Session 039 (2026-08-25) — F-D side thread ii′-6-R COMPLETE
Mission
Execute the ii′-6-R side thread (user directive 2026-08-25): the focused re-proof of ONLY the right part of the horizontal segments in Lagarias's Thm 6.1 (the broken (6.12) stepping, Finding F-D), per the persisted mission prompt D-073 §0. Deliverables D1–D4: the E-032 diagnostic, the repaired proof with explicit constants + audits, the cross-validation E-033, and the two draft paragraphs.
What happened (in order)
- 6-R-0 — E-032 diagnostic (commit ded9603, ALL CHECKS PASS).
Built
E-032-horizontal-true-integral.py(dps 50): the TRUE complex right-tail integral ∫₃^{2√n} k_n(w−1)(−ζ′/ζ(w)) dσ at T = c√n+ε (zero-avoiding ε from a deterministic scan). −ζ′/ζ computed two independent ways: truncated Dirichlet series with CERTIFIED tail < 1e-11 (primary, σ ≥ 5) and mpmath (σ ∈ [3,5)), cross-validated at 5 spot sigmas: max discrepancy 6.0×10⁻¹⁰. Trapezoid h = 0.05/0.02, h-delta ≤ 5×10⁻⁵. Envelope reproducibility vs E-031 ✓ (refined: 1.026e4 at 10⁴, 4.707e10 at 10⁵). VERDICT: HIDDEN CANCELLATION — decisive: | n | |I_right| | envelope | ratio | |I_II| (full) | |I_II|/√n | | 10³ | 7.18 | 50.8 | 0.141 | 7.44 | 0.235 | | 10⁴ | 16.70 | 1.03e4 | 1.6e-3 | 19.21 | 0.192 | | 10⁵ | 26.23 | 4.71e10 | 5.6e-10 | 30.89 | 0.098 | The per-side claim |I_II| = O(√n) is empirically TRUE (decreasing ratio!) but envelope-unprovable at T = √n. Mid-segment [−5/2, 3] almost no cancellation (0.7–0.95) — honest bound. Repair heights: envelope collapses (≈1.0 at c = 1.5, ≈0.35 at c = 2), ratio ≈ 1 — the repair needs no cancellation. Per the mission's STOP rule, reported to the user. User direction: repair proof only, the T = √n salvage parked as an open question (D-073 §8 D4). - 6-R-1 — the repaired proof (D-073 §§3–7) + E-033 (commit
de1a805). §3 dependency audit: of Thm 6.1's 13 ingredients ONLY
the (6.12) stepping is proof-breaking; items 4/8/12 benign
quantitative changes; AUDIT NOTE: the printed "(1+4/(16+n))ⁿ"
intermediate under-bounds the true max at σ = 3 by ~21% — a
harmless chain slip, e⁴+1 stays valid (NOT a second bug). §4
Lemma 4.1 (corrected stepping) + Cor 4.2 (explicit R(c))
[PROVED, elementary, self-contained]: P(σ+1)/P(σ) ≤ q = e^{1/c²}/2
for σ ≥ 3; R(c) = |ζ′/ζ(2)|·[e^{4/c²}/(4(1−q)log 2) + 1/(2 log 2)].
§5 c-choice: AUDIT CORRECTION — recommend c = 1.5 (reverses
D-072 §6's "2 margin/1.5 range" framing): the E-033 fold
computation shows the margin is I_III-LENGTH-DOMINATED (4T[…] ~ T·log
T ~ 87% of Bn at 10⁵), so smaller c wins BOTH the base margins
(181 256 vs 109 370 at 10⁵) AND the range (N₁ = 4×10²⁴ vs
2.25×10²⁴); c = 2 documented as the ultra-conservative option
(R = 1.97 < broken 3.3, q-margin 0.36). §6 consequence audit:
Thm 6.1 statement UNCHANGED (O-constant c-dependent); Thm 1.1
height ((T₀−1)/c)²; NO new perimeter. §7 E-033 ALL CHECKS
PASS: V1 Z₀ certified [0.5699210, 0.5703661] (charged upper edge;
first run failed on an over-tight script window — certified bracket
was correct; fixed, honesty note in D-073 §7); V2 q-table incl.
q(1) = 1.359 (the bug) and c* = 1.2011224; V3 R covers the E-032
measured truth 3.2–6.3× slack; V4 kernel charges (5.65 ≤ 6.92 at
1.5, 2.06 ≤ 3.72 at 2); V5 fold fit ≤ 0.3925/0.3883; V6 margins
0 on [8×10⁴, N₁(c)] (min 111 917 at c = 1.5); V7 full fold.
- 6-R-2 — drafts (D-073 §8). D3 the letter repair paragraph (one-line fix T = c√n + ε, c > 1/√log 2, with the E-032 note that the T = √n per-side statement appears true) + D4 the finding-note per-side verdict paragraph (with the OPEN QUESTION: the hidden-cancellation mechanism — not stationary-phase, |Φ′| = O(1)). DRAFTS ONLY — correspondence files untouched (freeze protocol).
Outcomes / state
- ii′-6-R side thread COMPLETE — all three success criteria met (repaired proof with explicit constants + audits, E-032 verdict recorded, main line can execute repair (R) from D-073 without further analysis).
- NGUYEN.md §5 ii′-6-R checked off; STATE.md rewritten: NEXT ACTION = main line executes repair (R) at c = 1.5 (the D-071 8-step protocol re-run: Bn_c at c = 1.5, convexity re-proof, base-point re-certification — values IMPROVE — then ii′-6-c with the fixed stepping; N₁ = 4×10²⁴; paper correction note at ii′-7 covers F-A + F-B + the F-D repair).
- db registrations: E-032 (9 rows, dps 50), E-033 (29 rows, dps 50).
- Commits: ded9603 (6-R-0), de1a805 (6-R-1/6-R-2 + closure).
Banked (technical)
- mpmath ζ vs certified-tail Dirichlet series agree to 6×10⁻¹⁰ at t ~ 632, dps 50, σ ∈ [3,5] — the "complex-ζ OK to ~1e-10" rule of thumb holds at these heights; the Dirichlet-series-with-certified- tail is the right primary for σ ≥ 5 (K(σ) collapses: 642 at σ = 5, 35 at σ = 8, 10 at σ = 12).
- Trapezoid on analytic integrands with tiny endpoints is near-spectral: h = 0.05 vs 0.02 delta ≤ 5×10⁻⁵ over ~2×10⁹-magnitude peaks (the Euler–Maclaurin error lives at the endpoints).
- The zero-avoiding ε-scan (20-point grid, zetazero at dps 50) is deterministic and cheap — reuse for future contour numerics.
- Detached-spawn pattern worked again (5–8 min run, log file, poll).
Next
Session 040: main line executes repair (R) at c = 1.5 — the ii′-6-a redo per D-073 §5 (STATE.md NEXT ACTION has the 8-step rundown with the exact Bn_c formula and the E-033-validated margins).