D-020 — E-010 analysis: zeros of Suzuki's W(a,θ;z) — lattice law, no γ-locking yet

Date: 2026-08-17 | Session: 003 | Task: 3.4c cont. | Status: complete Inputs: E-010 (numerics), D-018/D-019 (scoping/analysis), Suzuki arXiv:2606.09096v1 (eq. 1.11, Thm 1.5, Cor 1.6).

1. What E-010 computed

Deficiency vectors v_± from the E-009b-validated Galerkin machinery: (T_a v_±)(x) = e^{±x} solved as bordered mean-zero systems (Q − λM)c = b_±; W(a,θ;z) per eq. 1.11 using v₋(x) = v₊(−x) (symmetry verified to 7e-14; T-norms agree to 1e-13; θ=0 real gauge exact). Zeros on (0, 60] by sign-change + brentq with direct W evaluation. Configs: a ∈ {2,4,6,7}, θ ∈ {0, π/2, π}, λ ∈ {−0.5,−1,−2}, h-convergence at a=4.

2. Findings

(F1) Lattice law. Zeros of W(a,θ;z) form a near-perfect lattice: z_j(a,θ) ≈ s(a)·j + κ(a,θ), s(a) = (π/a)(1 + ρ(a)), with ρ(a) ≈ 0.0063 (a=2), 0.0014 (a=4), 0.00066 (a=6), 0.00079 (a=7) — a g-induced renormalization of the free i d/dx lattice (spacing π/a), decaying roughly like C/a² with C ≈ 0.02–0.03. θ shifts the offset κ(a,θ) as expected for a boundary-phase condition (θ=0: κ≈0; θ=π/2: +0.20; θ=π: +0.34 at a=4 — consistent with κ sweeping one full spacing over θ ∈ [0, 2π)).

(F2) λ-independence. Zeros at a=4 move by ≤ 2e-4 between λ ∈ {−0.5, −1, −2} — consistent with the abstract expectation that the extension spectra do not depend on the arbitrary shift λ (small drift attributed to discretization; h-convergence: z₁ rel change 4e-4 between h = 0.01 and 0.005).

(F3) No γ-locking at a ≤ 7. Lattice residuals (z_j − fitted lattice) are concentrated at LOW z (|res| max at z ≲ 9) and are smooth; at the positions j ≈ aγ₁/π there is no enhanced structure for a = 4, 6, 7 (a=2 shows a single 6e-2 blip at j≈9 ≈ aγ₁/π — likely a small-a transient; does not persist). For θ ∈ {0, π/2, π}, no zero is attracted toward γ₁ = 14.13, γ₂ = 21.02 within the measured range.

(F4) Density bookkeeping (important for interpreting Cor 1.6). W(a,θ;·) has ~2aT/π zeros in [0,T] — far MORE than z²ξ(½−iz)/ξ′(½−iz) (~T log T/2π). Since e^{φ} cannot remove zeros unless φ has logarithmic singularities, the φ(a,z) of Cor 1.6 must effectively act as a canonical factor cancelling all but the zeta-aligned zeros (or moving them off compacts). Consequence for numerics: raw zero positions of W are NOT the right observable for the Cor 1.6 limit; the right one is a normalized ratio such as R_a(z) := W(a,θ;z) / W(a,θ;z₀) (or log-derivative W′/W) evaluated at z = γ_j, tracking whether R_a(γ_j) → 0 with a for the RIGHT θ(a). Requires: a candidate θ(a) rule (Q-S3 remains the crux).

3. Interpretation / honesty notes

E-010 validates the FULL pipeline (deficiency vectors, W, real zeros) against every internal check — the machinery is sound and reusable.
The negative content (no γ-locking at a ≤ 7, fixed θ) is EXPECTED in hindsight given F4: the zeta content of W is in the fine structure of the lattice perturbation and in the θ-dependence, not in raw positions at small a.
The a=2 blip near γ₁ is noted but discounted (no persistence at larger a; would need θ(a)-tracking to confirm).

4. Next steps on the Suzuki line (queued, prioritized)

Q-S3 attack (the crux): find a principled θ(a). Candidates: (i) match the normalization of Cor 1.6 directly — demand e^{φ}W → z²ξ/ξ′ in L² on a compact set and READ OFF θ(a), φ(a,z) (least-squares in θ, φ = polynomial in z of growing degree — the density bookkeeping F4 says φ must supply the cancellation); (ii) consult Suzuki's §7–8 for any canonical θ prescription; (iii) Kim et al. (2607.24830) may have a specific extension choice — check their numerical setup.
Zero-tracking diagnostic: follow the zero nearest γ₁ across a = 2..12 for several θ rules; measure |z(a) − γ₁| decay.
If (1) yields θ(a): test Cor 1.6 convergence directly on compacts.

5. Status

E-010 complete and archived. Suzuki line state: machinery validated; lattice law measured; the path to Cor 1.6 runs through Q-S3 (θ(a)). No obstruction — this is an OPEN, well-posed next problem, not a wall.