← Session 13 — Phase 4 completion + draft.tex author block
thinking (233 chars)
Done. Final summary. Also update memory? The session had workflow learnings: db.py INSERT OR REPLACE gotcha — recorded in STATE handoff (project-specific, belongs in STATE.md not memory). Nothing memory-worthy beyond that. Summarize.
All four repairs complete, paper updated, everything committed (6 commits this session, ending at `5a03bf9`). Summary:
## Repair stage complete
**4.5a — R-004 hardening (D-056 + E-028, ALL CHECKS PASS).** The R-bound is now proved by the reference-point method (s₀ = 2+it): the fatal |s| numerator becomes |s₀−s| ≤ 4.5 and the far sum becomes the telescoping Σ 1/(|t−γ|(1+|t−γ|)). Result: |R(s)| ≤ 10.5 log(|t|+2) + 48, verified at 15 sample points with 26× slack. D-039's digamma bound repaired (Binet split at t ~ |V|/2; correct max-modulus recurrence majorant) → |r(w)| ≤ 1.2/|w|². **Self-review caught a second bug en route**: D-036's closed-form C₀ dropped the log log term — erratum filed, constants rebuilt on the honest form. New N₀ = 80 000 (6.25× Misguich–Voros overlap); margin(N₁) = 2.458×10²⁶ unchanged. **R-004's tag is fully restored.**
**4.5b — convexity proved (D-057 + E-029, ALL CHECKS PASS).** Chain-rule reduction φ(T(n))″ = ψ[φ](T)/(4n) turns concavity into 11 one-variable sign conditions, each discharged by an elementary inequality; sympy cross-checked. R-004's analytic chain is now proved end-to-end.
**4.5c — valley theorem repaired (D-058).** B1′(i): isolated zeros get the originally-claimed floor rate, now proved. B1′(ii): general zeros via node-killing projections — and self-review caught another overclaim (rim-annulus survivors contribute Θ(a²) via Bernstein), so the honest general form carries an explicit annulus count E(γ,a). Plus the B3 sinc² argument fix, the m_γ fix, and the three-level detector restatement.
**4.5d — bookkeeping (D-059).** Punctured-compact reading of (1.12); R_τ := W/F − R₀ with the distributional representation as provenance; Lemma FD (form-domain identification — flat bumps admissible); L1 written out.
Paper (`draft.tex`) updated throughout. **Next stage** (STATE.md): Phase 5 planning — paper honesty audit (5.4) vs Lean skeletons (5.1) vs the symbol-space opening on the (★) route.
raw content_json (database row)
[{"type":"thinking","thinking":"Done. Final summary. Also update memory? The session had workflow learnings: db.py INSERT OR REPLACE gotcha — recorded in STATE handoff (project-specific, belongs in STATE.md not memory). Nothing memory-worthy beyond that. Summarize.","signature":""},{"type":"text","text":"All four repairs complete, paper updated, everything committed (6 commits this session, ending at `5a03bf9`). Summary:\n\n## Repair stage complete\n\n**4.5a — R-004 hardening (D-056 + E-028, ALL CHECKS PASS).** The R-bound is now proved by the reference-point method (s₀ = 2+it): the fatal |s| numerator becomes |s₀−s| ≤ 4.5 and the far sum becomes the telescoping Σ 1/(|t−γ|(1+|t−γ|)). Result: |R(s)| ≤ 10.5 log(|t|+2) + 48, verified at 15 sample points with 26× slack. D-039's digamma bound repaired (Binet split at t ~ |V|/2; correct max-modulus recurrence majorant) → |r(w)| ≤ 1.2/|w|². **Self-review caught a second bug en route**: D-036's closed-form C₀ dropped the log log term — erratum filed, constants rebuilt on the honest form. New N₀ = 80 000 (6.25× Misguich–Voros overlap); margin(N₁) = 2.458×10²⁶ unchanged. **R-004's tag is fully restored.**\n\n**4.5b — convexity proved (D-057 + E-029, ALL CHECKS PASS).** Chain-rule reduction φ(T(n))″ = ψ[φ](T)/(4n) turns concavity into 11 one-variable sign conditions, each discharged by an elementary inequality; sympy cross-checked. R-004's analytic chain is now proved end-to-end.\n\n**4.5c — valley theorem repaired (D-058).** B1′(i): isolated zeros get the originally-claimed floor rate, now proved. B1′(ii): general zeros via node-killing projections — and self-review caught another overclaim (rim-annulus survivors contribute Θ(a²) via Bernstein), so the honest general form carries an explicit annulus count E(γ,a). Plus the B3 sinc² argument fix, the m_γ fix, and the three-level detector restatement.\n\n**4.5d — bookkeeping (D-059).** Punctured-compact reading of (1.12); R_τ := W/F − R₀ with the distributional representation as provenance; Lemma FD (form-domain identification — flat bumps admissible); L1 written out.\n\nPaper (`draft.tex`) updated throughout. **Next stage** (STATE.md): Phase 5 planning — paper honesty audit (5.4) vs Lean skeletons (5.1) vs the symbol-space opening on the (★) route."}]