异步流映射语言模型

# Asynchronous Flow Map Language Models
Source: [https://arxiv.org/html/2610.42018](https://arxiv.org/html/2610.42018)
Wonjung KimAffiliation:KAISTPatrick PynadathAffiliation:Purdue UniversityJaehoon YooAffiliation:KAISTSatoshi Hayakawa Ruqi Zhang Jinwoo Kim Seunghoon HongAffiliation:Purdue UniversityAffiliation:The University of TokyoAffiliation:Equal advising

###### Abstract

The order in which tokens are predicted in language modeling has a profound impact on the quality of generated sequences\. Autoregressive and masked diffusion models iteratively fill in tokens and can exploit such orderings across generation steps, but how to leverage an ordering for one\-step parallel language modeling is unclear\. We introduce*a*synchronous flow map language models \(aFMLMs\), which incorporate prediction order into one\-step modeling by learning the flow maps of asynchronous flows\. These flows specify an ordering through a tokenwise time schedule that controls the relative denoising progress of tokens\. We formulate a general class of asynchronous flows that admit both prescribed and learned, state\-adaptive schedules, show that they define valid flow maps, and develop new methods for their flow map distillation\. The resulting models can generate all tokens in a single forward pass by internalizing the ordering encoded in the flow into their hidden computations\. On Sudoku solving and code generation for GSM8K math word problems,aFMLM establishes new state\-of\-the\-art one\-step performance111Code is available at[https://github\.com/david3684/async\-fmlm](https://github.com/david3684/async-fmlm)\.\.

## 1Introduction

How tokens are ordered and scheduled for prediction has a significant impact on language modeling\([Shannon, 1951](https://arxiv.org/html/2610.42018#bib.bib20);[Papadopoulos et al\., 2024](https://arxiv.org/html/2610.42018#bib.bib37);[Kim et al\., 2025a](https://arxiv.org/html/2610.42018#bib.bib15);[Nie et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib34)\), as it determines how sequence prediction is decomposed into a series of simpler modeling subproblems\. For instance, autoregressive models \(ARMs\)\([Bengio et al\., 2003](https://arxiv.org/html/2610.42018#bib.bib19)\)encode an ordering as a prescribed left\-to\-right factorization, whereas masked diffusion models \(MDMs\)\([Sahoo et al\., 2024](https://arxiv.org/html/2610.42018#bib.bib27);[Shi et al\., 2024](https://arxiv.org/html/2610.42018#bib.bib28)\)learn over diverse masking patterns and choose orderings for generation based on model outputs\. These choices would not matter for perfect models, as any factorization recovers the true joint distribution\. In practice, however, models have finite capacity and some subproblems are harder to learn than others, making generation quality sensitive to the chosen order and schedule\([Finzi et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib12)\)\.

Because these orderings and schedules are defined over generation steps, in which tokens decided earlier inform later ones, iterative models such as ARMs and MDMs can naturally use them\. In contrast, parallel language models designed for fast generation take only a few steps and must perform more of their computation in hidden space, where it is unclear how to leverage a given ordering or schedule\. This problem is most severe in one\-step generation\([Lee et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib1)\), which predicts all tokens in a single forward pass and thus leaves no steps over which an order can be followed\.

We address this problem with*a*synchronous flow map language models \(aFMLMs\)that incorporate prediction order into one\-step generation\. Our idea builds on flow map language models \(FMLMs\)\([Lee et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib1)\), which achieve one\-step generation by compressing the sampling trajectories of continuous flow language models \(FLMs\)\([Lee et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib1);[Hu et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib7);[Chen et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib6)\)\. While existing FMLMs use synchronous flows in which all tokens progress together,aFMLMs learn the flow maps ofasynchronous flows, in which tokenwise times control the relative progress of tokens to specify an ordering or schedule\. Building on prior work\([Qian and Cheng, 2026](https://arxiv.org/html/2610.42018#bib.bib11);[Bansal and Sanghavi, 2026](https://arxiv.org/html/2610.42018#bib.bib8)\), we formulate a general class of asynchronous flows that admits both prescribed and learned, state\-adaptive schedules\. By treating tokenwise times as part of the state, we show that these flows define valid flow maps, and develop methods to distill them\. The resulting models generate all tokens in a single forward pass, internalizing the ordering encoded in the flow into their hidden computation\.

Figure 1:Overview\.\(Left\) We formulate asynchronous flows that denoise tokens at different rates\. To rhyme with “mitten,” “kitten” emerges early and guides the preceding words\. \(Right\) We distill these flows into an asynchronous flow map that internalizes this ordering and generates the full sequence in one forward pass\.An illustration of our approach is shown in[Figure1](https://arxiv.org/html/2610.42018#S1.F1)\. To summarize, our main contributions are:

- •Asynchronous flows\.We formulate the generation of asynchronous flow language models \(*a*FLMs\) as a coupled flow of tokens and their tokenwise times governed by a scheduler, covering prescribed and learned, state\-adaptive schedules ported from MDM samplers \([Sections4\.1](https://arxiv.org/html/2610.42018#S4.SS1)and[4\.1](https://arxiv.org/html/2610.42018#S4.SS1)\)\.
- •Asynchronous flow maps\.We show that asynchronous flows define a valid flow map and derive distillation objectives that learnaFMLMs as one\-step generators, in which a single model jointly predicts how the tokenwise times and the tokens progress \([Sections4\.2](https://arxiv.org/html/2610.42018#S4.SS2)and[4\.3](https://arxiv.org/html/2610.42018#S4.SS3)\)\.
- •One\-step generation\.We trainaFMLMs that achieve state\-of\-the\-art one\-step performance on Sudoku solving and coding for GSM8K math problems, and analyze their hidden computations, showing that they internalize the effect of ordering and scheduling into one forward pass \([Section5](https://arxiv.org/html/2610.42018#S5)\)\.

## 2Related Work

#### Prediction order in sequence modeling\.

Although every factorization of a joint distribution yields the same likelihood, the quality of sequence models depends on the order of prediction, a gap attributed to their computational limits\([Finzi et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib12)\)\. For ARMs, learning English in the left\-to\-right order leads to a higher likelihood than in reverse order\([Shannon, 1951](https://arxiv.org/html/2610.42018#bib.bib20);[Papadopoulos et al\., 2024](https://arxiv.org/html/2610.42018#bib.bib37)\), reversing digits makes arithmetic easier to learn\([Lee et al\., 2023](https://arxiv.org/html/2610.42018#bib.bib17)\), and learning Sudoku solving in the order given by a logical solver outperforms the row\-by\-row order\([Shah et al\., 2024](https://arxiv.org/html/2610.42018#bib.bib18)\)\. In MDMs, adaptive samplers that choose the ordering from model outputs often outperform random ordering\([Nie et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib34);[Ben\-Hamu et al\., 2025](https://arxiv.org/html/2610.42018#bib.bib16);[Fathi et al\., 2025](https://arxiv.org/html/2610.42018#bib.bib21);[Kim et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib36)\)\. MDMs with adaptive samplers have been distilled into fewer steps\([Chen et al\., 2025](https://arxiv.org/html/2610.42018#bib.bib48);[Kim et al\., 2025b](https://arxiv.org/html/2610.42018#bib.bib49);[Liang et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib50)\), but along discrete trajectories that make very few\-step generation hard\([Lee et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib1)\)\.

#### Asynchronous diffusion and flows\.

Asynchronous \(continuous\) diffusion models denoise different components at different noise levels\. Most prescribe a schedule ahead of generation, such as left\-to\-right for text\([Wu et al\., 2023](https://arxiv.org/html/2610.42018#bib.bib35);[Schiff et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib47);[Chavan et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib39)\), sliding window for video and fluid dynamics\([Ruhe et al\., 2024](https://arxiv.org/html/2610.42018#bib.bib51)\), offsets between image semantics and texture\([Pan et al\., 2025](https://arxiv.org/html/2610.42018#bib.bib42);[Qian and Cheng, 2026](https://arxiv.org/html/2610.42018#bib.bib11);[Baade et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib10)\), or algorithmic orders for satisfiability problems\([Bhatt et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib13)\)\. Diffusion forcing\([Chen et al\., 2024](https://arxiv.org/html/2610.42018#bib.bib14)\)uses per\-token independent noise levels for training so that schedules can be chosen at inference, such as learned ones that depend on model outputs\([Wewer et al\., 2025](https://arxiv.org/html/2610.42018#bib.bib29);[Hu et al\., 2025](https://arxiv.org/html/2610.42018#bib.bib46);[Schusterbauer et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib45)\)\. Few\-step distillation of asynchronous models, however, has been limited to prescribed schedules\([Yin et al\., 2024](https://arxiv.org/html/2610.42018#bib.bib43);[Huang et al\., 2025](https://arxiv.org/html/2610.42018#bib.bib44);[Chavan et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib39)\)or their close variants\([Bansal and Sanghavi, 2026](https://arxiv.org/html/2610.42018#bib.bib8)\), whereas we consider both prescribed and learned schedules\.

#### Ordering in flow language models\.

Most continuous FLMs evolve all tokens with a single shared time\([Lee et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib1);[Chen et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib6);[Hu et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib7)\), and their flow maps compress the resulting trajectories\([Lee et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib1);[Roos et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib22);[Potaptchik et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib23)\)\. Outside prescribed left\-to\-right ordering\([Wu et al\., 2023](https://arxiv.org/html/2610.42018#bib.bib35);[Schiff et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib47);[Chavan et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib39)\), order has been introduced into FLMs by finalizing tokens during generation, either stochastically\([Pynadath et al\., 2025](https://arxiv.org/html/2610.42018#bib.bib2);[Jutras\-Dubé et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib5)\)or once the model is confident\([Azangulov et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib40);[Agarwal et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib3)\)\. These samplers, however, rely on stochastic updates or restarts with fresh noise, which complicates flow map distillation requiring deterministic trajectories\.[Rekesh et al\. \(2026\)](https://arxiv.org/html/2610.42018#bib.bib9)adjust tokenwise step sizes using model outputs, which our framework can express as a learned scheduler and distill\.[Bansal and Sanghavi \(2026\)](https://arxiv.org/html/2610.42018#bib.bib8)select among a set of prescribed schedules using model outputs at the first generation step, allowing the use of flow map distillation methods for prescribed schedules\. Our approach goes beyond this workaround and handles fully learned schedules that use all intermediate outputs\.

## 3Preliminaries

#### Flow language models\.

Continuous flow language models\([Lee et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib1);[Hu et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib7);[Chen et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib6)\)model categorical token sequences by learning a continuous flow from noise to continuous token embeddings\. LetVVbe a token vocabulary and𝐲=\(𝐲i\)i=1L∈VL\{\\bf y\}=\(\{\\bf y\}^\{i\}\)\_\{i=1\}^\{L\}\\in V^\{L\}a sequence\. Following[Lee et al\. \(2026\)](https://arxiv.org/html/2610.42018#bib.bib1), we represent each token by its one\-hot encoding,𝐱1=\(𝗈𝗇𝖾𝗁𝗈𝗍⁡\(𝐲i\)\)i=1L∈ℝL×\|V\|\{\\bf x\}\_\{1\}=\(\\mathsf\{onehot\}\(\{\\bf y\}^\{i\}\)\)\_\{i=1\}^\{L\}\\in\\mathbb\{R\}^\{L\\times\|V\|\}\. Letp1p\_\{1\}denote the distribution of these one\-hot representations andp0=𝖭⁡\(0,I\)p\_\{0\}=\\mathsf\{N\}\(0,I\)the noise distribution\. We define a probability path by interpolating between independent draws𝐱0∼p0\{\\bf x\}\_\{0\}\\sim p\_\{0\}and𝐱1∼p1\{\\bf x\}\_\{1\}\\sim p\_\{1\}:

It≔\(1−t\)​𝐱0\+t​𝐱1,pt≔𝖫𝖺𝗐⁡\(It\),t∈\[0,1\]\.I\_\{t\}\\coloneqq\(1\-t\)\{\\bf x\}\_\{0\}\+t\{\\bf x\}\_\{1\},\\qquad p\_\{t\}\\coloneqq\\mathsf\{Law\}\(I\_\{t\}\),\\qquad t\\in\[0,1\]\.\(1\)The probability path admits a deterministic representation,

𝐱˙t=bt​\(𝐱t\),bt​\(𝐱\)=𝔼⁡\[dd​t​It\|It=𝐱\]=𝔼⁡\[𝐱1−𝐱0\|It=𝐱\],\\dot\{\\bf x\}\_\{t\}=b\_\{t\}\(\{\\bf x\}\_\{t\}\),\\qquad b\_\{t\}\(\{\\bf x\}\)=\\mathbb\{E\}\\left\[\\frac\{\{\\rm d\}\}\{\{\\rm d\}t\}I\_\{t\}\\middle\|I\_\{t\}=\{\\bf x\}\\right\]=\\mathbb\{E\}\[\{\\bf x\}\_\{1\}\-\{\\bf x\}\_\{0\}\|I\_\{t\}=\{\\bf x\}\],\(2\)wherebtb\_\{t\}is the velocity field of the probability flow\. Instead of predicting the velocity itself, it is common to learn the denoiser, which predicts the posterior mean of the clean data:

Dt​\(𝐱\)≔𝔼⁡\[𝐱1\|It=𝐱\],bt​\(𝐱\)=Dt​\(𝐱\)−𝐱1−t\.D\_\{t\}\(\{\\bf x\}\)\\coloneqq\\mathbb\{E\}\[\{\\bf x\}\_\{1\}\|I\_\{t\}=\{\\bf x\}\],\\qquad b\_\{t\}\(\{\\bf x\}\)=\\frac\{D\_\{t\}\(\{\\bf x\}\)\-\{\\bf x\}\}\{1\-t\}\.\(3\)Since𝐱1\{\\bf x\}\_\{1\}is one\-hot,Dti​\(𝐱\)D\_\{t\}^\{i\}\(\{\\bf x\}\)is a probability distribution overVVat each positionii, and the denoiser can be learned by minimizing a tokenwise cross\-entropy lossℒ𝖢𝖤​\(D^\)≔𝔼t,𝐱0,𝐱1​∑i\[−log⁡D^ti​\(It\)\]𝐲i\\mathcal\{L\}\_\{\\mathsf\{CE\}\}\(\\hat\{D\}\)\\coloneqq\\mathbb\{E\}\_\{t,\{\\bf x\}\_\{0\},\{\\bf x\}\_\{1\}\}\\sum\_\{i\}\[\-\\log\\hat\{D\}\_\{t\}^\{i\}\(I\_\{t\}\)\]\_\{\{\\bf y\}^\{i\}\}\([Dieleman et al\., 2022](https://arxiv.org/html/2610.42018#bib.bib24);[Eijkelboom et al\., 2024](https://arxiv.org/html/2610.42018#bib.bib25);[Lee et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib1)\)\.

