The claim
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. Thus symmetry is not opposed to transport. Symmetry identifies special transport operations and, more importantly, the invariants against which other transports can be assessed.”
Where it went
Manuscript §4: “The more inclusive notion is a category or semigroup of stochastic transformations, within which invertible symmetry maps are special cases.” The second half, that symmetry gives the measuring stick for transports that are not symmetries, is the bridge the manuscript lacks between its §4 symmetry and its §6 functional (TERM-036). It is the owner’s U37 point carried through (CLAIM-113).
Note of 2026-10-10: restored, and the invertible core of stochastic maps
The nesting was restored after the manuscript dropped it: at A178 (“Exact symmetries as special transport cases”), at A203 §20, which displays the chain exact symmetry ⊂ structure-preserving transport ⊂ general directed stochastic transport, and at A218 Ch11.10 (“Exact equivalences as a special case of a broader transport theory”). A203 §20 also keeps the yardstick role: “symmetries identify invariants against which more general transformations can be evaluated.”
A qualification, from A198 §§8–9 and checked by reasoning. 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 that subcategory is small. 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. The nesting stands. Its innermost term is narrower than A198 suggests.
A198 §8 also says: “Different objects can have different automorphism groups, while still being related by isomorphisms.” That is false. 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.
Note of 2026-10-15: split under ADR-043
ADR-043’s full classification found this claim compound: the nesting of exact symmetries inside the stochastic transports, and its narrow invertible core (a stochastic matrix with a stochastic inverse is a permutation matrix), are demonstrated, while the main use this claim gives symmetry, supplying the invariants against which non-symmetric transports are assessed, is interpretive, and it is what the defeat condition tests. The nesting, the invertible core and the correction of A198 §8 in the note of 2026-10-10 are now CLAIM-318 (demonstrated), on which this claim rests. This claim keeps the yardstick role and the reading that symmetry is not opposed to transport. Its title states the nesting as the premise of that reading and is kept; its warrant stays interpretive.
Note of 2026-10-30: what holds it back
“Their main use” ranks what symmetries are for, and nothing in the record argues for that ranking beyond A110’s own words. The test the defeat condition names, whether symmetry invariants sort lossy transports as judges rank them, has not been run. CLAIM-249 (Active) also shows the invariants can be compared across realizations only after the group is identified across them by an anchored correspondence, which the claim’s “shared observational scenario” assumes and a translation between media does not supply.