Four Mathematical Characterizations of a Four-Vertex Six-Arrow Strongly Connected Quiver

A four-vertex six-arrow directed graph Q with vertex set {L, S, T, C}, arrow set {a₁: L→S, a₂: S→T, a₃: T→C, a₄: C→L, a₅: L→T, a₆: C→T}, degree profile (1, 2), (1, 1), (3, 1), (1, 2), and first Betti number β₁ = 3 admits four mathematical characterizations, each derived in a distinct subfield, three of which share one structural input (the in-degree-3 convergence at T). Characterization I derives Q from the operational ordering of the four foundations of mathematics (Logic, Set Theory, Type Theory, Category Theory) under their published bilateral dependencies (Curry-Howard, Lambek, Lawvere-Tierney). Characterization II derives Q as the minimum strongly connected directed graph on four vertices that closes a four-operation cycle on itself with one in-degree-3 convergence vertex and one out-degree-3 distribution vertex absent. Characterization III derives Q from the 2² dual-axis partition of four positions under (boundary, interior) × (input, output), where that axis pair is the unique one of the three partitions of {L, S, T, C} whose 2×2 grid embeds in Q (a consequence of the absent S–C edge; Stewart, 2026g), and the two non-cycle arrows are fixed by the in-degree-3 requirement at the interior-output position. Characterization IV derives Q from the graph-theoretic minimum that realizes cycle rank 3 with one four-gon, one shortcut, and one digon, the smallest cycle-space triple for which all three cycle-types are simultaneously present and independent. Each characterization is an internally complete derivation in its own subfield (foundational mathematics, graph theory, finite-set combinatorics, algebraic topology). The four subfields are distinct, and Characterizations II, III, and IV share one structural input, the in-degree-3 convergence at T (equivalently the absent S–C edge), which the framework derives as a single closure appearing in three vocabularies (Stewart, 2026g). All four return Q with the same vertex labeling, the same arrow set, and the same graph-theoretic invariants. This paper states each characterization, verifies that each lands on Q, and reads the structural consequence: a directed graph reachable from four mathematical starting points in distinct subfields, three of them sharing one structural input, is the object those starting points were each addressing. **Keywords:** quiver, directed graph, multi-characterization, foundational mathematics, graph theory, cycle space, strong connectivity, four-gon, dual-axis partition, Betti number