Q-3.4c implementation: Suzuki screw function + Weil-form operator spectrum
Work
E-009a (screw function): g(t) per Suzuki eq. 1.3 in mpmath (dps 30).
Validated vs eq. 2.2 (A = 0.7075463657, c₂ = −0.8750599754, residual
structure exact over three decades); evenness exact; Lerch convention
identity exact. Tabulated [0,10] @ 1e-3 → CSV/JSON/db.
E-009b (Galerkin spectrum of A_a): P1-hat discretization of
Q(u,v) = ∫∫g(x−y)u′v̄′ (Toeplitz stiffness via triangle-weighted
integrals S(k) of g), mean-zero constraint via QR projection,
generalized eigensolve.
- v1 used the E-009a 1e-3 table for the regular part → spurious
−1e-4 bottom eigenvalues (fake ¬RH signal!). Diagnosed: 1/h²
amplification of interpolation noise (D-019 F3 lesson).
- v2: exact piecewise Gauss–Legendre S(k) (split at kinks log n and
at the triangle peak; analytic |t|log|t| cusp) → S exact to ≤1e-15.
- Diagnostics en route: machine-validated the Weil explicit formula
conventions (Gaussian test closes to 8.9e-16; arXiv HTML rendering
of Suzuki's W has Λ(n)/n — correct weight with our conventions is
Λ(n)/√n; logged as D-001 NT-1). Caught and fixed a wrong-period
autocorrelation in the diagnostic itself (L = 2π−φ).
- ABSOLUTE VALIDATION: vᵀQv = 2.437005e-4 vs zero-side
2Σ|v̂(γ)|² = 2.437016e-4 (rel 4.5e-6) — form normalization pinned.
D-019 (analysis): λ_a table (cliff: 0.414 @ a=0.25 → 2e-4 @ 0.5
→ ≤1.4e-9 @ a=1 → ≤1.7e-12 @ a=5; all positive, Galerkin = upper
bounds; consistent with Yoshida λ_a > 0 ∀a ⟺ RH). Independently
reproduces Kim et al. 2607.24830's "positive, superexponentially
decaying lowest eigenvalue". Q-S1 verdict: L²-to-uniform gap = same
wall as W2, but the entire-type + Hurwitz lifting route is shaped
like a real attack (parked for Phase 4). E-010 proposed (W(a,θ;z)
zeros vs γ_j — deficiency vectors via our validated machinery).
Decisions
A_a spectrum is NOT the zeta spectrum; zeta-aligned operator is
𝒟̄_{a,θ} (first-order, T_a-inner-product). Numerics next target:
W(a,θ;z) directly (E-010).
g implementation correct; all E-009b checks pass (S vs mpmath 1e-15,
Weil value 4.5e-6, eigenvalues ≥ −noise-floor, h-convergence,
monotonicity in a).
Outcomes / state
E-009a, E-009b in db; D-019 filed + indexed + db; D-001 updated
(NT-1 validated explicit formula, NT-2 Suzuki tool entry, Part III
web log). NGUYEN.md §5 3.4 updated; ledger row added.
Fork point reached (§8.4a): E-010 (Suzuki W), 3.2-cont (prime-side
Li), 3.3 (de Bruijn–Newman), T4 (ξ′). Presented to user.
E-010: deficiency vectors v_± via bordered mean-zero solves on
the E-009b machinery; W(a,θ;z) per eq. 1.11; zeros on (0,60].
- ALL CHECKS PASS (v₋ symmetry 7e-14, T-norms equal, θ=0 real gauge,
h-convergence 4e-4, λ-independence ≤2e-4).
- FINDINGS (D-020): zeros form a renormalized lattice
s(a) = (π/a)(1+ρ(a)), ρ ~ C/a², C≈0.02–0.03; θ shifts lattice
offset; perturbation concentrated at low z; NO γ-locking at a ≤ 7
for θ ∈ {0, π/2, π}. Density bookkeeping (F4): raw zero positions
can't show the Cor 1.6 limit — e^{φ} must supply canonical-factor
cancellation; the crux is Q-S3 (choice of θ(a)).
- Bugs fixed en route: float() discarding Im V (complex cast);
over-narrow brentq refine bracket; runtime (build cache, Lerch
400-term series for t>0.05, sieve to e^14).
- Fork 1 COMPLETE (machinery validated; next Suzuki step = Q-S3
attack, D-020 §4 — needs a principled θ(a), possibly least-squares
read-off from Cor 1.6 target, or Suzuki §7–8 / Kim et al. setup).
Remaining forks: 3.2-cont (Li prime side), 3.3 (de Bruijn–Newman),
T4 (ξ′).