Compute application-specific stability constants

Compute the explicit constant K(β, λ, k, r) appearing in the Wasserstein-1 stability bound that controls the distance between the cascade multiplier distribution and the log-Poisson law under approximate hierarchical symmetry, for concrete physical parameter regimes such as fully developed turbulence with β = 2/3 and C = 2.

Background

The stability theorem proves that approximate satisfaction of A1 implies the multiplier law is within O(√ε) in Wasserstein-1 distance of a log-Poisson distribution and gives an explicit constant K(β, λ, k, r).

For practical use in physics and other applied fields, instantiating this constant for particular parameter choices would provide quantitative tolerances for how much A1 may be violated while remaining close to log-Poisson.

References

Several directions remain open. The stability bound (Theorem~\ref{thm:stability}) provides an explicit constant $K(\beta, \lambda, k, r)$. Computing this constant for specific physical systems (e.g., fully developed turbulence with $\beta = 2/3$, $C = 2$) would yield concrete tolerances for the degree to which A1 can be violated while remaining close to log-Poisson.

— Hierarchical symmetry selects log-Poisson cascades: classification, uniqueness, and stability  (2604.01632 - Freeburg, 2 Apr 2026) in Section 6, Concluding remarks