Principia Orthogona Volume Two: Contact Realization of Generative Transitions

Principia Orthogona, Volume II: Contact Realization of Generative Transitions

Version 2a — May 2026

Pablo Nogueira Grossi · G6 LLC, Newark NJ · ORCID: 0009-0000-6496-2186
Zenodo concept DOI (resolves to latest): https://doi.org/10.5281/zenodo.20159456
Series root: https://doi.org/10.5281/zenodo.19117399
AXLE: 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 A_dm³ = φ^{T*/4}. The impulsive momentum jump p⁺ − p⁻ = μn at the fold corresponds to the contact Hamiltonian correction H_diss = −γVe^{−βz} as β → ∞.
• Theorem B: |κ| ↑ κ* ⟺ μ_max < 0 ⟺ τ = √(c/κ_noise) ∈ (0,∞). The curvature threshold κ* and the embodiment threshold τ are two parameterizations of the same event. Explicit values: τ = 2, ε₀ = 1/3 (outer basin).
• Theorem C: The four dm³ bifurcations (Contact Hopf, Saddle-node, Neimark–Sacker, Slow-fast crossover) correspond bijectively to the Whitney A₁–A₃ singularity types of Volume I.

The dm³ toy model equations (exact, §4.3):

ṙ = r(1 − r²) + 2(r−1)·e^{−z}
θ̇ = 1
ż  = r² − 2(r−1)²·e^{−z}

Parameters: (μ_max, ω, β) = (−2, 1, 1). Limit cycle: Γ = {r=1}, T* = 2π.

What V2a adds over V1

V2a completes the reproducibility stack to the same standard as the Vol. I V3 deposit (10.5281/zenodo.20298665) and the GTCT deposit (10.5281/zenodo.20239928).

1. PrincipiaOrthogona_VolumeTwo_v2a.pdf — full paper with all 7 figures embedded at publication size, DejaVu fonts (correct Unicode rendering throughout), Lean 4 proof status appendix, corrected G-series table (5 volumes, not a trilogy).

2. MiniBeast_Book3_DepositEdition.pdf — clean deposit edition of Book 3 (The Mini-Beast): all six chapters, exact parameter table (p. 26), Coherence Bridge theorem, 14-week CEFR→TOGT program, series table.

3. VolumeTwo.lean — Lean 4 / Mathlib4 formal verification file:

   • 8 theorems proved without sorry: eigenvalue_at_zero, eigenvalue_neg_pos_z, embodimentThreshold_pos, toyModel_tau, toyModel_epsilon0, thm_C_singularity_bijection, thm_B_mu_iff_tau, thm_C_A1_surjective
   • 4 open obligations documented with proof strategies and difficulty ratings: eigenvalue_limit_filter (★★), thm_A_contact_realization (★★★★), thm_B_full_chain (★★★★★), thm_gronwall_asymmetry / AXLE Issue #13 (★★★)
   • 0 axioms beyond Mathlib4

4. figures.py — Python figure generator producing all 7 figures from the exact dm³ equations (4.1)–(4.3). Dependencies: numpy, matplotlib, scipy. Run: python3 figures.py

5. Individual figures (fig1–fig7, PNG):

   • fig1_phase_portrait.png — dm³ phase portrait + transverse eigenvalue λ(z)
   • fig2_threshold_equivalence.png — Theorem B three-panel chain
   • fig3_bifurcation.png — four dm³ bifurcations ↔ Whitney A₁–A₃ (v2a fixed)
   • fig4_stability_radius.png — ε₀ = 1/3 basin + Fokker–Planck convergence
   • fig5_coherence_bridge.png — six-domain coherence heatmap (Theorem 5.4)
   • fig6_operator_sequence.png — G = U∘F∘K∘C with Vol. I/II boundary
   • fig7_contact_3d.png — contact 3-manifold, exact dm³ trajectories

6. dashboard.html — interactive HTML dashboard: live RK4 phase portrait with sliders, Theorem B threshold controls, Lean proof status badges, operator sequence diagram.

Proved without sorry (8 facts in VolumeTwo.lean)

Open obligations (4)

Gronwall asymmetry correction (AXLE Issue #13): ε₀ = 1/3 is valid for the outer basin {r > r_att} 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 paper_pdf.py
# → writes PrincipiaOrthogona_VolumeTwo_v2a.pdf

Lean 4:

lake update && lake build VolumeTwo
# Dependencies: Mathlib4 (current stable)

Series context

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 LLC
Contact: pablogrossi@hotmail.com · ORCID: 0009-0000-6496-2186