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What Must Survive? Exact Task-Information--State Frontiers for Resource-Sufficient Learning

arXiv · AI, language, vision and robotics · article · Sep 18, 2026 · UTC

A system may be compressed before its downstream task is fully known. We ask how much retained state is then necessary and how much can be saved by limited advance task information. For a finite family of linear tasks, a task message is revealed before state formation and the exact task only afterwards. For an advice alphabet of size $K$, the exact frontier is \[ p^*(K)= \min_{\substack{\Pcal\text{ partition of }\U\\|\Pcal|\le K}} \max_{C\in\Pcal}\rank(T_C), \] with the $b$-bit frontier obtained by setting $K=\min(2^b,|\U|)$. Thus advance task information reduces state through partitions whose

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