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CLAIM-318: Within a shared observational scenario, exact structure-preserving symmetries are the invertible members of a nested family of stochastic transports, and among Markov kernels between finite sets the invertible ones are the permutation matrices, so the invertible core is the relabellings, a groupoid in which objects joined by an arrow have isomorphic automorphism groups

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Properties8
StatusActive
Rolethesis
Warrantdemonstrated
Tagsmathematics
Linepragmatic-transport
SupportsCLAIM-103 — Symmetry is not opposed to transport: exact symmetries are the invertible core of a nested family of transports, and their main use is to supply the invariants against which non-symmetric transports are assessed
ComplementsCLAIM-098 — Translation and cross-modal reconstruction need not be group actions: they form a category or semigroup of directed stochastic transformations, in which invertible symmetries are special cases
Cited as “rests on” byCLAIM-249 — Comparing characters across realizations needs one group identified on both, which is fixed only up to automorphism, and automorphisms permute the irreducible characters, so "the same character" is relative to a chosen correspondence: the collapse problem again, at the level of the symmetry group

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The claim

This is the nesting CLAIM-103 states, with the qualification its note of 2026-10-10 adds, filed on its own so that its warrant can be. A110 §25: “Stochastic communicative transports ⊃ Structure-compatible transports ⊃ Exact structure-preserving symmetries. The last inclusion should be read within an appropriately shared observational scenario: exact symmetries are generally invertible automorphisms, whereas structure-compatible transports need not be.” CLAIM-098 holds the coarser form from the manuscript §4: “The more inclusive notion is a category or semigroup of stochastic transformations, within which invertible symmetry maps are special cases.”

The invertible core is small. A198 §9 puts the exact symmetries in “a larger category of communicative transports, whose invertible subcategory forms a groupoid”. For Markov kernels between finite sets, a stochastic matrix whose inverse is also stochastic is a permutation matrix, because a nonnegative matrix with a nonnegative inverse is monomial. So among stochastic maps on finite observation spaces the invertible core is the relabellings, and the only characters it carries are permutation characters, which count fixed points.

In that groupoid, objects joined by an arrow have isomorphic automorphism groups. An invertible u : A → B induces an isomorphism Aut(A) ≅ Aut(B), by g ↦ ugu⁻¹. Objects with non-isomorphic automorphism groups lie in different connected components of a groupoid and are joined by no arrow. A198 §8’s “Different objects can have different automorphism groups, while still being related by isomorphisms” is therefore false.

What it does not say

  • It does not say what symmetries are for. That their main use is to supply the invariants against which non-symmetric transports are assessed is CLAIM-103’s conclusion, and it is interpretive.
  • It does not define the middle term. “Structure-compatible transports” are placed between the two by A110 §25; this claim fixes only the innermost term and the outermost one.
  • It does not cover kernels on infinite spaces, where the derivation given here, by monomial matrices, does not apply as stated.

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