The Arithmetic of the Iterated Four-Gon (Hamiltonian) in the Four-Gon Quiver's Path Algebra

Let Q, the four-gon quiver, be the four-vertex strongly connected quiver with arrow set Q₁ = {a₁: L → S, a₂: S → T, a₃: T → C, a₄: C → L, a₅: L → T, a₆: C → T} and let kQ be its path algebra over a field k. The cycle space of Q has dimension three and is generated by the four-gon (Hamiltonian) γ₁ = a₁a₂a₃a₄, the shortcut cycle γ₂ = a₅a₃a₄, and the digon γ₃ = a₃a₆ (*The Four-Gon Quiver: Path Algebra, Cycle Relations, and Representation Theory*, Stewart, 2026ag, Theorem 2.3). A depth-n iterated four-gon path in Q is a sequence of n consecutive four-gon traversals, each based at any of the four vertices that Q's strong connectivity makes available as an entry point. This paper proves that the number of distinct depth-n iterated four-gon paths in Q is 4ⁿ (Theorem 3.1), that the set of depth-n iterated four-gon (Hamiltonian) paths partitions strictly as a 4-fold refinement chain 4 → 4 × 4 → 4 × 4 × 4 → ... → 4ⁿ indexed by the entry-vertex sequence of the iteration (Theorem 4.1), and that the iterated four-gon count is distinct from the walk-length grading of kQ, whose graded dimensions are 4, 6, 8, 12, 18 at lengths 0 through 4 (Stewart, 2026ag, Computation 3.6). Section 6 shows that the 4ⁿ result rests on Q's closure on itself (*What I Am Will Be? A Convergence, a Topology of Neural Substrate, Structure, an Endomorphism: Consciousness Its Cells-Itself*, Stewart, 2026c, Figure 1): in Q, strong connectivity and the existence of a four-gon are two readings of the same closure, both exhibited by γ₁ itself (*Four Mathematical Characterizations of the Four-Gon Quiver*, Stewart, 2026da, Lemma 1.2). Section 5 names the formalism handoff at the convergence vertex T: under the representation-level constraint κ: dim(V_T) = 1 (Stewart, 2026ag, Definition 5.4), the iterated four-gon count's structural content passes from kQ to the representation category Rep(A, κ), whose indecomposables satisfying κ carry exactly four dimension vectors (*The Four Indecomposable Representations of the Four-Gon Quiver Under a Convergence Constraint*, Stewart, 2026ae, Theorem 4.1), and, via a bilinear form on H₁(Q; ℝ) developed in companion work, to a Cartan-matrix structure on the cycle-generator basis. **Keywords:** quiver, path algebra, four-gon quiver, iterated four-gon, Hamiltonian cycle, strongly connected digraph, adjacency matrix, graded dimension, free monoid, partition refinement