paper Review Profile

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

publishedby Adam MurphyCreated 9/6/2026Reviewed under Calibration v1.4· 1 review

3.7/ 5

AI Rating

A nonlinear negative-excursion statistic on the band-limited explicit-formula field of the Riemann zeros shows a strong coherence excess that blind two-engine tests attribute entirely to GUE/CUE spectral rigidity. Intervention experiments on the prime field are explained by comb geometry plus Gaussian dilution. The surviving finite-height residual is resolved by bandwidth tomography into a variance deficit −2Λ/(λ²ℓ²) below the Heisenberg scale (Λ = 1.57314…) and a discrete recovery at λ = 1 + log(q/p)/ℓ; Bogomolny's published pair-correlation formula reproduces the measured curve at three…

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This is a technically disciplined numerical-methods paper on band-limited statistics of the Riemann zeros, and the panel — four math/logic specialists, three sources specialists, and two science/novelty specialists — converged on a strongly consistent picture despite minor scoring spread. All four math reviewers independently reproduced the paper's headline quantitative substitutions (the −2Λ/(λ²ℓ²) prediction at ℓ=9.73, the λ=1+log(q/p)/ℓ step positions at all three heights, and the Appendix C finite-CUE cumulant expansion κ₄=−23/24+1/(24N)−25/(96N²)) and found no sign, dimensional, or algebraic errors in any displayed equation. However, three of the four math specialists converged on a shared, load-bearing concern: the box-filter derivation in Section 7.2 that converts the cited BBLM/Bogomolny pair-correlation corrections into the paper's own relative-variance observable ΔC₂/C₂ is stated as a one-line result rather than shown, omitting the ρ̄=ℓ/2π substitution and C₂∝λ² normalization; likewise the Section 7.5 numerical pipeline (taper, Fourier convention, truncation at ε∈[0.004,90], discretization) is not specified in enough detail to independently reproduce the claimed ~0.2% curve match. One math specialist (claude-opus-5) went further and reconstructed the missing steps, confirming they check out — this is a genuine specialist disagreement worth surfacing: the compressed derivation is flagged as unverifiable-from-text by three reviewers, while a fourth independently audited and validated it. Readers should treat the Section 7.2/7.5 compression as the single most important thing to fix, since it is precisely the load-bearing calculation behind the paper's central no-free-parameter claim. A secondary, unresolved tension noted only by claude-opus-5 (and not corroborated elsewhere, so treated as a flag rather than a confirmed error) concerns an apparent inconsistency in the empirical Δκ₄≈−(3.1–3.6)/ℓ² scaling law failing at the ℓ≈11.98 rung, plus a wording ambiguity around whether the asymptotic prime-sum constant equals Λ or −Λ. The paper also overstates equivalence between the prime-cutoff field ν_X and the sinc-comb F_L based only on a 0.9991 correlation, which specialists agree is empirical proximity, not derived equivalence — a labeling issue rather than a structural flaw for the main tomography result, which is separately defined on the comb. On completeness, specialists agree the narrative logic is essentially complete and unusually transparent (blind two-engine replication, SHA-256 sealing, retraction of a null-convention artifact in §4.3, explicit claim-provenance taxonomy in Appendix B), but the underlying code, raw data, and numerical pipeline are referenced only by filename/hash rather than exposed, which limits independent reproducibility of the precision claims. On falsifiability/verifiability, the panel's domain was contested between 'methods' and 'pure_mathematics'; this reflects the paper's genuinely hybrid character (a numerical-methodological instrument tested against a well-defined mathematical spectrum with published closed-form comparanda) rather than any defect in the work — both readings support a strong score, since the paper's predictions (step locations, λ^-2 scaling, prospective third-height test) are concretely falsifiable and its comparisons are recomputable from public zero tables and the cited Bogomolny formula. The paper is commendably honest that it does not claim discovery of the underlying arithmetic corrections, log-2 scale, or Bogomolny-Keating formula, correctly attributing these to prior literature (Berry 1988, Bogomolny-Keating 1995/96, Berry-Keating 1999, BBLM 2006, Forrester-Mays 2015) and situating its own contribution as a bandwidth-resolved diagnostic instrument plus a genuinely open, honestly unresolved fourth-cumulant anomaly.

