Exact Finite-Observation Collisions for Symmetric Zero Multisets

Abstract

We prove an exact non-discrimination theorem for finite Li-type observations of functional-equation symmetric zero multisets. For every prescribed finite order $m$, we construct two finite multisets in the open critical strip, invariant under complex conjugation and $s \mapsto 1-s$, such that one multiset is supported on the critical line, the other contains an off-critical-line quartet, and the first $m$ Li-type observation coordinates agree exactly. The construction is arithmetic: the interpolation matrix is rational and has an explicit Chebyshev--Vandermonde determinant, while a rational off-line quartet vector is cancelled by integer multiplicity changes. We also show that the collision survives after adjoining a common unconditional background with a Riemann--von Mangoldt-type counting law. The result is a theorem about finite observables on a broad symmetric class; it does not assert that the same finite data exhaust the Euler-product and coefficient structure of the Riemann zeta function.