Kolmogorov Constant C_K

τ-Formula

C_K = dim(τ³)/dim(T²) = 3/2 = 1.5

Derivation

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 $τ^3$.

Source

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