D-019 — Q-3.4c analysis: Suzuki operator numerics, findings, Q-S1 assessment

Date: 2026-08-17 | Session: 003 | Task: 3.4c analysis | Status: complete Inputs: E-009a (screw function), E-009b (Galerkin spectrum), D-018 (scoping), Suzuki arXiv:2606.09096v1 (re-fetched this session: eq. 1.11, Thm 1.4, Cor 1.6, §6 deficiency structure — logged D-001 Part III).

1. What was validated (machine-level)

(a) Screw function g(t) (E-009a). Suzuki's eq. 1.3 implemented in mpmath (dps 30). Validated against eq. 2.2: g(t) = ½|t|log|t| + A|t| + prime sum + r(t), r = O(t²); measured r(t)/t² → c₂ = −0.8750599754 and g(t)/t − ½log t → A = 0.7075463657 with residual exactly c₂·t over three decades (t = 1e-2..1e-4). Evenness exact. Lerch-convention identity Φ(1,2,1/4) = ψ′(1/4) exact. Tabulated [0,10] @ 1e-3 (CSV+JSON, db).

(b) Weil explicit formula conventions (session diagnostic). The classical formula was re-derived from the contour integral and machine-verified: for h even, decaying, ĥ(ξ) = ∫h e^{iξx}dx,

Σ_ρ ĥ(γ_ρ) = ∫h(x)·2cosh(x/2)dx − 2Σ_{n≥2} Λ(n)/√n·h(log n) + (1/2π)∫ [Re ψ(1/4+iτ/2) − log π] ĥ(τ) dτ

Gaussian test h = e^{−x²/2}: RHS−LHS = 8.9e-16 (zero side 2.1e-43). NOTE: the arXiv HTML rendering of Suzuki's Weil functional shows Λ(n)/n — the contour derivation gives Λ(n)/√n with the e^{−x/2} twist in the pole terms; our conventions above are the validated ones. → added to D-001 Part II.

(c) The Galerkin form IS the Weil form (E-009b). For v = sin(π(x+a)/a), a=3: vᵀQv = 2.437005e-4 vs zero-side 2Σ_{γ>0}|v̂(γ)|² = 2.437016e-4 (rel 4.5e-6, within truncation/quadrature budget). This pins the form's normalization AND sign absolutely.

2. Findings

(F1) λ_a — bottom of the Weil-form spectrum — measured: | a | 0.25 | 0.5 | 1 | 2 | 3 | 4 | 5 | |---|------|-----|---|---|---|---|---| | λ₁ (h=5e-3) | 0.414 | 2.07e-4 | 1.4e-9 | 7.4e-11 | 1.1e-11 | 5.6e-12 | 1.7e-12 | (h-refinement at a=3: 8.8e-11 @ h=.01 → 1.1e-11 @ .005 → 5.1e-13 @ .002; Galerkin = conforming ⇒ these are UPPER bounds on true λ_a.)

• A steep CLIFF between a = 0.25 and a = 1 (four+ orders of magnitude): type-a entire functions v̂ suddenly gain the freedom to dodge the zeros. Qualitatively consistent with Thm 1.4 (λ_a = log(1/a)+O(1), small-a regime) — the asymptotic has not yet set in at a = 0.25.
• All values POSITIVE, no zero-crossing: consistent with Yoshida/Suzuki (λ_a > 0 ∀a ⟺ RH). [CONSISTENCY CHECK, not evidence: RH numerically true to height 10⁴ in our table and 10¹⁰+ in the literature.]
• Tiny values below ~1e-11 are at/below the eigensolver backward-error floor (eps·‖Q‖/μ_min(M)); quote as upper bounds only.