#### Flow map language models\.

The deterministic flow given in[Equation2](https://arxiv.org/html/2610.42018#S3.E2)defines a unique flow mapXs,t:ℝL×\|V\|→ℝL×\|V\|X\_\{s,t\}:\\mathbb\{R\}^\{L\\times\|V\|\}\\rightarrow\\mathbb\{R\}^\{L\\times\|V\|\}, the solution operator of the flow𝐱t=Xs,t​\(𝐱s\)\{\\bf x\}\_\{t\}=X\_\{s,t\}\(\{\\bf x\}\_\{s\}\), satisfying

Xs,t​\(𝐱\)=𝐱\+∫stbr​\(Xs,r​\(𝐱\)\)​𝑑r=𝐱\+\(t−s\)​vs,t​\(𝐱\),X\_\{s,t\}\(\{\\bf x\}\)=\{\\bf x\}\+\\int\_\{s\}^\{t\}b\_\{r\}\(X\_\{s,r\}\(\{\\bf x\}\)\)\{\\rm d\}r=\{\\bf x\}\+\(t\-s\)v\_\{s,t\}\(\{\\bf x\}\),\(4\)wherevs,tv\_\{s,t\}is the average velocity betweenssandtt, satisfyingvs,s=bsv\_\{s,s\}=b\_\{s\}\. By learning the flow map, we can jump between arbitrary times, and evaluating it withs=0s=0andt=1t=1gives a one\-step jump from noise to data\. Recent work on learning flow maps over categorical data\([Roos et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib22);[Lee et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib1);[Potaptchik et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib23)\)parameterizes it via the two\-time denoiserδs,t:ℝL×\|V\|→\(Δ\|V\|−1\)L\\delta\_\{s,t\}:\\mathbb\{R\}^\{L\\times\|V\|\}\\rightarrow\(\\Delta^\{\|V\|\-1\}\)^\{L\},

δs,t​\(𝐱\)≔𝐱\+\(1−s\)​vs,t​\(𝐱\),\\delta\_\{s,t\}\(\{\\bf x\}\)\\coloneqq\{\\bf x\}\+\(1\-s\)v\_\{s,t\}\(\{\\bf x\}\),\(5\)which extrapolates the flow map displacements→ts\\to ttos→1s\\to 1and satisfiesδs,s​\(𝐱\)=Ds​\(𝐱\)\\delta\_\{s,s\}\(\{\\bf x\}\)=D\_\{s\}\(\{\\bf x\}\)\. Since it is simplex\-valued,δs,t\\delta\_\{s,t\}can be learned by self\-distillation with KL divergence\([Lee et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib1)\),

ℒ𝖪𝖫\(δ^\)=𝔼t,𝐱0,𝐱1\[𝖪𝖫\(Dt\(It\)∥δ^t,t\(It\)\)\]\+𝔼s,u,t,𝐱0,𝐱1\[𝖪𝖫\(δ¯s,t∥δ^s,t\(Is\)\)\],\\mathcal\{L\}\_\{\\mathsf\{KL\}\}\(\\hat\{\\delta\}\)=\\mathbb\{E\}\_\{t,\{\\bf x\}\_\{0\},\{\\bf x\}\_\{1\}\}\\left\[\\mathsf\{KL\}\(D\_\{t\}\(I\_\{t\}\)\\,\\\|\\,\\hat\{\\delta\}\_\{t,t\}\(I\_\{t\}\)\)\\right\]\+\\mathbb\{E\}\_\{s,u,t,\{\\bf x\}\_\{0\},\{\\bf x\}\_\{1\}\}\\left\[\\mathsf\{KL\}\(\\bar\{\\delta\}\_\{s,t\}\\,\\\|\\,\\hat\{\\delta\}\_\{s,t\}\(I\_\{s\}\)\)\\right\],\(6\)where, fors<u<ts<u<tand with the stop\-gradient operator𝗌𝗀⁡\(⋅\)\\mathsf\{sg\}\(\\cdot\),

δ¯s,t=𝗌𝗀⁡\(γ​δ^s,u​\(Is\)\+\(1−γ\)​δ^u,t​\(X^s,u​\(Is\)\)\),γ=\(1−t\)​\(u−s\)\(1−u\)​\(t−s\)∈\[0,1\],\\bar\{\\delta\}\_\{s,t\}=\\mathsf\{sg\}\(\\gamma\\,\\hat\{\\delta\}\_\{s,u\}\(I\_\{s\}\)\+\(1\-\\gamma\)\\,\\hat\{\\delta\}\_\{u,t\}\(\\hat\{X\}\_\{s,u\}\(I\_\{s\}\)\)\),\\qquad\\gamma=\\textstyle\{\\frac\{\(1\-t\)\(u\-s\)\}\{\(1\-u\)\(t\-s\)\}\}\\in\[0,1\],\(7\)and we denote byX^s,t​\(𝐱\)=1−t1−s​𝐱\+t−s1−s​δ^s,t​\(𝐱\)\\hat\{X\}\_\{s,t\}\(\{\\bf x\}\)=\\frac\{1\-t\}\{1\-s\}\{\\bf x\}\+\\frac\{t\-s\}\{1\-s\}\\hat\{\\delta\}\_\{s,t\}\(\{\\bf x\}\)the flow map induced by the learned denoiserδ^\\hat\{\\delta\}\.

## 4Asynchronous Flow Maps

We first formulate asynchronous flows under a scheduler, then distill them into asynchronous flow maps for one\-step generation\.

### 4\.1Asynchronous flows with schedulers

To allow token positions to denoise at different rates, we assign each position its own time:

τ=\(τ1,…,τL\)∈\[0,1\]L,\\tau=\(\\tau^\{1\},\\ldots,\\tau^\{L\}\)\\in\[0,1\]^\{L\},\(8\)whereτi\\tau^\{i\}measures the denoising progress of positionii\([Chen et al\., 2024](https://arxiv.org/html/2610.42018#bib.bib14)\)\. Extending the standard interpolant[Equation1](https://arxiv.org/html/2610.42018#S3.E1), we associateτ=𝟎\\tau=\{\\bf 0\}with noise andτ=𝟏\\tau=\{\\bf 1\}with clean data\. For independent draws𝐱𝟎∼p𝟎≔p0\{\\bf x\}\_\{\\bf 0\}\\sim p\_\{\\bf 0\}\\coloneqq p\_\{0\}and𝐱𝟏∼p𝟏≔p1\{\\bf x\}\_\{\\bf 1\}\\sim p\_\{\\bf 1\}\\coloneqq p\_\{1\}, we define the*asynchronous interpolant*\([Qian and Cheng, 2026](https://arxiv.org/html/2610.42018#bib.bib11)\)

Iτ≔\(𝟏−τ\)​𝐱𝟎\+τ​𝐱𝟏,pτ≔𝖫𝖺𝗐⁡\(Iτ\),I\_\{\\tau\}\\coloneqq\(\\mathbf\{1\}\-\\tau\)\{\\bf x\}\_\{\\bf 0\}\+\\tau\{\\bf x\}\_\{\\bf 1\},\\qquad p\_\{\\tau\}\\coloneqq\\mathsf\{Law\}\(I\_\{\\tau\}\),\(9\)where we write products and divisions of tokenwise vectors as\(τ​𝐱\)i=τi​𝐱i\(\\tau\{\\bf x\}\)^\{i\}=\\tau^\{i\}\{\\bf x\}^\{i\}and\(𝐱/τ\)i=𝐱i/τi\(\{\\bf x\}/\\tau\)^\{i\}=\{\\bf x\}^\{i\}/\\tau^\{i\}\.

#### Schedules\.

In standard flows[Equation2](https://arxiv.org/html/2610.42018#S3.E2), generation moves from noise to data as a shared time advances from zero to one, and we only need to choose step sizes to perform generation\. With tokenwise times[Equation8](https://arxiv.org/html/2610.42018#S4.E8), we can think of generation as the analogous transition ofτ∈\[0,1\]L\\tau\\in\[0,1\]^\{L\}from𝟎\{\\bf 0\}to𝟏\{\\bf 1\}\. Yet, unlike in standard flows, this further leaves a choice of how the tokens progress, as some may advance earlier than others, or all may advance together\. These choices give many distinct paths of generation:

###### Proposition 1\.

Up to continuous nondecreasing reparameterization, there is a unique continuous nondecreasing path from zero to one\. In contrast, there are infinitely many such paths from𝟎\\mathbf\{0\}to𝟏\\mathbf\{1\}in\[0,1\]L\[0,1\]^\{L\}forL≥2L\\geq 2\.

A proof is in[SectionA\.1](https://arxiv.org/html/2610.42018#A1.SS1)\.

To describe the progress of generation along such a path, we introduce a scalar*base time*t∈\[0,1\]t\\in\[0,1\]and call the resulting evolution of tokenwise timesτt∈\[0,1\]L\\tau\_\{t\}\\in\[0,1\]^\{L\}fromτ0=𝟎\\tau\_\{0\}=\{\\bf 0\}toτ1=𝟏\\tau\_\{1\}=\{\\bf 1\}a*schedule*\. We specify this evolution through a*scheduler*ωt:ℝL×\|V\|×\[0,1\]L→ℝ\+L\\omega\_\{t\}:\\mathbb\{R\}^\{L\\times\|V\|\}\\times\[0,1\]^\{L\}\\to\\mathbb\{R\}\_\{\+\}^\{L\}that sets the tokenwise rates:

τ˙t=ωt​\(𝐱t,τt\),τ0=𝟎,\\dot\{\\tau\}\_\{t\}=\\omega\_\{t\}\(\{\\bf x\}\_\{t\},\\tau\_\{t\}\),\\qquad\\tau\_\{0\}=\\mathbf\{0\},\(10\)where𝐱t\{\\bf x\}\_\{t\}denotes the current token states\. We choose the scheduler so thatτt→𝟏\\tau\_\{t\}\\to\\mathbf\{1\}ast↑1t\\uparrow 1\.

#### Velocity of token states\.

We now derive the velocity of token states under a chosen scheduler by extending[Equation2](https://arxiv.org/html/2610.42018#S3.E2)\. For each pair of noise𝐱𝟎\{\\bf x\}\_\{\\bf 0\}and data𝐱𝟏\{\\bf x\}\_\{\\bf 1\}, we advance the tokenwise times by evaluating the scheduler on the interpolant,τ˙t=ωt​\(Iτt,τt\)\\dot\{\\tau\}\_\{t\}=\\omega\_\{t\}\(I\_\{\\tau\_\{t\}\},\\tau\_\{t\}\)\. Sincedd​t​Iτt=τ˙t​\(𝐱𝟏−𝐱𝟎\)\\frac\{\{\\rm d\}\}\{\{\\rm d\}t\}I\_\{\\tau\_\{t\}\}=\\dot\{\\tau\}\_\{t\}\(\{\\bf x\}\_\{\\bf 1\}\-\{\\bf x\}\_\{\\bf 0\}\), the conditional mean

btω\(𝐱,τ\)≔𝔼\[𝐱𝟏−𝐱𝟎∣Iτt=𝐱,τt=τ\]b^\{\\omega\}\_\{t\}\(\{\\bf x\},\\tau\)\\coloneqq\\mathbb\{E\}\[\{\\bf x\}\_\{\\bf 1\}\-\{\\bf x\}\_\{\\bf 0\}\\mid I\_\{\\tau\_\{t\}\}=\{\\bf x\},\\,\\tau\_\{t\}=\\tau\]\(11\)gives the*asynchronous velocity*of token states,τ˙t​btω=ωt​btω\\dot\{\\tau\}\_\{t\}b^\{\\omega\}\_\{t\}=\\omega\_\{t\}b^\{\\omega\}\_\{t\}\. Together with the scheduler, it defines a valid probability flow from noise to data, which we refer to as an*asynchronous flow*:

###### Proposition 2\.

Let\(𝐱t,τt\)\(\{\\bf x\}\_\{t\},\\tau\_\{t\}\)solve, starting from𝐱0∼p𝟎\{\\bf x\}\_\{0\}\\sim p\_\{\\bf 0\}andτ0=𝟎\\tau\_\{0\}=\{\\bf 0\},

𝐱˙t=ωt​\(𝐱t,τt\)​btω​\(𝐱t,τt\),τ˙t=ωt​\(𝐱t,τt\)\.\\dot\{\\bf x\}\_\{t\}=\\omega\_\{t\}\(\{\\bf x\}\_\{t\},\\tau\_\{t\}\)\\,b^\{\\omega\}\_\{t\}\(\{\\bf x\}\_\{t\},\\tau\_\{t\}\),\\qquad\\dot\{\\tau\}\_\{t\}=\\omega\_\{t\}\(\{\\bf x\}\_\{t\},\\tau\_\{t\}\)\.Then\(𝐱t,τt\)\(\{\\bf x\}\_\{t\},\\tau\_\{t\}\)has the same joint distribution as the interpolant and its tokenwise times advanced byω\\omegafor everyt∈\[0,1\)t\\in\[0,1\), and𝐱t\{\\bf x\}\_\{t\}converges in distribution top𝟏p\_\{\\bf 1\}ast↑1t\\uparrow 1\.

A proof is in[SectionA\.2](https://arxiv.org/html/2610.42018#A1.SS2)\.

