QUESTION-027: Is there a relational signature of a communicative kind that is stable under admissible transformations, sufficient for a family of decisions and discriminating on designated contrasts, and can it be identified from the available probes, for decisions outside them, when the full latent model cannot?

Why it is a question

The question came in two halves, from two stretches of the exchange, and A214 joined them.

Stability, sufficiency and discrimination. A198 §X proposed to “identify minimal, transformation-stable, decision-sufficient relational signatures of communicative kinds”. A203 §36, headed “The most important unresolved mathematical problem”, asks for a signature χ_A of a kind A, given a family of realizations, a set of transformations 𝒯 and a family of decision tasks 𝒬, with three properties: transformation stability, decision sufficiency and discriminating power. Its box reads, transcribed from the display: “Find a relational signature χ that is maximally stable under 𝒯, sufficient for decisions in 𝒬, and discriminating on designated structural contrasts.” It uses “character” literally only “when χ is a trace character of a specified representation”.

Sufficiency without identification. After the factorization turns (U57–U59), A214 §7 separated identifiability from sufficiency: “A perfectly identifiable signature can be useless for a particular decision. A nonidentifiable latent model can nevertheless support accurate predictions when the unidentifiable distinctions do not matter for those decisions.” So “decision-sufficient structure may be recoverable even when the full underlying model is not identifiable.” A214 §11 put the four properties together, “Identifiable”, “Transformation-stable”, “Decision-sufficient” and “Empirically grounded”, and boxed the question: “Can task-sufficient structural information be identifiable across representations even when the underlying latent factorization is not?”

A218 kept it as RQ3, “When can decision-relevant structural information be identified even when a richer latent decomposition is not uniquely recoverable?”, and as its priority theorem target: “identifiability of task-sufficient structure without identifiability of the full latent model”. Its substantive form there is “to characterize when particular relational signatures satisfy it under sparse, multi-view, context-indexed observation, and whether their sufficiency survives admissible stochastic transport.” A207’s boxed question and A210 §7’s rewrite of it (“remain identifiable, decision-sufficient, and structurally stable across translations, observational regimes, and causal interventions”) are earlier forms.

What would count as an answer: for a stated class of transformations and a stated family of decisions outside the observed probes, a characterization of the signatures that are both decision-sufficient and identified from the probes, or a proof that no such signature exists when the latent model is not identified.

What is already known

Much of the unrestricted question is answered.

• Gauge-invariant functionals are identified. Any function of the factors that is invariant under the gauge is a function of the predicted matrix WH, so it is identified whenever the reconstruction is (CLAIM-148). That covers every predicted entry, the rank and the singular values.
• Matrix completion. Under incoherence and enough random samples, every missing entry is identified with no identifiability of the factors (Candès and Recht, doi:10.1007/s10208-009-9045-5; not held in either record). The task “predict any cell” is answered.
• Partial identification. In econometrics and causal inference, a target functional is often identified without the full model. The causal-effect identification that A214 itself invokes is an instance.
• A218’s theorem target is a definition. A signature s is identified from a probe system 𝒪 exactly when θ ∼_𝒪 θ′ implies s(θ) = s(θ′), that is, when s is constant on the observational equivalence classes. A218 says so itself: “That basic equivalence is mathematical bookkeeping, not a novel theorem.”
• In-probe decisions are automatic. When a decision’s risk depends on the model only through the law of the observed probes, P_θ(Y_𝒪), every optimal rule is constant on the equivalence classes, so a decision-sufficient signature is identified by construction. Predicting held-out ratings in collaborative filtering is the standard case. A218’s hypothesis H4, “some decision-sufficient signatures will remain usable despite ambiguity in richer latent models”, cannot fail for such decisions.

So the question has content only for decisions that reach outside the fitted relations: new probes, interventions, or a target system after transport. There the question is when the optimal decision is constant on the set of models consistent with the data, and whether a transport preserves that set. A214’s “a worthwhile mathematical research question in its own right” overstates the novelty of the unrestricted version.

