[{"type":"thinking","thinking":"The user asks for project status. I've read STATE.md. Give a concise status summary. No need for more tool calls — STATE.md is the resume point and is current (updated 2026-08-22, session 024 complete).","signature":""},{"type":"text","text":"**NGUYEN-RH-1 — status as of session 024 (2026-08-22):**\n\n## Big picture\n- **Phases 0, 1, 3, 4: COMPLETE.** Phase 2 (spectral/Track B) mapped to exhaustion at current technology.\n- **Headline result (T-1 / R-004):** Li coefficients λ_n > 0 for all n ≤ N₁ ≈ 9.0×10²⁴, conditional on computational RH to T₀ = 3×10¹² (Platt–Trudgian) — and crucially, **Lean-verified modulo a single named analytic hypothesis (SfBoundProp) + the declared computational perimeter**. No sorryAx on that path; 3 standard axioms only.\n- **Paper A** (finite-height Li + machine-verified assembly): 8 pp, compiles clean, submission-ready, currently **frozen** pending the ii′ work.\n\n## Active work: Phase 5B — closing SfBoundProp in Lean (tier ii′)\nThe one remaining hypothesis is being eliminated before submission, per your option-(a) directive. Progress through the sub-stage map:\n\n| Sub-stage | Status |\n|---|---|\n| ii′-0 design (D-064); liCoeff defect found; crux Sf = Res₁ found | ✔ |\n| ii′-1a liCoeff pairing fix + summability | ✔ |\n| ii′-1b residue machinery (rectInt'_Jlag) | ✔ |\n| ii′-2 elementary swaps | ✔ |\n| ii′-3a archimedean/ζ Taylor coefficients | ✔ |\n| ii′-4 zero-avoiding T + kernel bounds + zeroMult(−2) = 1 | ✔ |\n| ii′-5a design (D-067, route b″) / 5b (‖ζ‖ ≤ (3/4)\\|t\\|² on the Davenport disc) | ✔ |\n| **ii′-5c loose R-bound** — ‖ζ′/ζ(s) − Σ_near m/(s−ρ)‖ ≤ 100000·(log\\|t\\|)², no perimeter hypothesis | ✔ (session 024) |\n| ii′-5d FE-reflection + left digamma bound | **next** |\n| ii′-5e/5f partial-fraction identity + tight constants (A = 10.5, B₀ = 48) | pending |\n| ii′-3b/3c moments → crux assembly | pending |\n| ii′-6 L0 assembly → ii′-7 closure + audit + paper Appendix B | pending |\n\nEverything green: full Lean build 8753 jobs, all new declarations axiom-audited clean.\n\n## Next action\nSession 025: **ii′-5d** — reflect the loose R-bound to the left half-plane via the functional equation (zero-sums cancel exactly by FE symmetry; ΓR-bracket via digamma + StirlingVert), plus zeroMult(−2n) = 1 for all n as 5e prep.\n\nNo blockers. Roughly 4–7 sub-stages remain before SfBoundProp is closed and Paper A gets its tag upgrade."}]