(F2) A_a's spectrum is NOT the zeta spectrum (confirmed numerically and structurally). The zeta-aligned object is 𝒟̄{a,θ}: self-adjoint extensions of the FIRST-ORDER minimal operator 𝒟_a = i d/dx acting in ℋ(T_a) (completion of C_c^∞ under the T_a-norm, T_a = A_a − λI, λ < λ_a). Deficiency indices (1,1); deficiency vectors v± solve (T_a v_±)(x) = e^{±x} (Lemma 6.2). Eigenvalues of 𝒟̄{a,θ} = zeros of W(a,θ;z) = (z−i)∫{-a}^{a} v₊(a,x)e^{izx}dx + e^{iθ}(z+i)∫_{-a}^{a} v₋(a,x)e^{izx}dx (eq. 1.11) all REAL unconditionally (Thm 1.5). Cor 1.6: if for some θ(a), φ(a,z), e^{φ(a,z)}W(a,θ;z) → z²ξ(½−iz)/ξ′(½−iz) uniformly on compacts, then RH. No distinguished θ is identified in the paper.

(F3) Noise-floor lesson (logged for all future operator numerics): stiffness entries are 4th-differences of the second antiderivative — quadrature error in S(k) is amplified by 1/h² (4e4 at h=5e-3). The first E-009b version (1e-3-table interpolation of the regular part) produced spurious −1e-4 bottom eigenvalues that would have FALSELY signaled λ_a < 0 (i.e., a fake ¬RH). Fixed by exact piecewise Gauss–Legendre with analytic singular part; S(k) now exact to ≤1e-15. Rule: for Weil-form operators, budget quadrature error < h²·(target eigenvalue accuracy).

3. Q-S1 assessment (can our second-moment technology touch Cor 1.6?)

The Cor 1.6 hypothesis is UNIFORM convergence on compacts of entire functions. Our Thm 5.7 machinery (Montgomery second moment, X ≤ T bandwidth) controls L²/average discrepancies of explicit-formula compressions — the same L²-vs-uniform gap as the W2 wall (D-013). Concretely: W(a,θ;z) is built from v_±, i.e., from the resolvent of A_a applied to e^{±x}; its a→∞ asymptotics involve the SAME prime-sum compression (g's definition) that our framework handles in L². To get uniformity one would need either (i) L²→pointwise lifting via entire-function growth bounds (Phragmén–Lindelöf on W — plausible since W has controlled exponential type a), or (ii) a stronger-than-second- moment input — the wall again. PRELIMINARY VERDICT: not closable with current data class, but the (i) route (type-a entire + L² discrepancy + Hurwitz) is a shaped like a real attack and is UNEXPLORED. Parked as the lead candidate for any Track-B/C synthesis assault (Phase 4).

4. Q-S2/Q-S3 updates

• Q-S2 (canonical systems): ℋ(T_a) IS a de Branges-type space by construction; the screw-function ↔ canonical-system dictionary is Suzuki's §6 machinery. Link to Phase 2.4 confirmed alive.
• Q-S3 (θ-family): no distinguished θ in the paper. NUMERICALLY explorable: for each θ, zeros of W(a,θ;z) are computable once v_± are; the zeta-aligned θ(a) should reveal itself by zeros approaching γ_j as a grows. → E-010 (proposed, next session): solve (Q − λM)c = b_± with b_±,i = ⟨e^{±x}, φ_i⟩ (our validated Galerkin machinery), build v_±, evaluate W on a grid, find real zeros for a ∈ {2,4,6,8}, several θ; test convergence toward γ_1, γ_2, ... (E-002 table). Cost: one focused sub-stage. Risk: moderate (choice of λ shift; normalization of v_± — "T_a-norms agree" condition).

5. Status of 3.4c and forks

Q-3.4c implementation COMPLETE (E-009a/b + this analysis). The Suzuki line is now the strongest Track B entry point (D-018 §2 confirmed by contact with actual numerics). Open forks (per §8.4a, all queued): (1) E-010: W(a,θ;z) zeros vs γ_j [Suzuki line continuation]; (2) 3.2 continuation: prime-side Li coefficients (Bombieri–Lagarias exact formulas) beyond n = 116; (3) 3.3: de Bruijn–Newman heat-flow vs 2/3-density; (4) T4: ξ′-analogue (cheap secondary Track A target).