Kolmogorov Structure Function Exponents ζ_p

τ-Formula

ζ_p = (p/9)(1 − (2/3)^(p/3))

Derivation

In orthodox turbulence theory, intermittency is a ``correction’’ to K41—something that spoils the self-similar picture. In $τ$, intermittency is primary: it arises because the fiber $T^2$ can collapse to a point, concentrating energy on co-dimension-2 filaments. Intermittency is not a perturbation; it is a structural feature of the fibered product.

The collective dynamics predictions of Category $τ$ derive from a single structural fact: $(T^2) = 2$.

Prediction & $τ$-formula & $τ$-value & Obs.\ value & Registry She–L'ev\^eque $β$ & $(T^2)/(τ^3)$ & $2/3$ & $2/3$ (fitted) & V.T248 She–L'ev\^eque $C_0$ & $(T^2)$ & $2$ & $2$ (fitted) & V.D308 Exponent $-5/3$ & $-(τ^3 + T^2)/τ^3$ & $-5/3$ & $-5/3$ & V.T250 $C_K$ & $(τ^3)/(T^2)$ & $3/2 = 1.5$ & $1.5 ± 0.1$ & V.T251 Two-thirds law & $(T^2)/(τ^3)$ & $2/3$ & $2/3$ & V.R441 $ζ_p$, $p ≤ 12$ & $p/9 + 2[1-(2/3)^p/3]$ & (see table) & $<1%$ err & V.T249 Reconnection rate & $ι_τ^2\,v_A$ & $0.117\,v_A$ & $0.1 ± 0.03$ & V.T252 Alfv'en damping & $ι_τ^2\,ω_A$ & $0.117\,ω_A$ & $∼ 0.1$ & V.T253 NS decompact.\ & $α_n = 1 - 1/p_n^#$ & $→ 1$ & (open) & V.T254

• She–L'ev\^eque from fibration: $β$ and $C_0$ are structural constants of $τ^3 = τ^1 ×_f T^2$, derived from the fiber dimension (Theorem (thm:ch65-she-leveque)). Agreement $<1%$ for all $p ≤ 12$.

Source

This prediction is derived in the Physics Ledger (Chapter 65 — collective-dynamics), Books IV–V of Panta Rhei.