The Algebra of Cycle Recovery, A Derivation of the Four-Position Cycle at the Cycle-Comparison Scale

Four standard operations on a quiver's cycle space are named in chain-complex homology, graph theory, Lie theory, and finite reflection-group theory by their construction steps and their discoverers, and none of those names says what operation is being performed: the kernel of the boundary operator, basis selection for H₁(Q; ℝ), the Cartan-Killing correspondence, and the Weyl group. The four are one cycle executing in forced dependency order at the scale where a quiver's cycles are compared. The quiver is the four-gon quiver Q, the four-incident endomorphism's quiver, whose cycle space H₁(Q; ℝ) has three independent cycles. The first operation admits closed walks of Q as cycles in H₁(Q; ℝ) by testing whether their boundary is zero. The second places admitted cycles into a basis of linearly independent generators. The third recovers the body the cycles' coupling specifies by applying the Cartan-Killing correspondence to the Cartan matrix the bilinear form B produces on the basis, which returns sl(3, ℂ) ⊕ sl(2, ℂ), whose compact real form, the one taken, is su(3) ⊕ su(2). The fourth carries the body forward as the closed catalog of reflections-and-compositions the body permits, the twelve-element group S₃ × ℤ/2 generated by the Weyl reflections σγᵢ v = v − B(v, γᵢ)·γᵢ. The four operations are the four positions of the four-incident endomorphism's cycle at the algebraic scale, with the cycle traversing in the evaluation direction (*On Occurrence: The Four-Gon*, Stewart, 2026g). The four operations are catalogued under this paper's operation-names: plain algebra (position 1), linear algebra (position 2), the algebraic resolution (position 3), and the algebraic commitment (position 4). Each names what its operation does where each field's label names how the operation was built or who built it, and the two coexist as references to the same objects from different vantage points. The paper derives the four-step chain's closure as one cycle, the four failure modes under three criteria, and the impossibility of a fifth operation, and states the catalog tags that downstream papers operating on H₁(Q; ℝ) inherit. **Keywords:** quiver, path algebra, cycle space, boundary operator, first Betti number, linear algebra, basis selection, Cartan-Killing correspondence, semisimple Lie algebra, Weyl group, reflection group, bilinear form, Cartan matrix, A₂ × A₁, su(3) ⊕ su(2)