Finite Li-Statistic Collisions for Formal Symmetric Zero Multisets

Abstract

We prove three main theorems about the standard Li-type observation of finite symmetric zero multisets, record two main corollaries, and prove complementary arithmetic-rigidity results for algebraic Gaussian and compactly supported Weil-type observations. Let $\mathcal{H}_{\mathrm{sym}}$ be the class of finite multisets in the open critical strip symmetric under $\rho \mapsto \bar{\rho}$ and $\rho \mapsto 1-\rho$, and let $P(\mathcal{Z})=1$ iff every atom has $\operatorname{Re} \rho = 1/2$. Theorem (exact Li collision): for every $m \geq 1$ there exist $\mathcal{Z}_+, \mathcal{Z}_- \in \mathcal{H}_{\mathrm{sym}}$ with $P(\mathcal{Z}_+)=1$, $P(\mathcal{Z}_-)=0$, and identical standard Li observations $\operatorname{Li}_j(\mathcal{Z}_+) = \operatorname{Li}_j(\mathcal{Z}_-)$ for $j = 1, \ldots, m$, realised by explicit rational-arithmetic witnesses. Theorem (no uniform separation): no fixed finite initial segment of standard Li values enforces a strictly positive gap between $P=1$ and $P=0$ within $\mathcal{H}_{\mathrm{sym}}$. Theorem (incomparability): the standard Li map $\mathcal{O}_{\mathrm{finite}}^{(K)}$ and the sampled counting map $\mathcal{O}_{\theta}$ are incomparable in the information refinement preorder on $\mathcal{H}_{\mathrm{sym}}$. Theorem (Gaussian Weil rigidity): for a real algebraic even Hermite-Gaussian test function and nondegenerate algebraic on-line/off-line data, no nonzero integer multiplicities balance an off-line quartet against on-line pairs. The exact-collision phenomenon is therefore sensitive to the arithmetic of the observation encoding: rational Li values admit exact collisions, while these algebraic-exponential values do not. The compact-support case splits sharply. Odd algebraic B-splines give compactly supported, finite-smoothness Paley--Wiener tests with the same exact no-collision property under an explicit whole-exponent-set nondegeneracy condition; arbitrary real even $C_c^\infty$ tests include algebraic finite-node annihilators; and, more strongly, for every nondegenerate on-line/off-line geometry there is a real even $C_c^\infty$ test whose two observations are both exactly $2$, giving the integer collision $(1,1)$. Thus nonvanishing alone is not a compact-support rigidity criterion; the following restricted problem is left unresolved by the present paper. Corollary (Andersson): agreement with any fixed finite initial collection of Euler factors of $\zeta$ does not force a Helson zeta function's zeros to lie on the critical line (existential obstruction pair, using Andersson \cite{Andersson2024} Theorem 5 and Corollary 3). Corollary (joint obstruction): the combined observation $\mathcal{O}_{\mathrm{finite}}^{(K)} \times \mathcal{O}_{\theta}$ also admits an exact information obstruction. None of these results assume or imply the Riemann Hypothesis. The class $\mathcal{H}_{\mathrm{sym}}$ consists of finite formal multisets; the actual zero set of $\zeta$ is infinite and subject to additional analytic structure not imposed on $\mathcal{H}_{\mathrm{sym}}$. A consequence is that $\mathcal{O}_{\theta}$ alone does not determine $P$ (see Remark ).