Polylaminin, Microtubules, and the k-nacci Spine: A Mathematical Framework for Neural Regeneration, Cancer Resonance, and the Multifractality of the Fabric of Matter

We derive three results from a single algebraic object: the k-nacci recurrence w(n+k) = sumᵢ₌₀ᵏ⁻¹ w(n+i), whose dominant spectral radius η_k is the largest real root of P_k(η) = η^k − η^(k−1) − ⋯ − η − 1 = 0. Result 1 (Biological geometry): The cross shape of the laminin heterotrimer — three short arms and one long arm — is forced by the contact condition α ∧ dα ≠ 0 on a contact 3-manifold with three generative axes. The shape is derived, not fitted. Result 2 (Fractal self-assembly): Acid-induced polylaminin (polyLM / polyLN521) self-assembles in vitro into fractal honeycomb networks with measured Hausdorff dimensions d_H ∈ [1.55, 1.70]. We show that d_H = log b / log η₃, where η₃ ≈ 1.839286755 is the Tribonacci spectral radius (k=3) and b ∈ (2.57, 2.82) is the hexagonal lattice branching factor. The fractal is inevitable: any hexagonal chiral contact-manifold polymer with three generative axes produces this family of Hausdorff dimensions. No parameter is free. Result 3 (Fabric of matter): The multifractal singularity spectrum f(α) derived from the k-nacci pressure function is a property of the contact 3-manifold itself, not of any particular physical realization. Every system realizing the same admissible contact topology — from atomic crystal lattices to cosmic web filaments — inherits the same f(α). The spectrum is substrate-blind. Three falsifiable clinical predictions follow: (C.1) resonance-selective cancer cell disruption at ν_c ≈ 221.8 kHz, derived from first principles; (C.2) autophagy flip from cytoprotective to cytotoxic; (C.3) axonal density scaling as η₃^(Δd_H) ≈ 1.10 across polylaminin networks of different Hausdorff dimensions, testable in rodent SCI models. All polylaminin Hausdorff values refer to acid-induced polymerization (pH ≈ 4, Ca²⁺-dependent) in vitro. Native basement-membrane laminin is not claimed to be identically fractal. Laminin–MT coupling is mechanochemical via the dystroglycan–integrin axis; direct laminin–tubulin binding is not claimed. SCI regeneration claims are supported by preclinical data (rats, dogs) and early human safety data; polylaminin remains investigational. This is Deposit 13 of the Principia Orthogona series (ISBN 979-8-9954416-0-1). All computations are fully reproducible via the accompanying Python (knacci_spine.py), Lean 4 (knacci_spine.lean), and seven vector figures. All operator algebra is formally stated in the AXLE engine (github.com/TOTOGT/AXLE). MSC 2020: 53D10, 37C45, 92C05, 92C40. --------------------------------------------------------------------------------------------- Dedicated to Vic, Alice, Sarah — Giulia and David. Once tiny, always strong. --------------------------------------------------------------------------------------------- Polylaminin, Microtubules, and the k-nacci Spine: A Mathematical Framework for Neural Regeneration, Cancer Resonance, and the Multifractality of the Fabric of Matter Abstract We derive three results from a single algebraic object: the k-nacci recurrence w(n + k) = Xk−1i=0w(n + i), k ≥ 2, whose dominant spectral radius ηk is the largest real root of Pk(η) = ηk − ηk−1 − · · · − η − 1 = 0.Result 1 (Biological geometry). The cross shape of the laminin heterotrimer— three short arms and one long arm projecting from a central coiled-coil domain— is forced by the contact condition α ∧ dα ̸= 0 on a contact 3-manifold withthree generative axes. The shape is derived, not fitted. Result 2 (Fractal self-assembly). Polylaminin networks produced by acid-induced polymerization of laminin have measured Hausdorff dimensions dH ∈ [1.55, 1.70] [6, 8]. We show that dH = log b/ log η3, where b ∈ (2.57, 2.82) isthe hexagonal lattice branching factor and η3 ≈ 1.839286755 is the Tribonacci 1 Grossi, P.N. — Polylaminin, Microtubules, and the k-nacci SpineZenodo 10.5281/zenodo.19501831 spectral radius (k = 3). The fractal is inevitable: any hexagonal chiral contact-manifold polymer with three generative axes produces this family of Hausdorff dimensions.Result 3 (Fabric of matter). The multifractal singularity spectrum f(α)derived from the k-nacci pressure function is a property of the contact 3-manifolditself, not of any particular physical realization. Every system realizing thesame admissible contact topology — from atomic crystal lattices to cosmicweb filaments — inherits the same f(α). Three falsifiable clinical predictionsfollow: (C.1) resonance-selective cancer cell disruption at νc ≈ 221.8 kHz; (C.2)autophagy flip from cytoprotective to cytotoxic; (C.3) axonal density scaling asη∆dH3 ≈ 1.10 across polylaminin networks.All computations are fully reproducible via the accompanying Python (knacci spine.py), Lean 4 (knacci spine.lean), and seven vector figures. The paper is self-contained; no prior knowledge of TO/TOGT is assumed. MSC 2020: 53D10 (contact manifolds), 37C45 (dimension theory of dynamicalsystems), 92C05 (biophysics), 92C40 (biochemistry).