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CLAIM-113: Transformations and invariants define one another: translation preserves some structures exactly, carries some equivariantly, preserves some approximately and reorganizes others, and identity is studied through what a class of transformations leaves invariant

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Properties12
StatusActive
Version2
Rolethesis
Warrantinterpretive
Tagsphilosophy-of-language, mathematics
Linepragmatic-transport
Workswhat-survives-translation
Rests onCLAIM-040 — Symmetry is a kind of invariance, and Noether's theorem is the point: invariance under transformation yields conservation
SupportsCLAIM-108 — Communicative identity is characterized by the observational structures and relations invariant or equivariant under specified classes of transformation, and fidelity by their preservation, deformation or loss, CLAIM-017 — What survives translation is the invariants, and approximately preserved relations, of communicative transformation, CLAIM-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, CLAIM-147 — A trope is a family of observational profiles joined by admissible transformations around a core of preserved relations, so trope instances can separate structural preservation from pragmatic preservation
SupersedesCLAIM-015 — Classical structuralism studies relations within a system, whereas this theory makes the transport of relational systems its object and does not require structure to remain invariant
ComplementsCLAIM-148 — A matrix factorization is identified only up to an invertible change of latent coordinates, which weight decay narrows to rotations and nonnegativity to rescalings and permutations plus a data-dependent remainder; such a gauge is not a symmetry of the relational system, and automorphisms of the data act on the factors only when the factorization is identifiable
UsesTERM-004 — equivariance

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The claim

A105: “we should recognize that transformations and invariants define one another.” Given a class G of transformations, study Inv(G); given a structure F, study Aut(F), the transformations that preserve it. “An expression’s identity can be studied through the transformations under which its selected communicative structure remains invariant.” A more precise distinction “between transformations that preserve a structure, transformations that deform it, and transformations that preserve some structures while changing others”. A102 had already said “structure is not simply what remains invariant under transformation. Structure is also what makes transformations identifiable, comparable, and meaningful.”

Manuscript §4: “Structural transformations help identify structure through what they preserve … We therefore separate (i) exact preservation, (ii) representation-changing equivariance, (iii) approximate preservation, and (iv) structural reorganization.” §11: “The invariants of a specified transformation class identify what counts as continuity, while equivariant mappings make sense of changes of medium.”

What it does not say

That all of this is a group: the manuscript’s setting is a semigroup (CLAIM-098). The Aut(F) direction, studying identity from the structure’s side, did not reach the manuscript.

Prior statement in Piaget

Piaget, Structuralism, p. 11 (LIT-837, skimmed): “all known structures … are … systems of transformation”. The book makes transformation, not only relation, constitutive of structure.

Note of 2026-10-29: promoted

It rests on CLAIM-040’s first half, that symmetry is invariance under transformation: the owner pressed it at U37 against CLAIM-015, A105 conceded it, and this claim is CLAIM-015’s revision on that concession. It does not use CLAIM-040’s Noether half, which holds only in CLAIM-093’s limited form. Promoted on CLAIM-040 and TERM-004, both Active, with no objection filed. The Aut(F) direction it lacked is CLAIM-148’s. The semigroup setting of CLAIM-098 does not touch it: the invariants of a class of transformations are defined whether or not the class is a group. Its defeat condition is a universal over transformation classes, which by CLAIM-239 an argument can meet; none in the record does.

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