She–Lévêque Turbulence Exponents Exact from τ³ Dimensions
The She–Lévêque anomalous turbulence scaling exponents ζ_p = p/9 + 2[1–(2/3)^{p/3}] are derived exactly from the dimension of τ³ with no free parameters.
Overview
V.T248 proves that the She–Lévêque anomalous scaling exponents for fully developed turbulence follow exactly from the τ³ dimensional structure. The formula ζ_p = p/9 + 2[1–(2/3)^{p/3}] arises because τ³ has dimension 3, the fiber T² has dimension 2, and the base τ¹ has dimension 1 — the three dimensions directly fix the two parameters in the She–Lévêque formula. No fitting is involved: the exponents are a structural output of the fibered product τ³ = τ¹ ×_f T².
Detail
Turbulence is one of the most famous unsolved problems in classical physics. The She–Lévêque model (1994) provides empirically accurate predictions of anomalous scaling exponents ζ_p that describe how velocity structure functions scale with separation distance in fully developed turbulence. The She–Lévêque formula ζ_p = p/9 + 2[1–(2/3)^{p/3}] fits experimental data extremely well, but in the orthodox framework it is a phenomenological model with the two parameters (1/9 governing linear growth and 2/3 governing the geometric part) fitted to data.
Book V (V.T248) derives these exponents as a theorem from the τ-framework. The τ³ fibration τ³ = τ¹ ×_f T² has three naturally distinguished dimensions: dim(τ³) = 3, dim(T²) = 2, dim(τ¹) = 1. The linear coefficient 1/9 = 1/(dim(τ³))² and the geometric ratio 2/3 = dim(T²)/dim(τ³) emerge directly from counting. No physical parameters are fitted; the formula is a theorem, not a model. The Kolmogorov –5/3 energy spectrum and the constant C_K = 3/2 (V.T251) follow from the same dimensional structure, establishing a complete τ-account of the statistical geometry of turbulence.
The sub-1% precision of the exponent predictions (crown jewel score 50 in the physics audit, rank 9 overall) places this alongside sub-10 ppm particle physics results in structural depth, even though the relevant observational context is laboratory turbulence rather than collider experiments.
Result Statement
V.T248: She–Lévêque anomalous scaling exponents ζ_p = p/9 + 2[1–(2/3)^{p/3}] derived exactly from τ³ dimensions. Linear coefficient 1/9 = 1/dim(τ³)² and geometric ratio 2/3 = dim(T²)/dim(τ³). Zero free parameters.