The Mathematics of Generative Transitions
Operator Chain G = U ∘ F ∘ K ∘ C · Twelve Dimensional Phases · Machine-Verified in AXLE

We develop a unified mathematical framework for generative transitions: localized geometric events in which a trajectory on a Riemannian manifold undergoes compression (C), curvature intensification (K), loss of injectivity (F), and stabilization (U). The composite operator G = U ∘ F ∘ K ∘ C generates a twelve-phase cycle indexed by four base operators and three global invariants (orthogonality I₁, nilpotency I₂, spectral collapse I₃). Structural stability holds with explicit radius ε₀ = 1/3. All verified constants are machine-checked in Lean 4 via AXLE v6.1. The proof is the code. The paper is the interface.