Non-strict sparse coupling conjecture for weak Markov blankets

Let $J$ be the Jacobian of a multiscale, coupled random dynamical system, with state components $\boldsymbol{\eta}$ and $\boldsymbol{\mu}$, and let $H$ denote its Hessian form. A weak Markov blanket allows the relevant cross-directional interactions to be absent for most component pairs and at some timescale, rather than for every degree of freedom. Non-strict sparse coupling conjecture. Consider a class of systems where the flow in at least one direction is decoupled so that the system has a weak Markov blanket, namely,

$$J_{\eta^i\mu^j}J_{\mu^j\eta^i}=0$$

and

$$H_{\eta^i\mu^j}=0$$

for most $\eta\mu$ or $\mu\eta$ pairs. Then all sufficiently high-dimensional, multiscale, coupled random dynamical systems that admit a stationary solution admit a stationary solution satisfying this property at some timescale. Consequently, physical systems possessing a steady state that do not satisfy this property anywhere at steady state should form a measure-zero set relative to those that do.