The claim
The terms. Governance-response sufficiency of a coarse description T = f(S) for (I, Y) (TERM-067) asks that each response kernel K_i(· | s) = P(Y^(i) | S = s) depend on s only through T(s). Rubenstein et al.’s exact transformation (THEORY-848) asks that a coarse model reproduce the push-forward of each declared interventional law. Beckers and Halpern’s uniform transformation is the same for every distribution on contexts, and it holds exactly when some context map τ_U commutes with every allowed intervention, τ(M_L(u, i)) = M_H(τ_U(u), ω(i)) (THEORY-849); τ-abstraction adds surjectivity and fixes ω to the map τ induces. Beckers, Eberhardt and Halpern’s distance d_τ is 0 exactly for a τ-abstraction (THEORY-843).
The relation. Take the low-level context to be the pre-intervention state S, with any further noise independent of S; ω the identity on I; and the comparison on Y alone. In the deterministic case sufficiency is then the compatibility condition with the context map required to be T itself: Y^(i)(s) = g_i(T(s)) for every s and i. In the probabilistic case it is the kernel form, P(Y^(i) | S = s) = G_i(· | T(s)): a high-level model whose context is T and whose remaining noise is independent of S given T. The papers’ notions only ask that some τ_U exist, so they are weaker than this in that respect. They are stronger in others, which sufficiency does not ask for: a structural high-level model, conditions on composed interventions, more than one outcome. Neither notion is a special case of the other.
The witness (CASE-125). As a structural model: K = E_K, Y_t = E_t, H = E_H, Y_{t+1} = (1 − H)E_{t+1} + HK, with E_K, E_t, E_{t+1} fair independent bits, and declared interventions ∅, do(H = 0), do(H = 1). Let τ drop K.
- Exact. K is set by no declared intervention, so marginalising it is exact by Rubenstein et al.’s Theorem 10, with the coarse model Y_{t+1} = (1 − H)E′₁ + HE′₂ and E′₂ = K (THEORY-848).
- A τ-abstraction. With τ_U(u) = (u_t, u_H, (u_{t+1}, u_K)), for every context and every intervention on H the low level gives (u_t, h, (1 − h)u_{t+1} + hu_K), and so does the high level from τ_U(u). τ and τ_U are surjective, and do(H = h) maps to do(H = h). So it is a τ-abstraction (THEORY-849), and d_τ = 0 (THEORY-843).
- Not sufficient. Under do(H = 1) the response of the state (k, y) is the point mass at k, which differs between states with the same Y_t. So Γ(Y_t) = ρ(h = 1) bits (CLAIM-407). The high-level noise E′₂ is where K went.
Strong abstraction. If K is endogenous and every intervention ω_τ translates is allowed, then K ← k is allowed and maps to the empty intervention, since every high-level state stays reachable; (K ← 1, H ← 1) maps to H ← 1. In a context with u_K = 0, H ← 1 gives Y_{t+1} = 0 and (K ← 1, H ← 1) gives Y_{t+1} = 1, from the same high-level context and the same high-level intervention, which no τ_U can do. So strong abstraction rejects the coarse-graining (NOTE-1032, Bearing on the record, item 4). It does so because the hidden parameter can be set, the pattern of Beckers and Halpern’s Examples 3.16 and 3.18, not because states the description merges respond differently to the governance intervention. With K exogenous, as A16 has it (“a hidden mechanism parameter”), τ is the identity on endogenous variables and the coarse model is trivially a strong abstraction.
What the papers already contain. Relativity to a declared intervention set (Rubenstein et al. §5; Beckers and Halpern’s I_L), which is CLAIM-407’s I; the support caveat (Beckers and Halpern’s Examples 3.4 and 3.11, behaviour off the support left unconstrained), which is CLAIM-407’s restriction to the interventions ρ charges and the states of positive probability; and a macro variable merging micro-states with different interventional consequences (Rubenstein et al.’s cholesterol case). Not contained: exact closure under the ordinary dynamics together with failure under a mechanism intervention, a state-conditional deficiency, or Γ’s monotonicity under refinement (CLAIM-409); the papers’ transitivity results are a different property.
What it does not say
- Not a new theory of causal abstraction. A25: “We should not present governance-preserving coarse-graining as a new general theory of causal abstraction.” The relation above places a fixed-target sufficiency beside the papers’ notions; it builds no abstraction of its own.
- Not that the papers are wrong. Their criteria are about models of populations under interventions, and they do what they say. The point is that a description can pass them while losing the response of each individual state.
- Not about TERM-043’s causal component in general. That component compares push-forward laws (a graded form of Rubenstein et al.’s Definition 3), so it is zero for this case too; the claim says why the governance-response profile is not that component with τ the identity.
- Not cyclic models, uncountable intervention sets, or more than one outcome. THEORY-848 and THEORY-849 hold under their own hypotheses (acyclicity; countability).