Band-Limited Variance Tomography of the Riemann Zeros: Resolving Finite-Height Arithmetic Corrections at the Heisenberg Scale

Draft v0.2 — 2026-09-06. Author: Adam Murphy. Methodology statement. Claude (Anthropic) and GPT (OpenAI) were used as independent computational, derivational, and audit engines. Where possible, numerical specifications were frozen before exchange, results were sealed by SHA-256, and disagreements were adjudicated against code and data rather than model consensus.

Abstract

We investigate band-limited statistics of the Riemann zeta zeros using the explicit formula and an equivalent sinc-smoothed zero comb. A nonlinear negative-excursion statistic initially exhibits a strong excess over phase-scrambled nulls, but blind two-engine tests show that this effect is reproduced by GUE/CUE surrogates and reflects universal spectral rigidity rather than zeta-specific arithmetic. Intervention experiments on the corresponding prime field are likewise explained by deterministic band-limited geometry plus Gaussian dilution.

A smaller finite-height residual remains. By scanning the bandwidth λ across the Heisenberg scale, we resolve the variance correction into a deficit below λ = 1 and a discrete recovery above it. The leading deficit agrees with the Bogomolny–Bohigas–Leboeuf–Monastra correction −2Λ/(λ²ℓ²), with Λ = 1.57314… and ℓ = log(T/2π). The recovery occurs at the arithmetic locations λ = 1 + log(q/p)/ℓ predicted by the Bogomolny–Keating off-diagonal pair-correlation formula. Direct numerical evaluation of the full published pair-correlation expression reproduces the measured bandwidth dependence at three finite heights without fitted parameters, including the dominant q/p = 2 feature and finer recovery structure, to about 0.2% in the relative variance.

The same analysis leaves a distinct fourth-cumulant residual whose sign changes near λ ≈ 1.1. A simple finite-CUE promotion does not derive this effect. Thus the pair-correlation anomaly is accounted for by known finite-height arithmetic corrections, while the higher-cumulant sector remains open.

1. Introduction

A threshold-type statistic on the explicit-formula field of the zeros — the fraction of the band-limited counting density lying below zero — is nonlinear in the density and therefore sensitive, in principle, to more than pair correlations. We asked whether it sees anything beyond random-matrix universality. We followed an anomaly until it stopped being anomalous.

The prior chain, from the originals:

• Montgomery (1973): pair-correlation conjecture. Odlyzko (1987): spacings and pair correlations for the first 10⁵ zeros and for 10⁵ zeros at index 10¹², with closer GUE agreement at height and low-height deviations later traced to the primes.
• Berry (1988): semiclassical number variance from the diagonal (prime-power) sum with a GUE plateau beyond τ*. Predicts universality breakdown at L_max = ln(E/2π)/(2π ln 2) mean spacings and quasi-random oscillations of wavelength ln(E/2π)/(r ln p), amplitude (2πr²pʳ)⁻¹ — the below-Heisenberg prime spikes in position space. Agreement with Odlyzko's zeros at 10¹² to 0.003; theoretical peaks "noticeably lower" than data (his Fig. 3), the missing piece being the off-diagonal correction.
• Bogomolny–Keating (1995, 1996): off-diagonal terms from the Hardy–Littlewood conjecture. Berry–Keating (1999), eqs (4.27)–(4.28) and Fig. 6: the off-diagonal correction with b(ξ) ≡ Φ^(off) compared with Odlyzko's zeros near n = 10⁹ through the number variance, reproducing its oscillatory deviations.
• Bogomolny (2000): the off-diagonal form factor as a dense sum of δ-peaks at T_H + log(q/p) with weights (μ(q)/ψ(q))²(q/p); "the largest peaks correspond to the shortest pseudo-orbits"; smooth RMT emerges only after smoothing (eqs 92–96).
• Bogomolny–Bohigas–Leboeuf–Monastra (2006): Λ = 1.57314… and N_eff = ℓ/√(12Λ), tested on spacing distributions at 10¹²–10²². Forrester–Mays (2015): finite-size corrections against Odlyzko's dataset.

Claims and scope. We do not claim the first numerical observation of finite-height departures from random-matrix statistics, nor the discovery of the log 2 scale or of the Bogomolny–Keating arithmetic corrections. Our contribution is a band-limited variance statistic that resolves the finite-height pair-correlation correction locally in bandwidth, including individual pseudo-orbit structure near the Heisenberg corner, together with a quantitative comparison to the published pair-correlation formula across three finite heights. The fourth-cumulant residual is reported as an unresolved numerical observation and is not assigned an arithmetic mechanism.

