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Infinite language generation from positive examples succeeds if and only if finite witnesses preserve an infinite common core, unlocking an infinite complexity hierarchy verified entirely in Lean.
Even when an input activates only $k$ of $s$ hidden units, sample complexity cannot escape the total parameter count, incurring an inescapable $\tilde{\Theta}(\sqrt{sk/m})$ rate while eliminating all explicit dependencies on input dimension.
Closing the gap in multiclass PAC learning reveals that optimal excess risk can be achieved without prior knowledge of oracle risk, fundamentally shifting our understanding of classifier performance.
