Principia Orthogona, Volume One: The Mathematics of Generative Transitions

This record contains the four papers of the Principia Orthogona / Generative Contact Mechanics (GCM) research program by Pablo Nogueira Grossi (g6llc, Newark NJ, 2026).

Paper 1 — Principia Orthogona, Volume One: The Mathematics of Generative Transitions This volume develops a unified mathematical framework for generative transitions: localized geometric events in which a trajectory undergoes compression, curvature intensification, loss of injectivity, and stabilization. The central object is the operator sequence C → K → F → U, acting on trajectories in a Riemannian manifold. Main results include: a critical curvature threshold κ* defined intrinsically by the focal radius; a free-discontinuity variational principle; a symplectic preservation theorem for the fold map; and a singularity classification restricted to the Whitney A1–A3 hierarchy. HAL: hal-05555216.

Paper 2 — Principia Orthogona, Volume Two: Contact Realization of Generative Transitions This volume constructs the explicit contact-geometric realization of Volume One. Theorem A: the fold operator is the pre-contact limit of the dm3 operator. Theorem B (Threshold Equivalence): |κ| ↑ κ* ⟺ μ_max < 0 ⟺ τ ∈ (0,∞). Theorem C: the four dm3 bifurcations correspond to Whitney A1–A3 singularity types. HAL: hal-05559997.

Paper 3 — Generative Contact Mechanics: A Geometric Framework for Dissipative Systems with Structured Limit Cycles Introduces the dm3 system and a complete operator algebra (g-, L-, R-, U-operators) embedded in contact geometry. Four theorems establish existence and stochastic stability of hyperbolic limit cycles, categorical closure under unification with the prediction τ₁₂ ≤ min(τ₁, τ₂), a universal contact normal form (μ_max, ω, β), and C¹-structural stability with explicit radius ε₀ = 1/3. Submitted to Journal of Geometric Mechanics.

Paper 4 — The dm3 Operator: Explicit Toy Model and Global Dynamical Analysis Complete explicit instantiation of GCM on the contact manifold M = R²×R with canonical invariant triple (T*, μ_max, τ) = (2π, −2, 2) and stability radius ε₀ = 1/3. Four theorems establish: global attractor Γ₁₂, normally hyperbolic invariant torus for 1:2 resonance, four bifurcations (contact Hopf, saddle-node, Neimark–Sacker, slow-fast crossover), and stochastic concentration below embodiment threshold τ = 2. Submitted to SIAM Journal on Applied Dynamical Systems.