2. The statistic

ν_X(τ) = θ′(τ)/π − (1/π) Σ_{n=p^k≤X} Λ(n) n^{−1/2} cos(τ log n), with λ ≡ log X / log(τ/2π). Equivalent (correlation 0.9991) to the sinc-band-limited comb of the true zeros F_L(t) = Σ_γ sin(L(t−γ))/(π(t−γ)), L = λℓ, whose Fourier power is a box |q| < λ in units of the Heisenberg frequency. Statistics: negative-excursion fraction f₋; per-term phase-scrambled null (line spectrum preserved) defining C^L; analytic Gaussian null defining C^G; variance C₂, connected fourth cumulant C₄, excess kurtosis κ₄ = C₄/C₂².

Null conventions matter. The per-term scramble is non-Gaussian on short windows (characteristic function exactly ∏ J₀(k aₙ/πσ)); mixing C^L- and C^G-referenced quantities produced an apparent departure from GUE of the size of the convention gap, which we retracted (§4.3).

3. Methods and conduct

Two engines with independent implementations (Claude: mpmath zeros, Dumitriu–Edelman GUE, Haar CUE; GPT: Odlyzko tables, independent GUE construction). Blind rounds with frozen specifications, SHA-256-sealed results, pre-registered pass conditions and failure modes; deterministic damage sets (golden-ratio ranking) so both engines test identical structures without exchanging results. True zeros: the first 2,001,052 (Odlyzko) for heights ≤ 1.13×10⁶; a 10¹²-index sample for the high scout. Windows of 3000 zeros; GUE and CUE references at matched window size (finite-window kurtosis bias cancels in differences). Provenance in Appendix A.

4. How the signal was explained: universality

4.1 Coherence law. C^L ≈ 0.036 λ^p, p ≈ 3.2–3.5 (11 cells, τ = 600 → 2×10⁶); Blind Round 1: 16/16 cells agree, max 0.42σ. 4.2 Surrogate ladder. Poisson combs give C < 0; GUE combs give C > 0 matching the real field at λ ≤ 1 with no free parameters: C^L is a signed nonlinear rigidity meter. 4.3 Retracted departure. An apparent GUE departure at λ > 1.25 was a null-convention mismatch. Blind Round 2 (C^G only, matched combs, 18 cells, λ = 0.75–3, heights 150–1300): 18/18 agreement, cross-engine zeta-side agreement 4.9×10⁻⁶, pre-registered call CONSISTENT WITH ZERO in both engines, |Δ^G| ≲ 0.003–0.006. 4.4 GUE reproduces the law and the one-point distribution. Ten cells, X = 10²–10⁴, heights 400 – 2×10⁶: real C ≈ 0.034 λ^3.12, surrogate 0.036 λ^3.24, cell ratios 0.87–1.15. Skewness, excess kurtosis, and the characteristic-function deformation (peak k ≈ 2.4–2.6) are reproduced to |Δφ| ≤ 0.011 at λ = 1.

5. How the intervention phenomenology was explained: comb geometry plus dilution

Prefix retention, random holes, internal gaps, top removal, displaced bands, residue classes mod 3/4/5, at X = 10⁴ and X = 5000. All recoveries are reproduced by a zero-parameter model — deterministic kept-sum plus Gaussian noise at the scrambled remainder's variance, thresholded at zero. Blind audit (14 rows, criterion ≥ 11/14 within 2sd): Claude 13/14, GPT 14/14; deterministic model columns identical across codebases to three decimals; topology-linked systematic not triggered. Eliminated: small-prime scaffold, generic contiguity, "constraint closure", low-order cumulant explanations, bispectral coupling (excluded structurally by unique factorization).

6. What survived: a finite-height residual

6.1 Fourth cumulant vs height at λ = 1.5 (GUE reference κ₄ = −0.961 ± 0.003; CUE −0.970 ± 0.004): T ~ 424: Δκ₄ = −0.16 (single window, 1.6σ); T ~ 600: −0.17 (2.6σ); 10⁴: −0.058 ± 0.008; 10⁵: −0.034 ± 0.006; 10⁶: −0.041 ± 0.008; index 10¹² (scout): −0.015 ± ~0.1. Frozen-round verdict: marginal on the pre-registered low cells; the post-registration ladder is 5–7σ. Empirically Δκ₄ ≈ −(3.1–3.6)/ℓ².

6.2 Cumulant decomposition. At λ = 1.5, the finite-height kurtosis residual is localized numerically to C₄: the variance differs from GUE by less than about 1%, while the connected fourth cumulant differs by several to twenty percent depending on height (+21, +19, +6.7, +3.8% at ℓ = 4.21, 4.56, 7.90, 9.75). This statement is specific to the λ = 1.5 slice; §7 shows a nonzero two-point correction near the Heisenberg corner.

6.3 λ-dependence (ℓ = 7.82 and 9.73): Δκ₄ = +0.044/+0.060 at λ = 1.0 (with ΔC₂/C₂ = −4.3/−3.1%); −0.071/−0.043 at 1.25; −0.060/−0.043 at 1.5; −0.033/−0.008 at 2.0. Sign change near λ ≈ 1.1.

