AI 中文总结
本文基于魏尔显式公式构造有限实对称素-魏尔矩阵，构建希尔伯特-波利亚计划的有限维算术模型，数值验证N=13时可复现前三个ζ纵坐标，未声称证明黎曼假设。
AI 中文摘要
从魏尔显式公式的黎曼-Ξ特化形式出发，我们利用极点、阿基米德项及有限素幂数据构造有限实对称素-魏尔矩阵（记为S）。该构造是希尔伯特-波利亚计划框架内的有限维算术模型，该计划旨在实现非平凡ζ零点参数的自伴谱表示。其非对角元构成具有秩2位移恒等式的洛纳型差商矩阵。我们将谱商表述为固定零均值对比空间上的厄米确定广义特征值问题，该表示在正仿射重标下不变，避免了基矢归一化及病态斜投影算子，且保留有限商谱。对于由N个不同正纵坐标γ_k构建的维度匹配零侧矩阵，有理插值给出精确的对比束谱{±γ₁,…,±γ_N}及正奇偶平方谱{γ₁²,…,γ_N²}。由于纵坐标为输入，这是一个重构定理。假设黎曼假设（RH）成立，同一插值向量可证λ_min(S)→0。当N=L=13时，莱姆克的基态商与对比束在数值上一致，并以报告精度复现前三个ζ纵坐标。剩余问题是压缩度量受均匀控制的相对素零扰动定理，本文未声称证明RH。
英文摘要
Starting from the Riemann--$Ξ$ specialization of Weil's explicit formula, we construct finite real-symmetric Prime--Weil matrices $(\mathbf S)$ from pole, archimedean, and finite prime-power data. This construction is a finite-dimensional arithmetic model within the Hilbert--Pólya program, which seeks a self-adjoint spectral realization of the nontrivial zeta-zero parameters. Their off-diagonal entries form a Loewner-type divided-difference matrix with a rank-two displacement identity. We formulate the spectral quotient as a Hermitian definite generalized eigenproblem on the fixed zero-mean contrast space. This realization is invariant under positive affine rescaling, avoids ground-vector normalization and an ill-conditioned oblique projector, and preserves the finite quotient spectrum. For a dimension-matched zero-side matrix built from $N$ distinct positive ordinates $γ_k$, rational interpolation gives the exact contrast-pencil spectrum $\{\pmγ_1,\ldots,\pmγ_N\}$ and positive-parity square spectrum $\{γ_1^2,\ldots,γ_N^2\}$. Since the ordinates are inputs, this is a reconstruction theorem. Assuming RH, the same interpolation vector proves $λ_{\min}(\mathbf S)\to0$. At $N=L=13$, Lemke's ground-state quotient and the contrast pencil agree numerically and reproduce the first three zeta ordinates to the reported precision. The remaining problem is a relative prime-to-zero perturbation theorem with uniform control of the compressed metric. No proof of RH is claimed.
Comments69 pages
