Exact spectral moments of the CUE band Gram matrix: m_5, m_6, m_7, and the lambda structure

Status: probe output of the sine_gram hunt. Everything below is a statement about the CUE band-Gram model at band ratio lambda = d/N (equivalently, in the limit, the sine process of unit density with kernel sin(pi lambda x)/(pi x)), whose N -> infinity limit is what the 10 August 2026 paper ("more than two thirds", Remark 7.5(f)) calls m_k(lambda). Nothing here is a statement about the zeros of zeta, and nothing here bears on RH. Grades follow the repository ladder: the integers are exact, the polynomials and limits are measured identifications with the overdetermination stated per claim.

1. The object and the engine

T_{m,m'} = Tr U^{m-m'} for m, m' in a band of d consecutive integers, U ~ CUE(N); G = T/N; m_k(lambda) = lim_N E tr T^k / (N^k d) with lambda = d/N held fixed. E tr T^k is an integer, computed exactly (integer arithmetic throughout, no floats, no truncation) by fast_moments.py, an optimized reorganisation of the validated engine exact_finite_N.py: sum over sorted h-multiset classes of J(class, N) * A(class) / mult(class), where J is the exact CUE joint trace moment (set partitions, then signed permutation cycles, each cycle a lattice count max(0, N - spread)) and A is the ordering-summed band count (details and the validation list in the module docstring).

Validation passed before anything below was computed (fast_moments.py validate --slow):

• term counts equal the Fubini numbers 13, 75, 541, 4683, 47293 (k = 3..7);
• vectorized J equals exact_finite_N.joint_trace_moment on E |Tr U^h|^2 = min(|h|, N) (h = 1, 3, N-1, N, N+3, 2N) and on 180 random sum-zero tuples, k = 3..7, N in {5, 9, 12};
• full E tr T^k equals the naive engine at (k=2, N=7,9), (k=3, N=7,9,11), (k=4, N=7..15 odd, including the recorded values 53739, 190029, 519981, 1201031, 2459255), (k=5, N=7), (k=6, N=7);
• the negation-symmetry shortcut equals the unreduced sum at (k=5, N=7), (k=6, N=5);
• E tr T^1 = dN and E tr T^3 = 2N^4 - N^2 on the sweep range.

All integers are checkpointed in exact_trTk_values.json (lambda = 1; keys k -> N -> E tr T^k) and exact_trTk_values_lambda.json (lambda = p/q families computed at N = q t, d = p t; keys p/q -> k -> N).

2. Results at lambda = 1: the exact polynomials and the limits

Computed at every odd N in [5, 35] for k <= 6 (16 values each), and at odd N in [5, PLACEHOLDER_K7_NMAX] for k = 7. Protocol: fit a degree-(k+1) polynomial through the k+2 smallest grid points, demand exact integer agreement at every remaining computed value, then check even N values not on the fitting grid at all.

Evidence, per k:

• Regime start. For every k the degree-(k+1) fit through N = 5..(2k+7) (odd) reproduces every other computed value exactly. No pre-regime point was observed: on this grid the polynomial form holds from N = 5 on.
• Overdetermination. k=5: fitted on 7 points, reproduced exactly at the 9 remaining odd values N = 19..35. k=6: fitted on 8 points, reproduced at the 8 remaining odd values N = 21..35. k=7: PLACEHOLDER_K7_OVER.
• Parity. The same polynomials reproduce the directly computed even-N values N = 6, 8, 10, 12 for k = 2..6 (16 further exact checks; PLACEHOLDER_K7_EVEN). No odd/even quasi-polynomial split is present at lambda = 1 in this range.
• Monte Carlo consistency (control, not proof; CUE sampling at N = 41, d = 41). Evaluating the polynomials at N = 41: E tr G^4/d = 3.248463 vs MC 3.2449 +- 0.0056 (-0.6 sigma); E tr G^5/d = 5.607311 vs MC 5.597 +- 0.014 (-0.7 sigma); E tr G^6/d = 10.149479 vs MC 10.12 +- 0.033 (-0.9 sigma); PLACEHOLDER_K7_MC

The moment sequence of the limiting spectral law of the lambda = 1 band Gram matrix therefore begins

1, 4/3, 2, 13/4, 101/18, 640/63, PLACEHOLDER_K7_M, ...

(m_1..m_4 as published in the source paper; m_5 onward computed here.)

Not established / not attempted. m_8(1) was not computed: with this engine the k = 8 sweep needs roughly 10x the k = 7 per-class work on ~5x the classes per N, an estimated 50+ CPU-hours for a validated fit, beyond this session's budget. Nothing above proves the polynomial form for N outside the computed range; the limit identification rests on the (heavily overdetermined, exact) agreement on the computed grid.

