Relational Response Fields: A General Theory of Black-Box LLM Response Consistency and Recovery

Black-box language-model reliability is commonly pursued by sampling, prompting, voting, verifying, or iteratively revising individual answers. We ask a prior question: what determines whether a collection of black-box responses is recoverable at all? We represent responses to typed transformations of a query as a relational response field (RRF). Edge transports encode how valid responses must change under paraphrase, scaling, decomposition, refactoring, or other task symmetries; anchors encode independently trusted evidence such as execution or a verifier. For relation operator D, anchor operator A, and at most k corrupted response nodes, we identify γk(D,A) as the intrinsic difficulty of black-box response recovery. It is positive exactly when every k-node corruption is identifiable; it gives a deterministic stability bound proportional to 1/γk; and a matching two-point minimax lower bound shows that no estimator can improve this dependence. Thus consistency is not truth: relation-only methods are blind to null directions, including shared hallucinations. We derive sparse field-repair algorithms while separating information-theoretic identifiability from the stronger null-space conditions required by convex optimization. Controlled theorem tests and black-box mathematics/code experiments evaluate four theory-fixed consequences: consistency--truth separation, anchor phase transitions, redundancy saturation, and cross-model, cross-task prediction of repair difficulty. The results support γk(D,A) as a measurable property of a response-recovery instance, rather than a score attached to one repair heuristic.