research/algebra
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Essay
A monoid-aggregated summary evaluates a finite rooted cop-labeled tree bottom-up through a finite state set and a finite commutative child-aggregation monoid. Once the multiplicity observation map and the monoid are fixed, context equivalence has finite index and is exactly equality of a finite behavior vector. This note sharpens the resulting pumping and normal-form theory: the crude pigeonhole bound in the product monoid is replaced by an exact index–period bound on each behavior type’s child contribution, isolating support, modular, and saturation counting in the Boolean, cyclic, and threshold families. Combining exact sibling pumping with a size-minimality argument — no behavior vector may repeat along a root-to-leaf path — yields a finite universe of normal representatives, and an external tie-break selects one canonical representative per class. Worked computations for one-node trees, stars, unary chains, and split-versus-concentrated examples make the bounds concrete.