When Does State-Law Co-Determination Have Physical Content? Fixed-Point Representations, No-Signalling, and Model Discrimination

Abstract

Proposals that physical states and laws jointly determine one another are often presented through fixed-point equations. Yet any ordinary fixed-law theory can be rewritten in joint fixed-point form by adding a constant law-update map. Fixed-point form alone therefore carries neither empirical content nor a distinctive metaphysics. This article turns that objection into a constructive philosophy-of-physics programme. First, it proves a trivial-embedding result and identifies six non-vacuity conditions for substantive state-law co-determination: non-constant physical dependence, representation invariance, counterfactual rigidity, compositional closure, recovery of a declared baseline, and cross-context discriminability. Second, it extends the existence question beyond finite-dimensional compactness by formulating a conditional weak-* fixed-point result on the state space of a unital C*-algebra. Third, it supplies an explicit bipartite quantum-statistical model in which a correlation-dependent law parameter is selected jointly with a Gibbs state. The model has a complete tensor-product rule, an exact bifurcation condition, and a no-signalling operational stage based on local completely positive trace-preserving maps. Finally, it shows how empirical content arises only when one law-selection parameter must compress observations across multiple intervention contexts; a single context remains observationally equivalent to a fixed-law description. The resulting account replaces the metaphor of a “recursive universe” with a more disciplined claim: state-law closure is scientifically meaningful only when it imposes invariant, compositional, and risky relations across possible experiments.