When Does Adaptive Spectral Normalization Define a Fixed Convex Geometry?

Finite spectral updates can approximate the polar direction without implementing one fixed convex geometry. We classify this distinction for $q_i(s)=h(s_i/\alpha (s))$, where the scale depends on the spectrum. With one initial Frobenius normalization in dimension at least three, one cubic Newton–Schulz step admits a norm-derived direction, whereas repeated cubic steps and every positive-depth Taylor-quintic composition do not, even after positive scalar rescaling and despite preserving order. More generally, an odd real-analytic filter in dimension at least four, or an odd polynomial in dimension at least three, admits a positive $C^1$ integrating factor exactly when $h(t)=at+c{t}^{2k+1}$. We also classify adaptive normalizers whose displayed response belongs to one fixed convex subdifferential. Under stated regularity, boundary, and critical-point assumptions in dimension at least two, the principal modular family is exhaustive and yields unitarily invariant norms. Exact incompatibility has an important limit: a classical perspective construction gives one norm per depth whose gradient approximates the Taylor fields uniformly as depth increases at fixed dimension, including near rank loss. Additional shape assumptions yield a sharp filter-preserving conditioning frontier. Exact Jordan certificates and dependent replay of 4,728 unique stored spectra characterize geometric separation, not a training-performance penalty.