3. The lambda structure of m_k(lambda)

E tr T^k was computed exactly for the families N = q t, d = p t (so lambda = d/N = p/q exactly), t = 1..12 (t = 1..10 or 11 for the more expensive families), for the lambda values listed below; each family was identified as a degree-(k+1) polynomial in t by the same fit-and-check protocol (every family reproduced its 2-4 spare points exactly), and m_k(lambda) is the leading coefficient divided by q^k p.

Exact values (lambda_structure.py re-derives this table from the JSON and re-runs every check):

(Blank cells were not needed for any fit or check and were not computed.)

The data identify the following piecewise structure, with J = floor(k/2) and one breakpoint at lambda = 1/j for each j = 2..J:

m_k(lambda) = 1 + sum_{j=1..J} a_{k,j} lambda^{2j}

• sum_{j=2..J, lambda > 1/j} (j lambda - 1)^{2j+1} * g_{k,j}(lambda) / lambda

with g_{k,j} a polynomial of degree k - 2j. Identified exactly:

Fit-and-check bookkeeping (all matches are exact rational equalities, run by lambda_structure.py):

• k=2, k=3: one-coefficient fits reproduce all 8 remaining lambda values, including everything above 1/2 (the defect is identically zero, consistent with the Wick/Gaussian computation being exact for k <= 3).
• k=4: Wick fitted on {1/5, 1/4}, reproduced at {1/3, 2/5, 1/2}; g fitted on {6/11} alone, reproduced at {3/5, 5/8, 2/3, 3/4, 4/5, 1} (6 spare).
• k=5: Wick fitted on {1/5, 1/4}, reproduced at {1/3, 2/5, 1/2}; g fitted on {6/11, 3/5}, reproduced at {5/8, 2/3, 3/4, 4/5, 1} (5 spare).
• k=6: Wick fitted on {1/6, 1/5, 1/4}, reproduced at {2/7, 3/10, 1/3}; the j=3 defect fitted on {3/8}, reproduced at {2/5, 3/7, 4/9, 1/2} (4 spare); the j=2 defect fitted on {3/5, 5/8, 2/3}, reproduced at {3/4, 4/5, 1} (3 spare).
• Consistency across routes: the piecewise model evaluated at lambda = 1 returns 13/4, 101/18, 640/63, equal to the independent lambda = 1 sweeps.
• Independent check of the Wick pieces: the superseded continuum Wick engine (exact_moments.py, a quadrature over Gaussian pairing patterns, no finite-N input) reproduces 1 + 2 lam^2 + 0.26667 lam^4 (k=4), 1 + 3.3333 lam^2 + 1.3333 lam^4 (k=5) PLACEHOLDER_WICK6 to its working precision, i.e. exactly the W_k identified here from finite-N integers.

Observed patterns worth recording (observations, not theorems):

• The Wick piece breaks exactly where j = floor(k/2) frequencies can saturate the band simultaneously (lambda = 1/j), which is where the Gaussian (Diaconis-Shahshahani) regime for joint trace moments ends; this is the same mechanism that made the Wick value 49/15 wrong at m_4(1) (RESULTS.md, entries 1-4).
• The first defect beyond lambda = 1/j is -(j lambda - 1)^{2j+1} * 2 / ((2j+1)! lambda) for even k = 2j: coefficient 1/60 = 2/5! at j=2, 1/2520 = 2/7! at j=3.
• Everything is continuous at the breakpoints to high order (the defects vanish to order 2j+1), so finite-precision numerics near a breakpoint would never see this structure; exact arithmetic was load-bearing.

Honest scope of the lambda claims. The window forms are exact rational identifications on the sampled lambda grids with the stated spare-point counts; between sampled lambda values an additional breakpoint would go unseen. The sampled grids are: 6 values in (0, 1/3], 5 in (1/3, 1/2], 7 in (1/2, 1]. The forms are conjectured, with this evidence, to hold on the full windows; they are not proved anywhere, and for k = 7 the lambda structure was not measured at all (only lambda = 1).

4. Files

• fast_moments.py: the optimized exact engine, validation suite, sweep and analysis CLI (this file computed every integer cited here).
• exact_trTk_values.json: all exact E tr T^k at lambda = 1.
• exact_trTk_values_lambda.json: all exact E tr T^k for the p/q families.
• lambda_structure.py: re-derives the m_k(lambda) table from the JSON and re-runs every lambda fit-and-check above.
• exact_finite_N.py: the naive reference engine (unchanged; the instrument the fast engine was validated against).

Produced 2026-08-17 in the sine_gram hunt; not committed by this session.