Structural Identity, Proof Skeletons, and Entropic Dispersion in the Set-Theoretic Multiverse

Abstract This paper develops a structural analysis of proofs in ZFC that distinguishes between their inferential identity and their ordinal modes of justification across models of set theory. While forcing extensions preserve the validity of proofs, they disperse the ordinal grounds on which those proofs can be justified. I introduce the notion of a proof skeleton, isolating the inferential core of a proof from its semantic parameters, and prove that this skeleton is invariant under forcing. I then define the ordinal support spectrum of a proof and measure its dispersion via an entropic invariant. My main technical result establishes an upper bound on this entropy in terms of the proof-theoretic complexity of the proof itself. These results show that the set-theoretic multiverse is neither chaotic nor ar bitrary: dispersion is real but controlled. The multiverse supports a form of dis tributed rationality in which proofs retain their identity while acquiring a modal profile that cannot be expressed within low-entropy universes such as L.