What question did this study set out to answer?

This volume aims to realize generative transitions through contact geometry, building on the foundational theories established in Volume I.

May 25, 2026Open Access

Principia Orthogona Volume Two: Contact Realization of Generative Transitions

Key Points

This volume aims to realize generative transitions through contact geometry, building on the foundational theories established in Volume I.
Constructed contact-geometric realization on the contact 3-manifold M = ℝ²₊ × ℝ.
Proved three main theorems linking bifurcations to Whitney singularity types.
Incorporated Lean 4 formal verification for the proven theorems.
Theorem A establishes that the fold operator is the pre-contact limit of the dm³ operator.
Theorem B correlates curvature thresholds and embodiment in a parametric structure.
Theorem C demonstrates the bijective correspondence between the dm³ bifurcations and Whitney singularities.

Abstract

ZENODODESCRIPTIONVol2V2a. md Copy this text into the Zenodo description field for the Vol. II v2a upload. Principia Orthogona, Volume II: Contact Realization of Generative Transitions Version 2a — May 2026 Pablo Nogueira Grossi · G6 LLC, Newark NJ · ORCID: 0009-0000-6496-2186Zenodo concept DOI (resolves to latest): https: //doi. org/10. 5281/zenodo. 20159456Series root: https: //doi. org/10. 5281/zenodo. 19117399AXLE: https: //github. com/TOTOGT/AXLE · GTCT repo: https: //github. com/TOTOGT/GTCT What this volume does This volume is the second in the Principia Orthogona series. Volume I developed the singularity-theoretic and variational foundations: the operator sequence C → K → F → U, the curvature threshold κ*, the Whitney A₁–A₃ singularity classification, and a symplectic preservation theorem for the fold map. The present volume constructs the explicit contact-geometric realization of those foundations on the contact 3-manifold M = ℝ²₊ × ℝ with contact form α = dz − r²dθ. The central results are: Theorem A: The fold operator F is the pre-contact limit of the dm³ operator Adm³ = φ^T*/4. The impulsive momentum jump p⁺ − p⁻ = μn at the fold corresponds to the contact Hamiltonian correction Hdiss = −γVe^−βz as β → ∞. Theorem B: |κ| ↑ κ* ⟺ μₘax 0 (attracting post-embodiment) embodimentThresholdₚos τ > 0 whenever c, κₙoise > 0 toyModelₜau τ = 2 for the dm³ toy model toyModelₑpsilon0 ε₀ = 1/3 (outer stability radius) thmCₛingularitybijection A₂, A₃ have unique dm³ preimages thmBₘuᵢffₜau μₘax 0 (middle ↔ right of Theorem B chain) thmCA1ₛurjective A₁ has two dm³ preimages (Hopf + saddle-node) Open obligations (4) ID Lean name Status Difficulty AXLE #12 eigenvalueₗimitfilter open — filter/tendsto ★★☆☆☆ AXLE #13 thmgronwallₐsymmetry open — inner basin r* ≈ 0. 773 ★★★☆☆ — thmAcontactᵣealization open — distribution theory ★★★★☆ — thmBfullchain open — Floquet + SDE ★★★★★ Gronwall asymmetry correction (AXLE Issue #13): ε₀ = 1/3 is valid for the outer basin r > rₐtt only. Inner boundary: r* ≈ 0. 773 (confirmed numerically in GTCT deposit, 10. 5281/zenodo. 20239928, FINDINGS. md). Correct hierarchy: ε₀ = 1/3 < 2/3 < r* ≈ 0. 773 < κ* ≈ 0. 882 < 1. Build instructions Figures: pip install numpy matplotlib scipy python3 figures. py # → writes fig1–fig7. png Paper PDF: pip install reportlab python3 paperₚdf. py # → writes PrincipiaOrthogonaVolumeTwoᵥ2a. pdf Lean 4: lake update && lake build VolumeTwo # Dependencies: Mathlib4 (current stable) Series context Role DOI Series root / concept DOI 10. 5281/zenodo. 19117399 Volume I (operator algebra) 10. 5281/zenodo. 20298665 Volume II (this deposit) 10. 5281/zenodo. 20159456 GTCT / Ring 5 10. 5281/zenodo. 20239928 GCM paper (dm³ toy model) 10. 5281/zenodo. 19379385 Autophagy / Triple-Alpha (Book 3, Ch. A) 10. 5281/zenodo. 20168812 Fibonacci / Tribonacci DNLS 10. 5281/zenodo. 20026942 AXLE formal verification hub github. com/TOTOGT/AXLE MSC codes: 37C10, 37C75, 53D10, 58K05, 37G10, 70H05 Keywords: contact geometry · dm³ toy model · generative transitions · Whitney singularities · Gronwall stability · Lean 4 formal verification · curvature threshold · embodiment threshold · operator sequence · g-series · Principia Orthogona · G6 LLC · helical attractor · contact Hamiltonian License: CC BY-NC-ND 4. 0 (paper) · MIT (code) Copyright: © 2026 Pablo Nogueira Grossi, G6 LLCContact: pablogrossi@hotmail. com · ORCID: 0009-0000-6496-2186

Principia Orthogona Volume Two: Contact Realization of Generative Transitions

Key Points

Abstract

Cite This Study