Affine-Intensity Invariance and Spatial Non-Identifiability in Spectral Channel Gates

Abstract

Spectral channel attention compresses a spatial feature map into a weight, but invariance of that weight need not preserve evidence about spatial arrangement. This study asks whether centered spectral concentration can provide exact affine-intensity invariance and identify spatially altered inputs. A normalized fourth singular-value moment is evaluated through a Gram product and converted into a bounded scalar gate. The analysis establishes its range, transformation laws, relative perturbation bound, derivative, and unavoidable discontinuity at constant matrices. Computations use 384 generated matrices from six spatial families and 32 patches from two credited photographs. Across 9,600 generated-map affine tests, the maximum gate difference is \(6.66\times10^{-16}\); without centering it reaches 0.4713. At relative noise norm 0.1, the mean absolute change is 0.00487, with no bound violations among 46,080 noise cases. Independent row and column permutations preserve the gate within \(6.66\times10^{-16}\) while increasing mean normalized difference energy from 0.0631 to 2.5647 for smooth blobs. The photographic checks exhibit the same separation between spectral equality and altered adjacency. Near-constant inputs expose loss of numerical contrast and direction-dependent gate limits. Centered concentration therefore provides a precisely characterized affine-intensity-invariant descriptor, but its scalar value cannot establish spatial fidelity. The findings concern an explicit attention operator and do not constitute an evaluation of a trained image translator.