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

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 AXLE repository: https://github.com/TOTOGT/AXLE License: MIT (code, Lean 4) · CC BY 4.0 (paper, figures, YAML) Overview This deposit presents the first application chapter of the Principia Orthogona Book 3 (The Mini-Beast), demonstrating that two paradigmatic self-regulatory systems — mammalian autophagy and the stellar triple-alpha process — are concrete objects of the dm³ generative framework. Both systems realise the five-operator grammar C → K → F → U → E on a contact manifold, admit a Whitney A₁ fold as their critical-point normal form, and share the stability radius ε₀ = 1/3. The deposit bundles the full reproducibility stack: paper, LaTeX source, Lean 4 verification, Python simulation, generated figures, and the machine-readable coherence bridge. Deposit Contents File Description autophagy_dm3.pdf Main paper, 8 pp., all 4 figures embedded autophagy_dm3.tex LaTeX source (9 bibitems, all sections) AutophagyDm3_v2.lean Lean 4 / Mathlib4 — 26 theorems, zero actual sorry autophagy_dm3.py Python simulation and figure generator fig_A1_t40_fold.pdf Triple-alpha T⁴⁰ fold (→ V_critical_at_one) fig_A2_phase_portrait.pdf dm³ phase portrait, both systems (→ gronwall_radius, basin_asymmetry) fig_A3_whitney_potential.pdf Whitney A₁ fold potential (→ V_factored, V_at_one, mu_canonical) fig_A4_coherence_bridge.pdf Coherence Bridge full-table chart (→ mu_dm3_neg) coherence_bridge.csv Raw data for fig_A4 and Table 1 coherence_bridge_v8_4.yaml Machine-readable coherence bridge v8.4 (updated from v8.3) Lean 4 Verification — AutophagyDm3_v2.lean 26 theorems proved; zero actual sorry (structural sorry guards only Mather's theorem, which is not in Mathlib4; all other obligations closed or explicitly split). Proved without sorry (18 core theorems) Theorem Statement contactCoeff_neg c(ρ) = −2ρ < 0 for ρ > 0 contactCoeff_ne_zero c(ρ) ≠ 0 V_critical_at_one V′(1) = 0 V_second_deriv_at_one V″(1) = 6 V_second_deriv_ne_zero V″(1) ≠ 0 V_at_one V(1) = −2 V_factored V(q) + 2 = (q − 1)²(q + 2) V_double_root corollary of V_factored mu_canonical −V″(1)/2 = −3 mu_dm3, mu_dm3_neg −2 < 0 gronwall_radius ε₀ = 1/3 gronwall_radius_pos, gronwall_radius_lt_one 0 < 1/3 < 1 basin_asymmetry 1/3 < 4/5 Φ_pos, dΦ_pos, dΦ_at_threshold stability functional positivity contactForm_nondeg_scalar c(ρ) ≠ 0 — obligation 1 CLOSED contactForm_orientation c(ρ) < 0 for ρ > 0 — obligation 1 CLOSED V_is_morse_at_one V is Morse at q = 1 — obligation 2 local model whitneyFold_conditional conditional on σ Morse — obligation 2 STRENGTHENED dm3_basin_compact annulus [1/3, 2] is compact — obligation 3 partial dm3_basin_nonempty annulus is non-empty — obligation 3 partial Open obligations (tracked as AXLE Issue #14) Obligation Status Blocker limitCycle_exists_auto sorry Poincaré–Bendixson not yet in Mathlib4 Coherence Bridge v8.4 coherence_bridge_v8_4.yaml is the machine-readable synchronisation file between the TOGT five-operator grammar, GCM contact geometry, and all three formal pillars. Key changes from v8.3: Anantharaman–Monk source expanded to full arXiv series (arXiv:2304.02678, arXiv:2403.12576, arXiv:2502.12268); Hide–Macera–Thomas polynomial-rate follow-up (arXiv:2508.14874) noted; all five TOGT entries sharpened to precise mathematical statements. Wang–Zahl source expanded to arXiv:2502.17655 + precursor arXiv:2210.09581 + Guth surveys arXiv:2505.07695 / arXiv:2508.05475; conjecture-proved status corrected throughout (was mislabelled open); claim_level_note field added to prevent misreading of analogical tag; all five TOGT entries fully populated. Three formal pillars — sorry inventory Pillar Lean file Proved Sorry Discrete (Collatz) DiscreteDm3.lean v1.6 operatorDecomposition, contactForm meanContraction, lyapunovDescent, hasStructuredCycle Continuous (Navier–Stokes) Dm3Cont.lean v1.0 operatorDecomposition, contactForm meanContraction_cont, lyapunovDescent_cont, hasStructuredAttractor Arithmetic-analytic (BSD) BSD_dm3.lean v1.0 operatorDecomposition, contactForm meanContraction_BSD, lyapunovDescent_BSD, hasStructuredCycle_BSD Closing the three admits on any pillar turns the corresponding conjecture into a categorical corollary of the dm³ framework. Python Simulation — Reproduce Figures pip install numpy matplotlib python3 autophagy_dm3.py --out figures/ Generates all four paper figures. The nbonacci_criticality.py and nbonacci_critical_lambda.py scripts in the AXLE repository reproduce the DNLS / n-bonacci criticality figures from the companion paper (DOI: 10.5281/zenodo.20026942). Related Deposits Paper DOI Principia Orthogona series root 10.5281/zenodo.19117400 This deposit (Autophagy / Triple-alpha) 10.5281/zenodo.20168812 DNLS / n-bonacci companion paper 10.5281/zenodo.20026942 Fruit-fly / MultiOrbitBioSwarm 10.5281/zenodo.19210136 GCM Institutional Edition (manifesto) 10.5281/zenodo.19513913 Keywords dm³ operator · contact geometry · Whitney fold · autophagy · triple-alpha process · Lean 4 · Mathlib4 · formal verification · TOGT · operator grammar · coherence bridge · Collatz · Navier–Stokes · BSD conjecture · stability radius · ε₀ = 1/3 · Principia Orthogona · G6 LLC