A Quiver-Algebraic Construction of the Standard Model Gauge Algebra

The Standard Model gauge algebra su(3) ⊕ su(2) ⊕ u(1) is recovered as the Cartan-Killing image of a symmetric bilinear form on the first homology of a torus, where the torus is the product of the cycle space of the four-gon quiver and an independent temporal circle. The four-gon quiver has four vertices, six directed edges, first Betti number three, and one vertex of in-degree three. A symmetric bilinear form B on the rank-three cycle space, constructed in Stewart (2026e) from a signed-edge partition of the convergence vertex's exit, has the Cartan matrix of A₂ × A₁ as its matrix in the cycle-generator basis. Cartan-Killing on A₂ × A₁ returns the rank-three semisimple algebra su(3) ⊕ su(2). The rank-four extension required for the full Standard Model gauge algebra is supplied by a fourth generator γ_τ identified with the fundamental class of the temporal circle of the torus traced by the iterated path algebra endomorphism. The generator γ_τ is independent of the cycle space because the temporal circle is independent of the quiver's first homology, carries no involution relation where the spatial generators carry the ones imposed on them, and commutes with all spatial generators because temporal and spatial coordinates of the torus are independent. The absence of an involution on γ_τ forces the abelian character of the fourth factor. The bilinear form extends to a 4 × 4 block-diagonal matrix B̃ with diagonal entry B̃(γ_τ, γ_τ) = 4 derived from the dual-axis structure of the quiver's boundary vertex pair. The semisimple part su(3) ⊕ su(2) is the Cartan-Killing image of the spatial block (the compact real form, selected by B's positive-definiteness over ℝ). The abelian u(1) is the summand carried by the independent generator γ_τ adjoined to it, not a Cartan-Killing image (u(1) has no root system), giving su(3) ⊕ su(2) ⊕ u(1). The structural invariant k₁ = B̃(γ_τ, γ_τ) / B̃(spatial diagonal) = 4 / 2 = 2. The Weinberg angle sin²θ_W = 1/3 as a structural ratio follows from k₁ = 2 together with a coupling-strength correspondence (bilinear-form diagonal to gauge-coupling reciprocal) that this paper takes as an input from Stewart (2026q) rather than derives. The 3+1 dimensional structure follows from the rank-three spatial cycle basis plus the rank-one temporal generator, with the involution holding only on the spatial generators, conditional on the compactification of the iteration index that the temporal circle requires. The interior-to-record traversal mapping sin²θ_W from the structural ratio (1/3) to the measured value (≈0.231) is named as an open sub-problem. **Keywords:** path algebra, quiver, cycle space, Cartan matrix, A₂ × A₁, Standard Model gauge algebra, Standard Model gauge group, Cartan-Killing correspondence, symmetric bilinear form, first homology, torus, root system, gauge coupling, hypercharge, Weinberg angle, 3+1 dimensions