sine_gram: exact spectral moments of the band Gram matrix of the sine process

Status: ACTIVE. Opened 2026-08-17. Nothing here is a result until it survives the campaign's destruction stages; grades follow the repository ladder.

The object

For the sine process of unit density and band-width 0 < lambda <= 1, let G_ij = K_lambda(x_i - x_j), K_lambda(x) = sin(pi lambda x)/(pi x), and d = lambda N. The normalised spectral moments

m_k(lambda) = lim_N E tr G^k / d

are the quantities the 10 August 2026 paper ("more than two thirds", §7.5(f)) uses through its Christoffel-function bound: m_k(1) = 1, 4/3, 2, 13/4 for k = 1..4 are stated there, and each additional pair of moments sharpens the conditional ladder built on the hypotheses HL*(k0, lambda).

What happened so far (chronological, kept as the record of method)

1. A quick Wick-pairing engine (exact_moments.py) produced m_4(1) = 49/15, disagreeing with the paper's 13/4. Per the certainty ladder the first inference was a defect on our side.
2. A CUE Monte Carlo at lambda = 1 exactly (N = d = 41, 6000 samples) gave m_4 = 3.2449 +- 0.0056: consistent with 13/4 (0.9 sigma), inconsistent with 49/15 (+3.9 sigma).
3. The decisive route (exact_finite_N.py): at finite N the circular model is exactly computable in integer arithmetic. E prod_j Tr U^{h_j} over CUE(N) reduces, through the determinantal correlation functions and the Dirichlet kernel, to lattice-point counts max(0, N - spread) per permutation cycle. No floats, no truncation, no Wick approximation. Validated: E |Tr U^h|^2 = min(|h|, N) for all tested h including h > N; k = 3 gives exactly E tr G^3/d = 2 - 1/N^2 and extrapolates to the paper's 2.
4. Exact values for k = 4, N = 7..15 odd, Lagrange-extrapolated in 1/N from both ends: 3.2499827 and 3.2499630. The paper's 13/4 stands; the Wick engine was wrong. Diagnosis: the Gaussian (Diaconis- Shahshahani) regime for joint trace moments requires total positive frequency at most N; at lambda = 1 the two-pair patterns leave that regime, exactly where k = 4 first differs from k <= 3. exact_moments.py is retained as a cautionary artifact and is superseded by exact_finite_N.py.

Files

• exact_finite_N.py: the exact engine (integer arithmetic; the instrument of record for this study).
• mc_moments.py: CUE Monte Carlo control.
• position_space.py: independent graph-integral machinery (B-splines / cycle space); check on individual terms.
• exact_moments.py: the superseded Wick engine, kept with its defect documented above.

2026-08-17 (later): the programme's step 1 and 2 executed

fast_moments.py (an optimized, re-validated reorganisation of exact_finite_N.py; class-summed, vectorized, still pure integer arithmetic) computed E tr T^k exactly for k <= 7 at lambda = 1 and for k <= 6 across seventeen exact lambda = p/q families. Everything is checkpointed in exact_trTk_values.json / exact_trTk_values_lambda.json, and moments_report.md carries the full claims-and-evidence record. Headlines, each an exact-polynomial identification heavily overdetermined on the computed grid and consistent with the Monte Carlo control:

• m_5(1) = 101/18, m_6(1) = 640/63 (and k = 7 in the report), extending the paper's 1, 4/3, 2, 13/4.
• E tr T^5 = (101 N^6 - 115 N^4 + 32 N^2)/18 and E tr T^6 = (640/63)N^7 - (140/9)N^5 + (64/9)N^3 - (5/7)N, exact for every computed N (odd and even, from N = 5).
• A piecewise-in-lambda structure for m_k(lambda): an even "Wick" polynomial below lambda = 1/floor(k/2), plus defect pieces -(j lambda - 1)^{2j+1} g_{k,j}(lambda)/lambda switching on at each lambda = 1/j; identified exactly for k = 4, 5, 6 with spare-point checks (lambda_structure.py).

Measured so far, and the immediate programme

Monte Carlo at lambda ~ 1 (naive, finite-size uncorrected): m_5 ~ 5.6, m_6 ~ 10.2. The literature (as surveyed today) stops at m_4. Programme:

1. Exact m_k(1) for k = 5, 6, 7, 8 via the exact engine plus quasi-polynomial identification in N, cross-checked by Monte Carlo and by the graph route where feasible.
2. The same as polynomials in lambda (the band constraint enters only through the spread counts, so the N-quasi-polynomials carry lambda as the band ratio d/N).
3. The Christoffel function Lambda_m(0) of the limiting spectral measure from moments m_0..m_{2m}: each new pair of moments prices the next HL*-type hypothesis in the source paper's conditional ladder. New conditional constants, stated as arithmetic consequences of hypotheses the paper already names.
4. Identification attempt on the limiting spectral law of the lambda = 1 sine-Gram matrix (moment sequence 1, 4/3, 2, 13/4, ...), plus a novelty search for it in the random-matrix and time-frequency (prolate/Landau) literature.