Retuned translated probes and the conditional bound Λ < 0.116398 for the de Bruijn–Newman constant

Abstract

Assume that every nontrivial zero of the Riemann zeta function has real part at most 7/8. This paper proves the computer-assisted upper bound Λ ≤ 116397726408291/10^15 = 0.116397726408291 < 0.116398 for the de Bruijn–Newman constant in the Polymath normalisation. The alternate rational construction gives 0.116397726408312. The argument combines a retuned physical probe-height schedule, matched finite logarithmic coefficients and 44 exact continuous-domain translated-kernel certificates. It optimises the complete contraction rate, including the entire distinguished-pair subtraction. The source preserves common multiplicative phases and cancels shared convolution terms before estimating the unmatched terms. Its selected bounds cover 608 closed time cells, all six frequency classes including an unbounded final class, all eleven translated class pairs and every future native cutoff. All 608 retuned native cells in both interval constructions, followed by the joint and matched-logarithmic calculations, were freshly replayed during deposit preparation. The reconstructed mathematical source equals the frozen source exactly. The full stored-source audit also passed, and both 14,899-step main histories regenerate byte-for-byte and pass a separately implemented exact rational auditor. Both native constructions use the same Arb/FLINT library, with different precision, partitions and derivative constructions. The earlier finite-head walls are retained; the published finite-RH computation is an imported theorem and was not rerun. The main paper includes the global first-contact argument, full-domain kernel proof, exact kernel-constant table and corrected Gaussian remainder. The deposit supplies the expanded matched-source argument, clearly labelled prior analytic foundations, editable LaTeX, verification guide, execution receipt and complete frozen evidence. A separate Further Consequences note proves an implication from the stronger initial strip H(34/43) to Λ < 0.09 and identifies a sufficient but unproved all-scale Möbius covariance cap. Those statements are not premises of the 7/8 theorem. The 7/8 zero-free assertion is an explicit hypothesis; the numerical checks do not prove it. This is not an unconditional record claim, a proof of RH, a global-optimality result, external peer-review certification or end-to-end formalisation. The AI assisted with code & computation. Version 1.0 is the first deposit of this conditional result. Earlier 0.118 drafts were not deposited.