Reasoning on the Manifold: Bidirectional Consistency for Self-Verification in Diffusion Language Models

TL;DR

This paper introduces a geometric, training-free metric called Bidirectional Manifold Consistency (BMC) to verify and improve the reasoning accuracy of diffusion large language models by assessing the stability of generated sequences.

Contribution

It proposes a novel, unsupervised, geometric approach to verify and enhance reasoning in diffusion language models without additional training.

Findings

BMC effectively discriminates valid solutions without ground truth.

It enables rejection resampling to focus on complex reasoning tasks.

It provides dense geometric rewards for self-evolving models.

Abstract

While Diffusion Large Language Models (dLLMs) offer structural advantages for global planning, efficiently verifying that they arrive at correct answers via valid reasoning traces remains a critical challenge. In this work, we propose a geometric perspective: Reasoning on the Manifold. We hypothesize that valid generation trajectories reside as stable attractors on the high-density manifold of the learned distribution, whereas invalid paths exhibit off-manifold drift. To operationalize this, we introduce Bidirectional Manifold Consistency (BMC), a training-free, unsupervised metric that quantifies the stability of the generated sequence through a forward-masking and backward-reconstruction cycle. Empirically, we demonstrate BMC's versatility across the full reasoning lifecycle: (1) in Diagnosis, it serves as a robust discriminator of solution validity without ground truth answer; (2) in Inference, it enables rejection resampling to effectively concentrate computational resources on complex reasoning tasks; and (3) in Alignment, it functions as a dense geometric reward that transforms sparse outcome supervision into fine-grained guidance, empowering models to self-evolve beyond standard baselines. Our results establish intrinsic geometric stability as a robust indicator of correctness for dLLMs.