ii′-5c. Let me give a concise summary of the current state. Nothing new has happened since my last message — but the user probably wants the recap of where we are now, maybe in a cleaner form. Let me check if there's anything pending... Actually the last message I gave was the completion report. "Update me here" likely means: give me the current status/where we left off (perhaps they're returning). STATE.md is updated. Just summarize concisely.
## Status: Session 024 complete — ii′-5c DONE
**Just finished (committed c7cbae5 + 0755db5):** the loose Davenport-disc R-bound in Lean.
`logDeriv_riemannZeta_sub_nearZeros_le` — for `s.im = t`, `re s ∈ [1/2, 7/2]`, `|t| ≥ 10`, `ζ s ≠ 0`:
$$\|\zeta'/\zeta(s) - \textstyle\sum_{|\gamma-t|<1} m_\rho/(s-\rho)\| \le 100000\,(\log|t|)^2$$
The full chain: R-gap selection (×10-rescaled midpoint pigeonhole, R_c ∈ [2.3, 2.4], clearance `gapD = 1/(50(k+1))`) → Blaschke quotient `g = Cf r_c F` analytic + nonvanishing (Zeta23 `Cf`/`ZeroFactorization`) → `|g| ≤ (9/4)|t|²·gapD^{−M_c}` (max-modulus over the zero-free sphere) → `LogOfAnalyticFunction` + `BorelCaratheodoryDeriv` (450·M factor) → logDeriv chain + affine push-forward + near/extras split. Constant fold landed at 75642L² ≤ 100000L². Constant is deliberately loose per D-067 §6 — it's only the growth input for the identity stage.
**Build:** green, 8753 jobs. **Axioms:** all 32 new declarations — 3 standard axioms only, no sorryAx. **Perimeter:** unchanged (and notably this theorem needed no `ZeroCountProp` — the Jensen count sufficed).
## Where we are in tier ii′
Done: ii′-0, 1a, 1b, 2, 3a, 4, 5a, 5b, **5c**.
Remaining: **ii′-5d** (FE-reflection to σ ∈ [−5/2, 1/2] + left-plane digamma bound) → ii′-5e (partial-fraction identity; needs `zeroMult(−2n) = 1` for all n) → ii′-5f (tight analysis → the baked A = 10.5, B₀ = 48) → ii′-3b (moments) → ii′-3c (crux Sf = Res₁) → ii′-6 (L0 assembly) → ii′-7 (closure + paper Appendix B).
Paper A stays frozen and submission-ready throughout (freeze protocol).
**Next action (Session 025):** ii′-5d. Say the word and I'll start.
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[{"type":"thinking","thinking":"The user says \"Update me here\" — they want a status update on where things stand. The session just completed ii′-5c. Let me give a concise summary of the current state. Nothing new has happened since my last message — but the user probably wants the recap of where we are now, maybe in a cleaner form. Let me check if there's anything pending... Actually the last message I gave was the completion report. \"Update me here\" likely means: give me the current status/where we left off (perhaps they're returning). STATE.md is updated. Just summarize concisely.","signature":""},{"type":"text","text":"## Status: Session 024 complete — ii′-5c DONE\n\n**Just finished (committed c7cbae5 + 0755db5):** the loose Davenport-disc R-bound in Lean.\n\n`logDeriv_riemannZeta_sub_nearZeros_le` — for `s.im = t`, `re s ∈ [1/2, 7/2]`, `|t| ≥ 10`, `ζ s ≠ 0`:\n$$\\|\\zeta'/\\zeta(s) - \\textstyle\\sum_{|\\gamma-t|<1} m_\\rho/(s-\\rho)\\| \\le 100000\\,(\\log|t|)^2$$\n\nThe full chain: R-gap selection (×10-rescaled midpoint pigeonhole, R_c ∈ [2.3, 2.4], clearance `gapD = 1/(50(k+1))`) → Blaschke quotient `g = Cf r_c F` analytic + nonvanishing (Zeta23 `Cf`/`ZeroFactorization`) → `|g| ≤ (9/4)|t|²·gapD^{−M_c}` (max-modulus over the zero-free sphere) → `LogOfAnalyticFunction` + `BorelCaratheodoryDeriv` (450·M factor) → logDeriv chain + affine push-forward + near/extras split. Constant fold landed at 75642L² ≤ 100000L². Constant is deliberately loose per D-067 §6 — it's only the growth input for the identity stage.\n\n**Build:** green, 8753 jobs. **Axioms:** all 32 new declarations — 3 standard axioms only, no sorryAx. **Perimeter:** unchanged (and notably this theorem needed no `ZeroCountProp` — the Jensen count sufficed).\n\n## Where we are in tier ii′\nDone: ii′-0, 1a, 1b, 2, 3a, 4, 5a, 5b, **5c**.\nRemaining: **ii′-5d** (FE-reflection to σ ∈ [−5/2, 1/2] + left-plane digamma bound) → ii′-5e (partial-fraction identity; needs `zeroMult(−2n) = 1` for all n) → ii′-5f (tight analysis → the baked A = 10.5, B₀ = 48) → ii′-3b (moments) → ii′-3c (crux Sf = Res₁) → ii′-6 (L0 assembly) → ii′-7 (closure + paper Appendix B).\n\nPaper A stays frozen and submission-ready throughout (freeze protocol).\n\n**Next action (Session 025):** ii′-5d. Say the word and I'll start."}]