Current task: —

Updated 08-17-2026

D-010 — Thread T1: sine-process moments m_k(1) and Christoffel Λ₂(0)

Date: 2026-08-17 | Session: 001 | Task: 1.3 warm-up (thread T1) | Status: complete Tags: [VERIFIED] (hand) + [NUMERICAL-EVIDENCE] (E-004)

1. Hand verification of Λ₂(0) = 5/36 [VERIFIED]

Moment sequence (paper §7.2(f)): m₀..m₄ = 1, 1, 4/3, 2, 13/4. Moment matrix M = (m_{i+j})_{i,j=0..2}: det M = 1·(4/3·13/4 − 4) − 1·(13/4 − 8/3) + 4/3·(2 − 16/9) = 1/3 − 7/12 + 8/27 = 5/108. cofactor (M₀₀) = 4/3·13/4 − 4 = 1/3. (M⁻¹)₀₀ = (1/3)/(5/108) = 36/5. Christoffel at 0: Λ₂(0) = 1/(M⁻¹)₀₀ = 5/36 ✔ matches paper. (Consequence quoted in §7.2(f): HL*(4) ⟹ N^s_0/N ≥ 13/18 ≈ 0.7222 — the conditional next rung. Route from Λ₂ to 13/18 not yet re-derived; noted for task 1.3 proper.)

2. The dimensionless limit object (scaling insight)

The paper's construction in spacing-1 units is L-independent: with grid spacing = mean spacing = 1 and kernel S(x) = sin(πx)/(πx), G̃_{kk'} = Σ_j S(γ_j − k) S(γ_j − k'), γ = sine-process points. Derivation: L_w(γ−α_k)/2 = π(γ̃−k) when h = 2π/L_w = mean spacing; the normalization aL² cancels exactly. This is WHY the limit exists and why m₁ = 1 is the exact identity Σ_k S(x−k)² = 1 (sharp-window Lemma 2.1).

3. Monte Carlo confirmation (E-004) [NUMERICAL-EVIDENCE]

60 GUE(400) bulk samples, window W = 600 unfolded points, integer grid: | moment | simulated | paper | rel err | |---|---|---|---| | m₁ | 0.9939 ± 0.0025 | 1 | 0.6% | | m₂ | 1.3271 ± 0.0252 | 4/3 | 0.5% | | m₃ | 1.9945 ± 0.0829 | 2 | 0.3% | | m₄ | 3.2489 ± 0.2141 | 13/4 | 0.03% | | Λ₂(0) (from sim. moments) | 0.1453 | 5/36 = 0.1389 | 4.6% | (Residuals = window-edge effects ~1/√W + sampling noise; m₁ defect 0.6% matches edge-mass loss.) ⟹ paper's m_k(1) sequence CONFIRMED independently. Thread T1 CLOSED.

4. Limiting density identification

m_k = 1, 4/3, 2 match Uniform[0,2] (moments 2^k/(k+1)) through k = 3; m₄ = 13/4 = 3.25 vs uniform's 16/5 = 3.2 ⟹ the limiting spectral distribution is a NEAR-uniform perturbation on ~[0,2], not exactly uniform. (Eigenvalues of the sinc-Gram concentrate in [0,2] with slight mass redistribution; consistent with E-003's observed bulk + small eigenvalue tail.) Not a standard named density — recorded as characterization: "uniform[0,2] through 3rd moment, m₄ = 13/4".

5. Erratum-level observation (paper, inconsequential)

Paper §2.2 states d = N(T,2T) + O(L); the RvM formula gives N − d = (1−2log2)·(T/2π)·... = (2log2 − 1)(T/2π) ≈ 0.386 T/2π, which is O(T), not O(L) (E-003 measured N−d = 123 at T=2000 ✔). Relative error 0.386/L → 0, so d = N(1+o(1)) as used everywhere — the "+O(L)" should read "o(N)". No consequence for any proof step. Noted for our paper's errata appendix (flag to authors eventually).

6. What T1 buys for task 1.3