Two Routes to the Bilinear Form on H₁(Q; ℝ): The Axiom Derivation and the Pure-Math Derivation

The symmetric bilinear form B on H₁(Q; ℝ) that produces the Cartan matrix of A₂ × A₁ on the cycle-generator basis {γ₁, γ₂, γ₃}, and its rank-4 extension B̃ that produces the gauge algebra su(3) ⊕ su(2) ⊕ u(1) on the extended basis {γ₁, γ₂, γ₃, γ_τ}, are derivable by two routes. The signed-convergence route (*The Endomorphic Collapse Traverses the Foundations of Mathematics*, Stewart, 2026e, and *The Temporal Generator: The Fourth Generator of the Standard Model Gauge Algebra*, Stewart, 2026q) derives the signed/unsigned edge distinction on Q from the four-axiom method (Logic, Set Theory, Type Theory, Composition) applied at the scale of cycle comparison and the scale of basis extension. The pure-math route (*The Four-Gon Quiver, Whose Bilinear Invariants Produce A₂ × A₁*, Stewart, 2026ba) takes the same signed/unsigned distinction as a structural premise and derives B by direct modification of the canonical Tits form. Both routes converge on the same Cartan matrix A₂ × A₁ at rank 3 and the same extended form B̃ at rank 4, the rank-4 construction being carried in *A Quiver-Algebraic Construction of the Standard Model Gauge Algebra* (Stewart, 2026bi), whose γ_τ propositions are internal to quiver representation theory and standard Lie theory and whose temporal diagonal B̃(γ_τ, γ_τ) = 4 rests on the dual-axis count that paper names as external to it (Stewart, 2026bi, O2). The convergence is structural: the signed edge is the only input the bilinear form requires once Q is fixed, and that input splits into a location the graph forces (a₃, the unique exit of the unique in-degree-3 vertex, which either route reads off Q) and a sign-value the graph cannot author (*On Occurrence: The Four-Gon*, Stewart, 2026g, Consequence 7b), which the signed-convergence route derives from the substrate and the pure-math route takes as a premise. The two routes share the one authored datum, sourced once (Stewart, 2026c), and differ only in whether they re-derive that authoring or take it as given. The agreement answers the self-application objection on the algebra rather than on the argument: B's warrant is inherited from a route the corpus's own method did not perform, which holds open the gap between warrant and scope that a self-verdict requires (Stewart, 2026g, Consequence 7b). **Keywords:** quiver, cycle space, first homology, symmetric bilinear form, Tits form, Euler form, Cartan matrix, A₂ × A₁, Dynkin type, Cartan-Killing correspondence, root system, Weyl group, semisimple Lie algebra, su(3) ⊕ su(2) ⊕ u(1), Standard Model gauge algebra, quiver representation theory, positive-definite form, signed convergence