Year: 2026
Type: article
Abstract: The standard algebraic operations used to recover the gauge algebra su(3) ⊕ su(2) from a four-vertex quiver Q form a closed four-step cycle. The four operations are the kernel computation of the boundary operator on Q's chain complex, the basis selection for the first homology H₁(Q; ℝ), the Cartan-Killing correspondence applied to the Cartan matrix of a symmetric bilinear form B on H₁(Q; ℝ), and the Weyl group construction of the reflection group generated by σ_{γᵢ}(v) = v − B(v, γᵢ) γᵢ. Each operation has a dependency on the prior operation that is forced by the construction. The kernel computation produces the cycle space without which there is nothing to select a basis for. The basis selection produces the linearly independent generators without which there is no matrix to apply Cartan-Killing to. The Cartan-Killing correspondence produces the semisimple Lie algebra without which there are no roots whose reflections generate the Weyl group. The Weyl group produces the closed set of permitted reflections that closes the algebra's iteration. The four-step chain proceeds in the evaluation direction, recovering from each operation what the prior operations had already implied. The paper enumerates the chain, proves the forced dependencies, and forces the count of four by two independent routes. The first removes each operation in turn and shows the removal breaks that operation's position and no other, and exhibits the candidate fifth the construction itself supplies, the specification of the bilinear form B, which folds back into the basis selection at finer grain. The second inherits a structural input (a dual-axis enumeration on the four positions) admitting exactly four positions: two independent binary axes cross to 2² = 4, and no third axis is available. The construction's contribution is the explicit identification of the four operations as a closed cycle, the proof that the chain is forced by the dependencies, the removal argument that forces the count on the construction's own operands, the closure proof, and the result that the four failure modes have three distinct criteria, since the third and fourth operations each fail if and only if B is not positive definite. The Lie-algebraic content of the derivation, su(3) ⊕ su(2), is recovered as one specific instance of the four-operation chain applied to a four-vertex quiver's cycle space. The dual-axis structure that produces the four positions is derived in *On Occurrence: The Four-Gon* (Stewart, 2026g) and treated as a fixed input on the second route, where the two routes' agreement on the count is a convergence of derivations sharing no operands. **Keywords:** quiver, path algebra, cycle space, boundary operator, basis selection, Cartan-Killing correspondence, Weyl group, bilinear form, Cartan matrix, A₂ × A₁, positive-definiteness, su(3) ⊕ su(2)
Author: Stewart, Arthur
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7171523833