nucleation

CLAIM-350: A coarse-graining preserves a deterministic judgment exactly when the judgment is constant on its fibres: the judgment then factors through the coarse description, uniquely on its image, and otherwise two states the judgment tells apart have been merged

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Properties9
StatusActive
Version2
Rolegranted
Warrantdemonstrated
Tagsmathematics, mathematical-statistics
Linepragmatic-transport, formal-apparatus
SupportsCLAIM-348 — No coarse-graining is normatively adequate in general because it preserves stability, prediction or aggregate performance: a description adequate for an institution's performance can merge states its members' claims must distinguish, so adequacy is fixed by the evaluative target, and the target cannot be justified solely by the description that preserves it, CLAIM-407 — A coarse-graining exactly informationally closed under a process's ordinary dynamics need not preserve its responses to an intervention on the mechanism: for a declared intervention distribution ρ, independent of the state, the deficiency Γ = I(S; Y^(I) | T(S), I) can be positive, as in the hidden governance bit, where it is ρ(h = 1) bits; and Γ = 0 exactly when, for every intervention ρ charges, the response kernels are constant on T's fibres over the states of positive probability
ComplementsCLAIM-152 — Any individuation built on sufficiency is relative to the target it is sufficient for: fix the outcomes differently and a different partition wins, so no boundary drawn this way is universal
Cited as “rests on” byCLAIM-518 — Normative judgments about an organization's continuity, reform or obligations can depend on diachronic relations that descriptions of present states or terminal outcomes do not show, so specifying the relevant relations and histories can be necessary to apply them adequately; which continuations ought to follow is a further normative question, not a prerequisite of every normative judgment, CLAIM-535 — An organization that acts on what it observes has a policy acceptable in every circumstance exactly when, for each observation, the circumstances it cannot tell apart share an acceptable action; so every circumstance having an acceptable action does not give a policy, a finer observation never removes one, and where acceptable actions overlap the coarsest observations that suffice are several and incomparable, so which distinctions must be made legible depends on which acceptable action the organization commits to

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The claim

A21 §3, under the owner’s U21 (“flesh this out further, and look for more connections on the nucleation record”): “For a deterministic judgment (Y=J(X)) and deterministic coarse-graining (Z=C(X)), the following are equivalent: 1. There exists a function (j) such that (Y=j(Z)). 2. Whenever (C(x_1)=C(x_2)), we have (J(x_1)=J(x_2)).” Its gloss: “A coarse description is adequate for a normative judgment exactly when it does not merge states that the judgment must distinguish.” A20 had stated the condition one turn earlier as a one-way test. A22 made it the manuscript’s Construction A, “Elementary demonstrable result”.

Proof

Let C: X → Z and J: X → Y be any functions on one set of states X.

  • If J = j∘C, J is constant on C’s fibres. If C(x₁) = C(x₂), then J(x₁) = j(C(x₁)) = j(C(x₂)) = J(x₂).
  • If J is constant on C’s fibres, J = j∘C. For z in the image C(X), choose any x with C(x) = z and set j(z) = J(x). Another choice x′ has C(x′) = z = C(x), so J(x′) = J(x), and j is well defined. Then J(x) = j(C(x)) for every x.
  • j is unique on the image, and free off it. Two such maps agree at every z = C(x), since both equal J(x) there. Values at points of Z that no state reaches are unconstrained.

Equivalently, C preserves J exactly when the partition of X into C’s fibres refines the partition into J’s level sets. The coarsest coarse-graining that preserves J is therefore J’s own partition, x ↦ J(x). A different judgment has different level sets and so a different coarsest description, which is CLAIM-152’s target-relativity stated for judgments.

How it stands to CLAIM-152

CLAIM-152 holds that a sufficient statistic is sufficient for a target, so the partition that wins depends on the target (“fix the outcomes differently and a different partition wins”). For a target that is a deterministic function J of the state, “C is sufficient for J(X)” and “J is constant on C’s fibres” coincide (up to null sets, when X carries a distribution). So this claim is CLAIM-152’s judgment-valued special case. What it adds is the converse: CLAIM-152 says which partitions can fail, and this says exactly which succeed.

A21’s further statements, recorded, not claimed

A21 §18: “probabilistic and decision-theoretic versions require preservation of the corresponding conditional distributions or achievable risks.” Each needs its own statement:

  • Probabilistic. For finite X with every state of positive probability, C is sufficient for a random judgment Y (P(Y | X) = P(Y | C(X))) exactly when the map x ↦ P(Y | X = x) is constant on C’s fibres. That is the proof above applied to that map: if P(Y | X = x) = g(C(x)), averaging over the fibre gives P(Y | C = z) = g(z); conversely, if P(Y | X = x) = P(Y | C = C(x)), it is a function of C(x). Checked here for the finite case only.
  • Decision-theoretic. Preservation of achievable risk over a declared family of decisions is Blackwell’s order (THEORY-156), not this lemma. A bounded decision-loss criterion (A21 §12, D(C) = E[L(J(X), Ĵ(C(X)))]) is a definition, and choosing its loss L is, as A21 says of one such choice, “itself normative”.
  • Causal. A21 §6’s condition C_*(P_ℓ^{do(a)}) = P_L^{do(ā)}, for a declared family of interventions, defines intervention-consistent coarse-graining. It is a definition, not a theorem, and it holds only relative to the macro-interventions ā, whose meaning QUESTION-115 asks about.

What it does not say

  • That the judgment is correct. A22: it “establishes the adequacy of an abstraction for a specified judgment, not the correctness of the judgment’s underlying standard.”
  • That a description adequate for one judgment is adequate for another. That is the point of CLAIM-348 and its construction CASE-110.
  • That observational adequacy is causal adequacy. A state space with a distribution can satisfy the probabilistic version while interventions tell its merged states apart (A22’s Construction B; CLAIM-127).
  • Anything about bounded judges. A receiver that cannot compute j can do better with a different description (CLAIM-199).

Note of 2026-10-21: two proposed TERMs that are this claim’s condition

A12 of the exchange of 2026-10-20 proposed a TERM for “identifiability relative to an observational regime” with the criterion π(s₁) = π(s₂) ⟹ V(s₁) = V(s₂). A13 proposed another for “target-preserving coarse-graining”, J = j∘T iff T(x) = T(x′) ⇒ J(x) = J(x′). Both are this claim’s condition, with the coarse-graining read as an observation map in the first. No TERM is filed. This claim’s title is the definition, and its “Equivalently … refines the partition into J’s level sets” gives the partition form.

The same exchange gave this claim’s “What it does not say” (Construction B, CLAIM-127) its first parameterized form: a closed coarse-graining that loses an intervention response, filed as CLAIM-407 with CASE-125. The note of 2026-10-21 on CLAIM-397 uses this claim’s condition to widen a non-identifiability result to every functional of a law.

The causal-abstraction papers were read the same day. This claim’s causal bullet defines intervention-consistent coarse-graining “for a declared family of interventions”; Rubenstein et al.’s exact transformation is that notion as a theorem-bearing definition, and its column condition for averaging linear variables is THEORY-755’s column-equitability (THEORY-847). It is distributional, a match of push-forward laws, not the fibre-wise factorization this claim proves, and the hidden governance bit separates them (THEORY-848, CLAIM-399). Beckers and Halpern’s compatibility condition is this claim’s shape one level up, on contexts (THEORY-849).

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