发表机构
VIDRAFT AI Research · QuantumOS（VIDRAFT AI Research · QuantumOS）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该论文给出铃木韦伊二次型算子的数值实现，通过P1有限元离散化等方法。得出谱的阿基米德定律等多个结果，实现韦伊正性准则算子形式，虽未证黎曼假设，但对经典恒等式有忠实数值实现。
AI 中文摘要
本文首次给出了铃木韦伊二次型算子的数值实现，该算子是将谱正性与黎曼假设（RH）联系起来的希尔伯特 - 波利亚计划的候选者。铃木2026年的构造是纯理论的，这里通过P1有限元离散化和理查森外推法实例化该算子。关键结果包括：在无素数区域，谱遵循封闭的阿基米德定律\(A_k(a)=\log(1/a)+\log(k - 2)+B_0 + O(a)\)，\(B_0=\log q - 2\log 2\)，精度验证到30位；梅林双极点论证证明了首项系数\(B(\nu)\)并表明\(B_0\)仅取决于导体\(q\)，与阿基米德参数无关；总谱强度遵循素数定理\(S(a)\sim(2a)^3/6\)；非平凡零点不是特征值而是出现在素数符号的显式公式误差项中；最佳匹配线\(\sigma^*(a)\)向临界线下降；韦伊正性准则以算子形式实现；最低特征值\(\lambda_1(a)\)严格为正，超指数衰减并平滑通过第一个素数阈值；首次计算了特征函数\(W(a,0;z)\)，所有零点被确认为实数；即使在直接检测受阻的情况下，GUE统计的间接痕迹也出现在矩结构中。作者强调这项工作并未证明RH。所有结果都是阿基米德且普遍的，其意义在于对经典恒等式的忠实数值实现而非新的算术。
英文摘要
This paper presents the first numerical realization of Suzuki's Weil-Quadratic-Form operator, a candidate for the Hilbert--Pólya program linking spectral positivity to the Riemann Hypothesis (RH). Suzuki's 2026 construction was purely theoretical; here, the operator is instantiated via P1 finite-element discretization and Richardson extrapolation. Key results include: (R1) In the prime-free regime, the spectrum follows a closed Archimedean law $A_k(a) = \log(1/a) + \log(k-2) + B_0 + O(a)$, with $B_0 = \log q - 2\log 2$, confirmed to 30-digit precision. (R2) A Mellin double-pole argument proves the head coefficient $B(ν)$ and shows $B_0$ depends only on the conductor $q$, independent of the Archimedean parameter. (R2b) The degree $d$ of an L-function appears directly as the logarithmic slope of the spectrum. (R3) Total spectral intensity follows the prime number theorem, $S(a) \sim (2a)^3/6$. (R4) Nontrivial zeros are not eigenvalues but occur in the explicit-formula error term of the prime symbol. (R5) The best-match line $σ^*(a)$ descends toward the critical line. (R6) Weil's positivity criterion is realized in operator form: bounded residual growth corresponds to all zeros on the line, while an injected off-line zero causes exponential blow-up. (R7) The lowest eigenvalue $λ_1(a)$ is strictly positive, decays superexponentially, and passes smoothly through the first prime threshold. (R8) The characteristic function $W(a,0;z)$ is computed for the first time, with all zeros confirmed real. (R9) Indirect traces of GUE statistics appear in the moment structure, even where direct detection is blocked. The authors emphasize that this work does not prove RH. All results are Archimedean and universal, with significance lying in the faithful numerical realization of classical identities rather than new arithmetic.
Comments18 pages
