Found on 2026-10-10 by a search the owner asked for: a genuine symmetry, acting on the situation and not on labels, with anchored identification on both sides, and one realization the symmetry changes and one it does not, for CLAIM-137 and CLAIM-249. CLAIM-132’s worked model of the swap was built on CASE-004, a stipulated scene. This case is stipulated on a documented basis. Part A is documented. Parts B and C are constructed by the record, and they say so.
The case
A. Documented: reciprocal and nonreciprocal address. Roger Brown and Albert Gilman, “The Pronouns of Power and Solidarity”, in T. A. Sebeok (ed.), Style in Language (MIT Press, 1960), pp. 253–276, read in the transcription at https://pages.mtu.edu/~rlstrick/rsvtxt/pronoun.htm:
- p. 255: “The power semantic is similarly nonreciprocal; the superior says T and receives V.”
- p. 258: “Solidarity is the name we give to the general relationship and solidarity is symmetrical. The corresponding norms of address are symmetrical or reciprocal”.
- p. 275: the T of contempt “can, of course, also be used by a subordinate to a superior. As the social distance is greater, the overthrow of the norm is more shocking”. Their example is Mosca, who says “you” to Volpone until he turns on him with “Bane to thy wolfish nature.”
Let G = C₂ swap the two persons of a dyad. Under the solidarity semantic, address is mutual T or mutual V, and the swap leaves the configuration fixed: its stabilizer is C₂. Under the power semantic, the superior says T and the subordinate V. The swap gives the subordinate saying T to the superior, which Brown and Gilman record as the norm’s “overthrow”, a different act; the stabilizer is trivial. Radcliffe-Brown draws the same line for joking relationships, “symmetrical” (each teases the other) against “asymmetrical” (A jokes at B’s expense and B does not retaliate), in Structure and Function in Primitive Society (Cohen & West, 1952), ch. IV, p. 90.
B. Constructed: one dyad, two realizations. Take a French dialogue in which a master says tu to a servant and the servant says vous. Render it into modern English, where both say “you” (Brown and Gilman, p. 253: “Most of us in speaking and writing English use only one pronoun of address”). Anchor the swap on the persons in both. In French, the swap moves the address configuration. In the English rendering, it leaves the pronoun layer fixed, unless the asymmetry is carried elsewhere (titles, “sir”). So one realization is changed by the swap on this layer and the other is not. No attested French text and English translation were checked for this: Molière’s Dom Juan, with Dom Juan’s tu to Sganarelle, is a lead the search did not verify.
C. Constructed: a teasing round and its mirror image. Three friends a, b and c sit at a round table. In one round each teases the person on their left in turn: a→b, then b→c, then c→a. Let G = C₃ act on the persons by r: a ↦ b ↦ c ↦ a.
- r fixes the round’s configuration {a→b, b→c, c→a}, so the act is unchanged. It moves a one-way reprimand {a→b} to {b→c}, so that act is changed. The contrast of part A recurs within one medium.
- The round also has an order in time. On the clockwise round, r carries each turn to the next one (a→b to b→c). On the anticlockwise round, where each teases the person on their right (a→c, then c→b, then b→a), r carries each turn to the one before (a→c to b→a). Read through one fixed character of the cycle of turns, the clockwise round gives χ₁(r^k) = ω^k and the anticlockwise one χ₂(r^k) = ω^(2k), with ω = e^(2πi/3). These are CLAIM-249’s two nontrivial characters.
- Draw the clockwise round with an arrow from each teaser to the teased, and print the drawing mirror-reversed. If G is identified by a label of the medium, “rotate clockwise on the page”, the mirror turns that rotation into r⁻¹ = r², and the drawing is read with χ₁∘α = χ₂, where α(g) = g²: it appears to show the other round. If G is identified by the persons (faces, names), the drawing still shows a teasing b, and the character is χ₁.
A near-miss, documented. Malinowski, Argonauts of the Western Pacific (Routledge, 1922), ch. III §I (Project Gutenberg #55822): on the Kula circuit, “In the direction of the hands of a clock, moves constantly one of these kinds—long necklaces of red shell, called soulava … In the opposite direction moves the other kind—bracelets of white shell called mwali.” Reversing the orientation exchanges two different kinds of transaction, and the participants anchor the orientation in geography, not in a label. But rotating the ring is not a symmetry of the situation: the islands are different communities, so it is a symmetry of the bare cycle only. And Kula gifts are exchanges, not speech. It is the “labels, not situation” warning in documented form.
