CASTLE: CAUSALLY ATTRIBUTED SOFT CONJUNC TIONS FOR SOUND AND INFORMATIVE SAFETY CERTIFI CATES

A multi-predicate safety specification is almost always relaxed into a differen tiable conjunction, most often by the log-sum-exp conorm. The relaxation is then used as a penalty. Turning it into a certificate — a state predicate a runtime mon itor may act on — additionally requires choosing the value at which the conjunc tion is declared satisfied, and the standard choice, log K/β, is not the tightest admissible one. We make three contributions. (i) For the weighted conorm we derive the minimal admissible robustness threshold τω = log(1/ωk⋆)/β, pinned to the budget mass on the binding predicate; the resulting certificate is sound for every budget and carries no slack beyond the conorm’s own gap, whereas the uniform threshold of prior work adds log(Kωk⋆)/β of pure conservatism. (ii) We show that the quantity that should decide where budget goes is not bindingness but control authority, and introduce IARS, an interventional attribution that measures per-predicate action sensitivity conditional on the causal order. It is identically zero for agent-independent monitors and for restatements of an already-binding predicate. (iii) We prove that the resulting decomposition — a soft conjunction over attributable predicates conjoined with an exact nominal check on the rest — is sound, and we formalise the design principle that robustness margin should be spent only where the agent can act. Across three environments and three seeds, CASTLE raises the activation rate of a sound certificate from 0 .215 to 0 . 381 on a specification containing a non-attributable predicate — at zero certified-but-unsafe steps — while matching or improving the safety–performance frontier; removing the exact monitor reproduces 3253 certified-but-unsafe steps, while the product t-norm certifies 1463 such steps at its natural threshold and the Łukasiewicz cer tificate never activates at all.