Equipartition duality for multi-degree-of-freedom kime states

Let $n\ge 2$, let $s$ be an entropy value, and let $\rho_{\bm J}$ be an action marginal on $(0,\infty)^n$. Consider states on $\mathbb T^n\times(0,\infty)^n$ with entropy at least $s$ and action marginal $\rho_{\bm J}$; write $u_j$ for the within-degree-of-freedom uncertainties, and let the phase-equipartitioned product state denote the state uniform in the phase variables with the prescribed action marginal. Consider also the multi-degree-of-freedom kime-deformed semigroup

$$\partial_t\tilde\rho=\sum_j\bigl(-\omega_j\partial_{\theta_j}+\varepsilon\,\partial^2_{\theta_j}\bigr)\tilde\rho.$$

Equipartition duality. Among all such states, the phase-equipartitioned product state, unique when it exists, simultaneously (i) maximizes the entropy, (ii) minimizes every within-degree-of-freedom uncertainty $u_j$, and (iii) is the unique state at which the per-degree-of-freedom conjectured bound of Problem (b) is saturated for all $j$; moreover, it is the unique fixed point, with the given marginal, of the multi-degree-of-freedom kime-deformed semigroup.

This is the distinctive contribution of the kime formulation: the open problems are re-expressed in estimable coordinates without loss of mathematical content. The claim proposes a duality between maximum entropy at fixed invariants and minimum uncertainty at fixed entropy, extending the established one-phase equipartition result to the sharp joint statement for $n\ge 2$.