Separation

The state space model Mamba has recently emerged as a promising paradigm in computer vision, attracting considerable attention for its efficient handling of long-sequence tasks. Its inherent causal structure makes it particularly well suited for autoregressive pretraining. However, existing autoregressive pretraining methods in vision are largely limited to short-sequence settings and may not fully use Mamba's capacity to model longer contexts. To investigate this setting, we introduce SeparaTors for AutoRegressive pretraining (STAR), a new autoregressive pretraining method for Vision Mamba that explicitly marks the boundaries between different images. STAR increases the patch-token sequence length from 144 to 640 by packing four images and four separator clusters. This is approximately $4.4\times$ the ARM patch-token sequence length. The increase is achieved without changing the resolution of any individual image: we use $192\times192$ inputs for autoregressive pretraining and $224\times224$ inputs for downstream classification fine-tuning. With this long-sequence pretraining scheme, STAR-B achieves 83.5% EMA top-1 accuracy on ImageNet-1K after 1,600 epochs of pretraining. The learned representation also transfers beyond in-distribution classification: compared with ARM, STAR-B improves COCO box AP from 46.11 to 46.84 and mask AP from 40.74 to 41.45, while raising the mean top-1 accuracy across five ImageNet robustness benchmarks from 55.1% to 56.8%. Under the evaluated four-image setting, these results indicate that separator-based long-sequence pretraining improves recognition robustness and dense visual prediction relative to ARM.

Language generation in the limit asks for valid unseen elements from every exhaustive positive presentation of an unknown infinite language. We characterize this task for arbitrary families over a countable universe. Generation is possible exactly when each target can be assigned a finite positive witness so that the targets activated by any finite sample have an infinite common intersection. The necessary direction follows from a universal normalization: a search through unconfirmed histories converts any successful generator into one depending only on the observed set. We then ask how large compatible witnesses must be. Positive separation width records the smallest uniform size bound, with two further levels for unbounded finite witnesses and the absence of any compatible finite-witness assignment. Every level occurs. Countable families admit singleton witnesses, explicit families realize every finite width, and a union of two families with infinite common cores requires unbounded finite witnesses. Finally, countable-support and finite-profile obstructions explain why local combinatorial data cannot determine generation in the limit. The characterization and full width hierarchy are checked in Lean, including the simplified normalization and a direct diagonal capture lemma. The accompanying Lean development is maintained at https://github.com/xiaoyulics/language-generation-characterization