[{"type":"thinking","thinking":"The user wants to stop. Everything is already committed (21c7497). They want a clear summary of where the research stands with the new stage (tier ii′). Let me give a concise, complete status summary — the project is saved, and here's the state. No new work needed. Just confirm everything is committed and give the status.","signature":""},{"type":"text","text":"Everything is saved — all work committed to git (latest: 21c7497), nothing lives only in conversation. Here's the complete picture of where we stand.\n\n## The new stage: tier ii′ — closing the Lean contour bound before submission\n\n**The goal.** Upgrade Paper A's finite-height Li theorem (λₙ > 0 for n ≤ (3×10¹²−1)² ≈ 9×10²⁴) from \"Lean-verified *assuming* the contour bound `SfBoundProp`\" to \"Lean-verified with the contour bound **proved inside Lean**\". Perimeter that stays declared (honest, per D-062): Platt–Trudgian RH-to-3×10¹², finite Li verification to 5×10⁵, Trudgian's N(T)/S(T), M₀.\n\n**Protection:** Paper A is frozen and submission-ready at all times — every sub-stage ends committed; nothing moves until the final sub-stage ii′-7.\n\n## Progress so far (3 of ~9 sub-stages done)\n\n| Sub-stage | Status | What it delivered |\n|---|---|---|\n| **ii′-0** (session 014) | ✔ | Design D-064 + **two discoveries**: (1) the `liCoeff` encoding was *false as written* (junk sum — the audit question you asked caught it); (2) the crux identity **S_f = Res₁ exactly**, making the assembly tail-free |\n| **ii′-1a** (session 015) | ✔ | The defect **repaired**: `liCoeff` now uses the correct paired sum, summability *proved*; `LiVerifiedUpTo`/`SfBoundProp` are now true statements, not vacuous |\n| **ii′-1b** (sessions 016–17) | **2/3 done** | (i) the w=1 residue value: **ResW1 = Σⱼ C(n,j)·lcoeff(j−1)** proved by Leibniz; (ii) the principal part integrates to the residue (**rectInt'_ppart**); (iii) the full rectangle residue theorem for J — *next* |\n\nAll new Lean code: full build green, **only the 3 standard axioms** everywhere (audited live each session).\n\n## What's next\n1. **ii′-1b(iii)** — rectangle residue theorem for J (poles at zeros + trivial zero −2; assembly `sides = ResW1 + Σ zeros + triv`).\n2. **ii′-2** — the two elementary binomial swaps (λₙ and S_∞ into binomial form).\n3. **ii′-3** — the long pole: the Lagarias (4.10) Laurent relation (EF-reuse first, Hadamard fallback).\n4. **ii′-4/5/6/7** — zero-avoiding T + kernel bounds, the R-bound, final assembly, closure + paper Appendix B update.\n\nEstimate: ~9–16 sessions total; we've done the design and the first machinery tiers on schedule.\n\n## State of the mission\n- **Paper A**: submission-ready *right now* if you change your mind — and *stronger* than before, because the formal claims are no longer vacuous (the liCoeff fix).\n- **Archive**: STATE.md has the exact resume point (\"Session 017: ii′-1b(iii)\"); journal-014/015/016 have full narratives; NGUYEN.md Phase 5B checklist tracks all 9 sub-stages.\n\nWhenever you're ready, just say continue — the resume point is set."}]