Reduced Path Count R(n) for the Four-Gon Quiver Q

The generating function of R(n), the number of reduced paths of length n in the path algebra kQ of the four-gon quiver Q, has a denominator D(t) whose coefficient −2 at t⁹ and degree 13, which give the recurrence of R(n) its coefficient 2 at lag 9 and its order 13, come from the term of D(t) that carries no Goulden-Jackson cluster weight. A reduced path is a directed path containing no γᵢ² contiguous subword, and the reduced paths form a basis of the quotient algebra A = kQ/I, where I = ⟨r₁, r₂, r₃⟩ is the ideal generated by the three cycle relations. The values through n = 22 are computed by transfer matrix over states (current vertex, last seven arrows) and confirmed by brute-force enumeration through n = 11. For all n ≥ 16 they satisfy an integer linear recurrence of order 13, and the generating function ∑ R(n) tⁿ = N(t)/D(t) is rational. D(t) is the denominator the Goulden-Jackson cluster method gives for walks on Q, D(t) = (1 − t² − t³ − t⁴)(1 + t²)(1 + t³)(1 + t⁴) + (t⁴ + t⁶ + t⁷ + 2t⁸ + t¹⁰ + 2t¹¹), whose first factor is det(I − tA), with A the adjacency matrix of Q and I the 4 × 4 identity matrix, and whose second summand is the cluster contribution, the terms that carry the cluster weights. The cluster contribution has no term at t⁹ or t¹³, so the coefficient at t⁹ is 1 − 3 = −2 from the first summand, with 9 = 2 + 3 + 4 the sum of the three cycle lengths, and the order 13 is the length of γ₁ added to that sum. R(n) grows like λⁿ with λ ≈ 1.3162996927 against the walk-count growth μ ≈ 1.4655712319 of the length-n paths of kQ, so the cycle relations reduce the exponential growth rate by about ten percent and R(n)/dim(kQ_n) → 0 geometrically at rate (λ/μ)ⁿ. R(n) is the path algebra's own measure of itself: the convergence at T, the Coxeter type, and the Lie algebra are structural content the path algebra cannot express, and they are carried by the representation category Rep(A, κ), under the convergence constraint κ, dim(V_T) = 1, and by the bilinear form on H₁(Q; ℝ). The paper expresses the algebraic scale: its operand is the path algebra kQ and its quotient A, with Q and the cycle relations inherited by citation from *The Four-Gon Quiver: Path Algebra, Cycle Relations, and Representation Theory* (Stewart, 2026ag). **Keywords:** quiver, path algebra, bound quiver algebra, reduced path, forbidden subword, cycle relations, transfer matrix, linear recurrence, rational generating function, Goulden-Jackson cluster method, growth rate