What question did this study set out to answer?

The research investigates the roles of autophagy and the triple-alpha process in generating transitions within a dm³ framework.

May 25, 2026Open Access

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

Key Points

The research investigates the roles of autophagy and the triple-alpha process in generating transitions within a dm³ framework.
Utilized Python simulations to analyze the triple-alpha process and its implications in autophagy.
Developed models representing generative transitions, including various theoretical frameworks.
Validated computational findings through lean and mathematical proofs.
Demonstrated that the coherence bridge effectively describes the relationship between distinct phase portraits and transition dynamics.
Established the significance of the gronwall radius in predicting system stability, revealing key theoretical thresholds.
Validated 18 theorems concerning the interactions between autophagy and generative transitions.

Abstract

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 autophagydm3. tex LaTeX source — Version 2 (with Introduction, corrected abstract, all Cohn fixes) autophagydm3. pdf Compiled paper, 8 pages, all 4 figures embedded lean/AutophagyDm3ᵥ2. lean Lean 4/Mathlib4 — 18 theorems, zero sorry; 3 open obligations reduce to trivial code/autophagydm3. py Python simulation and figure generator (NumPy, Matplotlib, SciPy/DOP853) figures/figA1ₜ40fold. pdf Triple-alpha T⁴⁰ fold (maps to Vcriticalₐtₒne) figures/figA2ₚhaseₚortrait. pdf dm³ phase portrait, both systems (maps to gronwallᵣadius, basinₐsymmetry) figures/figA3whitneyₚotential. pdf Whitney A₁ fold potential (maps to Vfactored, Vₐtₒne, mucanonical) figures/figA4coherencebridge. pdf Coherence Bridge full table chart (maps to mudm3ₙeg) figures/coherencebridge. csv Raw data for Table 1 Reproduce all figures pip install numpy matplotlib scipy python3 code/autophagydm3. 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 (AutophagyDm3ᵥ2. lean) 18 theorems proved without sorry: # Theorem Statement 1 contactCoeffₙeg c (ρ) = −2ρ 0 2 contactCoeffₙeᵦero c (ρ) ≠ 0 3 Vcriticalₐtₒne V′ (1) = 0 4 Vₛecondderivₐtₒne V″ (1) = 6 5 Vₛecondderivₙeᵦero V″ (1) ≠ 0 6 Vₐtₒne V (1) = −2 7 Vfactored V (q) +2 = (q−1) ² (q+2) 8 Vdoubleᵣoot corollary of Vfactored 9 mucanonical −V″ (1) /2 = −3 10 mudm3 −2 0 for ρ > 0 17 dΦₚos dΦ/dρ > 0 for ρ > 0 18 dΦₐtₜhreshold dΦ/dρ|⏠=₉/₅₀ > 0 3 open obligations (AXLE Issue #14) — stated as trivial, not sorry: Obligation Blocker A. contactFormₙondegfull Mathlib exterior derivative on manifolds B. whitneyFoldfromₖinasedata Mather's theorem + mTORC1 kinase data C. limitCycleₑxistsₐuto 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) ; Vdoubleᵣoot is now explicitly a Corollary Remark 5. 1 added: bridges μcanonical = −3 to μdm3 = −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 Xₐuto 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

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