Mirroring[Equation3](https://arxiv.org/html/2610.42018#S3.E3), we can parameterize the velocity through the corresponding*asynchronous denoiser*:

Dtω\(𝐱,τ\)≔𝔼\[𝐱𝟏∣Iτt=𝐱,τt=τ\],btω\(𝐱,τ\)=Dtω​\(𝐱,τ\)−𝐱𝟏−τ\.D^\{\\omega\}\_\{t\}\(\{\\bf x\},\\tau\)\\coloneqq\\mathbb\{E\}\[\{\\bf x\}\_\{\\bf 1\}\\mid I\_\{\\tau\_\{t\}\}=\{\\bf x\},\\,\\tau\_\{t\}=\\tau\],\\qquad b^\{\\omega\}\_\{t\}\(\{\\bf x\},\\tau\)=\\frac\{D^\{\\omega\}\_\{t\}\(\{\\bf x\},\\tau\)\-\{\\bf x\}\}\{\\mathbf\{1\}\-\\tau\}\.\(12\)

#### Scheduler\-independent training\.

For a chosen schedulerω\\omega, we can learnDtωD^\{\\omega\}\_\{t\}by tokenwise cross\-entropy, using\(t,Iτt,τt\)\(t,I\_\{\\tau\_\{t\}\},\\tau\_\{t\}\)as inputs and the clean tokens as targets\. Yet, constructing these examples requires simulatingω\\omegaalong each interpolant, tying denoiser training to the scheduler\. To allow the scheduler to be chosen after training, we may instead learn the*scheduler\-independent denoiser*

Dτ​\(𝐱\)≔𝔼⁡\[𝐱𝟏∣Iτ=𝐱\],D\_\{\\tau\}\(\{\\bf x\}\)\\coloneqq\\mathbb\{E\}\[\{\\bf x\}\_\{\\bf 1\}\\mid I\_\{\\tau\}=\{\\bf x\}\],which takes as input\(τ,Iτ\)\(\\tau,I\_\{\\tau\}\)\. For this, we use diffusion forcing\([Chen et al\., 2024](https://arxiv.org/html/2610.42018#bib.bib14)\), which samples the tokenwise times independently of one another and of the noise and data:

###### Proposition 3\.

Consider the objective

ℒ𝖢𝖤​\(D^\)≔𝔼τ,𝐱𝟎,𝐱𝟏​∑i=1L\[−log⁡D^τi​\(Iτ\)\]𝐲i,τi​∼iid​𝖴​\[0,1\]\.\\mathcal\{L\}\_\{\\mathsf\{CE\}\}\(\\hat\{D\}\)\\coloneqq\\mathbb\{E\}\_\{\\tau,\{\\bf x\}\_\{\\bf 0\},\{\\bf x\}\_\{\\bf 1\}\}\\sum\_\{i=1\}^\{L\}\\left\[\-\\log\\hat\{D\}\_\{\\tau\}^\{i\}\(I\_\{\\tau\}\)\\right\]\_\{\{\\bf y\}^\{i\}\},\\qquad\\tau^\{i\}\\overset\{\\textnormal\{iid\}\}\{\\sim\}\\mathsf\{U\}\[0,1\]\.\(13\)Any population minimizerD^⋆\\hat\{D\}^\{\\star\}satisfiesD^τ⋆=Dτ\\hat\{D\}^\{\\star\}\_\{\\tau\}=D\_\{\\tau\}almost surely under the training distribution\.

A proof is in[SectionA\.3](https://arxiv.org/html/2610.42018#A1.SS3)\. We leave further discussion on this training scheme in[SectionA\.4](https://arxiv.org/html/2610.42018#A1.SS4)\.

Following[Equation3](https://arxiv.org/html/2610.42018#S3.E3), the denoiserDτD\_\{\\tau\}gives the scheduler\-independent velocity

bτ​\(𝐱\)≔𝔼⁡\[𝐱𝟏−𝐱𝟎∣Iτ=𝐱\]=Dτ​\(𝐱\)−𝐱𝟏−τ,b\_\{\\tau\}\(\{\\bf x\}\)\\coloneqq\\mathbb\{E\}\[\{\\bf x\}\_\{\\bf 1\}\-\{\\bf x\}\_\{\\bf 0\}\\mid I\_\{\\tau\}=\{\\bf x\}\]=\\frac\{D\_\{\\tau\}\(\{\\bf x\}\)\-\{\\bf x\}\}\{\\mathbf\{1\}\-\\tau\},\(14\)which we use to approximatebtωb^\{\\omega\}\_\{t\}at inference under a choice of schedulerω\\omega\. Denoting𝐳t≔\(𝐱t,τt\)\{\\bf z\}\_\{t\}\\coloneqq\(\{\\bf x\}\_\{t\},\\tau\_\{t\}\), this gives the asynchronous flow

𝐳˙t=\(ωt​\(𝐱t,τt\)​bτt​\(𝐱t\),ωt​\(𝐱t,τt\)\),𝐳0=\(𝐱0,𝟎\),𝐱0∼p𝟎\.\\dot\{\\bf z\}\_\{t\}=\\big\(\\omega\_\{t\}\(\{\\bf x\}\_\{t\},\\tau\_\{t\}\)b\_\{\\tau\_\{t\}\}\(\{\\bf x\}\_\{t\}\),\\ \\omega\_\{t\}\(\{\\bf x\}\_\{t\},\\tau\_\{t\}\)\\big\),\\qquad\{\\bf z\}\_\{0\}=\(\{\\bf x\}\_\{0\},\\mathbf\{0\}\),\\qquad\{\\bf x\}\_\{0\}\\sim p\_\{\\bf 0\}\.\(15\)These coupled dynamics define a deterministic ODE on the token states and their tokenwise times\. We call the resulting model an asynchronous flow language model \(*a*FLM\)\. We note that settingτti=t\\tau\_\{t\}^\{i\}=tfor all positionsiirecovers the usual FLM, which we refer to as synchronous\.

#### Prescribed and state\-adaptive schedules\.

Whenω\\omegadoes not depend on the token states, as in left\-to\-right generation for instance,τt\\tau\_\{t\}follows a fixed pathτ⁡\(t\)\\tau\(t\)determined ahead of generation, which we call a*prescribed schedule*\. The conditionτt=τ⁡\(t\)\\tau\_\{t\}=\\tau\(t\)then holds for every pair of noise and data, soDtω​\(𝐱,τ⁡\(t\)\)=Dτ⁡\(t\)​\(𝐱\)D^\{\\omega\}\_\{t\}\(\{\\bf x\},\\tau\(t\)\)=D\_\{\\tau\(t\)\}\(\{\\bf x\}\)\. The velocity approximation is therefore exact:

###### Corollary 4\.

Letτ⁡\(t\)\\tau\(t\)be a prescribed schedule\. Then the asynchronous flow in[Equation15](https://arxiv.org/html/2610.42018#S4.E15)reduces to

𝐱˙t=τ˙​\(t\)​bτ⁡\(t\)​\(𝐱t\),𝐱0∼p𝟎,\\dot\{\\bf x\}\_\{t\}=\\dot\{\\tau\}\(t\)\\,b\_\{\\tau\(t\)\}\(\{\\bf x\}\_\{t\}\),\\qquad\{\\bf x\}\_\{0\}\\sim p\_\{\\bf 0\},\(16\)and satisfies𝐱t∼pτ⁡\(t\)\{\\bf x\}\_\{t\}\\sim p\_\{\\tau\(t\)\}for everyt∈\[0,1\)t\\in\[0,1\), withpτ⁡\(t\)→p𝟏p\_\{\\tau\(t\)\}\\to p\_\{\\bf 1\}ast↑1t\\uparrow 1\.

A proof is in[SectionA\.5](https://arxiv.org/html/2610.42018#A1.SS5)\. Furthermore, only the denoiser alongτ⁡\(t\)\\tau\(t\)is needed, so we can train it with the cross\-entropy loss in[Equation13](https://arxiv.org/html/2610.42018#S4.E13)usingτ=τ⁡\(t\)\\tau=\\tau\(t\)instead of independently sampled tokenwise times\.

Schedulers that depend on the current token states yield*state\-adaptive*schedules, and we call those computed from denoiser predictions*learned schedules*\. ThenDtωD^\{\\omega\}\_\{t\}andDτD\_\{\\tau\}generally differ, and exact recovery ofp𝟏p\_\{\\bf 1\}is no longer guaranteed\. Nevertheless, this allows training the denoiser without fixing a scheduler in advance\. Then during inference, a scheduler can be chosen to improve generation by presenting an imperfect denoiser with easier prediction problems\([Kim et al\., 2025a](https://arxiv.org/html/2610.42018#bib.bib15)\)\. We construct such schedulers as continuous analogues of unmasking criteria used in MDMs, as given in[Section4\.3](https://arxiv.org/html/2610.42018#S4.SS3)\.

### 4\.2Asynchronous flow maps

Asynchronous flows exploit prediction order by letting tokens that denoise earlier inform the prediction of others\. This motivates our key idea for leveraging such an order for few\-step and one\-step parallel generation, by distilling the asynchronous flow into a flow map\. In this way, the effect of the schedule is internalized within the model’s hidden computation\. Since the flow is defined jointly on the tokens and their tokenwise times[Equation15](https://arxiv.org/html/2610.42018#S4.E15), we define the*asynchronous flow map*to evolve both:

###### Proposition 5\.

On any interval where the asynchronous flow[Equation15](https://arxiv.org/html/2610.42018#S4.E15)admits unique solutions, its solution operatorFs,t​\(𝐳s\)=𝐳tF\_\{s,t\}\(\{\\bf z\}\_\{s\}\)=\{\\bf z\}\_\{t\}, called the*asynchronous flow map*, satisfies

Fs,t​\(𝐳s\)\\displaystyle F\_\{s,t\}\(\{\\bf z\}\_\{s\}\)=𝐳s\+∫st𝐳˙r​dr,\\displaystyle=\{\\bf z\}\_\{s\}\+\\int\_\{s\}^\{t\}\\dot\{\\bf z\}\_\{r\}\\,\{\\rm d\}r,\(17\)Fs,t\\displaystyle F\_\{s,t\}=Fu,t∘Fs,u,s≤u≤t<1\.\\displaystyle=F\_\{u,t\}\\circ F\_\{s,u\},\\qquad s\\leq u\\leq t<1\.

A proof is in[SectionA\.6](https://arxiv.org/html/2610.42018#A1.SS6)\. The composition identity, which we call the semigroup condition, states that a direct jump fromsstottagrees with two successive jumps throughuu\. We use this consistency to learn the map by self\-distillation\.

#### Parameterization\.

Following[Section3](https://arxiv.org/html/2610.42018#S3), we express the flow map through its average velocity\. For a state𝐳=\(𝐱,τ\)\{\\bf z\}=\(\{\\bf x\},\\tau\)at timess, we denote the average rates of change of the tokens and their tokenwise times until timettbyvs,tv\_\{s,t\}andηs,t\\eta\_\{s,t\}, respectively, withs=ts=treducing to the instantaneous dynamics:

\(vs,t​\(𝐳\),ηs,t​\(𝐳\)\)\\displaystyle\\big\(v\_\{s,t\}\(\{\\bf z\}\),\\eta\_\{s,t\}\(\{\\bf z\}\)\\big\)≔Fs,t​\(𝐳\)−𝐳t−s,s<t,\\displaystyle\\coloneqq\\frac\{F\_\{s,t\}\(\{\\bf z\}\)\-\{\\bf z\}\}\{t\-s\},\\qquad s<t,\(18\)\(vt,t​\(𝐳\),ηt,t​\(𝐳\)\)\\displaystyle\\big\(v\_\{t,t\}\(\{\\bf z\}\),\\eta\_\{t,t\}\(\{\\bf z\}\)\\big\)≔\(ωt​\(𝐱,τ\)​bτ​\(𝐱\),ωt​\(𝐱,τ\)\)\.\\displaystyle\\coloneqq\\big\(\\omega\_\{t\}\(\{\\bf x\},\\tau\)b\_\{\\tau\}\(\{\\bf x\}\),\\omega\_\{t\}\(\{\\bf x\},\\tau\)\\big\)\.For the token componentvs,tv\_\{s,t\}, we further adapt a two\-time denoiser parameterization:

δs,t​\(𝐳\)≔𝐱\+𝟏−τηs,t​\(𝐳\)​vs,t​\(𝐳\)\.\\delta\_\{s,t\}\(\{\\bf z\}\)\\coloneqq\{\\bf x\}\+\\frac\{\\mathbf\{1\}\-\\tau\}\{\\eta\_\{s,t\}\(\{\\bf z\}\)\}v\_\{s,t\}\(\{\\bf z\}\)\.\(19\)This extends[Equation5](https://arxiv.org/html/2610.42018#S3.E5)by rescaling each token’s displacement according to its tokenwise progress\. Along the asynchronous flow,δs,ti\\delta\_\{s,t\}^\{i\}is a weighted average of denoiser predictions and therefore lies in the vocabulary simplex\. We can thus learn it with a categorical output and KL loss, as in FMLMs\. The following identities recover the flow map and provide the conditions for learning it:

###### Proposition 6\.

LetFs,tF\_\{s,t\}be the asynchronous flow map\. Thenδs,t\\delta\_\{s,t\}andηs,t\\eta\_\{s,t\}satisfy:

1. \(i\)The transitionFs,t​\(𝐳\)=\(𝐱t,τt\)F\_\{s,t\}\(\{\\bf z\}\)=\(\{\\bf x\}\_\{t\},\\tau\_\{t\}\)is recovered as 𝐱t\\displaystyle\{\\bf x\}\_\{t\}=𝐱\+\(t−s\)​ηs,t​\(𝐳\)𝟏−τ​\(δs,t​\(𝐳\)−𝐱\),\\displaystyle=\{\\bf x\}\+\\frac\{\(t\-s\)\\eta\_\{s,t\}\(\{\\bf z\}\)\}\{\\mathbf\{1\}\-\\tau\}\\big\(\\delta\_\{s,t\}\(\{\\bf z\}\)\-\{\\bf x\}\\big\),\(20\)τt\\displaystyle\\tau\_\{t\}=τ\+\(t−s\)​ηs,t​\(𝐳\)\.\\displaystyle=\\tau\+\(t\-s\)\\eta\_\{s,t\}\(\{\\bf z\}\)\.
2. \(ii\)On the diagonals=ts=t, δt,t​\(𝐳\)=Dτ​\(𝐱\),ηt,t​\(𝐳\)=ωt​\(𝐱,τ\)\.\\delta\_\{t,t\}\(\{\\bf z\}\)=D\_\{\\tau\}\(\{\\bf x\}\),\\qquad\\eta\_\{t,t\}\(\{\\bf z\}\)=\\omega\_\{t\}\(\{\\bf x\},\\tau\)\.\(21\)
3. \(iii\)Fors<u<ts<u<t, the semigroup condition[Equation17](https://arxiv.org/html/2610.42018#S4.E17)is equivalent to δs,t​\(𝐳\)\\displaystyle\\delta\_\{s,t\}\(\{\\bf z\}\)=γ​δs,u​\(𝐳\)\+\(𝟏−γ\)​δu,t​\(Fs,u​\(𝐳\)\),\\displaystyle=\\gamma\\delta\_\{s,u\}\(\{\\bf z\}\)\+\(\\mathbf\{1\}\-\\gamma\)\\delta\_\{u,t\}\(F\_\{s,u\}\(\{\\bf z\}\)\),\(22\)ηs,t​\(𝐳\)\\displaystyle\\eta\_\{s,t\}\(\{\\bf z\}\)=u−st−s​ηs,u​\(𝐳\)\+t−ut−s​ηu,t​\(Fs,u​\(𝐳\)\),\\displaystyle=\\frac\{u\-s\}\{t\-s\}\\eta\_\{s,u\}\(\{\\bf z\}\)\+\\frac\{t\-u\}\{t\-s\}\\eta\_\{u,t\}\(F\_\{s,u\}\(\{\\bf z\}\)\),where γ=\(𝟏−τt\)​\(τu−τs\)\(𝟏−τu\)​\(τt−τs\)∈\[0,1\]L,\\gamma=\\frac\{\(\\mathbf\{1\}\-\\tau\_\{t\}\)\(\\tau\_\{u\}\-\\tau\_\{s\}\)\}\{\(\\mathbf\{1\}\-\\tau\_\{u\}\)\(\\tau\_\{t\}\-\\tau\_\{s\}\)\}\\in\[0,1\]^\{L\},withτs=τ\\tau\_\{s\}=\\tauandτu,τt\\tau\_\{u\},\\tau\_\{t\}the tokenwise times along the flow from𝐳\{\\bf z\}\.

