The Path Algebra of The Endomorphism

This paper constructs the path algebra kQ of the four-node quiver Q (nodes L, S, T, C, six directed edges, three independent cycles, first Betti number three), takes the quotient by the binary cycle relations (the convergence at T is held not as an algebra-level relation but as the representation-level constraint κ: dim(V_T) = 1), and computes what the resulting algebra determines. Q is the directed graph derived in *The Endomorphic Collapse as the Foundations of Mathematics* (Stewart, 2026d), whose four nodes are the four foundations of mathematics (logic, set theory, type theory, category theory) in operational order and whose six edges are the directed 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. The graded components of iterated Hamiltonian cycles recover the counts 4ⁿ. The ordering of the four foundational operations (distinction, membership, identification, 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_κ; 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_κ yields quantum group structure. **Keywords:** path algebra, quiver, representation theory, Tits form, Cartan matrix, Coxeter group, Dynkin diagram, Gabriel's theorem, Lie algebra, gauge algebra, Hall algebra, cluster algebra, quiver variety