math.NT — Number Theory

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

Contributed by Kunal Tyagi

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 $N_{d}/N = 5/6$ and the simple fraction at $2/3$. A natural proposal to break this degeneracy is to adjoin, at a second bandwidth $\lambda' = 1/2 + o(1)$ inside the Rudnick–Sarnak range, the unconditionally computable signed third trace together with two spectral-capacity rows: an all-$V$ 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 $\delta_0 = 0$ identically, an exact LP equality with an explicit, hand-checkable witness law. It holds over the full adversary grid for the distinct-zeros benchmark ($940$ records, $|\delta_0| < 10^{-12}$ everywhere, corner $5/6$); for the simple-fraction benchmark (corner $2/3$) 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 $3F - 2$ of the Frobenius row plus the second integrality level, hence blind to doubles-versus-pairs at every depth. On the $\psi_1$-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 $2\sqrt{2}\sqrt{C_{\mathrm{led}} \varepsilon_g}\,N = o(N)$, 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 ($w \le 1$, plus deep probes at $w \in \{1.5, 2\}$): proved unconditionally for single-pair columns to $w \le 2.8$, which covers that whole family, and for multi-pair columns at $w \le 0.82$ ($w \le 0.98$ modulo one crowding cap whose ledger constant is interval-certified; on the deepest sliver $w \in (0.98, 1]$ the coverage is adversarial-numerical with the unconditional $8/9$ backstop), with an unconditional $8/9$-strength spectral backstop at all depths. Mixed-depth multi-pair columns involving the deep probes $w \in \{1.5, 2\}$ 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.