From Logic, Sets, Types, and Categories to su(3) ⊕ su(2): A Path-Algebraic Construction

A symmetric bilinear form B on the cycle space H₁(Q; ℝ) of the four-gon quiver Q, constructed 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. The bilateral correspondences of mathematical logic, Curry-Howard (Curry & Feys, 1958; Howard, 1969/1980), Lambek (1972), and Lawvere-Tierney (Lawvere, 1970; Tierney, 1972), relate four foundational systems (propositional logic, set theory, type theory, category theory) by structure-preserving translations in both directions. The correspondences supply translations between the four foundations but do not specify a directional sequence in which the four foundations compose into a single closed cycle. This paper takes the directional sequence on the four foundations fixed by their operational dependencies in *The Forced Operational Ordering of Logic, Sets, Types, and Categories: The Foundations of Mathematics* (Stewart, 2026r), encodes it as the four-gon quiver Q, constructs the path algebra kQ in the standard sense (Assem et al., 2006), takes the quotient by the binary cycle relations, holds the convergence constraint at the in-degree-three vertex at the representation level, and computes the algebra's bilinear invariants. The partition is a structural input the present construction takes as given rather than derives, and B is unique up to diagonal scaling given that partition and the cycle-pair classification by basepoint (*From the Tits Form to A₂ × A₁: A Signed-Edge Bilinear Form on the Four-Gon Quiver*, Stewart, 2026dg, Proposition 4.1), so the construction below is complete given three declared inputs, the directional sequence, the binary cycle relations, and the partition. The Cartan-Killing correspondence on A₂ × A₁ returns sl(3, ℂ) ⊕ sl(2, ℂ), and its compact real form is su(3) ⊕ su(2). The construction produces, as a consequence of the directional sequence among four foundations and the signed-edge partition, the rank-three semisimple block of the Standard Model gauge algebra. The construction's mathematical content is the derivation of A₂ × A₁'s Cartan matrix from the four-gon quiver's bilinear invariants under the signed-edge partition of the convergence vertex's exit that *The Endomorphic Collapse Traverses the Foundations of Mathematics* (Stewart, 2026e) derives. The construction's philosophical content is this paper's proposal that the directional sequence on the four foundations, taken together with the bilateral correspondences and the signed-edge partition, contains the gauge algebra of the Standard Model's strong and electroweak isospin sectors as one of its consequences. The paper expresses the algebraic scale: its operand is the path algebra kQ of the four-gon quiver Q and its quotient A = kQ/I by the binary cycle relations, with the directional sequence inherited by citation from Stewart (2026r). The remaining u(1) factor required for the full Standard Model gauge algebra is not supplied by H₁(Q; ℝ) and is treated in *The Temporal Generator: The Fourth Generator of the Standard Model Gauge Algebra* (Stewart, 2026q), by extension to the temporal circle of the iterated path-algebra endomorphism's torus. The Hall algebra of A and the quiver variety of Q have not been computed, and the categorification of Q's bilinear data is open. **Keywords:** Curry-Howard correspondence, Lambek correspondence, Lawvere-Tierney correspondence, topos theory, path algebra, quiver, Tits form, Cartan matrix, A₂ × A₁, Cartan-Killing correspondence, su(3) ⊕ su(2), foundations of mathematics, gauge algebra, Standard Model