On Occurrence

The framework proposes four incidents as the foundational device of existence, the conditions anything must satisfy to occur at any scale in any domain. The four are Signal, Context, Coincidence, and Potentials, corresponding to four axioms, Existence, Space, Time, and Energy, from which their sequential ordering, dual-axis organization, and irreversibility are derived. Two independent binary distinctions (boundary versus interior and input versus output) cross to produce exactly four positions (2² = 4). Each incident is a necessary condition for the next, and no incident can be derived from the remaining three.

The four foundations of mathematics each formalize one incident in this sequence. Logic formalizes Signal, the gating distinction between something and nothing. Set theory formalizes Context, placement in a relational field. Type theory formalizes Coincidence, the resolution that determines identity. Category theory formalizes Potentials, the composition of output into input. Published correspondences (Curry, 1958; Howard, 1969/1980; Lambek, 1972; Lawvere, 1964) connect the four into a directed cycle that closes under composition, an endomorphism, a structure-preserving map from the system to itself.

The same four-node directed graph is derivable independently from the published thalamocortical anatomy (Stewart, 2026c) and from the operational ordering of the foundations of mathematics (Stewart, 2026e). The bilinear invariants of that graph, computed with zero free parameters at any step from the anatomy through the quiver to the cycle space, return the Standard Model gauge algebra su(3) ⊕ su(2) ⊕ u(1) (Stewart, 2026e, 2026g). The framework derives the direction of time and the structural distinction between past, present, and future from the irreversibility of completed forms.

Frameworks developed independently across different centuries, disciplines, and methodologies each arrived at the same four-part structure. None derived it from the others. The framework proposes they converged because they reached the same substrate. Apparent three-part systems compress two structurally distinct operations into one position; the missing fourth is recoverable in each case. Three testable predictions are advanced, each falsifiable by a single counterexample. The complete tenant map of twenty-six converging frameworks is presented in the companion supplement (Stewart, 2026, Supplement). The analysis of eight apparent three-part systems is presented in a companion paper (Stewart, 2026, Apparent Threes).

**Keywords:** formal ontology, foundational device, occurrence, four-incident form, cross-domain convergence, endomorphism, dual-axis organization, 2² structure, Curry-Howard-Lambek correspondence, category theory, quiver path algebra, Hamiltonian cycle, gauge algebra, bilinear invariant, recursion, irreversibility