截断韦伊二次型的有限吉南德 - 韦伊字典与阿基米德尾序
A finite Guinand-Weil dictionary and archimedean tail order for the truncated Weil quadratic form
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中文总结 AI 辅助
研究截断韦伊二次型，证明两个有限定理。一是实偶伽辽金系数向量确定带限吉南德 - 韦伊测试函数，其在zeta非平凡零点上的零和等于二次值；二是给出阿基米德尾的性质及双边认证规则。
中文摘要 AI 辅助
韦伊二次型在素数截断c>1和频带N处的康恩斯 - 范·苏伊勒科姆与康恩斯 - 孔萨尼 - 莫斯科维奇截断产生有限伽辽金矩阵，其谱是韦伊正性的有限秩窗口。我们证明了关于此截断的两个精确有限定理。首先，每个实偶伽辽金系数向量v以封闭形式确定一个带限吉南德 - 韦伊测试函数g_v，其在zeta的非平凡零点上的零和恰好等于二次值<v, Q v>：截断形式的每个值都是零点上的精确和。该构造通过维度为2N + 1的精确源商进行，并且允许一个非坍缩的极中性子族。其次，在伽辽金带之外，省略的阿基米德尾是一个完全正的柯西 - 斯蒂尔杰斯增量。这产生了一个双边认证规则，其明确预算B_T ~ (2N + 1) rho log(T) / (pi^2 T)，其中T是阿基米德截断，rho = 2 pi / log c：有限截断正性证明无截断正性，低于 -B_T的有限截断特征值证明无截断负性，并且在[-B_T, 0)带中的负特征值不证明任何东西。通过蛮力截断在c = 100处解析10^-59的谱尺度将需要T约为10^63；无截断区间LDL^T分解可直接解析它。该字典在前512个zeta零点上得到验证，并通过三条独立的计算路径；所有脚本和工件都随论文一起提供。本文未提出黎曼假设、素数计数、下一个素数或因式分解主张。
英文摘要
The Connes-van Suijlekom and Connes-Consani-Moscovici truncations of the Weil quadratic form, at a prime cutoff c>1 and frequency band N, produce finite Galerkin matrices whose spectra are the finite-rank window on Weil positivity. We prove two exact finite theorems about this truncation. First, every real even Galerkin coefficient vector v determines, in closed form, a band-limited Guinand-Weil test function g_v whose zero sum over the nontrivial zeros of zeta equals the quadratic value <v, Q v> exactly: every value of the truncated form is an exact sum over the zeros. The construction factors through an exact source quotient of dimension 2N+1 and admits a non-collapsing pole-neutral subfamily. Second, beyond the Galerkin band the omitted archimedean tail is a totally positive Cauchy-Stieltjes increment. This yields a two-sided certification rule with an explicit budget B_T ~ (2N+1) rho log(T) / (pi^2 T), where T is the archimedean cutoff and rho = 2 pi / log c: finite-cutoff positivity certifies cutoff-free positivity, a finite-cutoff eigenvalue below -B_T certifies a cutoff-free negative, and a negative eigenvalue in the band [-B_T, 0) certifies nothing. Resolving a spectral scale of 10^-59 at c=100 by brute cutoff would require T of order 10^63; a cutoff-free interval LDL^T factorization resolves it directly. The dictionary is verified over the first 512 zeros of zeta and by three independent computational routes; all scripts and artifacts ship with the paper. The paper makes no Riemann Hypothesis, prime-counting, next-prime, or factoring claim.
