Session 002 — 2026-08-17 (post-shutdown resume; Track C begins)

Resume verification

Computer shut down after Session 001; resumed cleanly — git tree clean at d1ffd4c, all artifacts intact. Cross-checked SQLite catalog vs files: 13 derivations, 2 graveyard, 5 experiments all consistent. Fixed one gap: sessions table was empty (db.py has no add-session command) — session 001 registered directly; STATE.md thread list housekeeping (T1/T3/T5 closed) committed as 466594f.

Work: task 3.1 setup — Track C literature landscape → D-014

Web research (per §4A.3, logged D-001 Part III): the Weil-form positivity frontier is ACTIVE in 2026:

• Suzuki 2606.09096: screw-function unification (Yoshida/Bombieri/CC/CCM), all unconditional; conjectured Hilbert–Pólya operator as a→∞ limit of self-adjoint operators on [−a,a]. ← Track C↔B bridge, 2 months old.
• Groskin 2607.02828v3: truncated form values = exact zero sums; certification budget B_T ~ (2N+1)ρ log T/(π²T), indeterminate zone [−B_T, 0). Verified on 512 zeros.
• Kim et al. 2607.24830v2: numerical Suzuki operator; lowest eigenvalue positive, superexponentially decaying; off-line injection → blow-up.
• Connes 2602.04022: primes<13 extremization approximates 50 zeros; approximants proved on-line.

Key positioning insight: the Anthropic paper's G̃ IS a Weil-form truncation (Gabor Galerkin at critical density; values = exact zero sums — Groskin's "dictionary" property holds by construction). The 2026 truncation literature lacks the unconditional second-moment input; the paper lacks the truncation/certification language. ⟹ our niche.

Cross-pollination questions formulated (task 3.4 precise versions):

• Q-3.4a: does the unconditional n₊ lower bound shrink Groskin's indeterminate zone [−B_T, 0)?
• Q-3.4b: positivity horizon — quantify λ_min decay vs support (E-006); if superexponential (cf. Kim et al.), a complexity wall for the positivity route — obstruction-map measurement.
• Q-3.4c: Suzuki's operators as concrete self-adjoint candidates for Track B spectral comparison (feeds Phase 2.4).

Next sub-stage: E-006 (positivity horizon numerics, designed in D-014 §4).

E-006 run + analysis → D-015

Result: eigenvalue floor of the Gabor-truncated Weil form is FLAT at ~1e-9–1e-10 (relative) across bandwidths λ ∈ {0.5,1,1.5,2} — instrument resolution limit, not the mathematical floor. Two solid conclusions: (1) truncation is genuinely ill-conditioned even at bandwidth 1 (~50/1834 eigenvalues < 1e-8 relative — structural, consistent across λ); (2) float64 eigensolvers cannot power positivity certificates — confirms why Groskin uses cutoff-free LDLᵀ (his 1e-59/T~1e63 example is exactly this wall). Q-3.4b refined: empirical floor decay needs extended precision (E-007 designed: mpmath, d~150–300).

Sanity check passed: Phase 1 conclusions used bulk statistics only — unaffected by the tiny-eigenvalue floor.

Side event: user asked about git — confirmed 100% local (no remotes).

Next: user to choose (a) E-007 [recommended], (b) Q-3.4a analysis, (c) task 3.2 Li coefficients.

User decision: do all three, one by one, then gather thoughts

(b) Q-3.4a → D-016 (done while E-007 runs): VERDICT: No — the unconditional n₊ bound cannot shrink Groskin's indeterminate zone [−B_T, 0). Reason: scale mismatch — the zone lives at o(1) eigenvalue scale (o(N)-event question), our 2/3 input is a bulk statement; same invisibility wall as G-001 (W4/W5 mechanism). General filter recorded: "does the input see o(N) events? If no, it cannot touch positivity certifications." Salvage: (1) unconditional negative-index bound n₋ ≤ (1/6+o(1))N — Thm A restated in Bombieri language (candidate minor result; literature check queued); (2) truncated form as quantitative off-line-zero DETECTOR with depth-to-eigenvalue calibration (to quantify if Track C continues).

(a) E-007 running: extended-precision (dps 40) floor measurement, Hann window with EXACT closed-form Fourier transform (no interpolation noise), T=200, λ ∈ {0.5,1,1.5,2}, mpmath.eigsy.

(c) next: E-008 Li coefficients design.

Triad completed (a)+(c) — D-017 written

(a) E-007/E-007b (extended-precision floor): Hann window with exact closed-form FT eliminated interpolation noise. T=200 λ-block at λ≥1.5 contaminated by dimensional deficiency (d > N′ — intrinsic: at λ>1 the critical grid outnumbers the zeros); excluded. Clean interleaved points (T=200/400, a = 1.73..4.15): floor decays ~10^{−0.72a}·mean — EXPONENTIAL in support (not clearly superexponential, pace Kim et al.'s Suzuki operator), plus dimension dependence. Certification cost: ~0.72a digits per support a — severe but polynomial-grade. Q-3.4b answered: conditioning is NOT the positivity route's true wall (the o(N) blindness is, D-016).

(c) E-008 (Li coefficients part 1): λ_n to n=2000 with explicit tail bounds; λ_1 validated (0.022961 + 5.4e-4 tail vs 0.0230957…); certified range n ≤ 116 only — instrument limit of 10⁴ zeros logged; n~10⁵ needs prime-side Bombieri–Lagarias formulas (logged as 3.2 continuation). Debugging note: first run's check failures were tolerance-design errors (arbitrary 1e-4 vs the honest tail bound); fixed by validating against the tail bound itself.

All three triad items done. Consolidated summary: D-017 §4. Next candidates: Q-3.4c (Suzuki operators — most interesting), 3.2 cont., 3.3 de Bruijn–Newman, T4. Time 16:10 — near break-prompt threshold.

Q-3.4c scoped → D-018; FORK RULE adopted; session end (~16:40)

New standing rule (user directive): at every fork, do ALL paths one by one, then gather thoughts and evaluate together. Added to NGUYEN.md §8.4a, committed 7011f82.

Q-3.4c scoping (Suzuki 2606.09096, full-text extraction):

• Screw function g(t) extracted EXACTLY (eq 1.3: exponentials + prime sum + digamma + Hurwitz–Lerch; all mpmath-computable).
• Thm 1.1 [proved]: Weil-form operator A_a = Friedrichs extension of D*G_aD on [−a,a]. Thm 1.5 [proved, unconditional]: self-adjoint extensions have all-real spectra = zeros of explicit entire W(a,θ;z).
• Cor 1.6 [the bridge]: IF W(a,θ;z) → z²ξ(½−iz)/ξ′(½−iz) (normalized, compact-uniform) THEN RH. All of RH compressed into one convergence.
• Assessment: strongest concrete Track B entry point seen; superior to Berry–Keating (which has no self-adjointness theorem).
• Questions opened: Q-S1 (can our second-moment technology touch the Cor 1.6 convergence? — unexplored intersection), Q-S2 (de Branges/ canonical-systems link → Phase 2.4), Q-S3 (θ-family).
• Implementation plan: E-009a (g in mpmath + validation), E-009b (Galerkin A_a, bottom spectrum vs zeros), D-019 (analysis). 2 sub-stages, moderate risk (t=0 quadrature care).

Session 002 ends here (user time budget). Clean state, all committed.