What Survives Coarse-Graining Four Components of Cross-Level Explanation, and the Bounds on Contingent Bits

Abstract

Abstract Asking whether a lower-level science can explain a higher-level one usually produces an argument rather than a result, because the question conflates objects that behave differently. This paper separates them, and downgrades V1’s central equation to a pair of bounds — the part downgraded being the part that carried the most and was the most fragile. The fidelity of a cross-level explanation splits along two axes into four components, not two. The axes are the form of the content — whether it says what cannot happen or what value something takes — and the licence to transmit: what a statement’s invariance under coarse-graining hangs on. The constraint component is exclusionary in content and licensed by what a coarse-graining is defined to do; it passes with fidelity one. The universality component is specificational in content and licensed by the flow of irrelevant operators to zero; it likewise does not decay, its transmission condition is basin membership rather than contingency, and it fails by a step rather than by attrition. The local-constraint component is exclusionary in content and unlicensed: its own premise is an irrelevant operator, so after coarse-graining the statement has no referent at the target level. Only the fourth component — everything else that specifies what actually happens — decays with accumulated contingency. A two-component scheme misrecords the universality component as specificational and predicts a decay that does not occur, and cannot see the local-constraint cell at all. Defining the central quantity requires fixing three things, and V1 fixed none of them: a reference ensemble on possible worlds, a loss function, and a conversion between bits and log-volume. The third is new here; it was previously assumed to exist. Additivity yields two bounds with opposite directions and different premises. The lower bound Q ≥ 2^(−Σb) follows from the chain rule and the fact that conditioning does not increase entropy, and needs no Markov assumption. The upper bound Q(n,k) ≤ 2^(−H∞(L_n | L_{n−1})) follows from the data-processing inequality and does require the chain of levels to be Markov. The two endpoints of reduction are therefore fixed by two different quantities — a cumulative one below, a single-step one above — and not, as V1 claimed, by the same number. Min-entropy and Shannon entropy cannot be interchanged. The guessing probability is governed by conditional min-entropy, but min-entropy admits no usable chain rule, and Σb written in min-entropy is not an upper bound on anything. A numerical counterexample is given; b stays in Shannon entropy and min-entropy appears only on the Q side. V1’s tightest sentence becomes conditional, and yields a different criterion in exchange: what a synthesizing discipline can achieve beyond Σb is at most the slack in the bound plus the bits it reads off empirically, and only the first kind of gain is portable.