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Paper · August 3, 2026

Exact Certificates, Moment Ledgers, and a Three-Wall Barrier Map for Zero-Density Estimates at the Guth-Maynard Frontier

Machine-checkable dual certificates for the classical zero-detection game, an entropy ledger that closes the higher-moment route, and a barrier map around the Guth-Maynard bound. Published on Zenodo.

Read on Zenodo ↗doi:10.5281/zenodo.21764256 ↗
Cite this paper
@misc{guvenc2026guthmaynard,
  author    = {G\"uven\c{c}, Baturalp},
  title     = {Exact Certificates, Moment Ledgers, and a Three-Wall
               Barrier Map for Zero-Density Estimates at the
               Guth-Maynard Frontier},
  year      = {2026},
  month     = aug,
  publisher = {Zenodo},
  doi       = {10.5281/zenodo.21764256},
  url       = {https://doi.org/10.5281/zenodo.21764256}
}

Baturalp Güvenç · Wiener Labs · Published August 3, 2026 · Zenodo, CC BY 4.0

Abstract

We study the exponent-level structure of zero-density estimates for the Riemann zeta function around the Guth-Maynard bound

A(σ)≤153+5σ.A(\sigma) \le \frac{15}{3+5\sigma}.

First, the classical zero-detection pipeline is formalized as an explicit two-player length-selection game and solved exactly. The Ingham, Huxley, Guth-Maynard and Jutila exponents all arise as closed-form straddle-closure values, each accompanied by a machine-checkable dual certificate. For the stated input library the value

15(1−σ)3+5σ\frac{15(1-\sigma)}{3+5\sigma}

is exactly optimal on [710,3950][\tfrac{7}{10}, \tfrac{39}{50}], Proposition 12.1 of Guth-Maynard is provably neutral, and the diagonal edge of the certificate is matched by an explicit example whenever M2−2σ≤TM^{2-2\sigma} \le T.

Second, an entropy ledger shows that within the Guth-Maynard Fourier framework every additional leg of the cyclic Gram moment costs a factor T11/3T_1^{1/3} at both critical configurations, which closes the r≥4r \ge 4 route.

Third, the extremal rational-concentration scenario is verified to be a simultaneous fixed point of four independent constraint families, including a Rudin-Chang bound over the Q\mathbb{Q}-dissociated log-prime system; a universal height law B=T11/3B = T_1^{1/3} emerges across the whole range.

Finally, a single Alignment Hypothesis on linear-form correlations of mollified Möbius coefficients is isolated and its exact price computed: it yields

A ⁣(34)≤420191A\!\left(\tfrac{3}{4}\right) \le \frac{420}{191}

but leaves the uniform exponent 3013\tfrac{30}{13} invariant.

All exponent arithmetic is verified in exact rational arithmetic. No new unconditional zero-density exponent is claimed; the contribution is the certification of optimality for the stated inputs, the barrier map, and the conditional results. Verification code accompanies the paper.

Where it sits

Zero-density estimates bound N(σ,T)N(\sigma, T), the number of zeros ρ=β+iγ\rho = \beta + i\gamma of ζ(s)\zeta(s) with β≥σ\beta \ge \sigma and ∣γ∣≤T|\gamma| \le T, in the form

N(σ,T)≪TA(σ)(1−σ)+ε.N(\sigma, T) \ll T^{A(\sigma)(1-\sigma)+\varepsilon}.

The 2024 Guth-Maynard breakthrough moved the frontier at σ=710\sigma = \tfrac{7}{10} for the first time since Ingham. This paper does not push the frontier further; it maps the walls around it. Three independent barriers are certified, each with an exact ledger of what any assault on it must pay, and the one hypothesis that would move a wall is priced exactly.

The full paper, together with the exact-arithmetic verification code, is openly available on Zenodo under CC BY 4.0.

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