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A summation-form MI lower bound over all active classes, a cascade critical cost theorem and a per-class budget corollary, nonzero where the uniform-budget bound vanishes, and an accuracy corollary.
Hyper-Fold is introduced, a rank-K separable convolutional backbone approaching this ceiling at message-passing cost, suggesting that a sufficiently expressive 3D backbone recovers information that fusion architectures previously borrowed from evolution-scale pretraining.
Protein structure modeling rests on a single computational primitive: the interaction between what a residue is (sequence content) and where it sits (three-dimensional geometry). What is the expressive limit of this layer class? We show that the complete bilinear operator over content-geometry outer products--the suffi...
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