Diversity vs Degrees of Freedom in Gaussian Fading Channels

The standard definitions of degrees of freedom (DOF) and diversity both normalize by $\log\rho$. When this ruler is wrong, both measurements give zero or become undefined, yet intuitively DOF and diversity ought to be channel properties, not artifacts of a normalization choice. We formalize this for Gaussian fading channels. For fixed-$H$ MIMO, DOF and diversity are both ranks of the bilinear map~$HX$ with different variables free: $\varepsilon$-covering the image of~$X\!\mapsto\!HX$ gives DOF on the $\log\rho$ gauge; expanding across all dimensions of the fading map gives diversity on the linear~$\rho$ gauge. Covering produces logs; expansion produces linear growth; so in every model studied here the two gauges differ. These geometric definitions do not yield tradeoff curves. We bridge the gap with Bhattacharyya packing, obtaining gauge-DOF and B-diversity as workable proxies -- finite and informative on every gauge, including those where the classical diversity order is zero. Three gauge classes emerge: $\log\rho$, $\log\log\rho$, and $(\log\rho)^\beta$, $\beta\in(0,1)$. The main result is a cross-gauge tradeoff for noncoherent fast fading: capacity lives on $\log\log\rho$, but B-diversity lives on $\log\rho$, exponentially larger, with matching upper and lower bounds. For coherent MIMO, block fading, and irregular-spectrum channels, the same approach recovers or extends known scaling laws.