Title: Principia Orthogona, Volume I: The Mathematics of Generative Transitions

Title: Principia Orthogona, Volume I: The Mathematics of Generative Transitions Description: Principia Orthogona, Volume I: The Mathematics of Generative Transitions This volume develops a unified mathematical framework for generative transitions: localised geometric events in which a trajectory undergoes compression, curvature intensification, loss of injectivity, and stabilisation, governed by the operator sequence G = U ∘ F ∘ K ∘ C. The deposit bundles the full reproducibility stack: paper, LaTeX source, Lean 4 verification, Python simulation, generated figures, and the machine-readable coherence bridge. The framework rests on six minimal assumptions and produces: constructive operator definitions with explicit formulas; five structural theorems including existence, non-commutativity, and finite branching; seven analytical invariants; four normal forms; a singularity classification restricted to the Whitney A₁–A₃ hierarchy; a free-discontinuity variational principle; and a symplectic Hamiltonian structure with a distributional generator at the fold. The second edition adds a fifth operator E (Generative Time Circuit) with ż ≥ 0, a term-by-term structural correspondence with Perelman's proof of the Poincaré conjecture via Ricci flow with surgery (Conjecture 15.1), and the dimensional threshold N = 3 as the minimum dimension for non-trivial contact geometry, connecting it to c = 3 in the Collatz map (Conjecture 16.1). Theorems A–D machine-checked in Lean 4, zero axioms beyond Mathlib4. Part of the Principia Orthogona / Generative Contact Mechanics series · G6 LLC, Newark NJ, 2026. Series root: https://doi.org/10.5281/zenodo.19117399 · AXLE: https://github.com/TOTOGT/AXLE · DM3-lab: https://github.com/TOTOGT/DM3-lab Contact: pgrossi888@outlook.com · g6llc@proton.me · ORCID: 0009-0000-6496-2186