nucleation

CLAIM-264: Two objects can agree on every probe in a restricted family without being isomorphic, so equivalence under a probe family is coarser than identity unless the family is rich enough

4 min read

Properties10
StatusActive
Version2
Rolethesis
Warrantdemonstrated
Tagsmathematics, epistemology, individuation
Lineformal-apparatus
GroundsTHEORY-032 — The Yoneda lemma determines an object only up to isomorphism, from its whole hom-functor, among objects the category already has; as a formal version of "an object is determined by its relations" it presupposes the relata, LIT-221 — Yoneda lemma
SupportsCLAIM-142 — Two communicative realizations can agree on every probe in a restricted family without being structurally identical, so observational equivalence is relative to the probes, and only independently justified probes evaluated on held-out cases are evidence that structure was preserved, CLAIM-190 — The anchoring measures are computed in the grammar that shapes the argument, so by themselves they cannot certify that drift fell: drift must also be judged under standards the record did not supply, CLAIM-187 — Whether two systems compute the same way can be tested by quotienting the full admissible symmetry group and measuring what remains: a residual barrier marks a different mechanism, and a feature with no counterpart blocks the comparison, CLAIM-322 — Agreement under a probe family is evidence that two objects share structure only to the degree that the family was chosen independently of the comparison and is evaluated on cases it was not chosen on, so a family selected to make two objects look alike shows only that it can be selected
ComplementsCLAIM-302 — Equivalence under feasible tests can be much coarser than statistical equivalence: there are ensembles almost disjoint from uniform that no polynomial-size test tells apart from it, and efficiently samplable ensembles of this kind exist if pseudorandom generators do
Cited as “rests on” byCLAIM-193 — A linear probe family can tell apart only realizations whose representations differ in normalized kernel, and kernels fix a representation only up to the symmetry of what is observed: any distinction beyond that is supplied by the analyst

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The claim

Restrict an object’s profile to a full subcategory 𝒬 of probes, A ↦ Hom(−, A)|_𝒬. Three things can be asked of this functor, and they come apart.

  • It is fully faithful exactly when 𝒬 is dense, as the one-point set is in Set. Then objects with isomorphic profiles are isomorphic, and every map of profiles comes from a unique morphism.
  • In a category with finite limits, it is conservative exactly when 𝒬 is a strong generator in Kelly’s sense, as ℤ is in groups. Conservative means a morphism f: A → B is an isomorphism when every Hom(Q, f) is a bijection. It says nothing about two objects whose profiles are isomorphic by a map that no morphism induces. Through ℤ, a group’s profile is its underlying set with the power maps g ↦ gⁿ. For p an odd prime, (ℤ/p)³ and the Heisenberg group of 3 × 3 upper unitriangular matrices over ℤ/p both have order p³ and exponent p. Choosing a generator of each of their (p³ − 1)/(p − 1) subgroups of order p and sending gᵏ to hᵏ gives a bijection that commutes with every power map, so the profiles are isomorphic, and one group is abelian and the other is not.
  • When 𝒬 is not a strong generator in such a category, the functor is not conservative: some morphism that is not an isomorphism induces a bijection on every probe.

Identifying objects up to isomorphism does not need density. In finite-dimensional vector spaces over a finite field, the field alone is not dense, yet its profile, the underlying set with scalar multiplication, has qⁿ elements and so fixes the dimension. Whether a given family is rich enough is a question about the family and the class of objects, which is why the claim says only that agreement can fall short of isomorphism. THEORY-032 adds three conditions from the nLab reading (LIT-221). The identifying profile is the hom-functor with its action by composition, not a table of hom-sets. The lemma needs identities. And it presupposes the objects it identifies.

The same holds for distributions read through probes. (φ_i)#P = (φ_i)#Q for every i does not in general imply P = Q, since matching each probe’s marginal leaves the joint across probes unconstrained. A rich enough family does separate: all one-dimensional linear projections determine a distribution (Cramér–Wold). For finite relational structures, homomorphism counts from every finite structure determine isomorphism (Lovász).

The rule for reading agreement under a probe family as evidence of shared structure is CLAIM-322’s, split from this claim on 2026-10-15.

Where the record applies it

CLAIM-142 applies it to communicative realizations, with a realist reading of communicative identity (CLAIM-046) and the remedy of anchored correspondences (CLAIM-011). CLAIM-190 applies it to the anchoring measures of a record. CLAIM-193 and CLAIM-187 apply its relativity to learned representations. Each keeps its own premises for the application.

What it does not say

  • It does not say any particular domain has a category whose objects the probes approximate. That is a premise of each application.
  • It does not say which probes to choose, or when a family is rich enough for a task (QUESTION-031).
  • It says nothing about contextuality: being representable and having a global section are independent.

Note of 2026-10-15: a restriction forced by feasibility

A15 of the exchange of 2026-10-14 (workbench entry of 2026-10-15) recommended amending this claim with computationally bounded probes. The result is filed apart, as CLAIM-302, because it is conditional on an assumption this claim does not need: if pseudorandom generators exist, there are distributions almost disjoint in total variation that no efficient test tells apart. “Computational indistinguishability gives it a non-arbitrary source of restricted observational families: feasibility.” This claim’s restricted families are chosen; CLAIM-302’s is imposed by the observer’s resources.

It bears on TERM-037, an observable as “a reproducible elicitation or coding procedure”. A15: “For computational observers, the procedure includes a computation.” The term is not revised. Equivalence under feasible probes is relative to the encoding and to any auxiliary information, as this claim’s evidential half would predict. The status is unchanged.

Note of 2026-10-15: split under ADR-043

ADR-043’s full classification found this claim compound: its exact half (the restricted Yoneda functor, Cramér–Wold, marginals) is demonstrated, and its evidential half, that only probes chosen independently of the comparison and evaluated on held-out cases are evidence that structure was preserved, is interpretive. The evidential half is now CLAIM-322 (interpretive), which rests on this claim. This claim keeps the exact half and stays demonstrated. Its title, which stated the evidential half, is shortened to the exact half, and the second disjunct of its defeat condition, about chosen probes, moves to CLAIM-322.

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