Self-Regulation: Autophagy and the Triple-Alpha Process as dm³ Generative Transitions

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 NJORCID: 0009-0000-6496-2186Zenodo (this deposit): https://doi.org/10.5281/zenodo.20221723Series root: https://doi.org/10.5281/zenodo.19117399AXLE repository: https://github.com/TOTOGT/AXLELicense: 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_root is 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 trivial not sorry 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