← Session 31 — ii′-5f: tight R-bound 10.5/48 (piece 3 + pinch)
thinking (404 chars)
Let me also check the git log to verify the current state matches STATE.md, and maybe the results index briefly. Actually, the git status snapshot at conversation start shows recent commits ending at 6748b6e "Session 035 closure" — consistent with STATE.md saying Session 035 complete. Let me give a concise status summary.
Let me quickly verify git log to confirm nothing new since session 035 closure.
Git confirms state is consistent (HEAD = 6748b6e, session 035 closure). Here's where we are:
## Project status — NGUYEN-RH-1
**Big picture.** Phases 0, 1, 3, 4 complete; Phase 2 (spectral) mapped to exhaustion at current technology. The flagship result is **R-004**: λ_n > 0 for all n ≤ N₁ ≈ 9.0×10²⁴ (finite-height Li theorem), Lean-verified modulo `SfBoundProp` + the declared computational perimeter. Paper A (that theorem + machine verification) is submission-ready and **frozen** while we close `SfBoundProp` in Lean first — that's tier **ii′**, the active program (your directive from 2026-08-21).
**Where we are in tier ii′** (sub-stages 1a–7):
| Stage | Status |
|---|---|
| ii′-1a/1b/2/3a/4 | ✅ done (liCoeff fix, residue machinery, swaps, archimedean/ζ coefficients, contour bounds) |
| **ii′-5e** | ✅ **DONE (session 034)** — the partial-fraction identity `ζ′/ζ(s) = B + Σ_ρ m(1/(s−ρ)+1/ρ) − 1/s − 1/(s−1) + ½log π − ½ψ(s/2)` is Lean-proved modulo only the perimeter |
| **ii′-5f** | 🔶 **IN PROGRESS** — tight constants `\|R(s)\| ≤ 10.5·log(\|t\|+2) + 48` from the identity |
| ii′-3b/3c | pending (moments + crux Sf = Res₁ assembly) |
| ii′-6 | pending (L0 assembly; D-069 Finding 2 mapped an H(T) near-sum gap to face here) |
| ii′-7 | pending (closure: `sfBoundProp`, 3-perimeter-hypothesis final theorem, audit, paper Appendix B) |
**Within ii′-5f** (session 035 state):
- 5f-0 ✅ — perimeter repair (ZeroCountProp strengthened to mult-weighted) + new perimeter additions `ZeroCountProp2` + `NoLowZeros`
- 5f-i ✅ — `Rexpr_decomp` (six-piece reference-point decomposition) + pieces 1, 2, 4, 5 proved in LiTightR.lean
- 5f-ii — piece 6 ✅ (`digamma_diff_le`, ≤ 6/t); **piece 3 remaining** (the far-sum interval machine — the big one)
- 5f-iii — assembly + pinch lemma + capstone `logDeriv_riemannZeta_sub_nearZeros_le_tight`
## Next action (Session 036)
**ii′-5f-ii piece 3** in `research/lean/NguyenRH1/NguyenRH1/LiTightR.lean`: the far difference series Σ_far m·(s₀−s)/((s−ρ)(s₀−ρ)) with per-term bound 4.5/(k√(1+k²)) on intervals k ≤ |t−γ| < k+1; per-interval counts from `ZeroCountProp`; the γ<0 side via `isNontrivialZero_conj`; heights < 14 empty by `NoLowZeros`; the sum S₃ = Σ 1/(k√(1+k²)) ≤ 1.3703 (k ≤ 3 exact rational sqrt bounds + k ≥ 4 telescope giving 1/3); k > t tail ≤ 0.5828. Target: piece 3 ≤ 12.333·C₀ + 6.014, feeding the assembly 5.94·L + 11.813·LL + 46.7 ≤ 10.5·L + 48 with pinch margin ≈ 1.9.
The full piece-3 proof map is in journal-035 ("Status / next" section) + D-069 §3. Ready to start piece 3 whenever you are.
