[{"type":"thinking","thinking":"The user says they walked out and are on Telegram now, asking for complete updates. They want a full status update — presumably everything since they stepped away, which covers the rest of session 015 (ii′-1a completion). Let me give a complete, self-contained update they can read on their phone: what was done, what's the state, what's next. Keep it complete but readable — not too long, but covering everything important. No new work requested, just the update.","signature":""},{"type":"text","text":"Complete update — everything is committed and the project is in a clean, resumable state. Here's where things stand:\n\n## What was completed today (sessions 014 + 015)\n\n**Session 014 — tier ii′ designed, with two major discoveries:**\n\n1. **Found a real encoding defect in the Lean work** (this is what your audit question uncovered): `liCoeff` was defined as an *unordered* infinite sum that is provably **not absolutely summable** (terms decay like n/|γ|, and Σ 1/|ρ| over the zeros diverges). In Lean, that sum silently evaluates to 0 — so the two hypotheses `LiVerifiedUpTo` and `SfBoundProp` were actually **false as encoded**, making the session-012 theorems vacuous. The mathematics in Paper A was always fine; this was purely a Lean encoding bug that the session-013 audit couldn't see (it checks axioms, not whether hypotheses are true).\n\n2. **The crux identity turned out simpler than feared**: S_f(n) = S_∞(n) + 1 − λ_n equals the w=1 residue Σⱼ C(n,j)c_{j−1} **exactly** (from Lagarias's (4.10), verified numerically against E-015's certified values). This makes the whole assembly **tail-free**: |S_f| ≤ |λ_n(T)| + |contour sides| + |trivial term| = B(n). The scariest sub-task from the old plan dissolves into two bounded pieces.\n\n3. **D-064 design doc written**: full route decisions R1–R5, lemma dependency graph, sub-stage map ii′-1a…7 (est. 9–16 sessions), feasibility confirmed (rectangle residue theorem already vendored and axiom-clean; Borel–Carathéodory/Jensen/digamma all present).\n\n**Session 015 — first sub-stage executed and closed:**\n\n- **NGUYEN.md got the \"PHASE 5B\" checklist** you asked for (ii′-0…ii′-7 with freeze protocol), committed.\n- **ii′-1a COMPLETE**: the defect is fixed. `liCoeff` now uses the classical paired symmetric sum (½·mult·(f ρ + f ρ̄)), which is manifestly real and — the hard part — **provably absolutely summable** in Lean (~340 new lines of proofs: the tail bound |Re f(ρ)| ≤ 2n²/‖ρ‖², comparison with Zeta23's already-proved Σ mult/(1+‖γ‖²) < ∞, plus a finite-ball argument for small zeros).\n- **Full package rebuild green** (8741 jobs) with **zero downstream changes needed** — and now the two hypotheses are *true* statements instead of vacuous ones.\n- **Axiom audit clean**: only the 3 standard axioms everywhere, including both main theorems.\n\n## Safety state (your standing requirement)\n\nPaper A is **untouched and submission-ready at this exact moment** (freeze protocol in force: no edits to paperA/paperB/draft/references until the final sub-stage ii′-7). Every sub-stage ends committed and resumable. If we change course at any point, nothing has moved — and in fact the submission is now *stronger* because the formal claims are no longer vacuous.\n\n## What's next\n\n- **Session 016 = sub-stage ii′-1b** (1–2 sessions): the residue machinery — general finite-order residue, rectangle residue theorem extended beyond simple poles, the w=1 residue by the Leibniz identity (verified algebraically in session 014), residues at zeros.\n- Then: ii′-2 (elementary binomial swaps), ii′-3 (the (4.10) Laurent relation — the long pole, 2–4 sessions), ii′-4, ii′-5 (R-bound), ii′-6 (assembly), ii′-7 (closure + audit + paper Appendix B update).\n\nEverything is committed: session 014 = e56cb66, checklist = 7a395b2, session 015 = be1ffb2. Say the word and I'll start ii′-1b."}]