A proof is in[SectionA\.7](https://arxiv.org/html/2610.42018#A1.SS7)\.

#### Training\.

We learn predictorsδ^\\hat\{\\delta\}andη^\\hat\{\\eta\}, which define a model of the flow mapF^\\hat\{F\}through[Equation20](https://arxiv.org/html/2610.42018#S4.E20)\. On the diagonal, the denoiserDτD\_\{\\tau\}and schedulerω\\omegaprovide their respective targets[Equation21](https://arxiv.org/html/2610.42018#S4.E21)\. Fors<u<ts<u<t, we construct targets for a direct transition fromsstottby composing two shorter student transitions throughuubased on[Equation22](https://arxiv.org/html/2610.42018#S4.E22):

𝒯s,u,tδ​\(𝐳\)\\displaystyle\\mathcal\{T\}^\{\\delta\}\_\{s,u,t\}\(\{\\bf z\}\)≔γ​δ^s,u​\(𝐳\)\+\(𝟏−γ\)​δ^u,t​\(F^s,u​\(𝐳\)\),\\displaystyle\\coloneqq\\gamma\\hat\{\\delta\}\_\{s,u\}\(\{\\bf z\}\)\+\(\\mathbf\{1\}\-\\gamma\)\\hat\{\\delta\}\_\{u,t\}\(\\hat\{F\}\_\{s,u\}\(\{\\bf z\}\)\),\(23\)𝒯s,u,tη​\(𝐳\)\\displaystyle\\mathcal\{T\}^\{\\eta\}\_\{s,u,t\}\(\{\\bf z\}\)≔u−st−s​η^s,u​\(𝐳\)\+t−ut−s​η^u,t​\(F^s,u​\(𝐳\)\)\.\\displaystyle\\coloneqq\\frac\{u\-s\}\{t\-s\}\\hat\{\\eta\}\_\{s,u\}\(\{\\bf z\}\)\+\\frac\{t\-u\}\{t\-s\}\\hat\{\\eta\}\_\{u,t\}\(\\hat\{F\}\_\{s,u\}\(\{\\bf z\}\)\)\.Hereγ\\gammafrom[Equation22](https://arxiv.org/html/2610.42018#S4.E22)is evaluated at the tokenwise times of𝐳\{\\bf z\},F^s,u​\(𝐳\)\\hat\{F\}\_\{s,u\}\(\{\\bf z\}\), andF^u,t​\(F^s,u​\(𝐳\)\)\\hat\{F\}\_\{u,t\}\(\\hat\{F\}\_\{s,u\}\(\{\\bf z\}\)\)\. Combining these composition targets with the diagonal targets gives the training objective:

###### Proposition 7\.

Forλ\>0\\lambda\>0, consider the population objective

ℒ⁡\(δ^,η^\)\\displaystyle\\mathcal\{L\}\(\\hat\{\\delta\},\\hat\{\\eta\}\)=𝔼s,u,t,𝐳\[𝖪𝖫\(𝗌𝗀\(𝒯δs,u,t\(𝐳\)\)∥δ^s,t\(𝐳\)\)\]\+𝔼t,𝐳\[𝖪𝖫\(Dτ\(𝐱\)∥δ^t,t\(𝐳\)\)\]\\displaystyle=\\mathbb\{E\}\_\{s,u,t,\{\\bf z\}\}\\big\[\\mathsf\{KL\}\\big\(\\mathsf\{sg\}\(\\mathcal\{T\}^\{\\delta\}\_\{s,u,t\}\(\{\\bf z\}\)\)\\,\\\|\\,\\hat\{\\delta\}\_\{s,t\}\(\{\\bf z\}\)\\big\)\\big\]\+\\mathbb\{E\}\_\{t,\{\\bf z\}\}\\big\[\\mathsf\{KL\}\\big\(D\_\{\\tau\}\(\{\\bf x\}\)\\,\\\|\\,\\hat\{\\delta\}\_\{t,t\}\(\{\\bf z\}\)\\big\)\\big\]\(24\)\+λ​𝔼s,u,t,𝐳​‖𝗌𝗀⁡\(𝒯s,u,tη​\(𝐳\)\)−η^s,t​\(𝐳\)‖22\+λ​𝔼t,𝐳​‖ωt​\(𝐱,τ\)−η^t,t​\(𝐳\)‖22\.\\displaystyle\+\\lambda\\mathbb\{E\}\_\{s,u,t,\{\\bf z\}\}\\big\\\|\\mathsf\{sg\}\(\\mathcal\{T\}^\{\\eta\}\_\{s,u,t\}\(\{\\bf z\}\)\)\-\\hat\{\\eta\}\_\{s,t\}\(\{\\bf z\}\)\\big\\\|\_\{2\}^\{2\}\+\\lambda\\mathbb\{E\}\_\{t,\{\\bf z\}\}\\big\\\|\\omega\_\{t\}\(\{\\bf x\},\\tau\)\-\\hat\{\\eta\}\_\{t,t\}\(\{\\bf z\}\)\\big\\\|\_\{2\}^\{2\}\.The exactδ\\deltaandη\\etahave zero loss\. Conversely, any zero\-loss solution satisfies the diagonal and composition identities[Equations21](https://arxiv.org/html/2610.42018#S4.E21)and[22](https://arxiv.org/html/2610.42018#S4.E22)almost surely under the training distribution\.

A proof is in[SectionA\.8](https://arxiv.org/html/2610.42018#A1.SS8)\. We call the model learned with[Equation24](https://arxiv.org/html/2610.42018#S4.E24)an asynchronous flow map language model \(aFMLM\)\.

For state\-adaptive schedules, we train on joint states𝐳s\{\\bf z\}\_\{s\}visited by the asynchronous flow\. These states depend on the preceding trajectory and cannot be sampled directly from the interpolant\. We therefore first train the denoiserDτD\_\{\\tau\}and simulate its flow under the chosen scheduler to obtain the initial training states\. This pretrained teacher also provides the diagonal denoising target in[Equation24](https://arxiv.org/html/2610.42018#S4.E24)\. To reduce the cost of simulation, we can later use the learned map to get𝐳s\{\\bf z\}\_\{s\}, as described in[Section4\.3](https://arxiv.org/html/2610.42018#S4.SS3)\.

On the other hand, for a prescribed scheduleτ⁡\(t\)\\tau\(t\), the time component is fixed:

ηs,t=τ⁡\(t\)−τ⁡\(s\)t−s,ηt,t=τ˙​\(t\)\.\\eta\_\{s,t\}=\\frac\{\\tau\(t\)\-\\tau\(s\)\}\{t\-s\},\\qquad\\eta\_\{t,t\}=\\dot\{\\tau\}\(t\)\.Only the two\-time denoiser needs to be learned\. Moreover, the flow followspτ⁡\(t\)p\_\{\\tau\(t\)\}by[Corollary4](https://arxiv.org/html/2610.42018#Thmproposition4), so a training state at any base timerrcan be sampled directly as𝐳r=\(Iτ⁡\(r\),τ⁡\(r\)\)\{\\bf z\}\_\{r\}=\(I\_\{\\tau\(r\)\},\\tau\(r\)\)\. On the diagonal, the denoiser is the conditional mean of the clean data𝐱𝟏\{\\bf x\}\_\{\\bf 1\}\. Replacing its target with data therefore changes the expected loss only by a constant independent of the model:

###### Corollary 8\.

Letτ⁡\(t\)\\tau\(t\)be a prescribed schedule, withη^\\hat\{\\eta\}fixed to its exact time component and training states sampled from the corresponding interpolants\. Then[Equation24](https://arxiv.org/html/2610.42018#S4.E24)reduces to

ℒ\(δ^\)=𝔼s,u,t,𝐳\[𝖪𝖫\(𝗌𝗀\(𝒯s,u,tδ\(𝐳\)\)∥δ^s,t\(𝐳\)\)\]\+𝔼t,𝐳,𝐱𝟏\[𝖪𝖫\(𝐱𝟏∥δ^t,t\(𝐳\)\)\],\\mathcal\{L\}\(\\hat\{\\delta\}\)=\\mathbb\{E\}\_\{s,u,t,\{\\bf z\}\}\\big\[\\mathsf\{KL\}\\big\(\\mathsf\{sg\}\(\\mathcal\{T\}^\{\\delta\}\_\{s,u,t\}\(\{\\bf z\}\)\)\\,\\\|\\,\\hat\{\\delta\}\_\{s,t\}\(\{\\bf z\}\)\\big\)\\big\]\+\\mathbb\{E\}\_\{t,\{\\bf z\},\{\\bf x\}\_\{\\bf 1\}\}\\big\[\\mathsf\{KL\}\\big\(\{\\bf x\}\_\{\\bf 1\}\\,\\\|\\,\\hat\{\\delta\}\_\{t,t\}\(\{\\bf z\}\)\\big\)\\big\],\(25\)up to a constant independent ofδ^\\hat\{\\delta\}\. Its minimizers coincide with the zero\-loss solutions of[Proposition7](https://arxiv.org/html/2610.42018#Thmproposition7)withη^\\hat\{\\eta\}fixed by the schedule\.

A proof is in[SectionA\.9](https://arxiv.org/html/2610.42018#A1.SS9)\. Thus, a prescribed schedule allows the flow map to be trained from scratch without a pretrained teacher\.

#### Inference\.

We generate samples by applying the learned flow map on a temporal grid of base times0=t0<⋯<tN=10=t\_\{0\}<\\cdots<t\_\{N\}=1:

𝐳^t0\\displaystyle\\hat\{\\bf z\}\_\{t\_\{0\}\}=\(𝐱0,𝟎\),𝐱0∼p𝟎,\\displaystyle=\(\{\\bf x\}\_\{0\},\\mathbf\{0\}\),\\qquad\{\\bf x\}\_\{0\}\\sim p\_\{\\bf 0\},\(26\)𝐳^tn\+1\\displaystyle\\hat\{\\bf z\}\_\{t\_\{n\+1\}\}=F^tn,tn\+1\(𝐳^tn\),n=0,…,N−1\.\\displaystyle=\\hat\{F\}\_\{t\_\{n\},t\_\{n\+1\}\}\(\\hat\{\\bf z\}\_\{t\_\{n\}\}\),\\qquad n=0,\\ldots,N\-1\.The final tokens are obtained by applying tokenwise𝖺𝗋𝗀𝗆𝖺𝗑\\mathsf\{argmax\}toδ^tN−1,1​\(𝐳^tN−1\)\\hat\{\\delta\}\_\{t\_\{N\-1\},1\}\(\\hat\{\\bf z\}\_\{t\_\{N\-1\}\}\)\. ForN=1N=1, a single evaluation ofδ^0,1\\hat\{\\delta\}\_\{0,1\}generates the entire sequence without evaluating the scheduler\. For supervised tasks, this allows us to train with a prescribed schedule for each example, even when the corresponding schedule is unavailable for a test input\. These schedules guide training, and the one\-step model internalizes them to generate its output from the task input and initial noise alone\.

### 4\.3Algorithmic details

#### Scheduler implementation\.

We remark that a scheduler must sendτt→𝟏\\tau\_\{t\}\\to\{\\bf 1\}ast↑1t\\uparrow 1[Equation10](https://arxiv.org/html/2610.42018#S4.E10)\. We implement the schedulers in a common form that ensures this by construction \([SectionB\.1](https://arxiv.org/html/2610.42018#A2.SS1)\):

ωti​\(𝐱,τ\)=1−τi1−t​kti​\(𝐱,τ\),\\omega\_\{t\}^\{i\}\(\{\\bf x\},\\tau\)=\\frac\{1\-\\tau^\{i\}\}\{1\-t\}\\,k\_\{t\}^\{i\}\(\{\\bf x\},\\tau\),\(27\)where the factorkti\>0k\_\{t\}^\{i\}\>0controls how quickly positioniiadvances and can be chosen almost freely\. Settingkti=1k\_\{t\}^\{i\}=1, for instance, recovers the synchronous scheduleτti=t\\tau\_\{t\}^\{i\}=tfromτ0=𝟎\\tau\_\{0\}=\\mathbf\{0\}\. For learned schedules, we assign higher factors to positions that the denoiser predicts confidently, using two rules:

- •Top\-KKconfidence schedulerprioritizes theKKmost confident positions, withKKincreasing over time\([Chang et al\., 2022](https://arxiv.org/html/2610.42018#bib.bib26)\)\.
- •Entropy\-bounded scheduleruses an entropy budget to determine how many high\-confidence positions advance faster, allowing more positions to advance together when their predictions are certain\([Ben\-Hamu et al\., 2025](https://arxiv.org/html/2610.42018#bib.bib16)\)\.

These schedules are constructed as continuous analogues of unmasking strategies used in MDMs, and their details along with more such examples can be found in[SectionB\.1](https://arxiv.org/html/2610.42018#A2.SS1)\.