Internal Consistency

4/5

The logical architecture is unusually disciplined: a claimed anomaly is progressively decomposed (universality → comb geometry + dilution → finite-height two-point correction), and each stage's status is tracked in Appendix B under the observed/predicted/reproduced/derived/not-derived taxonomy. Definitions of λ, ℓ, L, C₂, C₄, κ₄ are stable across sections. Boundary behaviour is handled coherently: the λ<1 deficit, its exact cancellation above λ=1, and the λ=1.5 slice where ΔC₂/C₂ ≈ 0 (§6.2) are mutually consistent with §7.1. The retracted GUE departure (§4.3) and the refusal to count the finite-CUE promotion (§6.4, Appendix C, because it predicts a −27% variance deficit that is measured to be ≈0) are examples of the paper enforcing its own consistency criteria against its own results. Deductions for three local tensions: (a) the empirical law Δκ₄ ≈ −(3.1–3.6)/ℓ² (§6.1) does not accommodate the ℓ≈11.98 rung, where −0.041×144 ≈ 5.9, an outlier that goes unremarked and also breaks the monotone ladder (−0.058, −0.034, −0.041); (b) §7.2 states 'the same Λ is the asymptotic value of Σ_{n≤X}Λ(n)²/n − ½log²X' while immediately noting it was 'first encountered here as an unexplained constant ≈ −1.5', i.e. the asymptotic value is −Λ, not Λ — a sign/wording inconsistency in a bridging remark; (c) at λ = 1.0 the measured ΔC₂/C₂ = −4.3% (ℓ=7.82) sits below the leading prediction −5.1% even though §7.3 places the first step at λ ≈ 1.07–1.09, so the deficit should still be full there; this is not reconciled. None of these disturbs the core two-point conclusion.

Mathematical Validity

3/5

No definite algebraic or dimensional error is established from the displayed equations. In particular, the Appendix C check is arithmetically consistent at the stated order: with N_eff=ℓ/sqrt(12Λ), ℓ≈7.90 and Λ=1.57314 give N_eff≈7.90/sqrt(18.87768)≈1.82, hence −1/(2N_eff)≈−0.275, consistent with the reported approximately −27%. The cited BBLM and Bogomolny expressions may properly be used as literature inputs. However, the paper's load-bearing new mathematical/computational step is compressed: it gives the final box-filter result and reports numerical Fourier integration of R₂, but does not derive or specify enough of the transform-to-variance map to verify the claimed relative-variance curve. In particular, the treatment of the constant density term, connected two-point function, Fourier normalization, taper, truncation at ε=0.004 and 90, and finite-window normalization is absent. Since the asserted 0.1–0.3% quantitative match depends on that conversion, mathematical validity cannot exceed 3 under the required central-unverified-derivation cap. This score does not dispute the cited pair-correlation formula itself.

Falsifiability

4/5

Using the empirical-falsifiability rubric for a methods submission, the work provides several specific, quantitative, presently testable claims: a coefficient and λ^-2 scaling below λ=1; height-dependent pseudo-orbit step locations; a no-adjustable-parameter finite-height variance curve; and a proposed fourth-cumulant sign change. The paper also identifies operational failure modes, including failure of locations to scale as 1+log(q/p)/ℓ and systematic covariance-aware discrepancies at new heights. Testability is reduced from the maximum because the central comparison already rejects exact agreement at high differential precision, while the stated numerical Fourier resolution and omitted next-order terms leave the accuracy target for the approximate formula incompletely specified.

Clarity

4/5

The paper is strongly organized around what was observed, what was eliminated by controls, what follows from prior theory, and what remains unresolved. The explicit retraction of a null-convention mismatch and the Appendix B observed/predicted/reproduced/derived taxonomy materially improve scientific communication. Some dense passages still require substantial specialist familiarity: the normalization connecting the explicit-formula field, sinc comb, box variance, and relative variance is compressed, and the full numerical implementation is referenced by filenames and hashes rather than presented in a readily independently executable form. Nevertheless, a graduate-level reader in analytic number theory or random-matrix spectral statistics can follow the main argument and limitations.

Novelty

3/5

The submission is unusually candid that the underlying arithmetic corrections, log-2 scale, and Bogomolny–Keating/Bogomolny formulas are established prior work. Its plausible new contribution is the band-limited tomography presentation: it resolves the known finite-height pair-correlation structure locally in bandwidth, uses a prospective height prediction for the q/p=2 feature, and separates this two-point result from an open fourth-cumulant observation. This is an interesting and potentially useful synthesis and diagnostic instrument, but it is primarily a new numerical interrogation of known formulas rather than a new arithmetic mechanism or a clearly distinct predictive framework.

