# Backpropagating Verifier Rewards Through Continuous Flow Language Models

Authors: Xu Han, Jiajing Hu, Li-Ping Liu.

Trains a critic to score intermediate continuous token representations, enabling reward gradients through the denoising trajectory before final discrete decoding. Evaluates this post-training approach on four reasoning tasks.

First public: 2026-10-13.
ArXiv initial submission: 2026-10-13.

[Original paper](https://arxiv.org/abs/2610.19080) · [Reading-list entry](https://latent-implicit-thinking.github.io/papers/2610.19080/)
[Preprint BibTeX](https://latent-implicit-thinking.github.io/papers/2610.19080/citation.bib) · [Preprint CSL-JSON](https://latent-implicit-thinking.github.io/papers/2610.19080/citation.csl.json) · [JSON record](https://latent-implicit-thinking.github.io/papers/2610.19080/record.json)

## Approaches

- [Continuous-state reasoning](https://latent-implicit-thinking.github.io/topics/continuous-states/): Use vectors—lists of numbers—to carry intermediate reasoning from one step to the next, instead of selecting a text token at every step. Some methods use the model's hidden states; others combine several candidate token representations.
- [Alternative generation methods](https://latent-implicit-thinking.github.io/topics/alternative-reasoning-schedules/): Change how intermediate steps are generated, revised, or searched—for example, by refining several steps in parallel. These are related approaches: some still produce written chains of thought.

Summary and categories are editorial. Read the original paper to verify claims and limitations. A citation to a paper does not, by itself, establish that a later method builds on it.

Reading-list version 0.9.5; literature checked through 2026-10-18.