N29 — Kolmogorov constant $C_K = 3/2$

N29: Prediction

CK = 3/2.

$C_K = 3/2$ (V.T250), the first parameter-free derivation of the Kolmogorov constant. Observed: $C_K = 1.5 ± 0.1$. Experiment: atmospheric turbulence, wind tunnels. Timeline: ongoing.

Derivation Context

The numerator $5 = (τ^3) + (T^2) = |gen| + (T^2)$ counts the total number of dissipation channels: three generation modes from $H_1(τ^3;ℤ) ≅ ℤ^3$ plus two fiber directions on $T^2$. The denominator is the spatial dimensionality of the fibered product.

In the K41 derivation, the exponent $5/3$ emerges from dimensional analysis: $[E(k)] = L^3 T^-2$, $[] = L^2 T^-3$, $[k] = L^-1$, so $E(k) ^2/3 k^-5/3$ by matching dimensions. This derivation gives the correct answer but does not explain why the dimensions work out to produce $5/3$. The $τ$-decomposition (eq:ch65-53-decomposition) provides the structural reason: $5/3$ is the ratio of dissipation channels to spatial dimensions in the fibered product. The exponent encodes the topology of $τ^3$.