bibkey: shi2026finitepencils authors: Yaoming Shi year: 2026 title: Construction of Finite Hilbert–Pólya Matrices from Weil’s Explicit Formula doi: null url: https://arxiv.org/abs/2609.04908v1 claim: A finite Hermitian definite quotient pencil has real spectrum under a simple least eigenvalue and a positive compressed metric; transfer to the full arithmetic spectrum still requires uniform relative perturbation estimates. strata_touched: [] license: citation-only triage: anchor

Finite Weil pencils and their relative-transfer boundary

The primary source is Shi, arXiv:2609.04908v1, submitted 4 September 2026. The source identifies its manuscript as Version 12. Theorem 4.1 and the relative-transfer discussion in Sections 1.4–1.5 and 4.8–4.9 were inspected. This note does not independently verify the full preprint, its numerical experiments or its proofs.

What the finite construction supplies

For the arithmetic matrix ${\mathbf{S}}_{N,L}$ the source subtracts its least eigenvalue ${ϵ}_{N,L}$ to form ${\mathbf{W}}_{N,L}={\mathbf{S}}_{N,L}-{ϵ}_{N,L}I$. It uses the fixed contrast space ${\mathcal{V}}_{N,L}^0=\ker {\mathbf{\delta }}_{N,L}^{\mathsf{T}}$, where ${\mathbf{\delta }}_{N,L}$ represents evaluation at zero in the chosen Fourier coordinates. Thus coefficient sum zero is this evaluation constraint; it is not the theta model’s $\nu$-mean constraint. An orthonormal basis matrix ${\mathbf{C}}_{N,L}$ gives the metric and differentiation form

$${\mathbf{G}}_{N,L}={\mathbf{C}}_{N,L}^{\ast }{\mathbf{W}}_{N,L}{\mathbf{C}}_{N,L},{\mathbf{K}}_{N,L}={\mathbf{C}}_{N,L}^{\ast }{\mathbf{W}}_{N,L}{\mathbf{D}}_{L,N}{\mathbf{C}}_{N,L}.$$

Theorem 4.1 assumes a simple ${ϵ}_{N,L}$ and ${\mathbf{G}}_{N,L}\succ 0$. It obtains a Hermitian definite pencil $\mathbf{K}y=\mu \mathbf{G}y$, a self-adjoint realization in that metric and real generalized eigenvalues. Its scale invariance and contrast construction are existing source results to reuse, rather than a new project method. Shifting by the least eigenvalue does not prove that the unshifted arithmetic form is nonnegative.

Section 4.8 also defines an unshifted pencil using ${\mathbf{G}}^{\mathrm{un}}={\mathbf{C}}^{\ast }\mathbf{SC}$ and ${\mathbf{K}}^{\mathrm{un}}={\mathbf{C}}^{\ast }\mathbf{SDC}$. Its definite-pencil conclusion requires ${\mathbf{G}}^{\mathrm{un}}\succ 0$. It does not supply unconditional all-scale positivity of that metric. The section states that relating the shifted and unshifted low-energy limits needs relative estimates in the unshifted metric.

Reconstruction and the estimate still required

Section 1.4 describes an exact finite zero-side reconstruction from $N$ supplied distinct positive real ordinates. It identifies the resulting pencil spectrum with their signed ordinates. This theorem does not obtain those ordinates from unproved full arithmetic convergence, or exclude off-line zeros.

Sections 1.5 and 4.9 leave a whole-pencil relative perturbation theorem and uniform control of the compressed metric as the transfer problem. The positive zero-tail decomposition used in Section 4.9 assumes RH. Numerical agreement at selected parameters is not that transfer estimate. The paper explicitly claims no proof of RH.

The original theta variance metric is another concrete relative-estimate target. Its ground, measure, odd derivative space and metric $B_L$ are independently fixed by the original form. No parameter map identifying Shi’s contrast metric or least-eigenvalue shift with that $B_L$ is supplied here. Consequently the source gives neither the theta endpoint $c=1/2$ nor a cofinal sequence $c\uparrow 1/2$, RH, full Robin or Lean certification.

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