The Endomorphic-Collapse Is the Foundations of Mathematics From Inside: The Path to Algebra

This paper constructs the formal mathematics of the endomorphic collapse derived in Stewart (2026d). The directed graph G = (V, E) with four vertices and six edges is defined, its strong connectivity and cycle structure are proved, and the path algebra kQ is constructed over it. The six arrows are derived as foundational dependencies between the four foundations of mathematics logic, set theory, type theory, and category theory (in that order), with the three established correspondences (Curry-Howard, Lambek, Lawvere-Tierney) cited as the bilateral evidence that translations between foundations exist. The endomorphism is the Hamiltonian cycle γ₁ = a₁ · a₂ · a₃ · a₄ in kQ, the composition of the four cascade…