Demand Geometry and Sparse Interfaces for Continuous Music Tokens

Continuous music tokenizers support reconstruction and downstream modeling, but different tasks may benefit from different low-dimensional views of the same tokens. For frozen tokenizers, we study tradeoffs among active input width, shared capacity, and demand-specific utility. A minimax certificate quantifies unavoidable compromise and rules out exact common optimality for the tested reconstruction and prediction operators. Using target-canonical normalization, we define demand relations invariant to invertible linear input and target transformations under fixed whitening supports and ridge strengths. For each tokenizer, Atom Atlas assigns each known demand $K$ of $M$ shared orthonormal atoms. Within a fixed union subspace, chosen dense-view projectors admit exact realization by shared atom subsets if and only if they commute pairwise. Support-aware certificates bound operator utility loss from approximate realization for fitted operators. Demand relations recur across three tokenizers on Moises and two on Slakh with a different teacher. Across six primary MUSDB/Moises settings at $M/K=4$, Atlas retains 84–97% of the dense-over-shared reduction in worst-demand operator regret. With matched nonlinear predictors on Moises, Atlas improves future-token prediction and prediction of frequency-band teacher embeddings over the shared view. On the SAME-L band task, native Atlas retains 85.3% of the improvement from the shared view to native dense, supporting prediction directly from stored atom coordinates.