6.4 A natural finite-CUE effective-size promotion gives an apparent match to κ₄ at two heights, but it simultaneously predicts a large variance deficit (−27% at N_eff ≈ 1.8) that is absent. We therefore do not count this as a derivation; details in Appendix C.

7. Form-factor tomography and the Bogomolny–Keating comparison

We present this section in the order the inferences were made, because the order shows that the final comparison was not reverse-engineered.

7.1 The variance deficit and its cancellation. ΔC₂/C₂(λ) at ℓ = 7.82 and 9.73: −4.3/−3.1% at λ = 1.0, ≈ 0 at λ = 1.25, 1.5, 2.0. Since the box variance is the cumulative integral of the form factor over |q| < λ, the integrated finite-height correction is negative below the Heisenberg frequency and compensated above it.

7.2 The leading-order prediction. BBLM's correction −(Λ/π²ρ̄²) sin²(πs) has Fourier support at q = 0, ±1. Through the box filter: ΔC₂/C₂ = −2Λ/(λ²ℓ²) for λ < 1, and 0 for λ > 1. Derived independently by the two engines (from BBLM's expansion and from the finite-CUE kernel |K_N|² = |K_∞|² + sin²(πs)/(3N²) with N_eff). Dense scan, ℓ = 9.73, λ = 0.8/0.9/0.95: predicted −5.2/−4.1/−3.7%, measured −4.8 ± 0.7/−4.1 ± 0.5/−3.6 ± 0.3%; ≈ 0 above λ ≈ 1.3. The same Λ is the asymptotic value of Σ_{n≤X}Λ(n)²/n − ½log²X (via (ζ′/ζ)′(s) = 1/(s−1)² − (γ₀²+2γ₁) + … and the Mellin residue), first encountered here as an unexplained constant ≈ −1.5 in the prime sum.

7.3 The corner is a staircase, and its first step moves with height. Fine scans (steps of 0.02 in λ) at ℓ = 7.82 and 9.73 show the deficit holding to λ ≈ 1.06–1.08 and then dropping by ~70% within one step, at 1.08→1.10 and 1.06→1.08 respectively, followed by a shelf of −1 to −2% and a second recovery near 1.20–1.26. The two positions are consistent with λ₂ = 1 + log 2/ℓ = 1.089 and 1.071. This relation was inferred from those two heights and used to predict the third: at ℓ = 11.98, λ₂ = 1.0579. The measured drop lies between 1.05 and 1.06. The 10⁴ and 10⁵ rungs are the discovery set; the 10⁶ rung is a prospective confirmation.

7.4 The steps are Bogomolny's pseudo-orbit peaks. Bogomolny's off-diagonal form factor places δ-peaks at λ = 1 + log(q/p)/ℓ with weights (μ(q)/ψ(q))²(q/p); q/p = 2 is the largest, followed by 6, 3, 3/2, 5, 6/5, 10/3, 7. At ℓ = 11.98 the predicted positions 1.058 (2), 1.092 (3), 1.134 (5), 1.150 (6) coincide with the measured dominant step, the partial recovery at 1.08–1.10, and the sharp recovery at 1.14–1.16 (60-window scan, 0.15% resolution). We do not attribute fixed fractions of the recovery to individual terms: the series is dense and only its smoothed sum is meaningful.

7.5 The full published formula. R₂ = ρ̄² − (1/4π²)∂²_ε ln[|ζ(1+iε)|²Φ^(diag)(ε)] + (1/4π²)|ζ(1+iε)|²e^{2πiρ̄ε}Φ^(off)(ε) + c.c., Φ^(diag) = exp(Σ_p Σ_{m≥2} 2(1−m)/(m²pᵐ) cos(mε log p)), Φ^(off) = ∏_p(1 − (1−p^{iε})²/(p−1)²) (Bogomolny 2000, eqs 74–76, 92–93). Evaluated numerically on ε ∈ [0.004, 90], Fourier-transformed, and integrated through the box. The formula satisfies R₂(0) = 0 on its own (Y₂(0⁺) = 0.9999); the Φ^(diag) factor is what enforces it (omitting it violates the exclusion property by 2c₀/ℓ²). The pipeline reproduces zero deviation on pure GUE input and −2Λ/λ²ℓ² on GUE + BBLM input.

