Why it is a question
The owner, at U3 of the exchange of 2026-10-20: “I think we have something like 5-ish distinctions here and so “matrixing” as I suggest would be at least 5! cases to consider, which isn’t feasible for a single paper … what if we censor or negate each virtue one at a time? that reduces our combinations from n! to n.”
A3 corrected the count: “for (n) binary virtues, there are (2^n) possible presence/absence configurations, rather than (n!)”. That is 32 for five, and the leave-one-out design is five cases beside the baseline. It asks whether each condition is independent of the conjunction of the others, which is stronger than asking whether any two are equivalent. A3 fixed the five conditions: predictive sufficiency (P), dynamical autonomy (D), memory closure (M), approximation fidelity (A) and computational tractability (C). It kept apart censoring a condition (not requiring it) from negating it (requiring that it fail), and three outcomes: constructive independence, logical dependence and independence only under further conditions. And: “An impossible configuration is itself a substantive result.”
Two warnings from A3 belong to the question. “P and D must have different conditioning sets … Otherwise the distinction risks collapsing by definition.” And: “Without a fixed target, we could make an ablation succeed simply by moving the goalposts.” So the regime is fixed first (TERM-065).
The owner observed at U4 that the recommendation had fallen from five ablations to three. A4 split that into structural reduction (some conditions are definable from others) and epistemic selection (three are easy to isolate with results the record holds), and found “the evidence supports B more directly than A”. A5 proposed a revised form of the question: “remove one source of information or one class of operations, then determine which properties remain identifiable” (the macro law only, no history, no interventions, bounded computation). That form is part of this question.
What the record holds
Each item is pairwise. None is a leave-one-out witness.
- ¬D with M. CLAIM-389: in a VAR(1), zero Mori–Zwanzig memory without dynamical independence. P holds there only for prediction from Y’s own past: Y’s own past leaves residual variance 0.75 and the full past 0.59, a transfer entropy of 0.120 nats (checked on 2026-10-21). A3 offered it as a witness of every condition but D. It is not one, and A and C are not addressed in it.
- D with ¬M. Impossible in VAR(1) (CLAIM-389). Realizable in VAR(2), as CLAIM-389’s “What it does not say” reports. Checked on 2026-10-21: the autonomous AR(2) Y_t = 0.5 Y_{t−1} + 0.3 Y_{t−2} + ε has Mori memory c₁ = 0.147, c₂ = −0.0315, non-zero, with autocovariance of Hankel rank 2 and memory of Hankel rank 1, as CLAIM-397 says. A3 offered CLAIM-398 for this case. In VAR(1) every CLAIM-398 example with memory is also dependent, and its finite augmentation makes M hold under A3’s own definition.
- Retention without feasible recovery. CLAIM-304 and CLAIM-302. That is a one-shot encoding, for which D, M and A are undefined.
- Accuracy against dependence. THEORY-771, CLAIM-390, CLAIM-386 and CLAIM-388, all in QUESTION-221, which holds the non-implication open.
- Not identifiable from a restricted law. From the coarse variable’s own law: CLAIM-397 and CLAIM-389’s note of 2026-10-20. From the passive joint law: CLAIM-396, on the direction of coupling.
- The same form on other lists. QUESTION-093 asks it of four autonomy criteria for agency (“can it have the first without the others?”), and QUESTION-115 of four individuation criteria (“where these criteria disagree”). Axiom-independence precedents: CLAIM-343 and CLAIM-353.
- Which conditions are parameters. Split off as QUESTION-324, Answered. CLAIM-401, after A4 and A5: P and A need a declared target, C restricts the operations, and M depends on the projection (QUESTION-222, THEORY-752). CLAIM-152 and CLAIM-348 give the target-relativity.
What would count as an answer
For one stated model class, linear Gaussian VAR first, with the target, projection, metric, horizon and budget declared: for each condition, a construction meeting the other four and failing it, or a proof that the other four entail it. And a report of which conditions dissolved into parameters of the regime on the way. An answer in A5’s revised form says, for each restricted access regime, which of the conditions remain identifiable.
Note of 2026-10-25: predictive closure proposed for a population
A36 §8 of the exchange of 2026-10-24, continued, proposed predictive closure, P(g(X_{t+1}) | X_t) = P(g(X_{t+1}) | g(X_t)), as the test of whether (m, a, r) is a sufficient reduction of a population of perspectives. That is this question’s memory closure under a declared projection. CASE-124’s witness 2 already shows a failure of it for (m, a). A42 marked the derivation of the three macro-variables from the population “Not established”. In the linear network of CASE-160 the influence-weighted coordinate is exactly closed. The question is unchanged.
Note of 2026-10-26: pairwise separations on one kernel
CASE-164 adds pairwise non-implications on one exactly known 192-state kernel, like those the record already holds, and no leave-one-out witness.
- Dynamical independence without sufficiency for an intervention target. The agents with phase are informationally closed on the declared law and lose 0.5051 bits about the next rule integrities under a declared intervention mixture, CLAIM-407’s shape.
- Closure on a support without global strong lumpability (defect 0.42). Lumpability is not on this question’s list, but THEORY-822 sets it beside dynamical independence.
- A sample budget against the exact quantity. At 300 training windows a fitted gain is negative where the exact defect is 0.0930 bits (dmarx/PerspectiveLab@9f82bfd).
Its refinement witness (authority lumpable, authority with one record bit not, the full state lumpable again) is in strong lumpability, not in informational closure. A strong one-step failure of Markov closure in the lab’s population (v1.4, dmarx/PerspectiveLab@7e2edc2 experiments/predictive_closure/) does not by itself show a positive infinite-history defect. In CASE-163, what a probe budget identifies is a further budget-relative verdict, of a mechanism’s class and parameters. The question is unchanged.
Note of 2026-10-29: lumpability does not fix the level effective information picks
Two items of this date, context and not witnesses.
- Eberhardt and Lee, LIT-1839 (NOTE-1413). Their example (17) has a strongly lumpable partition, so the block label is closed and Markov (THEORY-822) and the coarse-graining commutes with time evolution. Yet EI picks that partition at one step and the micro at two. Strong lumpability does not make the EI-optimal choice stable under a change of time step. That is a property of EI, not one of this question’s five conditions.
- Pfante, Bertschinger, Olbrich, Ay and Jost 2014, LIT-1837, which Rosas et al. (NOTE-1030) credit with the equivalence of informational closure and commutativity for Markov spatial coarse-grainings, is registered and Unavailable. The record still holds that result second-hand, through THEORY-820 and THEORY-822.
The question is unchanged and stays Open.
Note of 2026-10-29: predictive sufficiency without the causal reading
Causal feature learning, LIT-1833 (NOTE-1420) and LIT-1834 (NOTE-1415), context and not a witness. It separates two sufficiency conditions for a declared target: the observational partition, sufficient for predicting it, and the causal partition, sufficient for it under intervention. Its coarsening theorem says that, outside a set of measure zero, the causal partition is a coarsening of the observational one. The overview’s toy shows the converse failing: “Warm”, a confounded correlate, is a partition sufficient for prediction that is not causal. So predictive sufficiency for a declared target can hold where the causal reading fails. The causal reading is not one of this question’s five conditions, so this is not a leave-one-out configuration. It suggests that the first condition’s declared target needs a declared regime too, observation or intervention. Its subsidiary variables, coarser cause–effect pairs that keep sufficiency only for a coarser declared target, are the target-relativity the first condition already names.
The question is unchanged and stays Open.