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[{"type":"thinking","thinking":"Let me also check the git log to verify the current state matches STATE.md, and maybe the results index briefly. Actually, the git status snapshot at conversation start shows recent commits ending at 6748b6e \"Session 035 closure\" — consistent with STATE.md saying Session 035 complete. Let me give a concise status summary.\n\nLet me quickly verify git log to confirm nothing new since session 035 closure.","signature":""},{"type":"text","text":"Git confirms state is consistent (HEAD = 6748b6e, session 035 closure). Here's where we are:\n\n## Project status — NGUYEN-RH-1\n\n**Big picture.** Phases 0, 1, 3, 4 complete; Phase 2 (spectral) mapped to exhaustion at current technology. The flagship result is **R-004**: λ_n > 0 for all n ≤ N₁ ≈ 9.0×10²⁴ (finite-height Li theorem), Lean-verified modulo `SfBoundProp` + the declared computational perimeter. Paper A (that theorem + machine verification) is submission-ready and **frozen** while we close `SfBoundProp` in Lean first — that's tier **ii′**, the active program (your directive from 2026-08-21).\n\n**Where we are in tier ii′** (sub-stages 1a–7):\n\n| Stage | Status |\n|---|---|\n| ii′-1a/1b/2/3a/4 | ✅ done (liCoeff fix, residue machinery, swaps, archimedean/ζ coefficients, contour bounds) |\n| **ii′-5e** | ✅ **DONE (session 034)** — the partial-fraction identity `ζ′/ζ(s) = B + Σ_ρ m(1/(s−ρ)+1/ρ) − 1/s − 1/(s−1) + ½log π − ½ψ(s/2)` is Lean-proved modulo only the perimeter |\n| **ii′-5f** | 🔶 **IN PROGRESS** — tight constants `\\|R(s)\\| ≤ 10.5·log(\\|t\\|+2) + 48` from the identity |\n| ii′-3b/3c | pending (moments + crux Sf = Res₁ assembly) |\n| ii′-6 | pending (L0 assembly; D-069 Finding 2 mapped an H(T) near-sum gap to face here) |\n| ii′-7 | pending (closure: `sfBoundProp`, 3-perimeter-hypothesis final theorem, audit, paper Appendix B) |\n\n**Within ii′-5f** (session 035 state):\n- 5f-0 ✅ — perimeter repair (ZeroCountProp strengthened to mult-weighted) + new perimeter additions `ZeroCountProp2` + `NoLowZeros`\n- 5f-i ✅ — `Rexpr_decomp` (six-piece reference-point decomposition) + pieces 1, 2, 4, 5 proved in LiTightR.lean\n- 5f-ii — piece 6 ✅ (`digamma_diff_le`, ≤ 6/t); **piece 3 remaining** (the far-sum interval machine — the big one)\n- 5f-iii — assembly + pinch lemma + capstone `logDeriv_riemannZeta_sub_nearZeros_le_tight`\n\n## Next action (Session 036)\n\n**ii′-5f-ii piece 3** in `research/lean/NguyenRH1/NguyenRH1/LiTightR.lean`: the far difference series Σ_far m·(s₀−s)/((s−ρ)(s₀−ρ)) with per-term bound 4.5/(k√(1+k²)) on intervals k ≤ |t−γ| < k+1; per-interval counts from `ZeroCountProp`; the γ<0 side via `isNontrivialZero_conj`; heights < 14 empty by `NoLowZeros`; the sum S₃ = Σ 1/(k√(1+k²)) ≤ 1.3703 (k ≤ 3 exact rational sqrt bounds + k ≥ 4 telescope giving 1/3); k > t tail ≤ 0.5828. Target: piece 3 ≤ 12.333·C₀ + 6.014, feeding the assembly 5.94·L + 11.813·LL + 46.7 ≤ 10.5·L + 48 with pinch margin ≈ 1.9.\n\nThe full piece-3 proof map is in journal-035 (\"Status / next\" section) + D-069 §3. Ready to start piece 3 whenever you are."}]