D-045 — Fork 1: symbol identification — the zero-comb law

Date: 2026-08-19 | Session: 008 | Fork 1 of the D-044 follow-up fork Status: complete. Empirical basis: E-022b grids + extended-profile run (a=8, z ∈ [0.25, 14]) + plateau micro-regression.

1. The law (measured, then derived structurally)

K_a(z,z) → 2a on the plateau — the FREE Paley–Wiener kernel diagonal ⟨e_z, e_z⟩ = 2a — with narrow valleys at the true zeros: K_plateau/(2a): 0.973, 0.983, 0.993, 0.994, 0.997 (a = 4…12) → 1 (the D-042 "6/π ≈ 1.9" was the finite-a shadow of the limit 2). Extended profile (z < γ₁, no valleys): K ≈ 1.997a flat. Valley floors: K(γ,γ) ≈ 1.04 ± 0.06 — SAME for all γ, a-stationary. So the empirical symbol: 1/β_a ≈ 1 on the plateau, dipping to ≈ 1/(2a)·(1.04) at each γ with width ~ 1.5/a. In the a→∞ limit this is a function = 1 a.e. with zeros at Γ — i.e. β_a(ξ) → 1 + (zero comb at Γ): the screw-function operator's symbol converges to (flat background 1)

• (comb of mass-~1 bumps at the true zeros). This IS the explicit formula in operator-symbol form — the Hilbert–Pólya content of the model spaces, made visible by the K-diagonal.

2. Consequences (consolidates D-042/D-044)

• The linear-K law is simply the free-PW diagonal; the Weil-form operator T_a acts ≈ trivially on plateau modes and concentrates its response at the zero frequencies.
• Residual background density = (1/π)·(1/ρ̄) with ρ̄ → 2: R → Z/(2π) (≈ 44.6 at Z = 280; measured 48–58 — the gap is the finite-a prefactor 0.97–0.995 and floor drift; supersedes "Z/6").
• Valley mass = 1/K(γ,γ) ≈ 1/1.04 ≈ 0.96 ≈ m_γ = 1 ✓ (4% finite-a).
• The equilibrium is generic; the consolidation criterion is unchanged (D-044 §4): the λ-shift kills the plateau while floors stay pinned.

3. What this adds beyond D-044

1. The plateau constant is identified: 2 (free PW), not 6/π.
2. The symbol is identified STRUCTURALLY: 1 + zero-comb — the explicit formula as an operator symbol. A closed form beyond this (a named entire-function ratio) does NOT exist at finite a: the measured plateau flatness to ±1.5% rules out every |Ξ|-ratio candidate (they oscillate mid-gap); the comb forms only as a → ∞.
3. Plateau micro-regression: corr(logK, log|u|) = −0.22, corr with log z = −0.28, residual std 1.2% — no significant residual structure: the symbol's oscillatory content is invisible off-Γ.

4. Open remnants (honesty)

• The delta-mass per γ (≈1) is measured via the floor gauge, not derived; deriving it = the explicit formula in this representation — expected but not written out.
• The valley shape function (Lorentzian-like dip of width ~1.5/a) is measured, not derived.
• The symbol picture remains at the model-space/kernel level; the rigorous Toeplitz-symbol calculus for the exponentially growing g (analytic functional) is the formalization gap (flagged in D-044 §5).