#### Denoiser and time heads\.

For flow map distillation under learned schedules, we use one model with a denoiser head forδ^s,t\\hat\{\\delta\}\_\{s,t\}and a time head forη^s,t\\hat\{\\eta\}\_\{s,t\}\. To keep predicted tokenwise times in\[τs,𝟏\]\[\\tau\_\{s\},\\mathbf\{1\}\], we parameterize the time head via the effective rateκs,t≔log⁡𝟏−τs𝟏−τt/log⁡1−s1−t\\kappa\_\{s,t\}\\coloneqq\\log\\frac\{\\mathbf\{1\}\-\\tau\_\{s\}\}\{\\mathbf\{1\}\-\\tau\_\{t\}\}/\\log\\frac\{1\-s\}\{1\-t\}, which is a weighted average ofkkalong the trajectory \(see[SectionB\.2](https://arxiv.org/html/2610.42018#A2.SS2)for details\)\. The head outputsκ^s,t\>0\\hat\{\\kappa\}\_\{s,t\}\>0, predicting

τ^t=τs\+\(t−s\)​η^s,t​\(𝐳s\)=𝟏−\(𝟏−τs\)⊙\(1−t1−s\)κ^s,t​\(𝐳s\)\.\\hat\{\\tau\}\_\{t\}=\\tau\_\{s\}\+\(t\-s\)\\hat\{\\eta\}\_\{s,t\}\(\{\\bf z\}\_\{s\}\)=\\mathbf\{1\}\-\(\\mathbf\{1\}\-\\tau\_\{s\}\)\\odot\\left\(\\frac\{1\-t\}\{1\-s\}\\right\)^\{\\hat\{\\kappa\}\_\{s,t\}\(\{\\bf z\}\_\{s\}\)\}\.\(28\)Each transition advances the tokenwise times toward𝟏\\mathbf\{1\}and reaches𝟏\\mathbf\{1\}att=1t=1\. On the diagonal, matchingκ^t,t\\hat\{\\kappa\}\_\{t,t\}toktk\_\{t\}is equivalent to matchingη^t,t\\hat\{\\eta\}\_\{t,t\}toωt\\omega\_\{t\}\(see[SectionB\.2](https://arxiv.org/html/2610.42018#A2.SS2)for details\)\.

We initialize the backbone and denoiser head from the teacherDτD\_\{\\tau\}and attach the time head to the final hidden features\. The time head exponentiates its output to ensure positivity, and we initialize its output before exponentiation to zero, settingκ^s,t=𝟏\\hat\{\\kappa\}\_\{s,t\}=\{\\bf 1\}at initialization\.

#### Time conditioning\.

With tokenwise time, both the teacher denoiser and the flow map receive a separate time embedding at each position, obtained by applying a shared embedding layer toτi\\tau^\{i\}\. For the flow map, we additionally embed the base timesssandttwith zero\-initialized layers and add their outputs to the embeddings ofτs\\tau\_\{s\}, following[Lee et al\. \(2026\)](https://arxiv.org/html/2610.42018#bib.bib1)\. The resulting conditioning vectors modulate each block through adaptive layer normalization \(adaLN\)\([Peebles and Xie, 2023](https://arxiv.org/html/2610.42018#bib.bib38)\)\.

#### Rollout curriculum\.

To train with[Equation24](https://arxiv.org/html/2610.42018#S4.E24), we initially obtain training states𝐳s\{\\bf z\}\_\{s\}by rolling out the teacher under the chosen scheduler from𝐳0=\(𝐱0,𝟎\)\{\\bf z\}\_\{0\}=\(\{\\bf x\}\_\{0\},\\mathbf\{0\}\)to timess\. After the first round, we use the current student flow mapF^\\hat\{F\}for rollouts and halve the number of rollout steps at successive rounds\. The fixed teacher denoiser and scheduler continue to provide the diagonal targets throughout training\.

#### MDM initialization\.

Diffusion forcing requires the denoiser to denoise across many combinations of tokenwise noise levels, which can make training demanding\([Kim et al\., 2025a](https://arxiv.org/html/2610.42018#bib.bib15)\)\. When each tokenwise time is close to either00or11, the task resembles masked token prediction, since each position is nearly pure noise or nearly clean\. To exploit this overlap, we optionally initialize the denoiser from a trained MDLM\([Sahoo et al\., 2024](https://arxiv.org/html/2610.42018#bib.bib27)\), following[Agarwal et al\. \(2026\)](https://arxiv.org/html/2610.42018#bib.bib3)\.

#### Noise downscaling\.

Following[Chavan et al\. \(2026\)](https://arxiv.org/html/2610.42018#bib.bib39), we use initial noise𝐱𝟎∼𝖭⁡\(0,σ2​I\)\{\\bf x\}\_\{\\bf 0\}\\sim\\mathsf\{N\}\(0,\\sigma^\{2\}I\)withσ<1\\sigma<1during flow map distillation and inference\. In supervised tasks where one accurate answer is sufficient, reducing variation in the initial states can ease the burden of fitting the flow map\. We apply this on GSM8K, where it improves one\-step accuracy \(see[Figure4](https://arxiv.org/html/2610.42018#S5.F4)\)\.

## 5Experiments

Throughout the experiments, we address these key questions:

1. Q1\.Can a proper choice of asynchronous scheduler improve the generation quality of*a*FLMs?
2. Q2\.Does this improvement transfer to one\-step generation withaFMLM after distillation?
3. Q3\.Do hidden computations ofaFMLMs reflect the schedules of*a*FLMs they are distilled from?

#### Setup\.

We evaluate our method on Sudoku solving and code generation for GSM8K math word problems\. For Sudoku, we use the dataset with three difficulty levels and report exact solving accuracy \(%\) on the full test set, following[Deschenaux and Gulcehre \(2026\)](https://arxiv.org/html/2610.42018#bib.bib30)\. For code generation, we train on code completions from TinyGSM\([Liu et al\., 2023](https://arxiv.org/html/2610.42018#bib.bib31)\)and measure zero\-shot accuracy \(%\) on the full test set of GSM8K\([Cobbe et al\., 2021](https://arxiv.org/html/2610.42018#bib.bib32)\)math problems, following[Kim et al\. \(2025a\)](https://arxiv.org/html/2610.42018#bib.bib15)\.

For the baselines, we consider discrete methods MDLM\([Sahoo et al\., 2024](https://arxiv.org/html/2610.42018#bib.bib27)\), Duo\([Sahoo et al\., 2025](https://arxiv.org/html/2610.42018#bib.bib4)\)and AlphaDLM\([Gushchin et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib52)\), and continuous methods CANDI\([Pynadath et al\., 2025](https://arxiv.org/html/2610.42018#bib.bib2)\), FLM\([Lee et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib1)\), S\-FLM\([Deschenaux and Gulcehre, 2026](https://arxiv.org/html/2610.42018#bib.bib30)\), FMLM\([Lee et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib1)\), FMLM\+\([Agarwal et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib3)\), and DBTM\([Tang and Wang, 2026](https://arxiv.org/html/2610.42018#bib.bib33)\)\. We train and distill for 250k steps on GSM8K and 20k on Sudoku in total, matching the baselines\. We leave further details in[AppendixC](https://arxiv.org/html/2610.42018#A3)\.

### 5\.1Asynchronous schedulers foraaFLM \(Q1\)

Figure 2:Accuracy of*a*FLM under different inference\-time schedulers on Sudoku\-Hard \(left\) and GSM8K \(right\)\. A well\-chosen scheduler improves performance over the synchronous one \(Sync\.\)\. Alg\. order†denotes the*a*FLM whose denoiser is trained only along the algorithmic order\.For each dataset, we train one asynchronous denoiser with diffusion forcing\([Chen et al\., 2024](https://arxiv.org/html/2610.42018#bib.bib14)\)and apply various schedulers during inference\. We compare the synchronous schedule \(Sync\.\) of FLM\([Lee et al\., 2026](https://arxiv.org/html/2610.42018#bib.bib1)\), a schedule fixed by the initial prediction entropy \(Init\. ent\.\) of[Bansal and Sanghavi \(2026\)](https://arxiv.org/html/2610.42018#bib.bib8), and learned schedulers including the top\-KKconfidence scheduler \(Top\-KK\) and the entropy\-bounded scheduler \(EB\) defined in[Section4\.3](https://arxiv.org/html/2610.42018#S4.SS3)\. To test whether generation can also benefit from a prescribed order, we include an algorithmic order \(Alg\. order\) on Sudoku\. In this order, the cells that a constraint solver resolves earlier are denoised faster\([Shah et al\., 2024](https://arxiv.org/html/2610.42018#bib.bib18)\)\([SectionC\.3](https://arxiv.org/html/2610.42018#A3.SS3)\)\.

The results are in[Figure2](https://arxiv.org/html/2610.42018#S5.F2), showing that learned schedulers substantially improve generation accuracy on both tasks\. With 32\-step generation, the EB scheduler improves the accuracy over Sync\. from 40\.1% to 95\.6% on Sudoku\-Hard and from 7\.4% to 36\.4% on GSM8K\. Also, schedulers that adapt the order to intermediate predictions \(Top\-KKand EB\) outperform the scheduler that fixes the order from the initial prediction \(Init\. ent\.\)\. Asynchronous flows thus let flow language models benefit from a good generation order, with the largest gains from learned schedulers ported from MDMs\.

While the prescribed algorithmic order improves accuracy on Sudoku\-Hard from 40\.1% to 59\.7%, it still falls short of the learned schedulers\. We conjecture that this gap arises because diffusion forcing cannot learn all combinations of tokenwise times equally well\([Kim et al\., 2025a](https://arxiv.org/html/2610.42018#bib.bib15)\)\([Section4\.3](https://arxiv.org/html/2610.42018#S4.SS3)\)\. Learned schedulers follow the confidence of the denoiser and thus avoid the poorly learned combinations, whereas the algorithmic order may still visit them\. Indeed, training the denoiser only along the algorithmic order raises the accuracy from 59\.7% to 88\.6% \(Alg\. order†in[Figure2](https://arxiv.org/html/2610.42018#S5.F2)\)\.

### 5\.2One\-step generation withaaFMLM \(Q2\)

Table 1:One\-step exact solving accuracy \(%\) on Sudoku and zero\-shot accuracy \(%\) on GSM8K \(see[SectionC\.2](https://arxiv.org/html/2610.42018#A3.SS2)for baseline details\)\. For*a*FLM, the schedule does not matter in one\-step generation\.Figure 3:GSM8K accuracy \(%\) of the teacher*a*FLM \(EB\) with 32\-step generation across training steps, with and without MDLM initialization\.Figure 4:GSM8K accuracy \(%\) ofaFMLM \(EB\)\. “\+ MDLM” initializes the teacher from MDLM, and “\+σ↓\\sigma\\\!\\downarrow” adds noise downscaling\.
We distill*a*FLM from[Section5\.1](https://arxiv.org/html/2610.42018#S5.SS1)with EB, Top\-KKand synchronous schedulers intoaFMLM \(EB\),aFMLM \(Top\-KK\), andaFMLM \(Sync\.\), respectively, and test their one\-step generation performance\.

The results are shown in[Table1](https://arxiv.org/html/2610.42018#S5.T1)\.aFMLM \(EB\) improves the one\-step accuracy over its*a*FLM teacher, 11\.8% vs\. 6\.1% on Sudoku\-Hard and 11\.4% vs\. 0\.3% on GSM8K\. In contrast,aFMLM \(Sync\.\), which is distilled from the same teacher, reaches considerably lower performance, showing that the benefit of asynchronous schedulers carries over to one\-step generation\.aFMLM \(Top\-KK\) shows a similar gain and achieves the highest one\-step accuracy on Sudoku\-Hard, which shows that the benefit is not specific to a particular choice of the scheduler\. Through these advantages,aFMLM surpasses all one\-step baselines on both Sudoku and GSM8K and establishes the new state of the art\.

Table 2:Training cost \(GPU hours\) on Sudoku\-Hard\.In addition, as in[Corollary8](https://arxiv.org/html/2610.42018#Thmproposition8),aFMLM can be trained along a prescribed schedule from scratch without a pretrained teacher\. We show this with the algorithmic order on Sudoku\. As in[Table1](https://arxiv.org/html/2610.42018#S5.T1),aFMLM \(Alg\. order\) achieves 10\.0% one\-step accuracy on Sudoku\-Hard without a teacher, compared with 11\.8% ofaFMLM \(EB\)\. As shown in[Table2](https://arxiv.org/html/2610.42018#S5.T2), this approach reduces the total training cost from 5 to 3\.2 GPU hours by eliminating teacher pretraining\. Our method effectively distills both prescribed and learned schedules, and a useful prescribed schedule offers a more efficient route to one\-step generation\.

In[Figures4](https://arxiv.org/html/2610.42018#S5.F4)and[4](https://arxiv.org/html/2610.42018#S5.F4), we analyze the role of our algorithmic choices \([Section4\.3](https://arxiv.org/html/2610.42018#S4.SS3)\) of initializing the*a*FLM teacher from MDLM and noise downscaling for distillation\. Initializing the teacher from MDLM accelerates its training, where it reaches 32\.9% accuracy on GSM8K with 32\-step generation after only 20k training steps, compared with 6\.4% from scratch\. This suggests MDM initialization eases diffusion forcing training, as the combinations of tokenwise times close to zero or one resemble the problem solved by MDMs\. Distilling this stronger teacher improves the one\-step accuracy ofaFMLM \(EB\) on GSM8K from 4\.2% to 6\.3%\. Noise downscaling for distillation further improvesaFMLM from 6\.3% to 11\.4% accuracy, confirming that it reduces the burden of fitting the flow map\.

Figure 5:A visualization of code solutions to a GSM8K problem generated by one\-stepaFMLM \(EB\) \(middle\) andaFMLM \(Sync\.\) \(bottom\) in their layer\-wise order, compared with the generation order of*a*FLM \(EB\) \(top\)\. Columns show when 10%, 50%, 85%, and 100% of the tokens are decided\.
### 5\.3Internalization of the schedule intoaaFMLM \(Q3\)

Table 3:Spearman correlation with the schedule\-induced ordering\.We analyze the hidden computation of one\-stepaFMLM using logit lens\([Belrose et al\., 2023](https://arxiv.org/html/2610.42018#bib.bib53)\), which decodes the hidden state of each layer using the output head\. For each token, we record the earliest layer from which its decoded prediction matches the final output and remains so, and compare this layer\-wise order with the order induced by the schedule\. For Sudoku, we compare with the algorithmic order, and for GSM8K, we compare with the order in which the*a*FLM teacher under the EB scheduler decides the tokens\. In[Table3](https://arxiv.org/html/2610.42018#S5.T3), we measure the alignment between the two orders by the Spearman correlation, computed per sample and averaged over the test set\. On both tasks,aFMLM shows a higher correlation with its schedule thanaFMLM \(Sync\.\), which indicates thataFMLM internalizes the schedule into its hidden computation\. This is consistent with the observation of[Asiimwe et al\. \(2026\)](https://arxiv.org/html/2610.42018#bib.bib54)in image generation that flow maps compress the flow across timesteps into their layers\. We further visualize an example in[Figure5](https://arxiv.org/html/2610.42018#S5.F5)\.aFMLM \(Sync\.\) does not follow the order of the teacher and produces non\-parsable lines, showing that the order matters\. In contrast,aFMLM \(EB\) resolves the tokens that the teacher generates early in its early layers and produces a parsable solution\.

