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The two-moment certificate is robust under Rudnick–Sarnak-range cubic augmentation with capacity control: An exact absorption certificate for the two-moment system at bandwidth one, with a hand-checkable witness

Primarily AI-generated textHuman understanding: some partsmath.NT — Number Theorymath.OC — Optimization and Control

Contributed by Kunal Tyagi ↗

Version 1 / Oct 15, 2026 / CC BY 4.0

Abstract

Alpöge and Furman prove unconditionally that more than two thirds of the zeros of the Riemann zeta function are simple and lie on the critical line. Their argument rests on a two-moment certificate — the trace and the Frobenius norm of the Weil–Gabor compression, read against block structure and integrality — and that certificate is provably exhausted: within the data class (tr, Frobenius, positive-index count, integer atoms) an on-line double and a shallow off-line pair are indistinguishable, which pins the distinct-zeros optimum at Nd/N=5/6N_{d}/N = 5/6 and the simple fraction at 2/32/3. A natural proposal to break this degeneracy is to adjoin, at a second bandwidth λ′=1/2+o(1)\lambda' = 1/2 + o(1) inside the Rudnick–Sarnak range, the unconditionally computable signed third trace together with two spectral-capacity rows: an all-VV eigenvalue-count ladder and a garnish-capacity fuzz row. This paper proves, at the model level, and certifies computationally, that the augmentation adds exactly nothing. The marginal value of the entire cubic block over the two-moment baseline is δ0=0\delta_0 = 0 identically, an exact LP equality with an explicit, hand-checkable witness law. It holds over the full adversary grid for the distinct-zeros benchmark (940940 records, ∣δ0∣<10−12|\delta_0| < 10^{-12} everywhere, corner 5/65/6); for the simple-fraction benchmark (corner 2/32/3) at its two decision cells together with cross-checks, converged clean at the stopping criterion; and inside the near-CUE data class licensed by the unconditional pair-correlation theorem of Baluyot–Goldston–Suriajaya–Turnage-Butterbaugh, where the absorption is by strict slack: there the cubic row does not bind. The structural explanation is a set of exact identities proved here. On isolated blocks the signed cubic row is the affine function 3F−23F - 2 of the Frobenius row plus the second integrality level, hence blind to doubles-versus-pairs at every depth. On the ψ1\psi_1-zero grid the bandwidth-one Frobenius row degenerates exactly to the multiplicity count while the half-bandwidth kernel stays position-sensitive, so the cubic budget is met by arrangement at zero cost. Spectral escape is capped sharply at 22Cledεg N=o(N)2\sqrt{2}\sqrt{C_{\mathrm{led}} \varepsilon_g}\,N = o(N), and a scalar cubic tail row with any divergent cutoff is unconditionally vacuous. The pair-interference channel, the one gap in the atom-only model theorem, is closed by an integrality theorem throughout the admissible pair-depth family (w≤1w \le 1, plus deep probes at w∈{1.5,2}w \in \{1.5, 2\}): proved unconditionally for single-pair columns to w≤2.8w \le 2.8, which covers that whole family, and for multi-pair columns at w≤0.82w \le 0.82 (w≤0.98w \le 0.98 modulo one crowding cap whose ledger constant is interval-certified; on the deepest sliver w∈(0.98,1]w \in (0.98, 1] the coverage is adversarial-numerical with the unconditional 8/98/9 backstop), with an unconditional 8/98/9-strength spectral backstop at all depths. Mixed-depth multi-pair columns involving the deep probes w∈{1.5,2}w \in \{1.5, 2\} rest on that backstop together with converged adversarial pricing. The two-moment certificate, and with it the bandwidth-one ceiling, is robust under Rudnick–Sarnak-range cubic augmentation with capacity control.

Provenance statement

The mathematics in this paper - the derivations, the computations, the verification suite, the Lean formalization and the text - was produced by Claude (Anthropic) working under the author's direction inside a multi-session research program; the author set the objectives, made the technical and editorial decisions, and is responsible for the content. Twelve statements are machine-checked in Lean (the axiom footprint is listed in the paper), the LP certificates are interval-checked, and the complete working record (code, logs, certificates, session journal) is public at https://github.com/x67ai/riemann-rh-program. The same version of this paper is deposited on Zenodo, DOI 10.5281/zenodo.22171688.

Tools used

Anthropic
Claude OpusVersion 4.x (agents), August 2026
Anthropic
Claude CodeVersion CLI, August 2026

References

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Show all 16 referencesShow fewer references
  1. F. Gonçalves, D. de Laat and N. Leijenhorst, Multiplicity of nontrivial zeros of primitive LL-functions via higher-level correlations , Math. Comp. 94 (2025), no. 354, 2041--2058; doi:10.1090/mcom/4005; arXiv:2303.01095. The closest published third-party work to this paper: the same higher-level-correlation-versus-multiplicity question, attacked by semidefinite programming rather than by a marked-configuration LP. Cited in Sections 1.2 and 3.2. Their equation (5), Nn(T)=∑kS(n,k)Mk(T)N_n(T) = \sum_k S(n,k) M_k(T) with S(n,k)S(n,k) the Stirling numbers of the second kind, is the change of basis whose n=3n = 3 case is the atom half of Theorem thm:blind here.arXivDOI
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  5. H. Cohn, D. de Laat and A. Salmon, Three-point bounds for sphere packing , arXiv:2206.15373 (v1, 30 Jun 2022; unpublished as of August 2026). The comparandum: in the sphere-packing analog the third-order augmentation does pay, improving on the two-point Cohn--Elkies bound in several dimensions; here it is worth exactly nothing.arXiv
  6. S. Karlin and W. J. Studden, Tchebycheff Systems: With Applications in Analysis and Statistics , Pure and Applied Mathematics, Vol. XV, Interscience Publishers (John Wiley & Sons), New York, 1966. The classical Chebyshev--Markov--Stieltjes/Christoffel background for the fact that odd moments do not improve one-sided bounds --- the general form of the cubic blindness of Theorem thm:blind .

Version history

  1. v1Submitted by Kunal TyagiInitial depositCurrentOct 15, 2026