Completeness

4/5

The paper is well-developed for its stated goals. It defines its statistic, null conventions, and key variables (ν_X, λ, ℓ, Λ, C₂, C₄, κ₄, f₋, C^L, C^G, F_L, R₂, Φ^(diag), Φ^(off), N_eff). It addresses boundary conditions and edge cases: the λ < 1 vs λ > 1 behavior of the variance correction, the Heisenberg corner staircase, the step positions at λ = 1 + log(q/p)/ℓ, and the finite-CUE promotion's failure. Limitations are explicitly stated: pointwise χ² values are descriptive only, cross-bandwidth correlations are reported, the covariance-aware test rejects exact agreement at ~0.05% differential level, and the fourth-cumulant residual is left open. The paper addresses its own stated goals: it follows the anomaly through universality, intervention phenomenology, and the finite-height residual, and it quantitatively compares to the published Bogomolny formula. The methodology statement and appendices provide provenance and claim classification. Minor gaps: the exact formal definition of the negative-excursion fraction f₋ is given in words rather than a precise equation; the numerical evaluation details of the Bogomolny formula (tapered Fourier transform, truncation at ε ∈ [0.004, 90], Δq ≈ 0.01) are summarized but not fully specified; the fourth-cumulant residual is reported but not resolved, which the paper explicitly acknowledges as open. These are secondary details; the core argument is fully developed. The paper does not skip the derivation of its own central result—it compares measurements to a published formula and describes its evaluation pipeline. Score 4 reflects minor gaps in secondary details while the core argument is complete.

11 derivation flags— equations with compressed or unverified steps identified by math specialist

Strengths

+Rigorous, pre-registered blind two-engine replication protocol with SHA-256 sealing that catches and transparently retracts a null-convention false positive (§4.3) before publication
+Disciplined claim-provenance taxonomy (Appendix B: observed/predicted/reproduced/derived/not-derived) that prevents overclaiming and clearly separates this paper's contribution from prior literature (Berry, Bogomolny-Keating, BBLM)
+A genuine prospective prediction — the λ≈1.058 step position at ℓ=11.98 predicted from two lower heights and then confirmed — rather than purely retrospective curve-fitting
+Self-falsification discipline: the finite-CUE promotion in Appendix C is explicitly rejected despite matching κ₄ at two heights, because it also predicts a −27% variance deficit that is measured to be ≈0
+Honest reporting of the covariance-aware (Hotelling) test rejecting exact agreement at the ~0.05% differential level, rather than resting on the more favorable but structurally biased pointwise χ² figures
+Independently reproducible arithmetic: all four math specialists confirmed the numerical substitutions in §7.2, §7.3-7.4, and Appendix C exactly reproduce the paper's printed values

Areas for Improvement

-Expand the compressed derivation in Section 7.2 to explicitly show the ρ̄=ℓ/2π substitution and C₂∝λ² normalization that convert the cited BBLM sin²(πs) correction into the box-filtered ΔC₂/C₂=−2Λ/(λ²ℓ²) result — three of four math specialists could not independently verify this step from the text alone, though one specialist did reconstruct and confirm it
-Fully specify the numerical Fourier-transform pipeline used in Section 7.5 (taper function, transform sign/2π convention, treatment of the ε=0.004 lower cutoff and ε=90 tail, Euler-product truncation, discretization/convergence tests) so the claimed ~0.1-0.3% curve agreement is independently reproducible from the paper alone rather than from referenced but unavailable code/CSV files
-Clarify or correct the wording in §7.2 asserting the prime-sum asymptotic constant equals Λ while separately describing it as ≈−1.5, an apparent sign/wording tension flagged by one specialist
-Address the reported inconsistency between the empirical Δκ₄≈−(3.1-3.6)/ℓ² scaling law and the ℓ≈11.98 datum, which one specialist calculated falls well outside that band and breaks monotonicity
-Soften the Section 2 claim that ν_X and F_L are 'equivalent' to an explicit statement of empirical near-equivalence (correlation 0.9991) pending a derived error bound or explicit-formula remainder estimate
-Release or append the actual numerical pipeline, frozen configuration files, window-selection rules, and covariance/Hotelling test construction rather than relying on filenames and truncated hashes as the reproducibility path
-Provide the determinantal/trace derivation (or a citation to one) for the Appendix C exact finite-CUE cumulant formulas, even though they are used only to reject rather than establish a mechanism
-When summarizing the ~0.2% agreement in the abstract, foreground the covariance-aware rejection of exact agreement rather than leaving it to be found only in the body

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.

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