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Aperture stores Fourier moments of the token's weighted support at the model's rotary frequencies, and it is proved that these moments have minimal real dimension among continuous states sufficient for the selected expected rotary interactions.
This work quantifies the compression cost of boundaries while separating it from the prediction quality of the resulting token units, and introduces boundary licences, which limit the vocabulary entries permitted to cross cuts and admit the same form of certificate.
It is shown that split invariance forces the local mass factor to be linear for any fixed positive attention kernel, provided that factor is nondecreasing.
Three properties of a world model are distinguished: the distribution it approaches, the rate of approach, and the conditional dynamics it learns, which derive an absolute convergence bound from finite initialization banks and control departure from the reference through conditional action-space divergence.
SpatialCLI is proposed, a framework that teaches VLMs to reason with spatial tools and progressively internalize the specialist perceptual capabilities they provide and introduces SpatialCLI-Bench, a 516-example benchmark for compositional perception across localization, segmentation, depth, and pose.
This work introduces A 2 -Judger, a novel MLLM-based A gentic instantiation of A uto Judger equipped with semantic-aware retrieval and dynamic memory that significantly improves sample efficiency while maintaining reliable evaluation results.
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