This paper constructs the formal mathematics of the endomorphic collapse derived in Stewart (2026e). The directed graph G = (V, E) with four vertices and six edges is defined, its strong connectivity and cycle structure are proved, and the path algebra kQ is constructed over it. The six arrows are derived as foundational dependencies between the four foundations of mathematics (logic, set theory, type theory, category theory), with the three established correspondences (Curry-Howard, Lambek, Lawvere-Tierney) cited as the bilateral evidence that translations between foundations exist. The endomorphism is the Hamiltonian cycle γ₁ = a₁ · a₂ · a₃ · a₄ in kQ, the composition of the four cascade arrows in operational order. The convergence point T, with in-degree three and out-degree one, is shown to require a many-to-one reduction for γ₁ to compose. The iterated endomorphism traces a torus. Binary cycle relations and convergence constraints generate a quotient algebra, and Coxeter analysis of the cycle generators yields type A₂ × A₁, producing the Lie algebra su (3) ⊕ su (2) ⊕ u (1) and the Standard Model gauge group SU (3) × SU (2) × U (1). Each formalism is extended at the point where it reaches its expressive limit, and the sequence of extensions, from counting to algebra to representation theory, is itself a demonstration of how mathematical structure is built. At the convergence point, the path algebra cannot express the many-to-one reduction as a path relation, and the formalism transitions to representation theory, where convergence is expressed as dim (VT) = 1. At the involution, representation theory cannot simultaneously realize the convergence constraint and the cycle relations on spaces of dimension greater than one, and the Coxeter structure is determined by the combinatorial overlap of the cycle generators in the quiver.
Arthur Stewart (Sat,) studied this question.