What it can show
That CLAIM-132’s distinction of stabilizers, the one-way reprimand moved by the swap and the mutual configuration fixed, is not an artifact of CASE-004’s stipulated scene. It matches a documented, cross-linguistic distinction of address norms, and competent judgement attaches to it (“the overthrow of the norm is more shocking”). For CLAIM-137 that is support at the level CLAIM-132 allows. The invariant here is a feature of the G-set, its stabilizer, and for C₂ the permutation character detects it: a C₂-set with f fixed points and m free orbits has character (f + 2m, f), which determines f and m.
Part B shows a rendering that identifies the power dyad with the solidarity dyad on one layer: a collapse in CLAIM-011’s sense, where a distinction the source kept apart is not kept apart by the target.
Part C gives CLAIM-249 a group on which its point can bite. C₂ has no nontrivial automorphism, so in parts A and B anchoring cannot fail. C₃ has one, g ↦ g², and it swaps χ₁ and χ₂. Anchored on the persons, only the identity survives, which is the reply in CLAIM-137’s note of 2026-10-10; anchored on a label of the medium, the mirror image reads as the other round.
What it cannot show
- The cross-realization condition is met only by construction. Part A documents two kinds of address, one changed by the swap and one fixed. It does not document one act realized twice, once in a medium the swap changes and once in a medium it fixes. Part B builds that, and no attested pair of texts stands behind it.
- The symmetry acts on possible configurations, not on any one situation. A master and a servant are not interchangeable in fact. The swap acts on the space of address configurations, as in CLAIM-132, and the “swapped” dyad is a counterfactual that Brown and Gilman attest only as a breach of the norm.
- A stabilizer is not a character. The case shows that the act’s identity follows the stabilizer. It does not show that a character, as opposed to a stabilizer, carries it. For C₂ the two coincide; in general they do not, as CLAIM-132’s permutation characters show.
- The strongest reply. A critic pressing CLAIM-138’s Newman point would say that the anchoring “on the persons” is where all the work is done, and that it is supplied by the analyst, not by any invariant: the character tells the rounds apart only after the persons have been identified across the media. That is CLAIM-249’s own conclusion, and part C shows it rather than answering it.
- Part C’s characters depend on a choice. Which round “carries ω” rests on a fixed character of the cycle of turns, so on time order being anchored in both realizations. A drawing has no time; its arrows stand in for it.
- What would close the gap. An attested source with nonreciprocal T/V address and an English translation that flattens it, such as a checked edition of Dom Juan with a published English version, read for whether the asymmetry survives in titles or elsewhere. Then a reader test, sketched here because the search report gives none, which could serve as items for the discriminating study (CASE-055): give readers of the source and of the rendering the original exchange and the swapped one, and ask “Is this the same kind of exchange, with the roles reversed, or a different kind?”. The construction predicts that source readers call the swapped power dyad a different act and the swapped solidarity dyad the same one, and that readers of the flattened rendering cannot tell the power dyad from the solidarity dyad on the pronouns.
Sources
None of these works is held in the record as a LIT. They are cited here by identifier. None is a machine-learning work.
- Brown, Roger and Albert Gilman (1960), “The Pronouns of Power and Solidarity”, in T. A. Sebeok (ed.), Style in Language (MIT Press), pp. 253–276; no DOI. Read directly in the transcription at https://pages.mtu.edu/~rlstrick/rsvtxt/pronoun.htm. Page numbers are inferred from its bracketed markers, which close each page: the article’s opening falls before the marker [253]. On that reading the quotations fall on pp. 253, 255, 258 and 275. The search report read the markers the other way and gave pp. 254, 257 and 274 for the last three. To be checked against the printed volume.
- Radcliffe-Brown, A. R. (1952), Structure and Function in Primitive Society (London: Cohen & West), ch. IV, reprinting “On Joking Relationships”, Africa 13(3), 1940, doi:10.2307/1156093: https://archive.org/details/1952-radcliffe-brown-1952-structure-and-function-in-primitive-society. Read directly in the OCR text; the page number is the search report’s.
- Malinowski, Bronisław (1922), Argonauts of the Western Pacific (London: Routledge), Project Gutenberg #55822. Read directly, ch. III §I. The search report gives p. 81 of the 1922 edition; unverified.
- Molière, Dom Juan (1665). Not read; a lead only.