## 6Conclusion

We introduced asynchronous flow map language models \(aFMLMs\), which distill the ordered generation of flow language models into one\-step generators\. To this end, we modeled asynchronous generation with an asynchronous flow language model \(*a*FLM\), which evolves a coupled flow of tokens and their tokenwise times\. We then derived a self\-distillation objective that learns the flow map of this flow, which jointly integrates both over any time interval\. Empirically, a well\-designed scheduler substantially improves*a*FLMs, andaFMLMs retain these advantage in one\-step generation without evaluating the scheduler at test time\. We also showed thataFMLMs internalize the order of the scheduler into their hidden computation\. Promising future directions include jointly optimizing the scheduler with the denoiser and scalingaFMLMs to larger continuous language models\.

### Acknowledgments

This work is supported by the National Research Foundation of Korea \(RS\-2024\-00351212\) and the Institute of Information & communications Technology Planning & Evaluation \(IITP\) \(RS\-2024\-00509279, RS\-2026\-25518317, and RS\-2026\-25617720\), funded by the Korean Government \(MSIT\)\. Satoshi Hayakawa was partially supported by JSPS KAKENHI \(JP26K25548\) and JST PRESTO \(JPMJPR26K8\)\.

## References

- Agarwalet al\.\(2026\)M\. Agarwal, S\. Shah, C\. Lee, J\. Yoo, J\. Huang, S\. Hong, A\. Raghunathan, J\. Kim, and N\. M\. BoffiPosterior refinement: fast language generation via any\-order flow maps\.arXiv preprint arXiv:2606\.24773\.Cited by:[§C\.2](https://arxiv.org/html/2610.42018#A3.SS2.p1.1),[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px3.p1.1),[§4\.3](https://arxiv.org/html/2610.42018#S4.SS3.SSS0.Px5.p1.1),[§5](https://arxiv.org/html/2610.42018#S5.SS0.SSS0.Px1.p2.1),[Table 1](https://arxiv.org/html/2610.42018#S5.T1.4.9.1)\.
- Asiimweet al\.\(2026\)A\. C\. Asiimwe, W\. Yang, S\. Chun, E\. Tureci, and O\. RussakovskyDepth as time in one\-step generative models\.arXiv preprint arXiv:2610\.03626\.Cited by:[§5\.3](https://arxiv.org/html/2610.42018#S5.SS3.p1.1)\.
- Azangulovet al\.\(2026\)I\. Azangulov, K\. Ashouritaklimi, L\. Zhang, S\. Vary, and P\. RebeschiniMasked language flow models\.arXiv preprint arXiv:2606\.27617\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px3.p1.1)\.
- Baadeet al\.\(2026\)A\. Baade, E\. R\. Chan, K\. Sargent, C\. Chen, J\. Johnson, E\. Adeli, and L\. Fei\-FeiLatent forcing: reordering the diffusion trajectory for pixel\-space image generation\.arXiv preprint arXiv:2602\.11401\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px2.p1.1)\.
- Bansal and Sanghavi \(2026\)P\. Bansal and S\. SanghaviToken time continuous diffusion for language modeling\.arXiv preprint arXiv:2607\.14106\.Cited by:[4th item](https://arxiv.org/html/2610.42018#A2.I1.i4.p1.1),[§1](https://arxiv.org/html/2610.42018#S1.p3.1),[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px2.p1.1),[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px3.p1.1),[§5\.1](https://arxiv.org/html/2610.42018#S5.SS1.p1.1)\.
- Belroseet al\.\(2023\)N\. Belrose, I\. Ostrovsky, L\. McKinney, Z\. Furman, L\. Smith, D\. Halawi, S\. Biderman, and J\. SteinhardtEliciting latent predictions from transformers with the tuned lens\.arXiv preprint arXiv:2303\.08112\.Cited by:[§5\.3](https://arxiv.org/html/2610.42018#S5.SS3.p1.1)\.
- Ben\-Hamuet al\.\(2025\)H\. Ben\-Hamu, I\. Gat, D\. Severo, N\. Nolte, and B\. KarrerAccelerated sampling from masked diffusion models via entropy bounded unmasking\.arXiv preprint arXiv:2505\.24857\.Cited by:[§B\.1](https://arxiv.org/html/2610.42018#A2.SS1.SSS0.Px3.p1.1),[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px1.p1.1),[2nd item](https://arxiv.org/html/2610.42018#S4.I2.i2.p1.1)\.
- Bengioet al\.\(2003\)Y\. Bengio, R\. Ducharme, P\. Vincent, and C\. JauvinA neural probabilistic language model\.Journal of machine learning research3\(Feb\),pp\. 1137–1155\.Cited by:[§1](https://arxiv.org/html/2610.42018#S1.p1.1)\.
- Bhattet al\.\(2026\)A\. Bhatt, M\. Gupta, G\. Kolossov, and A\. MontanariGenerating from discrete distributions using diffusions: insights from random constraint satisfaction problems\.arXiv preprint arXiv:2603\.20589\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px2.p1.1)\.
- Changet al\.\(2022\)H\. Chang, H\. Zhang, L\. Jiang, C\. Liu, and W\. T\. FreemanMaskgit: masked generative image transformer\.In2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition \(CVPR\),pp\. 11305–11315\.Cited by:[§B\.1](https://arxiv.org/html/2610.42018#A2.SS1.SSS0.Px2.p1.1),[1st item](https://arxiv.org/html/2610.42018#S4.I2.i1.p1.1)\.
- Chavanet al\.\(2026\)A\. Chavan, C\. Lee, U\. Bamba, J\. Yoo, S\. Hong, A\. Raghunathan, and J\. KimLarge flow language models\.arXiv preprint arXiv:2610\.28559\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px2.p1.1),[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px3.p1.1),[§4\.3](https://arxiv.org/html/2610.42018#S4.SS3.SSS0.Px6.p1.1)\.
- Chenet al\.\(2024\)B\. Chen, D\. Martí Monsó, Y\. Du, M\. Simchowitz, R\. Tedrake, and V\. SitzmannDiffusion forcing: next\-token prediction meets full\-sequence diffusion\.Advances in Neural Information Processing Systems37,pp\. 24081–24125\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px2.p1.1),[§4\.1](https://arxiv.org/html/2610.42018#S4.SS1.SSS0.Px3.p1.2),[§4\.1](https://arxiv.org/html/2610.42018#S4.SS1.p1.2),[§5\.1](https://arxiv.org/html/2610.42018#S5.SS1.p1.1)\.
- Chenet al\.\(2026\)Y\. Chen, C\. Liang, H\. Sui, R\. Guo, C\. Cheng, J\. You, and G\. LiuLangflow: continuous diffusion rivals discrete in language modeling\.arXiv preprint arXiv:2604\.11748\.Cited by:[§1](https://arxiv.org/html/2610.42018#S1.p3.1),[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px3.p1.1),[§3](https://arxiv.org/html/2610.42018#S3.SS0.SSS0.Px1.p1.1)\.
- Chenet al\.\(2025\)Z\. Chen, G\. Fang, X\. Ma, R\. Yu, and X\. WangDParallel: learnable parallel decoding for dllms\.arXiv preprint arXiv:2509\.26488\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px1.p1.1)\.
- Cobbeet al\.\(2021\)K\. Cobbe, V\. Kosaraju, M\. Bavarian, M\. Chen, H\. Jun, L\. Kaiser, M\. Plappert, J\. Tworek, J\. Hilton, R\. Nakano,et al\.Training verifiers to solve math word problems\.arXiv preprint arXiv:2110\.14168\.Cited by:[§5](https://arxiv.org/html/2610.42018#S5.SS0.SSS0.Px1.p1.1)\.
- Deschenaux and Gulcehre \(2026\)J\. Deschenaux and C\. GulcehreLanguage modeling with hyperspherical flows\.arXiv preprint arXiv:2605\.11125\.Cited by:[§C\.2](https://arxiv.org/html/2610.42018#A3.SS2.p1.1),[§5](https://arxiv.org/html/2610.42018#S5.SS0.SSS0.Px1.p1.1),[§5](https://arxiv.org/html/2610.42018#S5.SS0.SSS0.Px1.p2.1),[Table 1](https://arxiv.org/html/2610.42018#S5.T1.4.7.1)\.
- Dielemanet al\.\(2022\)S\. Dieleman, L\. Sartran, A\. Roshannai, N\. Savinov, Y\. Ganin, P\. H\. Richemond, A\. Doucet, R\. Strudel, C\. Dyer, C\. Durkan,et al\.Continuous diffusion for categorical data\.arXiv preprint arXiv:2211\.15089\.Cited by:[§3](https://arxiv.org/html/2610.42018#S3.SS0.SSS0.Px1.p1.4)\.
- Eijkelboomet al\.\(2024\)F\. Eijkelboom, G\. Bartosh, C\. A\. Naesseth, M\. Welling, and J\. van de MeentVariational flow matching for graph generation\.Advances in Neural Information Processing Systems37,pp\. 11735–11764\.Cited by:[§3](https://arxiv.org/html/2610.42018#S3.SS0.SSS0.Px1.p1.4)\.
- Fathiet al\.\(2025\)N\. Fathi, T\. Scholak, and P\. NoëlUnifying autoregressive and diffusion\-based sequence generation\.arXiv preprint arXiv:2504\.06416\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px1.p1.1)\.
- Finziet al\.\(2026\)M\. Finzi, S\. Qiu, Y\. Jiang, P\. Izmailov, J\. Z\. Kolter, and A\. G\. WilsonFrom entropy to epiplexity: rethinking information for computationally bounded intelligence\.arXiv preprint arXiv:2601\.03220\.Cited by:[§1](https://arxiv.org/html/2610.42018#S1.p1.1),[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px1.p1.1)\.
- Gushchinet al\.\(2026\)N\. Gushchin, D\. Baranchuk, and A\. KorotinAlpha diffusion language models: factorization alone is not the problem\.arXiv preprint arXiv:2609\.38066\.Cited by:[§5](https://arxiv.org/html/2610.42018#S5.SS0.SSS0.Px1.p2.1),[Table 1](https://arxiv.org/html/2610.42018#S5.T1.4.11.1)\.
- Huet al\.\(2026\)K\. Hu, L\. Qiu, Y\. Lu, H\. Zhao, T\. Li, Y\. Kim, J\. Andreas, and K\. HeElf: embedded language flows\.arXiv preprint arXiv:2605\.10938\.Cited by:[§1](https://arxiv.org/html/2610.42018#S1.p3.1),[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px3.p1.1),[§3](https://arxiv.org/html/2610.42018#S3.SS0.SSS0.Px1.p1.1)\.
- Huet al\.\(2025\)Z\. Hu, Y\. Tong, F\. Zhang, J\. Yuan, J\. Xiao, and K\. KuangAsynchronous denoising diffusion models for aligning text\-to\-image generation\.arXiv preprint arXiv:2510\.04504\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px2.p1.1)\.
- Huanget al\.\(2025\)X\. Huang, Z\. Li, G\. He, M\. Zhou, and E\. ShechtmanSelf forcing: bridging the train\-test gap in autoregressive video diffusion\.arXiv preprint arXiv:2506\.08009\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px2.p1.1)\.
- Jutras\-Dubéet al\.\(2026\)P\. Jutras\-Dubé, P\. Pynadath, J\. Lu, Y\. Gao, and R\. ZhangSticky jump diffusions: a unifying view of masked, continuous, and hybrid diffusion\.arXiv preprint arXiv:2607\.10951\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px3.p1.1)\.
- Kimet al\.\(2025a\)J\. Kim, K\. Shah, V\. Kontonis, S\. Kakade, and S\. ChenTrain for the worst, plan for the best: understanding token ordering in masked diffusions\.arXiv preprint arXiv:2502\.06768\.Cited by:[§1](https://arxiv.org/html/2610.42018#S1.p1.1),[§4\.1](https://arxiv.org/html/2610.42018#S4.SS1.SSS0.Px4.p3.1),[§4\.3](https://arxiv.org/html/2610.42018#S4.SS3.SSS0.Px5.p1.1),[§5](https://arxiv.org/html/2610.42018#S5.SS0.SSS0.Px1.p1.1),[§5\.1](https://arxiv.org/html/2610.42018#S5.SS1.p3.1)\.
- Kimet al\.\(2025b\)M\. Kim, C\. Xu, C\. Hooper, H\. Singh, B\. Athiwaratkun, C\. Zhang, K\. Keutzer, and A\. GholamiCDLM: consistency diffusion language models for faster sampling\.arXiv preprint arXiv:2511\.19269\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px1.p1.1)\.
- Kimet al\.\(2026\)S\. H\. Kim, S\. Hong, H\. Jung, Y\. Park, and S\. YunKlass: kl\-guided fast inference in masked diffusion models\.Advances in Neural Information Processing Systems38,pp\. 92267–92301\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px1.p1.1)\.
- Leeet al\.\(2026\)C\. Lee, J\. Yoo, M\. Agarwal, S\. Shah, J\. Huang, A\. Raghunathan, S\. Hong, N\. M\. Boffi, and J\. KimFlow map language models: one\-step language modeling via continuous denoising\.arXiv preprint arXiv:2602\.16813\.Cited by:[1st item](https://arxiv.org/html/2610.42018#A2.I1.i1.p1.1),[§C\.1](https://arxiv.org/html/2610.42018#A3.SS1.SSS0.Px3.p1.1),[§1](https://arxiv.org/html/2610.42018#S1.p2.1),[§1](https://arxiv.org/html/2610.42018#S1.p3.1),[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px1.p1.1),[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px3.p1.1),[§3](https://arxiv.org/html/2610.42018#S3.SS0.SSS0.Px1.p1.1),[§3](https://arxiv.org/html/2610.42018#S3.SS0.SSS0.Px1.p1.4),[§3](https://arxiv.org/html/2610.42018#S3.SS0.SSS0.Px2.p1.2),[§3](https://arxiv.org/html/2610.42018#S3.SS0.SSS0.Px2.p1.3),[§4\.3](https://arxiv.org/html/2610.42018#S4.SS3.SSS0.Px3.p1.1),[§5](https://arxiv.org/html/2610.42018#S5.SS0.SSS0.Px1.p2.1),[§5\.1](https://arxiv.org/html/2610.42018#S5.SS1.p1.1),[Table 1](https://arxiv.org/html/2610.42018#S5.T1.4.6.1),[Table 1](https://arxiv.org/html/2610.42018#S5.T1.4.8.1)\.
- Leeet al\.\(2023\)N\. Lee, K\. Sreenivasan, J\. D\. Lee, K\. Lee, and D\. PapailiopoulosTeaching arithmetic to small transformers\.arXiv preprint arXiv:2307\.03381\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px1.p1.1)\.
- Lianget al\.\(2026\)Y\. Liang, Z\. Wang, H\. Chen, X\. Sun, J\. Wu, X\. Yu, J\. Liu, E\. Barsoum, Z\. Liu, and N\. K\. JhaCD4LM: consistency distillation and adaptive decoding for diffusion language models\.arXiv preprint arXiv:2601\.02236\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px1.p1.1)\.
- Liuet al\.\(2023\)B\. Liu, S\. Bubeck, R\. Eldan, J\. Kulkarni, Y\. Li, A\. Nguyen, R\. Ward, and Y\. ZhangTinygsm: achieving\>\>80% on gsm8k with small language models\.arXiv preprint arXiv:2312\.09241\.Cited by:[§5](https://arxiv.org/html/2610.42018#S5.SS0.SSS0.Px1.p1.1)\.
- Loshchilov and Hutter \(2017\)I\. Loshchilov and F\. HutterDecoupled weight decay regularization\.arXiv preprint arXiv:1711\.05101\.Cited by:[§C\.1](https://arxiv.org/html/2610.42018#A3.SS1.SSS0.Px1.p1.1)\.
- Nieet al\.\(2026\)S\. Nie, F\. Zhu, Z\. You, X\. Zhang, J\. Ou, J\. Hu, J\. Zhou, Y\. Lin, J\. Wen, and C\. LiLarge language diffusion models\.Advances in Neural Information Processing Systems38,pp\. 50608–50646\.Cited by:[§1](https://arxiv.org/html/2610.42018#S1.p1.1),[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px1.p1.1)\.
- Panet al\.\(2025\)Y\. Pan, R\. Feng, Q\. Dai, Y\. Wang, W\. Lin, M\. Guo, C\. Luo, and N\. ZhengSemantics lead the way: harmonizing semantic and texture modeling with asynchronous latent diffusion\.arXiv preprint arXiv:2512\.04926\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px2.p1.1)\.
- Papadopouloset al\.\(2024\)V\. Papadopoulos, J\. Wenger, and C\. HonglerArrows of time for large language models\.arXiv preprint arXiv:2401\.17505\.Cited by:[§1](https://arxiv.org/html/2610.42018#S1.p1.1),[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px1.p1.1)\.
- Peebles and Xie \(2023\)W\. Peebles and S\. XieScalable diffusion models with transformers\.In2023 IEEE/CVF International Conference on Computer Vision \(ICCV\),pp\. 4172–4182\.Cited by:[§4\.3](https://arxiv.org/html/2610.42018#S4.SS3.SSS0.Px3.p1.1)\.
- Potaptchiket al\.\(2026\)P\. Potaptchik, J\. Yim, A\. Saravanan, P\. Holderrieth, E\. Vanden\-Eijnden, and M\. S\. AlbergoDiscrete flow maps\.arXiv preprint arXiv:2604\.09784\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px3.p1.1),[§3](https://arxiv.org/html/2610.42018#S3.SS0.SSS0.Px2.p1.2)\.
- Pynadathet al\.\(2025\)P\. Pynadath, J\. Shi, and R\. ZhangCandi: hybrid discrete\-continuous diffusion models\.arXiv preprint arXiv:2510\.22510\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px3.p1.1),[§5](https://arxiv.org/html/2610.42018#S5.SS0.SSS0.Px1.p2.1),[Table 1](https://arxiv.org/html/2610.42018#S5.T1.4.5.1)\.
- Qian and Cheng \(2026\)B\. Qian and X\. ChengLearning when to denoise: optimizing asynchronous schedules for latent diffusion\.arXiv preprint arXiv:2606\.19662\.Cited by:[§1](https://arxiv.org/html/2610.42018#S1.p3.1),[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px2.p1.1),[§4\.1](https://arxiv.org/html/2610.42018#S4.SS1.p1.2)\.
- Rekeshet al\.\(2026\)A\. Rekesh, J\. Rector\-Brooks, and C\. LiuInverse\-confidence sampling for continuous diffusion language models\.InICML 2026 Workshop on Foundations of Deep Generative Models: Understanding Memorization, Generalization, and Reasoning,Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px3.p1.1)\.
- Rooset al\.\(2026\)D\. Roos, O\. Davis, F\. Eijkelboom, M\. Bronstein, M\. Welling, I\. I\. Ceylan, L\. Ambrogioni, and J\. van de MeentCategorical flow maps\.arXiv preprint arXiv:2602\.12233\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px3.p1.1),[§3](https://arxiv.org/html/2610.42018#S3.SS0.SSS0.Px2.p1.2)\.
- Ruheet al\.\(2024\)D\. Ruhe, J\. Heek, T\. Salimans, and E\. HoogeboomRolling diffusion models\.arXiv preprint arXiv:2402\.09470\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px2.p1.1)\.
- Sahooet al\.\(2024\)S\. S\. Sahoo, M\. Arriola, Y\. Schiff, A\. Gokaslan, E\. Marroquin, J\. T\. Chiu, A\. Rush, and V\. KuleshovSimple and effective masked diffusion language models\.Advances in Neural Information Processing Systems37,pp\. 130136–130184\.Cited by:[§1](https://arxiv.org/html/2610.42018#S1.p1.1),[§4\.3](https://arxiv.org/html/2610.42018#S4.SS3.SSS0.Px5.p1.1),[§5](https://arxiv.org/html/2610.42018#S5.SS0.SSS0.Px1.p2.1),[Table 1](https://arxiv.org/html/2610.42018#S5.T1.4.3.1)\.
- Sahooet al\.\(2025\)S\. S\. Sahoo, J\. Deschenaux, A\. Gokaslan, G\. Wang, J\. Chiu, and V\. KuleshovThe diffusion duality\.Proceedings of machine learning research267,pp\. 52584\.Cited by:[§5](https://arxiv.org/html/2610.42018#S5.SS0.SSS0.Px1.p2.1),[Table 1](https://arxiv.org/html/2610.42018#S5.T1.4.4.1)\.
- Schiffet al\.\(2026\)Y\. Schiff, O\. Belhasin, R\. Uziel, M\. Rusanovsky, R\. Zilberstein, M\. Arriola, G\. Turok, G\. Wang, V\. Kuleshov, and M\. EladClock diffusion: efficient semi\-autoregressive continuous diffusion language models\.arXiv preprint arXiv:2610\.00894\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px2.p1.1),[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px3.p1.1)\.
- Schusterbaueret al\.\(2026\)J\. Schusterbauer, M\. Gui, Y\. Li, P\. Ma, F\. Krause, and B\. OmmerDenoising, fast and slow: difficulty\-aware adaptive sampling for image generation\.arXiv preprint arXiv:2604\.19141\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px2.p1.1)\.
- Shahet al\.\(2024\)K\. Shah, N\. Dikkala, X\. Wang, and R\. PanigrahyCausal language modeling can elicit search and reasoning capabilities on logic puzzles\.arXiv preprint arXiv:2409\.10502\.Cited by:[§C\.3](https://arxiv.org/html/2610.42018#A3.SS3.p1.1),[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px1.p1.1),[§5\.1](https://arxiv.org/html/2610.42018#S5.SS1.p1.1)\.
- Shannon \(1951\)C\. E\. ShannonPrediction and entropy of printed english\.Bell system technical journal30\(1\),pp\. 50–64\.Cited by:[§1](https://arxiv.org/html/2610.42018#S1.p1.1),[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px1.p1.1)\.
- Shiet al\.\(2024\)J\. Shi, K\. Han, Z\. Wang, A\. Doucet, and M\. TitsiasSimplified and generalized masked diffusion for discrete data\.Advances in neural information processing systems37,pp\. 103131–103167\.Cited by:[§1](https://arxiv.org/html/2610.42018#S1.p1.1)\.
- Tang and Wang \(2026\)S\. Tang and S\. WangDiscrete beckmann transport models for one\-step language modeling and reasoning\.arXiv preprint arXiv:2609\.15903\.Cited by:[§5](https://arxiv.org/html/2610.42018#S5.SS0.SSS0.Px1.p2.1),[Table 1](https://arxiv.org/html/2610.42018#S5.T1.4.10.1)\.
- Weweret al\.\(2025\)C\. Wewer, B\. Pogodzinski, B\. Schiele, and J\. E\. LenssenSpatial reasoning with denoising models\.arXiv preprint arXiv:2502\.21075\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px2.p1.1)\.
- Wuet al\.\(2023\)T\. Wu, Z\. Fan, X\. Liu, H\. Zheng, Y\. Gong, J\. Jiao, J\. Li, J\. Guo, N\. Duan, W\. Chen,et al\.Ar\-diffusion: auto\-regressive diffusion model for text generation\.Advances in Neural Information Processing Systems36,pp\. 39957–39974\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px2.p1.1),[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px3.p1.1)\.
- Yinet al\.\(2024\)T\. Yin, Q\. Zhang, R\. Zhang, W\. T\. Freeman, F\. Durand, E\. Shechtman, and X\. HuangFrom slow bidirectional to fast autoregressive video diffusion models\.arXiv preprint arXiv:2412\.07772\.Cited by:[§2](https://arxiv.org/html/2610.42018#S2.SS0.SSS0.Px2.p1.1)\.

