= 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>.
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