Neither substitutable for the other: non-commensurable contributions in human-machine collaboration

Abstract

AI-assisted science increasingly asks who contributed what, and answers with a share: the machine did eighty percent, so the human did twenty. We argue that the shared scale is the problem. Under an observation protocol the state of the world is not a point but an equivalence class, and work on it divides into two operations acting on different objects. Refinement moves the target itself and has no upper bound. Within-layer compression reduces the distance to that target and is capped by the layer. A machine that compresses perfectly has not moved the target; a human that refines perfectly has not closed the distance. The two return profiles are therefore not commensurable, and we prove that no monotone exchange rate converts one into the other. Three supporting results explain why the single-currency question also fails empirically: the best available detector has a ceiling fixed by the protocol, verification effort saturates at a floor that does not reward a better reviewer, and the sign of the leading information-theoretic contribution estimator is set by a nuisance parameter fitted out of distribution rather than by the collaboration. Across 117 machine-checked theorems with zero admitted axioms, human and machine contributions are irreplaceable by each other, and the appropriate question is not which did more but what kind of thing each did.