Every real primitive Dirichlet L-function has de Bruijn-Newman constant at most 9/32: a kernel-checked proof in Lean 4

Lean 4 sources, verification records and paper for the theorem that, for every real primitive Dirichlet character chi != 1 and every t >= 9/32, every zero of the de Bruijn-Newman heat flow of the completed L-function Lambda(s, chi) is real (classically Lambda_chi <= 9/32). The Lean theorem bb_real_primitive_meaning_oai is the conjunction of this statement with the identification Hchi chi 0 z = (1/8) N^((s+k)/2) completedLFunction chi s, s = 1/2 + iz/2, which says that the Lean heat flow is that of the classical completed L-function. It is proved in one environment (Lean v4.34.1, Mathlib d13f23b) with OpenAI's kernel-checked zero-free half-plane Re s > 7/8 for Dirichlet L-functions (openai/math at adc7f124). Axioms: propext, Classical.choice, Quot.sound. The exported environment was replayed by five checkers from three implementation lineages (leanchecker, leanchecker-paranoid, nanoda, con-leche, con-ron), each of which rejected a skipKernelTC negative control; those replays were local and unsandboxed; a sandboxed Comparator run against a separately written challenge also accepted it. Reusable formal components: de Bruijn's theorem for a generic even kernel, Gauss's sign theorem for real primitive characters in Mathlib's normalisation, and the Mellin bridge from the theta kernel to Mathlib's completedLFunction. Credit: de Bruijn (1950, Theorem 13) for the engine; Freire, Gonzalez-Riquelme, Munoz, Poblete and Salinas (arXiv:2610.27343, Remarks 3.1 and 3.4) for the conditional statement Lambda_L <= D^2/2 = 9/32 under a zero-free strip of half-width 3/4; OpenAI for the 7/8 half-plane (manuscript unrefereed, Lean proof kernel-checked); Gauss for the sign theorem; Mathlib for the L-function API. The version-1 theorem is not a new analytic bound and not progress on the generalized Riemann hypothesis (Lambda_chi <= 0). Not yet refereed. Licensing: Lean sources and scripts Apache-2.0; paper and prose CC-BY-4.0. Version 2.0.0 adds three bounds on the same constant in weaker trust classes, stated as such in the paper (section 7): (1) unconditionally, for every real primitive chi of conductor N >= 10^8, Lambda_chi <= T(l) + b(l)/2 < 9/32 with l = log(N/pi) and T, b explicit (0.27643 at N = 10^8; asymptotically sqrt(5) log log N / log N); no GRH, no zero-free half-plane, no zero verification, Siegel zeros allowed; (2) given OpenAI's 7/8 as in version 1, Lambda_chi <= 0.27912 < 9/32 for every real primitive chi; (3) unconditionally, Lambda_chi <= 0.49330 for every real primitive chi. Method: an Arb-certified lower bound for -Im(H'/H) at a fixed height, fed into the contraction inequalities of Planat (arXiv:2609.37164) and Hakimoglu-Brown (arXiv:2610.12803), with FGMPS's Gaussian form, strip theorem and effective approximation. These are NOT kernel-checked: Arb certificates reproduced by a separate AI auditor session (identical in every field except wall time), AI-written proofs reviewed by one AI reviewer, unrefereed preprints. One arithmetic step is a Lean theorem (C2.C2bound_lt_9_32, built once at Lean v4.35.0). The version-1 theorem and its verification record are unchanged. Still not progress on GRH.