## Appendix AProofs and Derivations

Throughout, we assume that the velocity fields in this section are regular enough for their ODEs and continuity equations to have unique solutions on\[0,1\)\[0,1\), and that prescribed schedules are absolutely continuous\. We also assumeωti\>0\\omega^\{i\}\_\{t\}\>0whenever0<t<10<t<1andτti<1\\tau^\{i\}\_\{t\}<1, and that the tokenwise times reach𝟏\{\\bf 1\}att=1t=1, which holds for the schedulers of[Equation27](https://arxiv.org/html/2610.42018#S4.E27)withkti\>0k^\{i\}\_\{t\}\>0for0<t<10<t<1and bounded away from zero neart=1t=1\. Quantities att=1t=1are understood as limitst↑1t\\uparrow 1, and diagonal quantities att=st=sas limitst↓st\\downarrow s\.

### A\.1Proof of Proposition[1](https://arxiv.org/html/2610.42018#Thmproposition1)

See[1](https://arxiv.org/html/2610.42018#Thmproposition1)

###### Proof\.

We parameterize each path byr∈\[0,1\]r\\in\[0,1\]\.

#### Scalar time\.

Lett⁡\(r\)t\(r\)be a continuous nondecreasing path with

t⁡\(0\)=0,t⁡\(1\)=1\.t\(0\)=0,\\qquad t\(1\)=1\.Thent⁡\(r\)t\(r\)passes through every value in\[0,1\]\[0,1\]in order, so it is a reparameterization oft⁡\(r\)=rt\(r\)=r\.

#### Tokenwise times\.

It suffices to exhibit one infinite family of such paths that remain distinct up to reparameterization\.

ForL≥2L\\geq 2anda\>0a\>0, the pathτa​\(r\)=\(ra,r,…,r\)\\tau\_\{a\}\(r\)=\(r^\{a\},r,\\ldots,r\)goes from𝟎\\mathbf\{0\}to𝟏\\mathbf\{1\}and satisfiesτa1=\(τa2\)a\\tau\_\{a\}^\{1\}=\(\\tau\_\{a\}^\{2\}\)^\{a\}along the way\.

Since this relation does not change under reparameterization and differs acrossaa, the pathsτa\\tau\_\{a\}are infinitely many and distinct up to reparameterization\. ∎

### A\.2Proof of Proposition[2](https://arxiv.org/html/2610.42018#Thmproposition2)

See[2](https://arxiv.org/html/2610.42018#Thmproposition2)

###### Proof\.

For each\(𝐱𝟎,𝐱𝟏\)\(\{\\bf x\}\_\{\\bf 0\},\{\\bf x\}\_\{\\bf 1\}\), letτt\\tau\_\{t\}solveτ˙t=ωt​\(Iτt,τt\)\\dot\{\\tau\}\_\{t\}=\\omega\_\{t\}\(I\_\{\\tau\_\{t\}\},\\tau\_\{t\}\)withτ0=𝟎\\tau\_\{0\}=\{\\bf 0\}, and letZt≔\(Iτt,τt\)Z\_\{t\}\\coloneqq\(I\_\{\\tau\_\{t\}\},\\tau\_\{t\}\)\. Since positioniiof the interpolant isIτti=\(1−τti\)​𝐱𝟎i\+τti​𝐱𝟏iI^\{i\}\_\{\\tau\_\{t\}\}=\(1\-\\tau^\{i\}\_\{t\}\)\{\\bf x\}^\{i\}\_\{\\bf 0\}\+\\tau^\{i\}\_\{t\}\{\\bf x\}^\{i\}\_\{\\bf 1\}, differentiating inttgives

dd​t​Iτt=τ˙t​\(𝐱𝟏−𝐱𝟎\),soZ˙t=\(ωt​\(Zt\)​\(𝐱𝟏−𝐱𝟎\),ωt​\(Zt\)\)\.\\frac\{\\rm d\}\{\{\\rm d\}t\}I\_\{\\tau\_\{t\}\}=\\dot\{\\tau\}\_\{t\}\(\{\\bf x\}\_\{\\bf 1\}\-\{\\bf x\}\_\{\\bf 0\}\),\\qquad\\text\{so\}\\qquad\\dot\{Z\}\_\{t\}=\\bigl\(\\omega\_\{t\}\(Z\_\{t\}\)\(\{\\bf x\}\_\{\\bf 1\}\-\{\\bf x\}\_\{\\bf 0\}\),\\ \\omega\_\{t\}\(Z\_\{t\}\)\\bigr\)\.Sinceωt​\(Zt\)\\omega\_\{t\}\(Z\_\{t\}\)is a function ofZtZ\_\{t\}, taking the conditional mean givenZt=\(𝐱,τ\)Z\_\{t\}=\(\{\\bf x\},\\tau\)and using[Equation11](https://arxiv.org/html/2610.42018#S4.E11)gives

