← Session 13 — Phase 4 completion + draft.tex author block
Phase 4 is complete and committed (`7a84ca2`). Summary of what happened:
## 4.3b — Independent-instance review (D-054): the gauntlet earned its keep
Four fresh-context review instances attacked T-1/T-2/T-3 and D-052; I hand-verified every claimed finding before acting on it. **All three theorem statements survive; all three tags were downgraded:**
- **T-1 (R-004, finite-height Li)** — two real proof breaks found and confirmed: D-039's digamma proof fails (Binet tail is constant for |W| ≳ 35; the recurrence estimate conflates an absolute bound with a coefficient), and — the load-bearing one — D-036 §1's far-sum assembly only gives O(|t| log t), not O(log t) (the Davenport reference-point trick is missing). The A/B₀ constants stand as E-021-*measured*, and the margins (67 705 at N₀, 2.46×10²⁶ at N₁) absorb any plausible constant error. Repair: **4.5a**.
- **T-2 (reduction)** — the ⇐ direction was already a named skeleton (L3/L4); D-052 had overstated it as proved. Also: Suzuki's (1.12) needs the punctured-compact reading, and τ̂± are outside Paley–Wiener–Schwartz. Tag split applied.
- **T-3 (valley theorem)** — upper bounds stand (with the K ≤ 1/m_γ simplicity fix); lower bounds are broken as written (Lemma D's 1/g² term was dropped — honest worst case is a constant floor ~0.65–0.9, not 1−1/(2a)); B3's sinc² has a factor-2 error. The reviewers' one "FATAL" (form-domain) was refuted on re-verification. Repair: **4.5c**.
## 4.4 — E-027 detector stress test (D-055, ALL CHECKS PASS)
Synthetic off-line zero pair injected into the Weil form (4 cases: a ∈ {4,8} × γ* ∈ {23.0, 51.4}):
- **Bare form: no threshold** — λ_min goes negative at the smallest δ > 0 (~10¹³ × control floor), exactly as strict Schwarz predicts. D-050's "only if δ > π/(2a)" is thereby scoped to the carrier direction.
- **Tuned-carrier threshold π/(2a): exact** — quotient crossings at +0.005% to +0.9% of δ_crit across all cases.
- **Pencil (T_a) onset is magnitude-limited**: δ ≈ 0.11 (a=4), 0.038 (a=8) — well below π/(2a).
- Negative directions localize at the pair (contrast 3.4–6.1).
**Next stage (STATE.md):** repairs 4.5a–4.5d to restore the full tags, then paper update / Phase 5.
raw content_json (database row)
[{"type":"text","text":"Phase 4 is complete and committed (`7a84ca2`). Summary of what happened:\n\n## 4.3b — Independent-instance review (D-054): the gauntlet earned its keep\n\nFour fresh-context review instances attacked T-1/T-2/T-3 and D-052; I hand-verified every claimed finding before acting on it. **All three theorem statements survive; all three tags were downgraded:**\n\n- **T-1 (R-004, finite-height Li)** — two real proof breaks found and confirmed: D-039's digamma proof fails (Binet tail is constant for |W| ≳ 35; the recurrence estimate conflates an absolute bound with a coefficient), and — the load-bearing one — D-036 §1's far-sum assembly only gives O(|t| log t), not O(log t) (the Davenport reference-point trick is missing). The A/B₀ constants stand as E-021-*measured*, and the margins (67 705 at N₀, 2.46×10²⁶ at N₁) absorb any plausible constant error. Repair: **4.5a**.\n- **T-2 (reduction)** — the ⇐ direction was already a named skeleton (L3/L4); D-052 had overstated it as proved. Also: Suzuki's (1.12) needs the punctured-compact reading, and τ̂± are outside Paley–Wiener–Schwartz. Tag split applied.\n- **T-3 (valley theorem)** — upper bounds stand (with the K ≤ 1/m_γ simplicity fix); lower bounds are broken as written (Lemma D's 1/g² term was dropped — honest worst case is a constant floor ~0.65–0.9, not 1−1/(2a)); B3's sinc² has a factor-2 error. The reviewers' one \"FATAL\" (form-domain) was refuted on re-verification. Repair: **4.5c**.\n\n## 4.4 — E-027 detector stress test (D-055, ALL CHECKS PASS)\n\nSynthetic off-line zero pair injected into the Weil form (4 cases: a ∈ {4,8} × γ* ∈ {23.0, 51.4}):\n- **Bare form: no threshold** — λ_min goes negative at the smallest δ > 0 (~10¹³ × control floor), exactly as strict Schwarz predicts. D-050's \"only if δ > π/(2a)\" is thereby scoped to the carrier direction.\n- **Tuned-carrier threshold π/(2a): exact** — quotient crossings at +0.005% to +0.9% of δ_crit across all cases.\n- **Pencil (T_a) onset is magnitude-limited**: δ ≈ 0.11 (a=4), 0.038 (a=8) — well below π/(2a).\n- Negative directions localize at the pair (contrast 3.4–6.1).\n\n**Next stage (STATE.md):** repairs 4.5a–4.5d to restore the full tags, then paper update / Phase 5."}]