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• For power-law risk schedules with p > 1, any classifier-based safety gate cannot achieve unbounded cumulative utility (sum TPR_n = infinity) while maintaining bounded cumulative risk (sum delta_n < infinity).result 95%
• A Lipschitz ball verifier can achieve zero risk (delta = 0) with a positive true positive rate (TPR > 0), escaping the classifier impossibility.result 90%
• The exact finite-horizon utility ceiling for a classifier grows as exp(O(sqrt(log N))) — subpolynomial — and is vastly slower than linear growth.result 90%
• At N = 10^6 steps with risk budget B = 1.0, a classifier extracts at most U* ~ 87 utility, while a verifier can achieve ~500,000 utility.result 85%
• The Hölder exponent beta used in the impossibility bound is minimax-optimal; no valid impossibility bound can use a strictly larger exponent.technical 85%
• Formal Lipschitz bounds for pre-LayerNorm transformers under LoRA enable LLM-scale verification.technical 80%
• A second independent proof via the NP counting method yields a 13% tighter bound than the Hölder-based proof.method 80%
• The finite-horizon utility ceiling U*(N,B) is achieved with equality if and only if the risk is uniformly allocated across steps (delta_n = B/N for all n).technical 75%

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