The Security Budget of Code-LLM Prompt Hardening: Provable Limits Under Pass-Only Acceptance

Abstract

We give a quantitative impossibility result for pass-only prompt hardening of code LLMs. For any deterministic prompt filter $h$ and a registered family of finite executable-equivalence task variables $\mathcal Y_{\mathrm{exec}}$, the shared filtered-prompt channel $\rmI(h(p);h(\tilde p))$ is lower-bounded by a worst-$Y$ Fano floor; on HumanEval and MBPP the universal pass-only floor evaluates to $\mathcal F^{\mathrm{op}}\ge 0.84$ and $1.20$ nats at $\eta=0.05$ task-collapse tolerance, and the identity row realizes $\mathcal F^{\mathrm{id}}\ge 1.67$ and $1.80$ nats. An estimator-invariance corollary lifts the floor to any deterministic embedding pipeline; a dataset-agnostic corollary states the floor in visible-spec entropy and is empirically witnessed by $164/164$ HumanEval+ and $224/224$ MBPP+ $V(p)$-invariance. We operationalize the floor as the \emph{Tri-Audit Protocol}, a two-axis reporting protocol that separates a prompt-side deductive registry attribute (Shannon nats on the visible-spec representation) from a model-side empirical proxy (KSG-1 primary, MINE secondary, on hidden states). A constrained best-of-family search over deterministic and guarded learned filters on CodeLlama-7B, Qwen2.5-Coder-7B/1.5B and DeepSeek-Coder-6.7B at $n=164$ yields the \emph{Cross-Model Tri-Audit Invariance}: of twenty-eight pass-preserving rows, twelve antecedent-preserving deterministic rows fail proxy-axis leakage reduction on every backbone with sign-invariant positive deviations, twelve antecedent-changed-of-record learned-canonicalizer rows fail proxy-axis leakage on every backbone, and four antecedent-violating rows are reported as registered-family collapse; no filter produces a shared Tri-pass on a nine-cell gate-sensitivity sweep. Pass@1 alone cannot certify code-LLM prompt hardening.