[{"type":"text","text":"## Side thread ii′-6-R — COMPLETE\n\nAll three sub-stages done and committed (`ded9603`, `de1a805`).\n\n**6-R-0 (E-032):** the T = √n per-side claim is empirically **true** via massive hidden cancellation (|I_II| = 7.4/19.2/30.9 vs envelope up to 4.7×10¹⁰) — but unprovable there. Salvage parked as an open question (your call).\n\n**6-R-1 (D-073 §§3–7 + E-033):** the repair is proved and cross-validated:\n- **Dependency audit:** of Thm 6.1's 13 ingredients, only the (6.12) stepping is proof-breaking. (Found one harmless extra looseness: the (1+4/(16+n))ⁿ intermediate under-bounds at σ=3 by ~21%, but e⁴+1 stays valid — not a second bug.)\n- **Lemma 4.1 + Cor 4.2 [PROVED, elementary]:** product ratio ≤ q = e^{1/c²}/2 < 1 per step; explicit right-tail bound R(c) = |ζ′/ζ(2)|·[e^{4/c²}/(4(1−q)log 2) + 1/(2 log 2)].\n- **Audit correction on c:** recommend **c = 1.5**, reversing D-072's \"2 for margin / 1.5 for range\" — the fold is I_III-length-dominated, so c = 1.5 wins *both* (margin(10⁵) = 181,256 vs 109,370; N₁ = 4×10²⁴ vs 2.25×10²⁴). c = 2 documented as the conservative option (R = 1.97 < the broken 3.3).\n- **E-033 ALL CHECKS PASS:** Z₀ certified, R covers the measured truth with 3.2–6.3× slack, fold fit ≤ 0.40, margins positive to N₁, full fold. (One first-run failure was a script-tolerance error, fixed, trail in git.)\n\n**6-R-2 (D-073 §8):** drafts for the Lagarias letter (one-line repair paragraph) and finding note (per-side verdict + open question) — correspondence files untouched per freeze protocol.\n\n**Consequence:** Lagarias's Thm 6.1 statement survives with a c-dependent constant; Thm 1.1 height becomes ((T₀−1)/c)². For us: N₁ = 4×10²⁴ at c = 1.5, criterion-3 intact, no new perimeter hypotheses.\n\n**Next action (session 040):** main line executes repair (R) at c = 1.5 — the ii′-6-a redo (Bn redefinition, convexity re-proof, base-point re-certification — values improve) then ii′-6-c with the fixed stepping. STATE.md has the full 8-step rundown.\n\nContinue now with session 040 (Lean work), or end the session here (I'll write journal-039 and run the end protocol)?"}]