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 autophagydm3. pdf Main paper, 8 pp. , all 4 figures embedded autophagydm3. tex LaTeX source (9 bibitems, all sections) AutophagyDm3ᵥ2. lean Lean 4 / Mathlib4 — 26 theorems, zero actual sorry autophagydm3. py Python simulation and figure generator figA1ₜ40fold. pdf Triple-alpha T⁴⁰ fold (→ Vcriticalₐtₒne) figA2ₚhaseₚortrait. pdf dm³ phase portrait, both systems (→ gronwallᵣadius, basinₐsymmetry) figA3whitneyₚotential. pdf Whitney A₁ fold potential (→ Vfactored, Vₐtₒne, mucanonical) figA4coherencebridge. pdf Coherence Bridge full-table chart (→ mudm3ₙeg) coherencebridge. csv Raw data for figA4 and Table 1 coherencebridgeᵥ8₄. yaml Machine-readable coherence bridge v8. 4 (updated from v8. 3) Lean 4 Verification — AutophagyDm3ᵥ2. 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 contactCoeffₙeg c (ρ) = −2ρ 0 contactCoeffₙeᵦero c (ρ) ≠ 0 Vcriticalₐtₒne V′ (1) = 0 Vₛecondderivₐtₒne V″ (1) = 6 Vₛecondderivₙeᵦero V″ (1) ≠ 0 Vₐtₒne V (1) = −2 Vfactored V (q) + 2 = (q − 1) ² (q + 2) Vdoubleᵣoot corollary of Vfactored mucanonical −V″ (1) /2 = −3 mudm3, mudm3ₙeg −2 0 — obligation 1 CLOSED Vᵢsₘorseₐtₒne V is Morse at q = 1 — obligation 2 local model whitneyFoldconditional conditional on σ Morse — obligation 2 STRENGTHENED dm3basincompact annulus 1/3, 2 is compact — obligation 3 partial dm3basinₙonempty annulus is non-empty — obligation 3 partial Open obligations (tracked as AXLE Issue #14) Obligation Status Blocker limitCycleₑxistsₐuto sorry Poincaré–Bendixson not yet in Mathlib4 Coherence Bridge v8. 4 coherencebridgeᵥ8₄. 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) ; claimₗevelₙote 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 meanContractioncont, lyapunovDescentcont, hasStructuredAttractor Arithmetic-analytic (BSD) BSDdm3. lean v1. 0 operatorDecomposition, contactForm meanContractionBSD, lyapunovDescentBSD, hasStructuredCycleBSD 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 autophagydm3. py --out figures/ Generates all four paper figures. The nbonaccicriticality. py and nbonaccicriticalₗambda. 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