D-023 — E-014 analysis: Cor 1.6 diagnostics (Q-S3) — density wall mapped; θ = π candidate

Date: 2026-08-17 | Session: 004 | Task: Q-S3 (Suzuki-line crux) Status: complete — parameter space mapped to a = 10; thinning mechanism NOT observed at a ≤ 10; canonical candidate θ(a) = π found (weak evidence).

1. Theory frame (what Cor 1.6 requires)

Cor 1.6 (Suzuki 2606.09096): IF for some θ(a), φ(a,z) (≠ +∞), e^{φ}W(a,θ;z) → z²ξ(½−iz)/ξ′(½−iz) uniformly on compacts, THEN RH. Constraints established before E-014:

• W has exponential type 2a and ALL REAL zeros (Thm 1.5), e^{φ} ≠ 0 ⟹ zeros of e^{φ}W = zeros of W.
• Target zero density ~ (log T)/2π; measured W density at a ≤ 8 (E-010): maximal lattice, spacing π/a ⟹ ~aT/π zeros in [0,T].
• As θ varies at fixed a, zeros sweep along ℝ (de Branges interlacing); count in a fixed window is θ-independent up to O(1). ⟹ For Cor 1.6, the zero density must COLLAPSE with a (zeros, confined to ℝ, must flow out to ±∞), and this can only come from the a-dependence of the deficiency vectors v± = T_a^{−1} e^{±x}.

2. E-014 findings (a ≤ 10, h = 0.01, λ = −1; all checks pass)

• (H1) Density is maximal and θ-independent. Zero count in [0,40] at a = 6: 75–76 across 16 θ values (spread 1; argument-principle count). ≈ aT/π exactly. Interlacing picture confirmed quantitatively.
• (H2) v± ARE boundary-localized — but it doesn't thin the spectrum. Edge-1 mass fraction (L2): 0.804 (a=4) → 0.825 (a=10), slowly rising; close to the free e^{±x} profile (1−e^{−2} = 0.865). So localization is essentially the FREE deficiency shape; the A_a distortion is a small correction at these a. No localization-driven thinning.
• (H3) Perfect lattice to a = 10; no flow reversal, no γ-locking. z₁(a) tracks π/a to ≤ 4e-4 at a = 4,6,8,10 (h-drift consistent with E-010). Zeros keep flowing IN toward 0 as a grows. The outflow Cor 1.6 needs is absent at a ≤ 10.
• (H4) Read-off selects θ = π at every a. Imaginary-axis fit (log|W(a,θ;iy)| + polynomial vs log|F(iy)|, y ∈ [1,20], F = z²ξ/ξ′): best θ* = π at ALL a ∈ {4,6,8,10}, with PERFECT phase agreement (sign of W(iy) matches sign F(iy) at all 96 points). BUT: residuals do not decrease with a (0.0177 → 0.0242) and θ-selectivity is weak (best vs median ~7–20%). The fit is dominated by the common e^{2ay} growth; θ = π is a consistent candidate, not yet evidence of convergence.

3. Verdict on Q-S3

The crux is now sharply posed: Cor 1.6 requires a spectral phase transition (maximal lattice → Γ-thinning) that has NOT begun at a ≤ 10. Everything at a ≤ 10 is free-lattice plus O(1/a²) renormalization (E-010 F1). The transition, if it exists, starts at a ≫ 10. θ = π is the candidate boundary condition to track (selected uniformly by the read-off with exact phase agreement). No obstruction found — but the numerically accessible window (sieve limit a ≤ 10) is too small to see the phenomenon. This is a SCALE limitation, not a wall.

4. Queued next steps (prioritized)

1. Q-S3a (analytic, cheap): prove/disprove v±(a,·) → A_∞^{−1}e^{±·} in a norm that controls W on compacts. If A_a^{-1} has a limit whose Fourier-data involves ξ, the thinning mechanism becomes visible analytically (this is essentially Suzuki's §7 program made concrete).
2. Q-S3b (numerics, medium): redesigned prime-sum backend (blockwise Λ with checkpointed prefix sums) to reach a ~ 12–14; track the θ = π zero flow and the read-off residual there.
3. Q-S3c (diagnostic, cheap): at θ = π, compute W′/W(iy) (Herglotz density of the zero distribution) for a = 4..10 and compare its fine structure against F′/F(iy) — detects sub-lattice-scale zero-redistribution before individual zeros resolve.

5. Honesty notes

• h = 0.01 discretization (z₁ drift 4e-4 vs h = 0.005, E-010); trends reported are 10–100× larger.
• P1's argument-principle count is exact up to the rectangle eps = 0.02; spread 1 across θ is the interlacing theorem made visible.
• H4's θ = π selection could be an artifact of the growth-dominated fit; the phase agreement (1.000 at all a) is the non-trivial part — log as [NUMERICAL] hint, not evidence of convergence.