← Session 22 — ii′-1: liCoeff pairing fix + residue machinery
thinking (128 chars)
The user is asking about the status of Phase 5B. I have the STATE.md and NGUYEN.md context. Let me give a concise status report.
## Phase 5B status — tier ii′ (close `SfBoundProp` in Lean before Paper A submission)
**Goal:** discharge the last analytic hypothesis (`SfBoundProp`) so `lambda_pos_finite_height_verified` rests only on the 3 declared perimeter hypotheses (Platt–Trudgian RHUpTo, LiVerifiedUpTo, Trudgian N(T)/S(T), M₀).
**Progress — 3 of 8 sub-stages done (sessions 014–016):**
| Sub-stage | Status | Content |
|---|---|---|
| ii′-0 Design | ✅ (session 014) | D-064 design doc. Two discoveries: (1) **liCoeff encoding defect** (unordered tsum → junk; hypotheses FALSE as encoded); (2) **crux identity Sf(n) = Res₁(n) EXACT** → assembly is tail-free |
| ii′-1a liCoeff fix | ✅ (session 015) | Paired `liSummand` + `liSummable` PROVED; rebuild green (8741 jobs); hypotheses now TRUE statements |
| ii′-1b R1 residues | 🔶 2/3 done (session 016) | (i) `ResW1_eq_sum` (w=1 residue = Σⱼ C(n,j)·lcoeff(j−1)) ✅; (ii) `rectInt'_ppart` (principal-part integral) ✅; **(iii) rectangle residue theorem for J — NEXT** |
| ii′-2 … ii′-7 | ⬜ | elementary swaps → (4.10) **long pole** (2–4 sess) → L3+L1 → R-bound → assembly → closure/audit |
**Session 017 next action** (STATE.md:72): ii′-1b(iii) — R := J − ppart n holomorphic off {zeros} ∪ {−2}, simple poles at zeros (residue mult·(1−(ρ/(ρ−1))ⁿ)) and at −2 (residue 1−(2/3)ⁿ), then vendored `residueTheorem_finset` to assemble sides(T) = ResW1 n + Σ + triv(n).
Everything axiom-audited so far: 3 standard axioms only. Freeze protocol in force — Paper A stays submission-ready/frozen until ii′-7. Estimate remaining: ~6–13 sessions, with ii′-3 (Lagarias (4.10)) the long pole.
No blockers. Ready to start ii′-1b(iii) whenever you are.
raw content_json (database row)
[{"type":"thinking","thinking":"The user is asking about the status of Phase 5B. I have the STATE.md and NGUYEN.md context. Let me give a concise status report.","signature":""},{"type":"text","text":"## Phase 5B status — tier ii′ (close `SfBoundProp` in Lean before Paper A submission)\n\n**Goal:** discharge the last analytic hypothesis (`SfBoundProp`) so `lambda_pos_finite_height_verified` rests only on the 3 declared perimeter hypotheses (Platt–Trudgian RHUpTo, LiVerifiedUpTo, Trudgian N(T)/S(T), M₀).\n\n**Progress — 3 of 8 sub-stages done (sessions 014–016):**\n\n| Sub-stage | Status | Content |\n|---|---|---|\n| ii′-0 Design | ✅ (session 014) | D-064 design doc. Two discoveries: (1) **liCoeff encoding defect** (unordered tsum → junk; hypotheses FALSE as encoded); (2) **crux identity Sf(n) = Res₁(n) EXACT** → assembly is tail-free |\n| ii′-1a liCoeff fix | ✅ (session 015) | Paired `liSummand` + `liSummable` PROVED; rebuild green (8741 jobs); hypotheses now TRUE statements |\n| ii′-1b R1 residues | 🔶 2/3 done (session 016) | (i) `ResW1_eq_sum` (w=1 residue = Σⱼ C(n,j)·lcoeff(j−1)) ✅; (ii) `rectInt'_ppart` (principal-part integral) ✅; **(iii) rectangle residue theorem for J — NEXT** |\n| ii′-2 … ii′-7 | ⬜ | elementary swaps → (4.10) **long pole** (2–4 sess) → L3+L1 → R-bound → assembly → closure/audit |\n\n**Session 017 next action** (STATE.md:72): ii′-1b(iii) — R := J − ppart n holomorphic off {zeros} ∪ {−2}, simple poles at zeros (residue mult·(1−(ρ/(ρ−1))ⁿ)) and at −2 (residue 1−(2/3)ⁿ), then vendored `residueTheorem_finset` to assemble sides(T) = ResW1 n + Σ + triv(n).\n\nEverything axiom-audited so far: 3 standard axioms only. Freeze protocol in force — Paper A stays submission-ready/frozen until ii′-7. Estimate remaining: ~6–13 sessions, with ii′-3 (Lagarias (4.10)) the long pole.\n\nNo blockers. Ready to start ii′-1b(iii) whenever you are."}]