A narrow window with a certified positive margin
The preprint reports an unconditional finite-scale result for a mathematical test tied to the Weil quadratic form. For every real, even test function confined to the interval from −0.8 to 0.8, the authors certify a positive lower margin at support 1.6. In plain terms, the quantity being tested cannot fall below 8.9 × 10−18 times the function’s squared L2 norm, a measure of its size in this calculation.
The same margin extends to arbitrary complex test functions in that window, removing the restrictions that the function be real or even. The result can also be stated in terms of autocorrelations: the associated Weil functional is nonnegative when those autocorrelations are supported from −1.6 to 1.6.
The spectral calculation also separates the lowest even and odd sectors. At support 1.6, the first even-sector value is bounded between 8.9 × 10−18 and 2.523 × 10−16, while the second even value is at least 2.085 × 10−12. The first odd-sector value lies between 8.206 × 10−15 and 2.347 × 10−14. On the reported window, the bottom state is therefore certified to be simple, meaning non-repeated, and even.
The upper bounds fall extraordinarily fast
The paper’s other main calculation searches for explicit test functions that make the same normalized quantity small. Across windows with half-width L from 0.5 to 2.0, unconditional interval-arithmetic evaluations produce certified upper bounds that fall from about 10−6 at L = 0.5 to 3.2 × 10−283 at L = 2.0. Each number is an upper endpoint for a geometric-side Rayleigh quotient, so it is a guaranteed ceiling for the value generated by that trial function, not automatically the exact infinite-dimensional minimum.
At L = 0.8, the two unconditional certificates place the window profile between 8.9 × 10−18 and 2.27 × 10−17. Both directions are certified for that finite window.
To describe the decline, the authors compare the certified upper bounds with a Landau–Widom-type form based on the number of zeta zeros below a chosen resolution height, divided by the logarithm of that count. A fit to four asymptotic points gives a constant of 20.13 ± 0.10, with in-sample residuals below 0.7 percent. A version with no fitted constant reproduces those four points with errors ranging from 1.4 percent to 2.6 percent.
Evidence for a pattern, not a proof of the law
The comparison with a competing model is part of the authors’ case for the proposed form. For the single-constant Landau–Widom model, the four out-of-sample residuals are −4.4 percent, −4.9 percent, −5.5 percent and −4.8 percent, with no reported trend. A three-parameter model based on pure zero counting instead produces residuals of +0.8 percent, +2.0 percent, +3.7 percent and +5.8 percent, rising steadily. The authors reject that competing model, but this remains model discrimination from a small finite set of points, not a proof of an asymptotic law.
That distinction matters because the upper-bound pipeline certifies only one side of the infinite-dimensional problem. It does not establish the reverse inequality or show that the trial functions sit close to the true infimum. The preprint therefore treats the identification of the fitted decay law with the true window profile as a conjecture.
The zeros-side calculations have a narrower role than the headline pattern might suggest. Multiprecision values for zeta zeros are used to propose trial vectors and to cross-check the geometric certificates, but they are not formally verified enclosures and do not form part of the unconditional certificates.
What the study leaves open
The preprint also proves a conditional statement under the Riemann Hypothesis, or RH. For sufficiently large L, its upper bound has an exponent proportional to minus L times e raised to L, giving a qualitative double-exponential decay scale. The theorem does not establish the full proposed e raised to 2L scale or derive the fitted constant, and it depends on RH.
The authors identify a separate computational barrier for certificates built from a pointwise envelope of the Weil symbol. Such certificates must cross a comb-alignment threshold, after which the required matrix sizes and quadrature-node counts grow doubly exponentially with L. The threshold cannot be lowered within that particular class of certificate, but the result does not rule out every other certification strategy.
The paper also retracts an earlier exploratory claim at half-width 1.19, corresponding to support 2.38. It says the earlier calculation substituted 4.6948 for the relevant comb-mass constant 7.0750, bounding the comb in the wrong direction and yielding no conclusion about the Weil form. The corrected work therefore does not certify positivity at support 2.38.
The work is a version-2 arXiv preprint dated 2 September 2026, with a manuscript date of 3 September 2026. It reports supplementary source code, software versions, zero data, certified matrices and Cholesky factors, run logs, error budgets, SHA-256 hashes and an independent verifier, giving other researchers material to inspect the calculations.
Paper data and sources
Original title: Weil positivity in compact windows: certified two-sided bounds and a Landau--Widom decay law
Authors: Marcus Chuk
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text