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

TL;DR

Eidoku introduces a neuro-symbolic verification method for LLM reasoning that detects hallucinations by checking structural consistency rather than relying on probability, improving the detection of false statements.

Contribution

This work reformulates LLM verification as a constraint satisfaction problem using a lightweight System-2 gate, avoiding heuristics and enhancing hallucination detection.

Findings

Successfully rejects structurally disconnected false statements

Enables deterministic rejection of high-probability hallucinations

Provides a neuro-symbolic sanity check for reasoning

Abstract

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.