Multiscale Residual Tokenization for Continuous Spectral Coordinates: Semantics and Physical Stability

Spectral coordinates are continuous, but many foundation-model pipelines require discrete vocabularies. We introduce \textbf{multiscale residual tokenization (MRT)}, which represents each coordinate by level-specific coarse-to-fine residual digits. On frozen NMR Sim$\rightarrow$Exp retrieval, MRT improves $^{13}$C Top-1 from $0.1363$ to \textbf{$0.2006$}, compared with $0.1071$ for fixed-width quantization, and improves $^1$H Top-1 from $0.0141$ to \textbf{$0.0281$}. We use controlled coordinate jitter to identify the perturbation scale at which this representation pays off. MRT reduces degradation relative to adaptive quantization, but the absolute comparison depends on perturbation scale: at the smallest tested jitter, post-perturbation kNN is tied on $^1$H and IR and lower on $^{13}$C; by jitter comparable to the finest residual scale, MRT overtakes adaptive on both NMR coordinates and marginally on IR. Using normalized jitter $\rho=\sigma/\Delta_{\mathrm{fine}}$, at $\rho=1.50$ on $^1$H, $\rho=1.00$ on $^{13}$C, and $\rho=1.00$ on IR, MRT is ahead by $0.0243$, $0.0202$, and $0.0032$ kNN Tanimoto, respectively. Paired bootstrap intervals on the jitter degradation difference exclude zero at every tested NMR and IR jitter level. A packed-token ablation removes most of this advantage, supporting the role of explicit level-wise geometry. These results identify a tradeoff between clean semantic geometry and stability in spectral tokenization.