Result. Across 56 points at three heights (22, 22, 12) the formula reproduces the measured relative-variance curve — deficit, dominant step, shelf, second recovery, and cancellation — with no adjustable parameters; the largest absolute discrepancy is 0.3% in relative variance, typical 0.1–0.2%. Pointwise χ²/n = 0.75, 0.55, 0.80 (and 0.77 for the 60-window fine scan). These pointwise values are descriptive only: adjacent bandwidths share spectral content and zero windows, and the window-to-window residuals at the 10⁶ rung are correlated at 0.80–0.97 across λ. A covariance-aware (Hotelling) test on the 60-window scan rejects exact agreement, at the differential precision of ~0.05% that the correlation affords. The residual is concentrated at the step edges, where the measured step is sharper than the predicted one; the numerical resolution of our evaluation of the formula (Δq ≈ 0.01 from the truncated Fourier integral) is the leading uncertainty there, and next-order terms in the asymptotic formula are the next candidate. The formula captures the structure to ~0.2%; it does not exhaust the data.

8. Discussion

We followed an anomaly until it stopped being anomalous. The coherence signal is random-matrix rigidity; the intervention phenomenology is comb geometry plus noise; the residual in the two-point sector is the finite-height arithmetic correction to the pair correlation, whose form Bogomolny and Keating wrote down three decades ago, seen here in frequency space where its pseudo-orbit terms separate into steps. Berry's and Berry–Keating's number-variance comparisons contain the same physics in position space; what this instrument adds is local resolution in bandwidth at low height and a quantitative test of the formula as printed.

The process is reported alongside the results because it was load-bearing. Three mechanisms (scaffold, constraint closure, cumulant-hierarchy arithmetic) and one apparent departure from GUE were eliminated by blind replication, sealed hashes, pre-registered failure modes, and better nulls before they could be published. The distinctions observed / predicted / reproduced / derived / not yet derived were maintained throughout (Appendix B).

But one statistic did not stop. The fourth cumulant's finite-height residual, with its sign change near λ ≈ 1.1, is reproduced neither by the sine-kernel process nor by the finite-CUE promotion that succeeds for the pair correlation. Its correlation-order origin, the kernel-only coefficient D₄(λ), and the zeta-specific four-point terms are the subject of a sequel.

Appendix A — provenance

Round 1: Claude 66f6f09c…, GPT 730322c9…. Round 2: Claude 01746051…0c35d, GPT f7212242…ee0e6. Dilution audit: Claude e531d447…, GPT 99f78a2b…. Height lever: Claude c8466e60…, CUE raw 95e01735…, GPT GUE leg a40cda08…. Data: claude_formfactor_tomography.csv, claude_heisenberg_corner.csv, claude_corner_1e6.csv, claude_fine_corner_1e6_60win.csv, claude_bk_exact_vs_data.csv. Exact BK implementation: claude_bk_pipeline.py.

Appendix B — provenance of claims: observed, predicted, reproduced, derived, not derived

Observed: coherence law; one-point distribution; height ladder of κ₄; variance tomography and staircase. Predicted before measurement: the 10⁶ q/p = 2 position (1.0579 → observed 1.05–1.06). Reproduced by surrogates with no free parameters: coherence law, one-point distribution, intervention recoveries. Derived from published theory with no free parameters: the λ < 1 deficit (BBLM), the step positions (Bogomolny 2000), the full tomography curve to ~0.2% (Bogomolny 2000 pair-correlation formula). Not claimed: first observation of finite-height corrections (Berry 1988; Berry–Keating 1999; BBLM 2006; Forrester–Mays 2015); discovery of the log 2 scale (Berry 1988); a literal "prime staircase" (the series is over rationals q/p); attribution of fixed recovery fractions to individual q/p terms; a derivation of the fourth-cumulant residual.

Appendix C — the finite-CUE promotion and why it is not counted

Exact determinantal cumulants (GPT) for CUE_N with a periodized box spectrum at λ = 1.5: C₂ = 2 − 1/N, C₄ = −23/6 + 4/N − 13/(6N²) + 1/N³, κ₄ = −23/24 + 1/(24N) − 25/(96N²) − …; validated against Monte Carlo (Claude) at even N = 2, 4, 6, 8 to ≤ 0.004; odd N differ by up to 0.05 through the band-edge mode convention. With N_eff = ℓ/√(12Λ): Δκ₄ = −0.051 (ℓ = 7.90; measured −0.058 ± 0.008), −0.032 (9.75; −0.034 ± 0.006), −0.019 (12.0; −0.041 ± 0.008), −0.001 (24.5). The same formula gives a variance deficit −1/(2N) ≈ −27% at N_eff ≈ 1.8, versus a measured ≈ 0. The 1/N terms are mode-count (Euler–Maclaurin) artifacts absent from the continuum kernel correction K_N = K_∞(1 + π²s²/6N² + …); the kernel-only N⁻² coefficient D₄(λ) has not been computed. Tiny-N Monte Carlo (N = 2–10, pooled moments) shows parity structure from band-edge quantization rather than a 1/N² law. The κ₄ agreement at two heights is therefore recorded but not counted as a derivation.