Four Algebraic Operations as One Closed Cycle, A Derivation from a Quiver's Cycle Space

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 proves that no fifth operation is possible in the chain because the structural input (a dual-axis enumeration on the four positions) admits 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, and the closure proof. 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 here as a fixed input. **Keywords,** path algebra, cycle space, boundary operator, basis selection, Cartan-Killing correspondence, Weyl group, bilinear form, Cartan matrix, A₂ × A₁, foundations of algebra