Why it is a question
A173 put it as a secondary question: “When do restricted probes separate communicative realizations?” A178 made it RQ3: “When does a selected family of observational probes distinguish relevant communicative realizations, and when do restricted probing profiles collapse distinct structures?” A214 §3 asked it again: “How rich must the observational relationships be to separate the communicative kinds we care about?”
It is open because CLAIM-142 makes observational equivalence (TERM-014) relative to the probe family, and says nothing about which family suffices. CLAIM-077 requires a transport to keep specified distinctions apart, and this question asks when the probes can even see them. CLAIM-050 makes fidelity relative to the receiver’s decisions, so “the realizations a task needs told apart” is fixed by a family of decisions, not by full structural identity.
An answer would be a condition on a probe family, checkable before the data are seen, under which two realizations that agree on every probe also agree on every decision in the stated family. A demonstration that some family separates some pair is not an answer.
What is already known
- The discrete case. For finite relational structures, homomorphism counts from every finite structure determine isomorphism (Lovász 1967, noted in THEORY-032 and NOTE-195). Restricted families give exact restricted answers: counts from trees characterize colour refinement, and counts from graphs of bounded treewidth characterize the k-dimensional Weisfeiler–Leman test. That last pair of results is given here from memory, without an identifier, and has not been checked.
- The categorical case. The restricted Yoneda functor along a family of probes is fully faithful exactly when the family is dense (CLAIM-142). A probe table without composition is not a Yoneda profile at all.
- The enriched case. For Lawvere metric spaces the Yoneda embedding is an isometry (Lawvere 1973), so “distance between profiles equals distance between objects” already holds there. Bradley, Terilla and Vlassopoulos model expressions by [0,1]-enriched copresheaves of continuation probabilities (arXiv:2106.07890, not held), which is prior art for a probabilistic version. A158 §6 proposed “an enriched or probabilistic analogue of Yoneda” without knowing of either.
What stays open
The stochastic, decision-relative version. Probes return distributions, not hom-sets; the families are finite and fixed by an experimenter; and the target is not isomorphism but sameness for a stated family of decisions, an order in Blackwell’s sense. No result the record holds says when such a family separates what the decisions need. A158’s probe-weighted distortion, a weighted sum of per-probe distances, is not a Yoneda construction: it ignores composition. QUESTION-022 asks the neighbouring question for transports: which stochastic maps carry compatible empirical models to compatible ones.
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.
- THEORY-175: Cosine similarities to matched anchor points are invariant exactly to the maps that preserve cosines, and…
- CLAIM-129: A factorization recovers only the structure visible in the relations it is fitted to, so a topic can…
- CLAIM-149: A probe family that supplies a transformation’s training signal cannot also certify what the…
- THEORY-019: Symmetry forces spectral degeneracy but degeneracy does not identify a symmetry: an operator commuting…
- THEORY-008: What a regularized linear readout can decode from a representation is a function of its normalized kernel,…
THEORY-008 gives half of an answer for linear probes: a regularized linear readout can decode only what is a function of the normalized kernel.
Note of 2026-10-25: choosing probes
From A29–A31 of the exchange of 2026-10-24, continued. Bayesian experimental design (Chaloner and Verdinelli 1995, LIT-1525, read in NOTE-1161) turns this question into a choice: pick the design ξ over contexts that maximizes a criterion of G_ξ = Σ_c ξ(c)G_c (log det, or a discrimination or prediction criterion fixed by the task). A29: “our objectives are often more specific. We might care about distinguishing the effect of authority from the effect of legibility, rather than estimating every parameter equally well.” In CASE-152, do(b = b₀) gives G_aa = b₀² > 0 (checked). In CASE-159, a rare context makes authority shares identifiable that routine probes leave invisible. Both separate by changing the experiment, not the coordinates (CLAIM-305). A caution from A29, with CLAIM-130: changing a participant’s standing to separate explanations “could also change which communicative act is performed”.
The reading places these. Σ_c ξ(c)G_c with log det is Bayes D-optimality on the expected Fisher information (THEORY-956 for the linear case, THEORY-954 for the nonlinear one). A29’s “distinguish authority from legibility” is c- or Ds-optimality, or Spezzaferri’s discrimination criterion (NOTE-1161, §8.3). The review gives no separation condition: it ranks probe families by a utility, and so it does not answer this question.
One decision family does have an answer. In a finite partial-monitoring game, two outcome laws that agree on the signals of every probe in a set D also agree on the comparison of actions i and j for every such pair iff ℓ_i − ℓ_j lies in the span of those probes’ signal row spaces (THEORY-962, from Bartók et al.). That condition can be checked before any data. It covers pairwise comparisons only, with deterministic or known-law signals, so the question stays Open. The question is unchanged.
Split, 2026-10-25 (ADR-049)
The linear case is QUESTION-272, Answered. When the task’s target and the probes’ outcome laws are linear in one state, a probe family separates what the task needs exactly when the target lies in the span of what the probes reveal: for pairwise comparisons of actions, the loss difference in the probes’ combined signal row space (THEORY-962); for the authority shares of a mixture of known profiles, affinely independent profiles over the probe contexts (CLAIM-452). This question keeps the rest: targets or outcome laws that are not linear in a common state, profiles that must themselves be estimated, and probes whose availability the organization’s own choices decide.
Note of 2026-10-26: a nonlinear case, with a known catalog
CASE-163 is a case of the remainder: nonlinear production laws, noisy readouts, and a catalog of four known mechanisms (dmarx/PerspectiveLab@9f82bfd). For a fixed candidate family the condition is the ordinary factorization one: a probe family d separates what a target intervention a needs exactly when any two candidates with the same law under d have the same law under a.
The case shows the collapse the question names. A complete audit pulse leaves self-dependent production and feedback only alike, so class accuracy is 75.0%, against 82.6% for a milder pulse. A short producer interruption added to it separates all four (100%). For the parameters of self-dependent production at nominal values, an audit pulse gives a local observation Jacobian of rank 2 for three parameters, and so does a pulse with an audit-producer interruption. Interrupting both producer teams as well gives rank 3.
All of this is local, the catalog is given, and each probe runs on a reset copy of the organization, so which probes are available is not the organization’s choice. The case is not linear in one state, so it is not QUESTION-272’s. The question is unchanged and stays Open.
The lab’s v2.4 and v2.5 (dmarx/PerspectiveLab@f392084 experiments/adaptive_interventions/ and adequacy_guard/) drop the reset: probes are chosen online, within a cost ceiling, on one organization they change. v2.5 adds alarms for when no candidate fits, scored prequentially and calibrated on 160 in-catalog episodes for a nominal 10% familywise rate. On 120 independent in-catalog episodes they fired 13.3% and 16.7% of the time. An alarm says the catalog fits badly, not which law is missing, and what it can see depends on the probes the policy has already chosen. The resulting fallback helped under one withheld law and hurt under another. So within a stipulated family, a probe set that separates in principle is not yet one that a policy which alters the system will reach.