Four Mathematical Characterizations of the Four-Gon Quiver
Arthur Stewart · Zenodo (CERN European Organization for Nuclear Research) · 2026
The four-gon quiver Q, the four-vertex six-arrow directed graph 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 a complete derivation within its own subfield (foundational mathematics, graph theory, finite-set combinatorics, algebraic topology) given one shared structural input: the in-degree-3 convergence at T, equivalently the absent S–C edge, which *On Occurrence: The Four-Gon* (Stewart, 2026g) derives as a single closure appearing in three vocabularies. The input enters each subfield in that subfield's own vocabulary, as the second convergent dependency into the resolution layer in Characterization I, as requirement R3 in Characterization II, as the pinned placement of the two additional arrows in Characterization III, and as the basepoint placement that raises T's in-degree in Characterization IV. 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, each expressing one shared structural input in its own vocabulary, is the object those starting points were each addressing. **Keywords:** quiver, directed graph, multi-characterization, foundational mathematics, graph theory, cycle space, strong connectivity, Hamiltonian cycle, dual-axis partition, Betti number