The claim
Manuscript §4: “translation and cross-modal reconstruction are often irreversible; their mappings need not be group actions. The more inclusive notion is a category or semigroup of stochastic transformations, within which invertible symmetry maps are special cases. We therefore separate (i) exact preservation, (ii) representation-changing equivariance, (iii) approximate preservation, and (iv) structural reorganization.”
What it does not say
That no symmetry applies: equivariance (TERM-004) is “often more apt than literal invariance” (§4). Nor does it supply a criterion for telling (ii) from (iv); CLAIM-004 was one.
A case from structuralist myth analysis
Santucci, Doja and Capocchi (LIT-807) say their myth variants form a group, but the operations they implement do not. Replacing one term with another that is already in the myth merges the two, and nothing can separate them again. The operations form a monoid, as this claim expects of transformations in general. This is the reader’s observation, not the paper’s.
Prior statement in Piaget
Piaget’s Structuralism (LIT-837, skimmed from the publisher’s preview, NOTE-618) makes the same distinction on p. 15. Logico-mathematical structures are fully reversible operations, which are groups. Linguistic and social transformations are “not entirely reversible”.
Note of 2026-10-10: the replies
CLAIM-260 scopes (iii). In total variation, with Markov kernels, approximate transports compose with ε₁ + ε₂. If TV(K₁#e_A, e_B) ≤ ε₁ and TV(K₂#e_B, e_C) ≤ ε₂, then
TV(K₂#K₁#e_A, e_C) ≤ TV(K₂#K₁#e_A, K₂#e_B) + TV(K₂#e_B, e_C) ≤ ε₁ + ε₂,
the first step by the triangle inequality and the second because every Markov kernel is nonexpansive in total variation. So in that setting approximate transports form a category in which distortions add. For other distortions, which may lack the triangle inequality or meet expansive kernels, the category claim waits on QUESTION-017. The manuscript’s text quoted above is unchanged.
Note of 2026-10-19: the warrant
The warrant is interpretive. The parts that can be proved are held by Active claims: CLAIM-260 for the category of kernels and the composition of approximate transports in total variation, and CLAIM-035 for irreversibility. This claim’s own content is a choice of model. Its defeat condition, that directed lossy transport “adds no predictive power over symmetry alone”, is a comparative null. As CLAIM-238 says of such conditions, it needs a margin fixed in advance before any study can bear on it.