Abstract

We establish three independent results concerning the fractal structure of the T144 biological timing framework. Result 1 (Algebraic self-similarity). The T144 cycle 40 = 36 + 4 generates an exact autosimilar tower: for all k ≥ 1, 40 × 4^(k−1) = 36 × 4^(k−1) + 4^k. The ratio execution/total is constant at 9/10 at every scale. The integer 4 is the fractal seed: it is the generator of the autosimilarity at every level of the tower.

Result 2 (Hausdorff dimension). Applying the Hausdorff-Besicovitch formula to the T144 cycle — 9 autosimilar copies, contraction ratio 4 — yields the Hausdorff dimension:

Dₙ(T144) = log₂(3) ≈ 1.585

This is the dimension of the ternary Cantor set, derived from first principles with no free parameter. Result 3 (Fractal corridor). The dimension log₂(3) lies strictly within the interval [4/3, 5/3], where 4/3 is the Hausdorff dimension of the 2D self-avoiding walk (Lawler-Schramm-Werner 2001) and 5/3 is the Kolmogorov-Obukhov turbulent cascade exponent (1941). The T144 APS invariant contributes 1 + η = 169/120 ≈ 1.408 to this corridor. The corridor has a precise biological interpretation: [4/3, 5/3] separates ordered homeostatic paths from turbulent pathological regimes in 2D phase space.

All three results are derived without fitting. The fractal seed 4 is the same integer that generates the 2-adic tower Z₄ ⊂ Z₈ ⊂ Z₁₆ = Aut(T144), establishing a direct link between the fractal self-similarity of the T144 cycle and the automorphism group of the T144 orbifold.

Keywords: fractal seed, Hausdorff dimension, log₂(3), self-avoiding walk, Kolmogorov 5/3, T144 framework, autosimilarity, 2-adic tower, Cantor set, fractal corridor, biological complexity