The Collapse Completeness Theorem: A Formal Algebra of Collapse-Based Computation

This paper presents a formal algebraic framework for collapse‑based computation: a class of systems in which every input either collapses to a unique fixed point or is annihilated, with no intermediate outcome. The algebra defines a collapse operator, an admissibility function, an exhaustive veto partition, a recovery operator, a bounded foresight operator, and a governance filtration. These components compose into a complete, deterministic algebra with a unique fixed point. The central result is the Collapse Completeness Theorem, which establishes the law of the excluded middle as a theorem of the algebra rather than an axiom. Supporting results include the Exhaustive Veto Theorem, identity extinction, universal fungibility of computational workers, constitutional immutability as a reachability property, and the conjugate duality between collapse convergence and Fibonacci growth. The algebra is presented as universal mathematics, applicable to any system—computational, biological, or physical—that admits a binary admissibility function and an exhaustive veto partition. Companion papers demonstrate that natural systems satisfy these conditions due to thermodynamic boundary constraints, and that the algebra provides a unifying lens across several open mathematical frontiers identified in Cartier’s survey of Grothendieck’s work.