The Availability of the Four-Gon Quiver Q: Finite Invariants, an Infinite-Dimensional Path Algebra, Infinite Representation Type With Four Indecomposables Under the Convergence Constraint, and a Bilinear Form of No Dynkin Type

The four-gon quiver Q, the directed graph on four vertices L, S, T, C with six arrows a₁: L → S, a₂: S → T, a₃: T → C, a₄: C → L, a₅: L → T, and a₆: C → T, is finite in every count the graph carries and does not close. Finiteness and its absence are each stated at the formalism that carries them, labeled by position as *One Bit Resurrects the Dead Form* (Stewart, 2026dh) labels them: the graph counted at L is finite, the path algebra at S is infinite-dimensional, representation theory at T is of infinite type before the convergence constraint κ (the condition dim(V_T) = 1 on the vector space a representation places at T) and finite under it, and the bilinear form at C has rank three. Each result is cited to the paper that proves it, none is proved here, and the paper does not operate on Q. Q is strongly connected with degree profile (1,2), (1,1), (3,1), (1,2), automorphism group trivial, and first Betti number three, so its cycle space has rank three with basis the four-gon γ₁ = a₁a₂a₃a₄, the shortcut γ₂ = a₅a₃a₄, and the digon γ₃ = a₃a₆. T is the unique vertex of in-degree three and a₃ is its unique exit. The pair {T, C} is the set of vertices every cycle traverses, and deleting either T or C leaves no closed walk. Q is the minimum strongly connected digraph on four vertices carrying one in-degree-three vertex with out-degree one and no out-degree-three vertex, and it is unique up to isomorphism among six-arrow realizations of that specification. Each of the four positions carries three structural layers, a shape, a place, and an operation, and every graph invariant, the degree profile, the Betti number, the cycle basis, the automorphism group, and minimality, belongs to the shape layer, which is the graph, the object a derivation must reach to arrive at Q. Q's canonical bilinear invariant selects no Dynkin type, its Euler form has no argument position for the sign that would repair it, the convergence at its unique in-degree-three vertex has no faithful statement in its own path algebra, and the quotient by the involutions γ₁² = e_L, γ₂² = e_L, γ₃² = e_T, each cycle squared to the trivial path at its basepoint, is infinite-dimensional. Boundedness and self-sufficiency are what stand in tension, rather than boundedness and openness, since a closed system supplying its own orienting datum is what *On Occurrence: The Four-Gon* (Stewart, 2026g, Theorem 1) forbids. The object is finite and its instances are not, unbounded in depth, in path count, and under mutation, and the finiteness is what generates the unboundedness, since a fixed position count is what supplies the closed cycle that iterates and the alphabet an arbitrary-length address is written in. Under κ, the unboundedness is in count and depth rather than in kind: Q is of infinite representation type before the constraint, and the representation category under it carries exactly four dimension vectors of indecomposables. Three derivations reach Q from premise-disjoint operands at three scales: four neural circuits at the neural scale, the four foundations of mathematics at the mathematical scale, and the four incidents of occurrence (Signal, Context, Coincidence, Potentials) at the scale of occurrence itself. The three-scale structure requires a citation convention: a derivation of Q is named with the scale it derives at, since what a reader must compare to check premise-disjointness is the operands and the scale is what fixes them. The paper is conducted at the mathematical scale, on the graph itself, and the neural-scale and occurrence-scale derivations enter by citation. **Keywords:** quiver, directed graph, cycle space, first Betti number, strongly connected digraph, graph automorphism, degree sequence, path algebra, Tits form, Euler form, Dynkin diagram, quiver representations, indecomposable representations, representation type, Gabriel's theorem, quiver mutation, foundations of mathematics