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

arXiv: https://arxiv.org/abs/2608.04552 · HTML: https://mlsanity.com/paper/2608.04552/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 58 references that are in the map.
- First posted on arXiv 5 Aug 2026.

## Builds on (most cited first, 8 of 8)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [Evaluating Large Language Models Trained on Code](https://mlsanity.com/paper/2107.03374/) (2021)
- [Qwen2.5 Technical Report](https://mlsanity.com/paper/2412.15115/) (2024)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Code Llama: Open Foundation Models for Code](https://mlsanity.com/paper/2308.12950/) (2023)
- [Phi-3 Technical Report: A Highly Capable Language Model Locally on Your Phone](https://mlsanity.com/paper/2404.14219/) (2024)
- [StarCoder 2 and The Stack v2: The Next Generation](https://mlsanity.com/paper/2402.19173/) (2024)
- [Metamorphic Testing: A New Approach for Generating Next Test Cases](https://mlsanity.com/paper/2002.12543/) (2020)

## Abstract (arXiv)

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.