The Path Algebra of The Endomorphism

Stewart (2026d), *The Endomorphic Collapse as the Foundations of Mathematics*, derives a directed graph Q on four nodes (L, S, T, C) with six directed edges, three independent cycles, and first Betti number three. The four nodes are the four foundations of mathematics logic, set theory, type theory, category theory used in that order. The six edges are the directed operational dependencies between them, with the bilateral equivalences established by Curry and Feys (1958), Howard (1969/1980), Lambek (1972), and Lawvere (1970) as the evidence that the translations exist. This paper constructs the path algebra kQ of that graph, takes the quotient by the binary cycle relations and the convergence constraint at T, and computes what falls out. The graded components of iterated Hamiltonian cycles recover the counts 4ⁿ. The ordering of the four foundational operations (distinction, membership, identity, composition) produces three adjacent-pair transpositions satisfying the Coxeter relations of type A₃ on the vertex ordering. The cycle generators do not satisfy Coxeter braid relations as identities in the quotient algebra kQ/I; the quotient reaches its expressive limit at the braid relation, forcing the next step. The Tits form of Q is indefinite with one isotropic generator, so Q alone does not select a Dynkin type. A bilinear form B, derived from the T/C asymmetry of Stewart (2026c) by applying the four foundational operations at the scale of cycle comparison, is the unique form (up to diagonal scaling) consistent with all four operations. Under B, the Cartan matrix of A₂ × A₁ is realized on the cycle-generator basis of H₁(Q; ℝ), giving su(3) ⊕ su(2) via Cartan-Killing. The cycle basis has rank 3. A u(1) factor would require a fourth generator the construction does not supply. The paper identifies five connections to existing research programs (quiver varieties, D-brane quivers, quantum groups, cluster algebras, categorification) and proposes five predictions, including that cluster mutation of Q produces a finite mutation class and that Ringel's Hall algebra construction applied to kQ/I yields quantum group structure. **Keywords:** path algebra, quiver, directed graph, endomorphism, foundations of mathematics, non-commutativity, Coxeter group, Gabriel's theorem, Lie algebra, su(3) ⊕ su(2), graded algebra, scale invariance, representation theory