← Session 28 — ii′-5e: partial-fraction identity (zero series + F entire)
thinking (222 chars)
The user asks for current status. Give a concise summary of where the project stands. Read nothing new needed — I have full context from this session. Provide the status: where we are in tier ii′, what's done, what's next.
## Current status (end of session 030)
**Big picture**: The project is in **Phase 5B (tier ii′)** — closing the analytic hypothesis `SfBoundProp` in Lean before Paper A submission. Paper A stays frozen/submittable throughout. The partial-fraction identity route (ii′-5) is the current long pole, and it's ~85% built.
**Tier ii′ pipeline state** (design D-064):
| Sub-stage | Status |
|---|---|
| ii′-1a/1b/2 | ✅ residue machinery + swaps (sessions 015–018) |
| ii′-3a | ✅ archimedean coefficients (session 020) |
| ii′-4 | ✅ L3 zero-avoiding T + L1 kernel bounds (021) |
| ii′-5b/5c/5d | ✅ loose R-bounds ≤ 100000/200000·(log\|t\|)² both half-strips (022–025) |
| ii′-5e-0,i,ii | ✅ zeroMult(−2n)=1, γ-gap circles, digamma bounds (026–027) |
| ii′-5e-iii | ✅ the zero series: summable, analytic off zeros, polar subtraction analytic at each zero (028) |
| ii′-5e-iv | ✅ **F entire** — `Gentire` via punctured-limit extension (029) |
| **ii′-5e-v** | 🔶 **IN PROGRESS** (this session): count bridges + Dirichlet bound + tsum-split machinery done; **remaining**: tail bound → seriesFun circle bound → logDeriv/elem assembly → Schwarz squeeze ⟹ **∃ B partial-fraction identity** |
| ii′-5f | ⬜ tight constants A = 10.5, B₀ = 48 (D-056's analysis) |
| ii′-3b/3c | ⬜ moments + crux assembly (Sf = Res₁) |
| ii′-6/7 | ⬜ L0 assembly + closure (SfBoundProp proved; final audit; paper Appendix B) |
**This session (030)** added `LiCircleBound.lean`: the unconditional mult-weighted zero count (`A0count` from Zeta23's RvM local count), window/dyadic interval counts, the Dirichlet bound ‖ζ′/ζ‖ ≤ C₂ for re ≥ 2, the tsum-split tool, and `lowZeroConst`. Build is green; everything committed (`cf1e28c`).
**Next action** (STATE.md): 5e-v-b chunk 2 — the dyadic tail bound + the main `seriesFun` circle bound, then the logDeriv/elem assembly ⟹ ‖Fexpr‖ ≤ C(log R)³ on the γ-gap circles, then the Schwarz squeeze ⟹ the ∃ B identity capstone. Full design with verified API is in `research/journal/session-030.md`.
**Nothing is blocked.** Theorem status is unchanged: T-1 (finite-height Li, R-004) is Lean-verified modulo SfBoundProp + the declared computational perimeter — exactly what tier ii′ is closing.
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[{"type":"thinking","thinking":"The user asks for current status. Give a concise summary of where the project stands. Read nothing new needed — I have full context from this session. Provide the status: where we are in tier ii′, what's done, what's next.","signature":""},{"type":"text","text":"## Current status (end of session 030)\n\n**Big picture**: The project is in **Phase 5B (tier ii′)** — closing the analytic hypothesis `SfBoundProp` in Lean before Paper A submission. Paper A stays frozen/submittable throughout. The partial-fraction identity route (ii′-5) is the current long pole, and it's ~85% built.\n\n**Tier ii′ pipeline state** (design D-064):\n\n| Sub-stage | Status |\n|---|---|\n| ii′-1a/1b/2 | ✅ residue machinery + swaps (sessions 015–018) |\n| ii′-3a | ✅ archimedean coefficients (session 020) |\n| ii′-4 | ✅ L3 zero-avoiding T + L1 kernel bounds (021) |\n| ii′-5b/5c/5d | ✅ loose R-bounds ≤ 100000/200000·(log\\|t\\|)² both half-strips (022–025) |\n| ii′-5e-0,i,ii | ✅ zeroMult(−2n)=1, γ-gap circles, digamma bounds (026–027) |\n| ii′-5e-iii | ✅ the zero series: summable, analytic off zeros, polar subtraction analytic at each zero (028) |\n| ii′-5e-iv | ✅ **F entire** — `Gentire` via punctured-limit extension (029) |\n| **ii′-5e-v** | 🔶 **IN PROGRESS** (this session): count bridges + Dirichlet bound + tsum-split machinery done; **remaining**: tail bound → seriesFun circle bound → logDeriv/elem assembly → Schwarz squeeze ⟹ **∃ B partial-fraction identity** |\n| ii′-5f | ⬜ tight constants A = 10.5, B₀ = 48 (D-056's analysis) |\n| ii′-3b/3c | ⬜ moments + crux assembly (Sf = Res₁) |\n| ii′-6/7 | ⬜ L0 assembly + closure (SfBoundProp proved; final audit; paper Appendix B) |\n\n**This session (030)** added `LiCircleBound.lean`: the unconditional mult-weighted zero count (`A0count` from Zeta23's RvM local count), window/dyadic interval counts, the Dirichlet bound ‖ζ′/ζ‖ ≤ C₂ for re ≥ 2, the tsum-split tool, and `lowZeroConst`. Build is green; everything committed (`cf1e28c`).\n\n**Next action** (STATE.md): 5e-v-b chunk 2 — the dyadic tail bound + the main `seriesFun` circle bound, then the logDeriv/elem assembly ⟹ ‖Fexpr‖ ≤ C(log R)³ on the γ-gap circles, then the Schwarz squeeze ⟹ the ∃ B identity capstone. Full design with verified API is in `research/journal/session-030.md`.\n\n**Nothing is blocked.** Theorem status is unchanged: T-1 (finite-height Li, R-004) is Lean-verified modulo SfBoundProp + the declared computational perimeter — exactly what tier ii′ is closing."}]