# Eidoku: A Neuro-Symbolic Verification Gate for LLM Reasoning via Structural Constraint Satisfaction

arXiv: https://arxiv.org/abs/2512.20664 · HTML: https://mlsanity.com/paper/2512.20664/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Placed in Reasoning benchmarks and evaluation, one of 17 directions in LLM reasoning.
- Builds on 8 of its 17 references that are in the map.
- First posted on arXiv 19 Dec 2025.

## Placed in

- LLM reasoning / [Reasoning benchmarks and evaluation](https://mlsanity.com/lr/reasoning-benchmarks-and-evaluation/)

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

- [Self-Consistency Improves Chain of Thought Reasoning in Language Models](https://mlsanity.com/paper/2203.11171/) (2022)
- [A Survey on Hallucination in Large Language Models: Principles, Taxonomy, Challenges, and Open Questions](https://mlsanity.com/paper/2311.05232/) (2023)
- [SelfCheckGPT: Zero-Resource Black-Box Hallucination Detection for Generative Large Language Models](https://mlsanity.com/paper/2303.08896/) (2023)
- [Chain-of-Verification Reduces Hallucination in Large Language Models](https://mlsanity.com/paper/2309.11495/) (2023)
- [INSIDE: LLMs' Internal States Retain the Power of Hallucination Detection](https://mlsanity.com/paper/2402.03744/) (2024)
- [SAC3: Reliable Hallucination Detection in Black-Box Language Models via Semantic-aware Cross-check Consistency](https://mlsanity.com/paper/2311.01740/) (2023)
- [GraphEval: A Knowledge-Graph Based LLM Hallucination Evaluation Framework](https://mlsanity.com/paper/2407.10793/) (2024)
- [InterrogateLLM: Zero-Resource Hallucination Detection in LLM-Generated Answers](https://mlsanity.com/paper/2403.02889/) (2024)

## Built on by (most cited first, 1 of 1)

- [SymDiag: Explainable Diagnosis for LLM Reasoning via Neuro-Symbolic Verification](https://mlsanity.com/paper/2608.08786/) (2026)

## Abstract (arXiv)

Large Language Models (LLMs) frequently produce hallucinated statements that are assigned high likelihood by the model itself, exposing a fundamental limitation of probability-based verification. This suggests that hallucination is often not a low-confidence phenomenon, but a failure of structural consistency. In this work, we reformulate the verification of LLM reasoning as a Constraint Satisfaction Problem (CSP) operating independently of the generation likelihood. Rather than optimizing for statistical plausibility, we model verification as a feasibility check based on structural violation cost -- the computational cost required to embed a candidate reasoning step into the contextual graph structure. We define a total cost function composed of three proxies: (i) graph connectivity (structural), (ii) feature space consistency (geometric), and (iii) logical entailment (symbolic). Crucially, verification is performed via a lightweight System-2 gate, Eidoku, which rejects candidates exceeding a context-calibrated cost threshold. The threshold is not learned but is derived from the intrinsic statistics of the context, avoiding ad hoc heuristics. We demonstrate that this approach successfully rejects ``smooth falsehoods'' -- statements that are highly probable yet structurally disconnected -- that probability-based verifiers are principally incapable of detecting. Our experiments on a controlled diagnostic dataset show that explicitly enforcing structural constraints allows for the deterministic rejection of this specific class of hallucinations, serving as a neuro-symbolic sanity check for generative reasoning.