What it is not

• It is not well posed as A203 states it. Maximal stability favours coarse signatures, and sufficiency and discrimination favour fine ones. “Maximally stable” subject to the other two needs a trade-off parameter, and A203 gives none. The problem has the shape of an information bottleneck (CLAIM-092, LIT-338) with an added invariance constraint. A214 §11’s boxed optimization, transcribed from the display, “minimizes relevant information loss under admissible transformations, subject to identifiability, discrimination, and held-out decision-performance constraints”, names no objective, rate or constraint set either.
• It is not CLAIM-125. It is the factorization-side analogue of CLAIM-125’s first open part. Both ask when decision-relevant information survives a map whose finer structure is not determined. CLAIM-125 asks it of transport between empirical models whose covers change. This question asks it of a latent model the probes do not identify.
• It is not one sense of “anchor”. Three senses meet here and must not be conflated. A214 §2’s “pragmatic anchors”, carried into A218 Ch6.4 as “Pragmatic analogues of anchor variables”, are observables selectively associated with one kind, by analogy with anchor words. CLAIM-011’s anchoring estimates the source–target correspondence on independently measured distinctions. THEORY-175’s anchors are matched reference points whose cosines define a representation.
• It is not the question of whether the signature is the kind. An identified, decision-sufficient signature is still at the second of CLAIM-135’s three steps.

Note of 2026-10-11: material in the record that may bear on this

A synthesis review of 2026-10-11 found these entries bearing on this question without being linked to it. Each is listed for what its title says; none has yet been assessed as an answer, and none is linked by answers.

• CASE-042: Collaborative filtering and topic models: latent relational structure recovered from interaction matrices,…
• CLAIM-137: A communicative kind is identified by an invariant that does not depend on any one realization, as a group…
• TERM-041: character-like structural signature, as a realization- independent invariant of a communicative kind,…
• CLAIM-132: Characters classify finite-dimensional representations of a finite or compact group up to isomorphism only…
• THEORY-191: In natural odour sources, the rank order of correlations between compound concentrations has the…
• THEORY-192: Pairwise correlations among rat CA1 place cells have the Betti-curve signature of points sampled uniformly…
• THEORY-112: Once the architecture’s symmetries are factored out, most of the linear barrier between independently…
• THEORY-116: A linear barrier between two minima that survives the removal of symmetries marks a difference in the…

Note of 2026-10-12: an answer and an assessment filed on 2026-10-10

The note of 2026-10-11 says none of the listed entries is linked by answers. That holds of the listed entries only. PR #121 filed CLAIM-209 as a proposed answer to this question, and the front matter carries it as answered_by. CLAIM-249, granted and Active, assesses the character route of CLAIM-137, TERM-041 and CLAIM-132: “the same character” is relative to a chosen correspondence. CLAIM-226 cites CASE-042 as a ground.

Note of 2026-10-14: two readings that bear on the unrestricted form only

LIT-950 and LIT-936, filed for this question, were read in full. Each bears on the unrestricted, in-probe form only, the form “What is already known” calls answered, and not on decisions outside the observed probes.

• LIT-950 (NOTE-746) proves which linear properties of a next-token predictor are shared by every model inducing the same distribution (Theorems 5 and 14), under its stated conditions and as sufficient conditions. The observation there is the whole conditional distribution on every sequence, so there are no unobserved probes. It is a proved instance of a signature identified when the latent model is not, in the case the question already counts as automatic. The account is filed as THEORY-234.
• LIT-936 (NOTE-755) measures a relational geometry that is invariant under every output-preserving reparameterisation and is identified from a context battery. Both hold because the signature is a function of the observed laws. Its two results that reach outside the probes, a transfer probe across models and a risk relation applied to unseen model families, are empirical and narrow, and neither is a theorem. The account is filed as THEORY-230.