The Endomorphic-Collapse as the Foundations of Mathematics-The Bridge Between Quantum Mechanics and General Relativity

Three correspondences between the foundations of mathematics have been independently discovered across the twentieth century. Curry and Feys (1958) and Howard (1969/1980) established that intuitionistic propositional logic corresponds to simply typed lambda calculus. Lambek (1972) extended the correspondence to category theory. Lawvere (1970) and Tierney brought set theory into the structure through topos theory. These results are established and published. What has not been stated is the compositional collapse. The three correspondences confirm that the four foundations are structurally identical, each translatable into the others. The operational order of the cascade composes them into a single directed cycle: logic into set theory, set theory into type theory, type theory into category theory, category theory back into logic. That cycle is an endomorphism of the four-node directed graph derived in Stewart (2026c), a composition that maps the graph's operational structure back to itself. The cycle closes because it collapses: the convergence point (type theory) receives three inputs and reduces them to one output, a mandatory many-to-one reduction without which no single resolved state exists for the morphism to carry forward. The iterated endomorphism traces a torus, with the spatial loop around the four nodes and the temporal loop through the recursion. This paper derives the collapse from the four operations derived in Stewart (2026g), here termed the axioms of occurrence, and proposes the endomorphism as the structure within which quantum mechanics and general relativity already occupy defined positions. The four positions of the endomorphism are assigned functional labels native to the physics domain: Ground (the binary record of what has and has not occurred), Field (the space of all simultaneous potentials), Collapse (the many to one reduction that resolves the field into one determination), and Propagation (the energy that carries the resolved state forward into the next cycle). The framework proposes these four positions correspond to the four foundations of mathematics: Ground to logic, Field to set theory, Collapse to type theory, and Propagation to category theory. The framework proposes general relativity formalizes the Ground, the binary record of what has and has not occurred (incident 1), and that quantum mechanics formalizes the Field (incident 2), the Collapse (incident 3), and the Propagation (incident 4). On the framework's account, both theories employ all four foundations as mathematical tools, because the endomorphism requires all four for any mathematical structure to exist. The framework proposes the distinction is between what each theory formalizes as its subject matter and what it employs as its apparatus. On this account, the incompatibility between QM and GR is the incompatibility between two partial formalisms, each built from a subset of the mathematical foundations as its subject, applied separately to different positions within a single structure. The many-to-one reduction at the convergence point reframes wave function collapse: collapse is not a problem to be solved or an interruption of unitary evolution but the structural operation that makes the endomorphism possible. Without the collapse, no single resolved state exists for the morphism to carry forward, the cycle cannot close, and the endomorphism does not complete. The framework proposes the unification is the endomorphic collapse itself. This is a claim about the structure any unification must instantiate. The framework proposes this structure already exists as published mathematics and that it has not been applied to physics because no one has identified the collapse as the structure both theories are inside of. **Keywords:** endomorphism, graph theory, algebraic topology, category theory, type theory, set theory, logic, Curry-Howard correspondence, Lambek correspondence, Lawvere-Tierney topos theory, foundations of mathematics, strongly connected directed graph, CW-complex, Betti number, torus topology, wave function collapse, quantum mechanics, general relativity, unification, scale invariance