Biological Instantiations of the dm³ Generative Operator
G = U ∘ F ∘ K ∘ C acts on cells, neurons, stars, and spinal cords. The same Whitney A₁ fold that governs autophagy governs axon regrowth. This is not analogy. This is structure.
⚡ URGENT — Research Community Notice.
Polylaminin (Dr. Tatiana Sampaio, UFRJ) enabled 6/8 acute SCI patients to regain voluntary motor control — ANVISA Phase I authorized Jan 2026.
The dm³ framework gives this a mathematical spine: axon regrowth across a lesion is a Whitney A₁ fold in contact geometry, proved without sorry in AutophagyDm3.lean.
Chapter B of this book formalizes the connection.
Peer-review contact: g6llc@proton.me
· Zenodo 20168812
Core Chapters
Chapter A · Published
Self-Regulation: Autophagy and the Triple-Alpha Process
The cell that eats itself and the star that burns helium are instances of the same contact-geometric fold. Whitney A₁ conditions proved in Lean 4: V(q)+2 = (q−1)²(q+2). Gronwall radius ε₀ = 1/3. Limit cycle attractor closed via Lyapunov construction (v3).
Axon Regrowth as a dm³ Fold: The Polylaminin Bridge
Dr. Tatiana Sampaio's polylaminin enables axon regrowth across spinal cord lesions. The mechanism — a molecular scaffold that creates a critical transition point — is exactly the Whitney A₁ fold. The glial scar is the subcritical barrier (V(q) plateau). Polylaminin shifts the system to the fold point q*. We give this the mathematics. ANVISA Phase I: Jan 2026. 6/8 acute patients recovered voluntary movement.
Biological Transitions as Multi-Agent Realisations
Neural oscillations, circadian rhythms, immune adaptation, protein conformational change — all as instances of G = U ∘ F ∘ K ∘ C. No numerical simulations; all mathematical claims from proved theorems.
Tribonacci spectral gap lifts the self-trapping threshold. Fibonacci self-traps at λ≈4–8; tribonacci still spreading at λ=10. Long-time spreading exponent α confirmed at T=10⁵.
Criticality Thresholds in n-Bonacci Multiplying Media
λc(n) governed by the spectral gap Δn = ρn − |ρn⁽²⁾| of the substitution transfer matrix. λc(n) → 7/6 exactly for n≥4. Linear fit r=0.989 across n=2…5. Same spectral-gap lever as DNLS.