The Representation Bandwidth: A Conservation Analysis under RL Alignment

RL alignment is widely believed to compress the representational capacity of a pretrained transformer. It does not. We introduce the representation bandwidth of a transformer weight matrix W, defined through the singular-value spectrum of W viewed as the adjacency of a weighted bipartite graph, and ask whether it changes under alignment. On three independently developed pretrained-aligned pairs (Qwen2.5-7B, Mistral-7B, Yi-1.5-6B) and one weak-training null baseline, relative Frobenius weight change spans 0.017% to 24.6% -- a 1,400x range -- yet the spectral bandwidth shift remains below 0.5% across all four, while the mean principal angle between base and aligned singular-vector bases scales monotonically with training intensity (0.92 deg -> 1.62 deg -> 7.99 deg -> 9.33 deg). Alignment is therefore consistent with a graph isometry: the spectrum is preserved and the singular-vector bases rotate. The earlier 'PR collapse' reports arose from a scalar formulation of participation ratio that is not well-posed: random-token probes yield a 30% coefficient of variation on frozen checkpoints, while fixed prompts yield 0%, and PR varies 2x across semantic query classes on a single model. Participation ratio is a bivariate function PR(model, query); the weight-side bandwidth it attempts to probe is conserved. We release a phased-array probe suite, a rotation analyzer, and four falsifiable conservation predictions for VLM, MoE, long-RL, and continual-alignment regimes; all code and data are at https://github.com/HenryZ838978/spectral-flow-probe, and complete rotation measurements reproduce in minutes on a single GPU.