 Finite spectral matching by polynomial invariants

Suppose a compact parameter set carries a locally finite countable multiset of positive quantities, and a fixed injective transform of each labelled quantity is a polynomial in finitely many ambient coordinates. Assume only finitely many labels can have bounded value anywhere in the compact set. Then a uniform finite initial multiset determines the entire multiset. For each source label there are finitely many possible matching labels. Equality for finitely many labels in both directions is a finite union of algebraic sets, obtained from injective matchings and polynomial equalities. These algebraic sets decrease as more labels are included, so they stabilize by the Hilbert basis theorem. A compatible infinite matching in each direction follows by the König infinity lemma, and local finiteness makes multiplicities equal. Squared traces of hyperbolic holonomy representations give the application to the Buser finite length spectrum theorem.