Null controls for the moments concentration pattern

Date: 2026-08-04 Code: zeta/surrogate.py, scripts/15_null_control.py, tests/test_surrogate.py Commits: 18967ff, 61e3cbf, bb634a8, e248d15 Answers: attacks A1–A3 of REDTEAM.md §3.

Verdict: the candidate pattern is explained. Concentration rising with moment order and with height is reproduced by controls carrying no arithmetic input, and it is present in a function that violates the Riemann Hypothesis. Per the repository's standing gate, a property the counterexample also has distinguishes nothing. Nothing here is evidence for or against RH.

What was measured

The statistic under audit is the share of the 2k-th moment integral carried by the largest one percent of grid intervals, plus the dispersion of disjoint block integrals. zeta.surrogate.interval_statistics re-expresses the interval algorithm of scripts/14_moment_experiment.py so it accepts any value array; it reproduces block_peak_sweep on real zeta data to 1.7e-16, so every column below runs the same code path. That check also independently reproduced the three top-one-percent shares that REDTEAM.md §1 flagged as unpinned (25.81 / 67.06 / 88.67 percent).

Grid throughout: 750 nominal zero gaps, 64 points per gap, 48,000 intervals, 8 blocks, seed 20260804.

Three controls

A1, first-order log-correlated surrogate. X(t) = sum_{p<=T} p^{-1/2} cos(t log p + theta_p), intensity exp(2X). Variance (1/2) sum 1/p and exact moments prod_p I_0(2k p^{-1/2}), both derived in code and measured against sampling rather than assumed.

A2, full random Euler product. Z(t) = prod_p (1 - p^{-1/2} e^{i(theta_p - t log p)})^{-1}, which keeps the higher prime-power terms the first-order model drops. Its exact intensity moment prod_p sum_m d_k(p^m)^2 p^{-m} reproduces the arithmetic factor a_k for k=1..4 to ten digits against the repository's independently computed arithmetic_factor.

A3, CUE characteristic polynomial. |Lambda_N(theta)| at N = round(log(T/2pi)), fixed by the height and not fitted. Exact moments from the Keating–Snaith product. Unlike both Euler surrogates this field has zeros on its own contour.

A3b, Davenport–Heilbronn. Z_dh from zeta/epstein.py; the mandatory counterexample gate. No CFKRS-style polynomial exists for it, so only shape statistics are computed.

Results

Top-one-percent share at t=1e6, with zeta for comparison:

Davenport–Heilbronn, its own heights, 150 gaps:

Tail profile of log|f|, t=1e6:

Selberg's asymptotic variance at this height is 1.3129; zeta overshoots it and both Euler surrogates undershoot it.

What each control settled

1. Concentration is generic. Both Euler surrogates produce the rise with k and with height using no arithmetic input at all. The qualitative observation therefore carries no zeta-specific content.

1. The arithmetic factor is not the explanation. Adding a_k left concentration essentially unchanged (93.76% against 94.36% at the eighth moment) — as it must, since the share is a ratio of integrals and any scale factor cancels exactly.

1. Neither is the variance. Zeta has a larger variance than both Euler surrogates but a smaller upper tail at every quantile. The excess variance is left-tail: log|zeta| diverges at every zero, and both Euler surrogates are zero-free by construction. Calibrating a zero-free null to zeta's variance would therefore widen its upper tail and worsen the fit — the opposite of the obvious move.

1. Random-matrix structure is the explanation. The CUE control, whose only free quantity N is fixed by the height, matches zeta within a couple of points at every moment order and matches the p99.9 tail to 3.3707 against 3.3605. Agreement improves with height, the direction random-matrix theory predicts.

1. The counterexample has the property too. Davenport–Heilbronn shows the same rise with moment order and with height, so under the repository's standing gate the pattern distinguishes nothing structural.

A byproduct worth keeping: a_k from the random Euler product times g_k from the Keating–Snaith product converges to the CFKRS leading coefficient that scripts/14_moment_experiment.py derives independently (ratio 1.0016 at N=40000, falling by a factor of ten per decade of N). Each control supplies one factor of the constant, and neither is read from a table.

Limits of these controls

• Concatenated CUE matrices carry no correlation beyond one matrix (about 12 gaps at t=1e6), so the CUE block-dispersion column is the weakest number here and should not be read as a window-variance model.
• The Davenport–Heilbronn heights (200, 2000) are far below the zeta heights and the two are not calibrated to each other; only the qualitative rise is being compared.
• One seed per configuration in the tables, with a separate five-seed check on the Euler surrogates confirming the ordering is not seed luck. The CUE and DH rows have not been seed-replicated.
• None of these dispersions or ranges is a confidence interval, an error bound or a certificate.

Consequence for the programme

REDTEAM.md §5 ordered the work as literature audit, then A1–A3, then a held-out quantitative test only if a residual survived. No residual survived: the pattern is generic, present in the counterexample, and quantitatively reproduced by CUE. The recommendation is to record the observation as explained and to make no entry in the conjectures/ ledger.

Follow-up: the rate of approach to CUE

The one question the controls left open was the finite-height rate at which zeta's concentration approaches its random-matrix counterpart. Measured across six heights with 200 CUE seeds each, so the comparison band is the empirical CUE distribution rather than a chosen threshold (--approach):

The gap shrinks monotonically in magnitude through 1e6 and then oscillates about zero. Zeta sits outside the CUE band only at 1e3, and only for the 6th and 8th moments. The residual closes.

Two honest limits on that table. The CUE band is wide — at 1e6 the central 95% for the 8th moment spans 88.69% to 99.96% — so "inside" is a weak test and the 2nd and 4th moments are inside at every height including 1e3, which is a statement about low power, not about agreement. And each zeta row is a single window: the 2nd and 4th moments drift above the CUE median at 1e7 and 1e8 (+7.73%, +10.64%), still comfortably inside the band, which on one realisation is window luck until replicated.

With that, the programme has no live residual. The candidate pattern is explained, the counterexample shares it, and the quantitative gap to the random-matrix control closes with height.