A Divisive Foundation for Mathematics: Generating Structure Through the Division of a Unit

AbstractThe foundations of modern mathematics rest on a decision so foundational it is rarelyexamined: to begin from nothing. In the Zermelo-Fraenkel framework, all mathematicalstructure is built upward from the empty set through aggregation and succession. Thispaper proposes a rigorous alternative: rather than constructing structure from absence, wegenerate it through the successive division of a single primitive object — a Primordial UnitU —governed by a minimal axiomatic system called Divisive Foundation Theory (DFT).The framework generates natural numbers as stabilized division patterns (fixed pointsof a structural iteration operator Σ), constructs the real continuum through division cutswithout presupposing a completed infinite set, and dissolves Russell’s Paradox as a structuralconsequence of the well-founded generative hierarchy. The framework is proved relativelyconsistent with ZFC (DFT is consistent if ZFC is), and its relationship to type theory,category theory, and constructive mathematics is examined precisely.Version 3.9 completes the alignment with the six-paper DFT program: Paper Zero v9.310 establishes syntropy as the outer-domain principle and derives the non-collapse condition as a theorem (Corollary 1). The Time Paper v2.0 11 derives temporal experienceand adds Prediction P1 (density-dependent negative-time scaling, §5.5). The CompanionPhysics Paper 12 proves all gravitational theorems unconditional via the Minkowski Bridge(Theorem 1), and its Proposition 1 derives G = ℏc/M2p from DFT primitives — a resultthat depends directly on Theorem 7.4.1e (algebraic closure of R2D) of the present paper.