# Discrete Beckmann Transport Models for One-Step Language Modeling and Reasoning

arXiv: https://arxiv.org/abs/2609.15903 · HTML: https://mlsanity.com/paper/2609.15903/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion reasoning and test-time scaling, one of 17 directions in LLM reasoning.
- Builds on 18 of its 58 references that are in the map.
- First posted on arXiv 14 Sep 2026.

## Placed in

- LLM reasoning / [Diffusion reasoning and test-time scaling](https://mlsanity.com/lr/diffusion-reasoning-and-test-time-scaling/)
- Diffusion models / [Discrete and text diffusion](https://mlsanity.com/diffusion/discrete-and-text-diffusion/)
- Knowledge distillation / [Knowledge distillation for LLMs](https://mlsanity.com/distill/knowledge-distillation-for-llms/)

## Builds on (most cited first, 18 of 18)

- [Training Verifiers to Solve Math Word Problems](https://mlsanity.com/paper/2110.14168/) (2021)
- [CTRL: A Conditional Transformer Language Model for Controllable Generation](https://mlsanity.com/paper/1909.05858/) (2019)
- [One Billion Word Benchmark for Measuring Progress in Statistical Language Modeling](https://mlsanity.com/paper/1312.3005/) (2013)
- [Continuous diffusion for categorical data](https://mlsanity.com/paper/2211.15089/) (2022)
- [TinyGSM: achieving >80% on GSM8k with small language models](https://mlsanity.com/paper/2312.09241/) (2023)
- [Flow Map Language Models: One-step Language Modeling via Continuous Denoising](https://mlsanity.com/paper/2602.16813/) (2026)
- [ELF: Embedded Language Flows](https://mlsanity.com/paper/2605.10938/) (2026)
- [LangFlow: Continuous Diffusion Rivals Discrete in Language Modeling](https://mlsanity.com/paper/2604.11748/) (2026)
- [Discrete Flow Maps](https://mlsanity.com/paper/2604.09784/) (2026)
- [SplitMeanFlow: Interval Splitting Consistency in Few-Step Generative Modeling](https://mlsanity.com/paper/2507.16884/) (2025)
- [Language Modeling with Hyperspherical Flows](https://mlsanity.com/paper/2605.11125/) (2026)
- [Continuous Diffusion Scales Competitively with Discrete Diffusion for Language](https://mlsanity.com/paper/2605.18530/) (2026)
- [α-Flow: A Unified Framework for Continuous-State Discrete Flow Matching Models](https://mlsanity.com/paper/2504.10283/) (2025)
- [Gumbel-Softmax Flow Matching with Straight-Through Guidance for Controllable Biological Sequence Generation](https://mlsanity.com/paper/2503.17361/) (2025)
- [Posterior Refinement: Fast Language Generation via Any-Order Flow Maps](https://mlsanity.com/paper/2606.24773/) (2026)
- [DeltaFlow: Noise-Adaptive Bidirectional Gated Delta Networks for Embedded Language Flows](https://mlsanity.com/paper/2608.01240/) (2026)
- [Beckmann Transport Models: From Autonomous Flows to One-Step Maps](https://mlsanity.com/paper/2608.01692/) (2026)
- [Expanding Flow Maps](https://mlsanity.com/paper/2607.21585/) (2026)

## Built on by (most cited first, 3 of 3)

- [ELF-REG: Scaling Continuous Diffusion Language Models to Reasoning Tasks](https://mlsanity.com/paper/2609.29102/) (2026)
- [Reasoning with Continuous Latent Diffusion](https://mlsanity.com/paper/2609.35694/) (2026)
- [Alpha Diffusion Language Models: Factorization Alone Is Not the Problem](https://mlsanity.com/paper/2609.38066/) (2026)

## Abstract (arXiv)

Discrete diffusion and flow models are a promising alternative to autoregressive language models, but compressing many-step sampling into fewer steps typically requires distilling a pretrained teacher model. This caps the student at the teacher's quality and requires a costly two-stage training pipeline. We introduce Discrete Beckmann Transport Models (DBTM), built on a time-independent flow whose autonomous transport map provably carries any point in the ambient space to a fixed point on the vertices of the simplex in a single step. We show that this fixed-point property is characterized by a conservation equation whose residual can be minimized directly from data, removing the requirement for a teacher flow and time conditioning. Under this construction, a partially trained map corresponds to the flow truncated at finite time, so generation reduces to iterating one map until it reaches a fixed point. We further extend the map to a partial-context interpolant where additional function evaluations act as refinement steps rather than ODE integration steps. On language modeling and reasoning tasks, DBTM enables one- and few-step generation that improves quality and accuracy over discrete diffusion and continuous flow baselines.