Session 005 — 2026-08-18

Plan (user-approved): do ALL queued directions one by one (Fork Rule)

Order: A3 → A1 → A2 → C → B, with D (Li n>2000) launched in background. A3 = Q-S3c Herglotz diagnostic; A1 = Q-S3a analytic; A2 = Q-S3b backend; C = ξ′-world ceiling; B = T4b scoping; D = Li n=3000 detached run.

Work done

D (background): E-015 launched

• Created E-015-li-coefficients-n3000.py (E-011 method, n_max=3000, dps=1450, own output files — avoids clobbering E-011's registered CSV).
• First detached launch FAILED: sandbox blocks DuplicateHandle (WinError 5) for multiprocessing spawn. Relaunched with sandbox disabled — healthy (main PID 23300 + 14 workers). ETA ~1.5–2 h.
• LESSON: detached multiprocessing jobs must be launched with the sandbox disabled (session-004 machinery note amended).
• Run still in progress at session stop (user called stop). If the machine shuts down, rerun: E-015 script with --force (crash-safe: outputs written only at completion).

A3 (Q-S3c): E-016 created, run, V3 fails — rich diagnostic findings

• E-016-herglotz-fine-structure.py: ρ_W(y) = −(1/π) Im(W′/W)(iy) via ANALYTIC Galerkin z-derivative (no finite differences) at θ=π and θ=0 control, a ∈ {4,6,8,10}, vs ρ_F(y) = (1/π)[2/y + ξ′/ξ − ξ″/ξ′].
• Bugs found and fixed during bring-up: (i) hat-derivative missing a factor h; (ii) ξ(1) removable pole at y=0.5 (mp.zeta(1) raises) — exact value ½ supplied; (iii) flint cross-check NaN at s=1 (arb pole) — point dropped; (iv) |W|-minima zero finder with brent polish was noisy at large z (spurious brackets where |W| grazes) — replaced by exact real-gauge sign-change + brentq (c=1 exactly ⟹ W(iy-real) pure imaginary, g(z)=z ImV−ReV).
• Results: V1 (plateau = a/π, ≤0.15%) PASS; V2 (ρ_F vs finite-diff of log|F|, 1.3e-8) PASS; V4 (mirror symmetry) PASS; xi vs flint 8.8e-17. V3 (Poisson reconstruction from located zeros + EM lattice tail vs analytic ρ_W) FAILS: rel ~0.3 at all a.
• Debug findings (analysis in D-024):
• θ=π zeros ARE real (sign changes confirmed) — Thm 1.5 survives Galerkin at θ=π.
• z₁(θ=π)·a ≈ 4.504, 4.499, 4.497, 4.496 at a=4,6,8,10 — a stiff ~4.5/a law, NOT the rigid half-shift π/2a. No θ=π zero in (0,π/a).
• ρ_W − recon: +a/π at y=0.5, monotone decrease, zero crossing y≈7, −0.99 at y=20 (a=4) — structure consistent with nonzero Im κ (exponential-factor gauge) and/or missing/extra zeros; root cause NOT pinned at stop time.
• d_pi ≈ d_zero (θ=π marginally WORSE on raw L2 over [1,20]: 1.2201 vs 1.2165 at a=4) — raw L2 is plateau-dominated; the informative metric must weight small y.
• Session interrupted 3× by network dropouts (no state loss); user stopped during V3 root-causing.

Decisions

• E-016 kept as experiment-with-failed-check (CSV/JSON written with checks recorded honestly); V3 root cause = first task next session.

Next session

1. Finish V3 root-cause (D-024 §4 lists the three candidates: Im κ ≠ 0 measurable via W′(0)/W(0); low-zero bookkeeping vs θ=0; large-z zero-finder noise contaminating recon).
2. Then resume approved order: A1 → A2 → C → B; collect E-015 (or rerun).