The objection
CLAIM-137 makes the spoken, translated and visual realizations of a teasing remark “like one representation written in different coordinates”, with a shared character-like invariant as what makes them one act. A character is a function on a group. To compare χ_A on a spoken realization with χ_B on a visual one, the same group element g has to be named on both sides: one group G acts on both media, and some transformation of the image is identified as the transformation g of the speech.
The identification is fixed only up to automorphism. Change it by an automorphism α of G, and the character read off becomes χ∘α. Inner automorphisms change nothing, because a character is a class function (CLAIM-132, item 1). Other automorphisms can move characters.
A worked case. Let G = {1, g, g²} be the cyclic group of order 3, and ω = e^(2πi/3). Its two nontrivial irreducible characters are χ₁(g^k) = ω^k and χ₂(g^k) = ω^(2k). The map α(g) = g² is an automorphism. G is abelian, so its only inner automorphism is the identity, and α is not inner. Then
χ₁(α(g^k)) = χ₁(g^(2k)) = ω^(2k) = χ₂(g
So α swaps χ₁ and χ₂. A realization identified with G one way “has” χ₁, and identified the other way “has” χ₂. The realization does not fix which. The same happens in any group with an automorphism that moves a character: in an abelian group, inversion sends each character χ to its complex conjugate, which differs from χ whenever χ takes a non-real value.
Which group? The transformations the line studies are not group elements. CLAIM-098: “The more inclusive notion is a category or semigroup of stochastic transformations, within which invertible symmetry maps are special cases.” CLAIM-103’s note adds that “A stochastic matrix whose inverse is also stochastic is a permutation matrix” (now held as CLAIM-318), so among stochastic maps on finite observation spaces the invertible core is the relabellings. So G has to be a symmetry group of the situation that the analyst chooses, such as CLAIM-132’s swap of the two participants, with its action on each medium specified independently. That is CLAIM-011’s requirement of an anchored correspondence, and CLAIM-138’s Newman point, once more, now at the level of the symmetry group.
For semigroups, traces say less. For representations of a monoid, traces fix at most the semisimplification. CLAIM-132’s example, read as an action of the monoid ℕ, shows it: n ↦ [[1, n], [0, 1]] and the trivial action on ℂ² both have trace 2 for every n, and they are not isomorphic.
What the record holds nearby. CLAIM-132’s title says characters “fix neither the group nor a permutation action”, and its approximate bound “needs an invertible S between representations of one group, which is not the translation case”. TERM-041 says the signature is “Not yet built”. Neither names the identification of G across realizations, or the automorphism ambiguity of that identification.
What it does not say
- It does not say the character analogy is useless. Within one medium, or once the identification of G across media is fixed independently, characters are a complete invariant in characteristic zero (CLAIM-132).
- It does not say the ambiguity always bites. The cyclic group of order 2 has no nontrivial automorphism, so CLAIM-132’s worked model of the swap is unaffected. The ambiguity bites for groups with automorphisms that move characters.
- It complements CLAIM-138, which makes the same point for relations in general.
Note of 2026-10-27: the edge says what the claim uses
The status review of this date (ADR-057, ADR-058) re-pointed rests_on from CLAIM-098 to CLAIM-318 (Active, demonstrated): “among Markov kernels between finite sets the invertible ones are the permutation matrices”. The body already says the point is “now held as CLAIM-318”. CLAIM-098’s contested part, that directed lossy transport is the better model, is not used.