RG critical-exponent conjecture for MaxCal kernel dynamics

Let $R_\Lambda:\mathcal{K}\to\mathcal{K}$ be the renormalization-group map, let $\beta(k)$ be its beta function, and let $k^*$ be an RG fixed point satisfying $\beta(k^*)=0$. Let $D^2_k\mathcal{S}^*|_{k^*}$ denote the Hessian of the optimized path entropy at the corresponding self-consistent kernel, with eigenvalues computed in the Hilbert–Schmidt metric. RG critical-exponent conjecture. The critical exponents at RG fixed points equal the eigenvalues of $D^2_k\mathcal{S}^*|_{k^*}$ in the Hilbert–Schmidt metric, so that universality classes correspond to stability basins of self-consistent kernels. This proposes a precise correspondence between renormalization-group universality and MaxCal kernel stability; the surrounding text gives no proof or evidence of resolution.