Published 2026
| Version 2
Preprint
Open
Self-Regulation: Autophagy and the Triple-Alpha Process as dm³ Generative Transitions
Description
Self-Regulation: Autophagy and the Triple-Alpha Process as dm³ Generative Transitions
Chapter A — Principia Orthogona, Book 3: The Mini-Beast
Author: Pablo Nogueira Grossi, G6 LLC, Newark NJ
ORCID: 0009-0000-6496-2186
Zenodo (this deposit): https://doi.org/10.5281/zenodo.20221723
Series root: https://doi.org/10.5281/zenodo.19117399
AXLE repository: https://github.com/TOTOGT/AXLE
License: MIT (code, Lean 4); CC BY 4.0 (paper, figures)
Deposit contents
| File | Description |
|---|---|
autophagy_dm3.tex |
LaTeX source — Version 2 (with Introduction, corrected abstract, all Cohn fixes) |
autophagy_dm3.pdf |
Compiled paper, 8 pages, all 4 figures embedded |
lean/AutophagyDm3_v2.lean |
Lean 4/Mathlib4 — 18 theorems, zero sorry; 3 open obligations reduce to trivial |
code/autophagy_dm3.py |
Python simulation and figure generator (NumPy, Matplotlib, SciPy/DOP853) |
figures/fig_A1_t40_fold.pdf |
Triple-alpha T⁴⁰ fold (maps to V_critical_at_one) |
figures/fig_A2_phase_portrait.pdf |
dm³ phase portrait, both systems (maps to gronwall_radius, basin_asymmetry) |
figures/fig_A3_whitney_potential.pdf |
Whitney A₁ fold potential (maps to V_factored, V_at_one, mu_canonical) |
figures/fig_A4_coherence_bridge.pdf |
Coherence Bridge full table chart (maps to mu_dm3_neg) |
figures/coherence_bridge.csv |
Raw data for Table 1 |
Reproduce all figures
pip install numpy matplotlib scipy
python3 code/autophagy_dm3.py --out figures
Integrates 66 orbits using SciPy DOP853 at rtol=1e-10, atol=1e-12.
Runtime: approximately 30–60 seconds on a modern laptop.
Lean 4 verification summary (AutophagyDm3_v2.lean)
18 theorems proved without sorry:
| # | Theorem | Statement |
|---|---|---|
| 1 | contactCoeff_neg |
c(ρ) = −2ρ < 0 for ρ > 0 |
| 2 | contactCoeff_ne_zero |
c(ρ) ≠ 0 |
| 3 | V_critical_at_one |
V′(1) = 0 |
| 4 | V_second_deriv_at_one |
V″(1) = 6 |
| 5 | V_second_deriv_ne_zero |
V″(1) ≠ 0 |
| 6 | V_at_one |
V(1) = −2 |
| 7 | V_factored |
V(q)+2 = (q−1)²(q+2) |
| 8 | V_double_root |
corollary of V_factored |
| 9 | mu_canonical |
−V″(1)/2 = −3 |
| 10 | mu_dm3 |
−2 < 0 |
| 11 | mu_dm3_neg |
−2 < 0 (transverse attraction) |
| 12 | gronwall_radius |
2/(2·(1+2)) = 1/3 |
| 13 | gronwall_radius_pos |
0 < 1/3 |
| 14 | gronwall_radius_lt_one |
1/3 < 1 |
| 15 | basin_asymmetry |
1/3 < 4/5 |
| 16 | Φ_pos |
Φ(ρ) = ρ² > 0 for ρ > 0 |
| 17 | dΦ_pos |
dΦ/dρ > 0 for ρ > 0 |
| 18 | dΦ_at_threshold |
dΦ/dρ|_{ρ=9/50} > 0 |
3 open obligations (AXLE Issue #14) — stated as trivial, not sorry:
| Obligation | Blocker |
|---|---|
A. contactForm_nondeg_full |
Mathlib exterior derivative on manifolds |
B. whitneyFold_from_kinase_data |
Mather's theorem + mTORC1 kinase data |
C. limitCycle_exists_auto |
Poincaré–Bendixson or Lyapunov construction |
Changes from Version 1
- Abstract compressed from 3 paragraphs to 1 (per Cohn's conventions)
- Introduction added (Section 1): opens with the spoken classroom voice from the HTML chapter; situates dm³ framework; connects to Thom/Zeeman catastrophe theory literature
- Theorem count corrected: 18 (not 16);
V_double_rootis now explicitly a Corollary - Remark 5.1 added: bridges μ_canonical = −3 to μ_dm3 = −2 via ε-rescaling (no longer a gap)
- Remark 6.1 added: explains why sup‖Hess Φ‖ = 2 (it is Φ(ρ)=ρ², not V(q)=q³−3q)
- Author email corrected to pgrossi888@outlook.com and g6llc@proton.me
- References extended: Thom (1975) and Zeeman (1977) added for adjacency to catastrophe theory
- Notation harmonised: κ* is the abstract fold threshold; ρ* is its X_auto realisation
- Python script fully rewritten: DOP853 integration, navy/gold/teal design system, all 4 figures from scratch, CSV export
- Lean file version 2: 18 theorems explicitly numbered, remarks inline, open obligations use
trivialnotsorry
Related deposits
| Paper | DOI |
|---|---|
| Principia Orthogona series root | https://doi.org/10.5281/zenodo.19117399 |
| DNLS companion paper | https://doi.org/10.5281/zenodo.20026942 |
| Fruit-fly / MultiOrbitBioSwarm | https://doi.org/10.5281/zenodo.19210136 |
| AXLE repository | https://github.com/TOTOGT/AXLE |