Radius Level and Radius Variation in Pre-Normalized Decoders: Sharp Normalization Distortion and Width Dependence

We study the worst-case cost of RMS normalization in the single-sign vector Rademacher process—one Rademacher sign per example, a Euclidean norm outside the signed sum—that arises when a bounded linear readout is peeled from a hidden-state class. We separate out two effects : variation of hidden-state radii across hypotheses, and distance of those states from zero. For fixed $\nu >0$ and ${\rho }_0>0$, on the annulus $1\le \text{rms}h\le 1+2\rho$ with $0\le \rho \le {\rho }_0$, the dimension-unconstrained sharp (with matching lower bound) order is ${\Theta }_{\nu ,{\rho }_0}(1+\rho \sqrt{m})$; at every fixed positive $\rho$ the width-aware sharp order is ${\Theta }_{\rho ,\nu }(1+\sqrt{\min \{m,d\}})$, so distortion saturates at ${\Theta }_{\rho ,\nu }(1+\sqrt{d})$ in a fixed-width architecture. A uniform joint envelope for bounded $\rho$ identifies the remaining gap; it matches throughout $0<\rho \le 1$ and $m\le d/\rho$. Conditionally on a bound for the raw hidden-state complexity, these factors enter a standard bounded-linear-readout generalization inequality. For radius level, we derive sufficient residual-amplitude conditions under which an embedding floor propagates to a population hidden-state floor at every depth, and give explicit attention and SwiGLU amplitude bounds. A five-checkpoint audit on public pre-RMS decoders shows that the tested uniform certificates fail at native scale, although sampled radii remain positive throughout depth.