Principia Orthogona Volume Two develops the contact-geometric realization of generative transitions — localized events in which a dynamical trajectory undergoes compression, curvature intensification, fold singularity, and stabilization — governed by the operator chain G = U ∘ F ∘ K ∘ C on a contact 3-manifold. Core results. The volume proves four canonical theorems: Theorem A (Contact realization). The fold operator F is realized as a Whitney A₁ singularity on the contact 3-manifold M = ℝ²₊ × ℝ with contact form α = dz − r²dθ. The fold locus Σ¹(F) is a Legendrian curve in (M, α). Theorem B (Threshold equivalence). The curvature threshold κ* and the embodiment threshold τ = 2 are equivalent: κ* = √(7/9), and the contact orbit reaches τ = 2 precisely when curvature intensification crosses κ*. Theorem C (Singularity–bifurcation correspondence). Each Whitney A₁ fold event corresponds to a saddle-node bifurcation in the associated contact flow, with bifurcation parameter the kinetic threshold K*. Theorem T1 (Entropy monotonicity). Along any contact orbit satisfying α(ẋ₀) = 0, the entropy functional z(t) = ∫_Γ S(x,t) dμ_α is monotone non-decreasing. Partial Lean 4 verification in VolumeTwo.lean; full closure is AXLE obligation O3. Canonical invariants (all computed in closed form). Invariant Value Period T* = 2π Lyapunov exponent μ_max = −2 Embodiment threshold τ = 2 Outer stability radius ε₀ = 1/3 Whitney fold threshold r★ ≈ 0.77594059 (certified by certify_rstar.py) Curvature threshold κ* = √(7/9) ≈ 0.8819 Formal verification. VolumeTwo.lean contains the Lean 4 / Mathlib4 formalisation of Theorems A–C and the supporting lemmas. All stated theorems are either proved without sorry or carry an explicit documented admit with a proof sketch. No hidden sorrys. The AXLE engine (github.com/TOTOGT/AXLE) is the companion formal verification repository. Deposit contents (17 files). File Role PrincipiaOrthogona_VolumeTwo_v3.pdf Typeset paper (primary) VolumeTwo.lean Lean 4 formalisation of Theorems A–C and T1 certify_rstar.py Interval-arithmetic certification of r★ = 0.77594059 figures.py Generates all seven figures from first principles paper_pdf.py Programmatic PDF assembly minibeast_pdf.py Bridge to Volume Three fonts.py, dashboard.html, README.md Supporting files fig1_phase_portrait.png Contact phase portrait, operator chain orbit fig2_threshold_equivalence.png κ* and τ equivalence diagram fig3_bifurcation.png Saddle-node bifurcation at K* fig4_stability_radius.png Lyapunov basin and ε₀, r★, κ* hierarchy fig5_coherence_bridge.png Bridge to Volume Three biological applications fig6_operator_sequence.png Full C→K→F→U chain diagram fig7_contact_3d.png 3D rendering of contact structure on M Series context. This is Volume Two of the Principia Orthogona series (Series ISBN 979-8-9954416-6-3). Volume One established the Riemannian foundations and the operator chain. This volume lifts the construction to the contact-geometric setting. Volume Three (The Mini-Beast) applies the framework to twelve biological domains. The companion paper Contact-Geometric Theory of Generative Transitions (Zenodo 10.5281/zenodo.20682934) extends the framework to nuclear matter and contains seven independent proofs of the Tribonacci constant η ≈ 1.8393. Reproducibility. Every figure is produced by figures.py. The value r★ = 0.77594059 is certifiable by running certify_rstar.py on any standard Python installation. The Lean formalisation compiles with lake build against Mathlib4. Series root: 10.5281/zenodo.19117399 · AXLE: github.com/TOTOGT/AXLE · Contact: grossiatwork@gmail.com · ORCID: 0009-0000-6496-2186