The Endomorphic-Collapse Is the Foundations of Mathematics From Inside: The Path to Algebra

This paper constructs the formal mathematics of the endomorphic collapse derived in Stewart (2026d). 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, and category theory (in that order), 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 its representation theory is classified. The symmetrized Euler (Tits) form of Q is computed on the cycle-support dimension vectors and shown to be indefinite with one isotropic generator, so Q is not of Dynkin type and its canonical bilinear invariants do not select a finite reflection group. Each formalism is extended at the point where it reaches its expressive limit, and the sequence of extensions, from counting to path algebra to representation theory to the quiver's canonical bilinear form, 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. At the braid relation, the quotient algebra kQ/I cannot produce (γ₁γ₂) ³ = eL by its reduction rules. At the Tits form, the canonical bilinear invariants of Q produce an indefinite symmetric matrix with negative diagonal and one isotropic generator, so no Dynkin Coxeter type is selected by Q's mathematics alone. The staircase closes once H₁ (Q; ℝ) is equipped with a bilinear form that weights signed-convergence edges (T's exit) asymmetrically from unsigned-cession edges (C's exit), a distinction derived in Stewart (2026c) from the functional anatomy of the endomorphism. That bilinear form B is derived from the T/C asymmetry by performing the four axioms of occurrence at the scale of cycle comparison: Axiom 1 supplies the logical distinction (signed/unsigned edges, L-based/T-based cycles), Axiom 2 partitions edges and cycles by the distinction, Axiom 3 assigns a type to each cycle pair (coupled vs. decoupled by basepoint identity), and Axiom 4 composes the type judgments into the bilinear morphism. B is shown unique, up to diagonal scaling, as the form consistent with all four axioms. Under this framework-derived form, the Cartan matrix of A₂ × A₁ is realized on the cycle-generator basis, producing the Lie algebra su (3) ⊕ su (2) via the Cartan-Killing correspondence. The Coxeter type is a consequence of Q equipped with the 2026c T/C asymmetry, not of Q alone, and the construction of B is itself an instance of the endomorphism's scaling logic applied to the construction of a bilinear form. The cycle-generator basis has rank three, and the Cartan-Killing correspondence on A₂ × A₁ returns a rank-three semisimple algebra. A fourth generator, required for the full Standard Model gauge algebra su (3) ⊕ su (2) ⊕ u (1), is not supplied by H₁ (Q; ℝ) and requires an independent structural input beyond the cycle basis. The companion paper Stewart (2026g) derives that fourth generator γ_τ from the temporal circle of the torus T² = S¹ × S¹, extends the bilinear form B to a fully determined 4 × 4 matrix B̃ with B̃ (γ_τ, γ_τ) = 4 (derived from the boundary pair's 2² self-containment), and returns the full Standard Model gauge algebra su (3) ⊕ su (2) ⊕ u (1) at rank 4, with normalization ratio k₁ = 2, sin²θW = 1/3 at the fundamental scale, and det (B̃) = 24 = 4!. What the paper does and what the paper is are the same act, because for an occurrence doing γ₁ and being γ₁ are one event. The construction performed below is one γ₁ traversal of the cycle whose bilinear invariant the construction returns. The gauge algebra of physics is what that bilinear invariant carries, read off from inside the cycle by the cycle running. Mathematics is one γ₁. Physics is what γ₁ produces. The paper is the act in which the two are seen to be the same cycle viewed from opposite sides of itself. **Keywords: ** path algebra, quiver representation, quotient algebra, Tits form, Euler form, Coxeter group, Cartan matrix, A₂ × A₁, endomorphism, foundations of mathematics, Curry-Howard correspondence, Lambek correspondence, Lawvere-Tierney, signed convergence, su (3) ⊕ su (2)