The Phillips-Toms conjecture for minimal homeomorphisms: radius of comparison equals half the mean dimension

Theorem 1.1: for every minimal homeomorphism $h$ of an infinite compact metrizable $X$, $\mathrm{rc}(C(X)\rtimes_h\mathbb Z)=\frac12\mathrm{mdim}(X,h)$ in $[0,\infty]$. The new part is $\mathrm{mdim}\le2\,\mathrm{rc}$, obtained by turning comparison witnesses into bundle data and compressing cube-valued maps via the companion's cobordism theorem, for every finite open cover and with no finite-dimensionality hypothesis on $X$. Corollary 1.2: zero mean dimension, the small boundary property, Jiang-Su stability, finite nuclear dimension and nuclear dimension at most one are equivalent for these crossed products. It does not treat $\mathbb Z^d$ or general amenable group actions (Niu's broader formulation).

Theorem 1.1: for every minimal homeomorphism $h$ of an infinite compact metrizable $X$, $\mathrm{rc}(C(X)\rtimes_h\mathbb Z)=\frac12\mathrm{mdim}(X,h)$ in $[0,\infty]$. The new part is $\mathrm{mdim}\le2\,\mathrm{rc}$, obtained by turning comparison witnesses into bundle data and compressing cube-valued maps via the companion's cobordism theorem, for every finite open cover and with no finite-dimensionality hypothesis on $X$. Corollary 1.2: zero mean dimension, the small boundary property, Jiang-Su stability, finite nuclear dimension and nuclear dimension at most one are equivalent for these crossed products. It does not treat $\mathbb Z^d$ or general amenable group actions (Niu's broader formulation).