D-040 — Task 2.10 analysis: spectral-measure convergence — the valley observable, weight-channel γ-encoding, and the fixed-θ saturation ceiling

Date: 2026-08-19 | Session: 008 | Task: 2.10 (D-035 deliverable D2) Status: complete. Experiments: E-022 (ALL CHECKS PASS), E-022b (ALL CHECKS PASS).

1. What was computed

For Suzuki's operator family at θ = π, a ∈ {4, 6, 8, 10, 12}: the Clark/de Branges spectral measure μ_a = Σ_k w_k δ_{z_k} — atoms at the (all-real, E-016 F1) zeros z_k of W(a,π;z), weights w_k = 1/K_a(z_k,z_k) with K_a(z,z) = Re(b_z^H v), v the bordered (mean-zero, L²_0 — Suzuki's space) Galerkin solution of T_a v = e^{−izx}. Galerkin validation: Schur-Toeplitz vs dense bordered solve 3.6e-14 (V0); weights positive (V1); h-refinement 1.1e-3 (V2); lattice count + z₁·a law (V3). Target measure: unit atoms at the first ~135 Riemann zeros γ ≤ 280.

2. Findings

• F1 (mass conservation). Total mass in (0, 280]: 184.0–184.1 for every a — an a-independent invariant.
• F2 (coarse weak metric flat). Gaussian-bump (σ = 8) discrepancy d_a ≈ 0.01053 → 0.01059, flat (slightly increasing). Cumulative-mass L∞ ≈ 0.025–0.030 flat. At coarse resolution the measure is already near the smooth zero-counting shape and shows no a-drift.
• F3 (THE VALLEY OBSERVABLE — new). K_a(z,z) sampled continuously in z has DEEP VALLEYS EXACTLY AT THE TRUE ZEROS γ: mean log K at γ's vs at gap midpoints differs by Δ(a) = 2.16 (a=4) → 2.51 (6) → 2.74 (8) → 2.87 (10) → 2.94 (12) (K-contrast e^Δ ≈ 9×–19×). corr(dist-to-nearest-γ, log K) = +0.45 (a=8), +0.37 (a=12). Valleys are at the γ's specifically, NOT at the model's own zeros (log K at lattice zeros ≈ +2.9 at a=12).
• F4 (weights carry the γ-information, not positions). Within fixed z-bins (controlling the γ-density confound that naive window-mass tests fail — shifted-γ null test), atoms within 0.1 of a γ carry e^{2.0}–e^{2.8}× more weight than atoms 0.3+ away. Detrended corr(dist-to-γ, log w) = −0.31 (a=4), −0.15 (a=12). This coexists with E-019's chance-level POSITION proximity: the spectral measure concentrates mass near the true zeros THROUGH THE WEIGHTS while the atom positions stay lattice-stiff.
• F5 (mass concentration). Fraction of μ_a within ±0.1 of a true zero: 0.634 (a=4) → 0.699 (6) → 0.725 (8) → 0.725 (10) → 0.724 (12) vs 0.096 uniform expectation.
• F6 (saturation). Δ(a) fits Δ_∞ − c/a with Δ_∞ = 3.33, c = 4.76 (max resid 0.032): the valley depth is DECELERATING and consistent with saturation at Δ_∞ ≈ 3.3 (mass fraction capped ~72–73%).

3. Interpretation (honesty-tagged)

• [NUMERICAL-VALIDATED] F1–F6 as measured facts (V0–V3 + within-bin and shifted-null controls).
• The measure-level reading of D-035's resolvent convergence: full consolidation (μ_a → atomic-at-γ's) needs Δ(a) → ∞. At fixed θ = π the depth saturates (Δ_∞ ≈ 3.3) — fixed-θ truncation has a saturation ceiling; a ≤ 12 shows no consolidation (consistent with D-029's no-transition and D-035 §6 "nothing is converging yet").
• Structural implication for 2.9: Cor 1.6's convergence is stated with a MOVING boundary condition θ(a) (D-023's open θ(a) problem). The measured fixed-θ saturation is exactly what one expects if the true resolvent convergence requires the θ(a) flow: at fixed θ the family D_{a,π} converges (if at all) to the wrong self-adjoint extension. The valley depth Δ(a) is the sharp observable to test any candidate θ(a): the right flow must show Δ(a) → ∞ (mass fraction → 1).
• This UPGRADES D2 from a flat negative (coarse metric) to a precise instrument: Δ(a) and the ±0.1 mass fraction are continuous, cheaply computable, γ-referenced convergence probes.

4. Caveats

• Weights are norm-dependent (λ = −1 shift, Galerkin gauge); all cross-a comparisons use identical conventions, and F3–F6 are ratio/contrast statements, gauge-robust.
• The γ target uses mp.zetazero float64 — fine for statistics (the E-021 precision caveats do not apply at this level).
• The Δ_∞ − c/a fit is a 5-point phenomenological model; a > 12 points would test it (backend supports a ≤ 12; a = 13+ deprioritized per D-029 but now has a concrete motivation if revisited).

5. Feeds

• 2.9-D1: the Krein-frame deficiency convergence must reproduce Δ(a) → ∞ along θ(a); the fixed-θ ceiling is a new constraint.
• 2.11 (D-038): the saturation scale informs the transition-scale lower bound.
• Paper: E-022/E-022b as the spectral-measure numerics exhibit.

ERRATUM (2026-08-19, D-043 §C)

F4's "weights carry the γ-information, not positions" is softened: positions DO track γ's (mean min-dist ≈ 0.65/a², quadratic locking — E-019's contrary claim was a wrong-direction metric). Correct reading: positions lock slowly AND weights concentrate in the K-valleys; the residual mass (excess plateau atoms) remains the obstruction.