The Path Algebra of The Endomorphism

The four foundations of mathematics, logic, set theory, type theory, and category theory, compose in a fixed operational order that forms a four-node directed graph. This paper takes the path algebra of that graph and asks what standard algebra says it determines, and the answer is the non-abelian gauge algebra of the Standard Model. The graph's own canonical invariant, its Tits form, is indefinite and selects no Dynkin type: on its own the graph is dead. The graph carries a structural asymmetry between its convergence point (which resolves with a sign) and its cession point (which does not), and a bilinear form derived from that one asymmetry, unique up to scaling, turns the dead Tits form into the Cartan matrix of A₂ × A₁, returning su(3) ⊕ su(2) via the Cartan-Killing correspondence. The gauge content sits in the asymmetry the graph supplies. This is a construction in the composition direction. Standard mathematics operates in the evaluation direction, decomposing composed structures into single values, and no existing formalism captures building structure across all four foundations as one algebraic object. The path algebra is that formalism. Along the way it recovers the counts 4ⁿ as the graded dimensions of the iterated four-gon (a Hamiltonian cycle), produces the Coxeter group A₃ from the ordering of the four operations, and reaches a clean expressive limit: the cycle generators do not satisfy the braid relation as an identity in the quotient algebra, which is exactly where the Coxeter and Lie-algebraic content passes to the bilinear form on H₁(Q; ℝ). The cycle basis has rank 3, saturating su(3) ⊕ su(2). The u(1) factor of the full gauge algebra requires a fourth generator this basis does not supply and is left as a stated open extension. The graph Q is the same one derived independently from four neural circuits in *What I Am Will Be? A Convergence, a Topology of Neural Substrate, Structure, an Endomorphism: Consciousness Its Cells-Itself* (Stewart, 2026c) and from the operational ordering of the foundations in *The Endomorphic Collapse as the Foundations of Mathematics-The Bridge Between Quantum Mechanics and General Relativity* (Stewart, 2026d), with the bilateral equivalences of Curry and Feys (1958), Howard (1969/1980), Lambek (1972), and Lawvere (1970) supplying its edges. The paper closes by handing the quiver to five active research programs (quiver varieties, D-brane quivers, quantum groups, cluster algebras, categorification). The cluster-algebraic question is already settled negatively: Q is of infinite mutation type, and the exchange matrix does not distinguish Q from the quiver with the digon deleted, so the invariant is blind at the seed to the generator carrying the A₁ factor. The remaining programs carry concrete, checkable predictions. **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