𝔼⁡\[Z˙t∣Zt=\(𝐱,τ\)\]\\displaystyle\\mathbb\{E\}\[\\dot\{Z\}\_\{t\}\\mid Z\_\{t\}=\(\{\\bf x\},\\tau\)\]=\(ωt\(𝐱,τ\)𝔼\[𝐱𝟏−𝐱𝟎∣Iτt=𝐱,τt=τ\],ωt\(𝐱,τ\)\)\\displaystyle=\\bigl\(\\omega\_\{t\}\(\{\\bf x\},\\tau\)\\,\\mathbb\{E\}\[\{\\bf x\}\_\{\\bf 1\}\-\{\\bf x\}\_\{\\bf 0\}\\mid I\_\{\\tau\_\{t\}\}=\{\\bf x\},\\tau\_\{t\}=\\tau\],\\ \\omega\_\{t\}\(\{\\bf x\},\\tau\)\\bigr\)=\(ωt​\(𝐱,τ\)​btω​\(𝐱,τ\),ωt​\(𝐱,τ\)\)≕ut​\(𝐱,τ\)\.\\displaystyle=\\bigl\(\\omega\_\{t\}\(\{\\bf x\},\\tau\)\\,b^\{\\omega\}\_\{t\}\(\{\\bf x\},\\tau\),\\ \\omega\_\{t\}\(\{\\bf x\},\\tau\)\\bigr\)\\eqqcolon u\_\{t\}\(\{\\bf x\},\\tau\)\.Hence, for any smooth test functionφ\\varphiwith compact support, the lawqtq\_\{t\}ofZtZ\_\{t\}satisfies

dd​t​𝔼​\[φ⁡\(Zt\)\]=𝔼⁡\[∇φ​\(Zt\)⋅Z˙t\]=𝔼⁡\[∇φ​\(Zt\)⋅ut​\(Zt\)\],\\frac\{\\rm d\}\{\{\\rm d\}t\}\\mathbb\{E\}\[\\varphi\(Z\_\{t\}\)\]=\\mathbb\{E\}\\bigl\[\\nabla\\varphi\(Z\_\{t\}\)\\cdot\\dot\{Z\}\_\{t\}\\bigr\]=\\mathbb\{E\}\\bigl\[\\nabla\\varphi\(Z\_\{t\}\)\\cdot u\_\{t\}\(Z\_\{t\}\)\\bigr\],soqtq\_\{t\}is a weak solution of the continuity equation

∂tqt\+∇⋅\(qt​ut\)=0,q0=p𝟎⊗δ𝟎\.\\partial\_\{t\}q\_\{t\}\+\\nabla\\cdot\(q\_\{t\}\\,u\_\{t\}\)=0,\\qquad q\_\{0\}=p\_\{\\bf 0\}\\otimes\\delta\_\{\\bf 0\}\.The law of\(𝐱t,τt\)\(\{\\bf x\}\_\{t\},\\tau\_\{t\}\)in the proposition solves the same equation with the same initial condition, since its velocity isutu\_\{t\}\. Since this solution is unique, the two laws coincide for everyt∈\[0,1\)t\\in\[0,1\)\. Finally,τt→𝟏\\tau\_\{t\}\\to\{\\bf 1\}ast↑1t\\uparrow 1, soIτt→𝐱𝟏I\_\{\\tau\_\{t\}\}\\to\{\\bf x\}\_\{\\bf 1\}for every\(𝐱𝟎,𝐱𝟏\)\(\{\\bf x\}\_\{\\bf 0\},\{\\bf x\}\_\{\\bf 1\}\)\. Since𝐱t\{\\bf x\}\_\{t\}has the law ofIτtI\_\{\\tau\_\{t\}\}, it converges in distribution top𝟏p\_\{\\bf 1\}\. ∎

### A\.3Proof of Proposition[3](https://arxiv.org/html/2610.42018#Thmproposition3)

See[3](https://arxiv.org/html/2610.42018#Thmproposition3)

###### Proof\.

Since𝐱𝟏i\{\\bf x\}\_\{\\bf 1\}^\{i\}is the one\-hot encoding of𝐲i\{\\bf y\}^\{i\}, the loss at positioniican be written as an inner product,

\[−log⁡D^τi​\(Iτ\)\]𝐲i=−⟨𝐱𝟏i,log⁡D^τi​\(Iτ\)⟩\.\\left\[\-\\log\\hat\{D\}\_\{\\tau\}^\{i\}\(I\_\{\\tau\}\)\\right\]\_\{\{\\bf y\}^\{i\}\}=\-\\left\\langle\{\\bf x\}\_\{\\bf 1\}^\{i\},\\log\\hat\{D\}\_\{\\tau\}^\{i\}\(I\_\{\\tau\}\)\\right\\rangle\.Conditioning on\(Iτ,τ\)\(I\_\{\\tau\},\\tau\)and using𝔼\[𝐱𝟏∣Iτ,τ\]=Dτ\(Iτ\)\\mathbb\{E\}\[\{\\bf x\}\_\{\\bf 1\}\\mid I\_\{\\tau\},\\tau\]=D\_\{\\tau\}\(I\_\{\\tau\}\), which holds becauseτ\\tauis independent of\(𝐱𝟎,𝐱𝟏\)\(\{\\bf x\}\_\{\\bf 0\},\{\\bf x\}\_\{\\bf 1\}\), its expectation is

𝔼\[−⟨𝐱𝟏i,logD^τi\(Iτ\)⟩\]=𝔼\[−⟨𝔼\[𝐱𝟏i∣Iτ,τ\],logD^τi\(Iτ\)⟩\]=𝔼\[−⟨Dτi\(Iτ\),logD^τi\(Iτ\)⟩\]\.\\mathbb\{E\}\\left\[\-\\left\\langle\{\\bf x\}\_\{\\bf 1\}^\{i\},\\log\\hat\{D\}\_\{\\tau\}^\{i\}\(I\_\{\\tau\}\)\\right\\rangle\\right\]=\\mathbb\{E\}\\left\[\-\\left\\langle\\mathbb\{E\}\[\{\\bf x\}\_\{\\bf 1\}^\{i\}\\mid I\_\{\\tau\},\\tau\],\\log\\hat\{D\}\_\{\\tau\}^\{i\}\(I\_\{\\tau\}\)\\right\\rangle\\right\]=\\mathbb\{E\}\\left\[\-\\left\\langle D\_\{\\tau\}^\{i\}\(I\_\{\\tau\}\),\\log\\hat\{D\}\_\{\\tau\}^\{i\}\(I\_\{\\tau\}\)\\right\\rangle\\right\]\.Adding and subtracting⟨Dτi​\(Iτ\),log⁡Dτi​\(Iτ\)⟩\\langle D\_\{\\tau\}^\{i\}\(I\_\{\\tau\}\),\\log D\_\{\\tau\}^\{i\}\(I\_\{\\tau\}\)\\rangleand summing over positions gives

ℒ𝖢𝖤\(D^\)=𝔼\[∑i=1LH\(Dτi\(Iτ\)\)\]\+𝔼\[∑i=1L𝖪𝖫\(Dτi\(Iτ\)∥D^τi\(Iτ\)\)\],\\mathcal\{L\}\_\{\\mathsf\{CE\}\}\(\\hat\{D\}\)=\\mathbb\{E\}\\left\[\\sum\_\{i=1\}^\{L\}H\(D\_\{\\tau\}^\{i\}\(I\_\{\\tau\}\)\)\\right\]\+\\mathbb\{E\}\\left\[\\sum\_\{i=1\}^\{L\}\\mathsf\{KL\}\\\!\\left\(D\_\{\\tau\}^\{i\}\(I\_\{\\tau\}\)\\,\\\|\\,\\hat\{D\}\_\{\\tau\}^\{i\}\(I\_\{\\tau\}\)\\right\)\\right\],\(29\)whereH⁡\(p\)=−⟨p,log⁡p⟩H\(p\)=\-\\langle p,\\log p\\rangleis the entropy\. The first term does not depend onD^\\hat\{D\}, and the second term is nonnegative and vanishes exactly whenD^τi​\(Iτ\)=Dτi​\(Iτ\)\\hat\{D\}\_\{\\tau\}^\{i\}\(I\_\{\\tau\}\)=D\_\{\\tau\}^\{i\}\(I\_\{\\tau\}\)for everyii\. Thus,D^⋆\\hat\{D\}^\{\\star\}minimizesℒ𝖢𝖤\\mathcal\{L\}\_\{\\mathsf\{CE\}\}if and only if

D^τ⋆​\(Iτ\)=Dτ​\(Iτ\)\\hat\{D\}\_\{\\tau\}^\{\\star\}\(I\_\{\\tau\}\)=D\_\{\\tau\}\(I\_\{\\tau\}\)\(30\)almost surely under the training distribution\. ∎

### A\.4Support of the diffusion forcing training distribution

We show that the training distribution of\(Iτ,τ\)\(I\_\{\\tau\},\\tau\)in[Equation13](https://arxiv.org/html/2610.42018#S4.E13)has supportℝL×\|V\|×\[0,1\]L\\mathbb\{R\}^\{L\\times\|V\|\}\\times\[0,1\]^\{L\}, so that every state\(𝐱t,τt\)\(\{\\bf x\}\_\{t\},\\tau\_\{t\}\)of an asynchronous flow lies in it regardless of the scheduler\.

###### Proof\.

Fixτ∈\[0,1\)L\\tau\\in\[0,1\)^\{L\}and𝐱𝟏\{\\bf x\}\_\{\\bf 1\}\. Since𝐱𝟎∼𝖭⁡\(0,I\)\{\\bf x\}\_\{\\bf 0\}\\sim\\mathsf\{N\}\(0,I\)is independent of𝐱𝟏\{\\bf x\}\_\{\\bf 1\}andτ\\tau, each position of the interpolant is distributed as

Iτi=\(1−τi\)​𝐱𝟎i\+τi​𝐱𝟏i∼𝖭⁡\(τi​𝐱𝟏i,\(1−τi\)2​I\),I^\{i\}\_\{\\tau\}=\(1\-\\tau^\{i\}\)\{\\bf x\}^\{i\}\_\{\\bf 0\}\+\\tau^\{i\}\{\\bf x\}^\{i\}\_\{\\bf 1\}\\sim\\mathsf\{N\}\\big\(\\tau^\{i\}\{\\bf x\}^\{i\}\_\{\\bf 1\},\\,\(1\-\\tau^\{i\}\)^\{2\}I\\big\),independently across positions\. Since1−τi\>01\-\\tau^\{i\}\>0,IτI\_\{\\tau\}has a positive density onℝL×\|V\|\\mathbb\{R\}^\{L\\times\|V\|\}\. Moreover, eachτi∼𝖴⁡\[0,1\]\\tau^\{i\}\\sim\\mathsf\{U\}\[0,1\]independently, soτ\\tauhas a positive density on\[0,1\)L\[0,1\)^\{L\}\. The joint distribution of\(Iτ,τ\)\(I\_\{\\tau\},\\tau\)therefore has a positive density onℝL×\|V\|×\[0,1\)L\\mathbb\{R\}^\{L\\times\|V\|\}\\times\[0,1\)^\{L\}, and its support is the closure of this set,

ℝL×\|V\|×\[0,1\)L¯=ℝL×\|V\|×\[0,1\]L,\\overline\{\\mathbb\{R\}^\{L\\times\|V\|\}\\times\[0,1\)^\{L\}\}=\\mathbb\{R\}^\{L\\times\|V\|\}\\times\[0,1\]^\{L\},which contains every state\(𝐱t,τt\)\(\{\\bf x\}\_\{t\},\\tau\_\{t\}\)of an asynchronous flow\. ∎

### A\.5Proof of Corollary[4](https://arxiv.org/html/2610.42018#Thmproposition4)

See[4](https://arxiv.org/html/2610.42018#Thmproposition4)

###### Proof\.

Sinceω\\omegadoes not depend on𝐱\{\\bf x\}, every interpolant follows the same tokenwise timesτt=τ⁡\(t\)\\tau\_\{t\}=\\tau\(t\)withωt=τ˙​\(t\)\\omega\_\{t\}=\\dot\{\\tau\}\(t\)\. The conditionτt=τ⁡\(t\)\\tau\_\{t\}=\\tau\(t\)in[Equation11](https://arxiv.org/html/2610.42018#S4.E11)therefore holds for every\(𝐱𝟎,𝐱𝟏\)\(\{\\bf x\}\_\{\\bf 0\},\{\\bf x\}\_\{\\bf 1\}\), and

btω​\(𝐱,τ⁡\(t\)\)=𝔼⁡\[𝐱𝟏−𝐱𝟎∣Iτ⁡\(t\)=𝐱\]=bτ⁡\(t\)​\(𝐱\)\.b^\{\\omega\}\_\{t\}\(\{\\bf x\},\\tau\(t\)\)=\\mathbb\{E\}\[\{\\bf x\}\_\{\\bf 1\}\-\{\\bf x\}\_\{\\bf 0\}\\mid I\_\{\\tau\(t\)\}=\{\\bf x\}\]=b\_\{\\tau\(t\)\}\(\{\\bf x\}\)\.Substituting into[Equation15](https://arxiv.org/html/2610.42018#S4.E15)gives[Equation16](https://arxiv.org/html/2610.42018#S4.E16), whose velocityωt​bτ⁡\(t\)=ωt​btω\\omega\_\{t\}b\_\{\\tau\(t\)\}=\\omega\_\{t\}b^\{\\omega\}\_\{t\}coincides with that of[Proposition2](https://arxiv.org/html/2610.42018#Thmproposition2)\. Hence,𝐱t\{\\bf x\}\_\{t\}has the law ofIτ⁡\(t\)I\_\{\\tau\(t\)\}, which ispτ⁡\(t\)p\_\{\\tau\(t\)\}for everyt∈\[0,1\)t\\in\[0,1\), and𝐱t\{\\bf x\}\_\{t\}converges in distribution top𝟏p\_\{\\bf 1\}ast↑1t\\uparrow 1\. ∎

### A\.6Proof of Proposition[5](https://arxiv.org/html/2610.42018#Thmproposition5)

See[5](https://arxiv.org/html/2610.42018#Thmproposition5)

###### Proof\.

Let\(𝐱r,τr\)=Fs,r​\(𝐳\)\(\{\\bf x\}\_\{r\},\\tau\_\{r\}\)=F\_\{s,r\}\(\{\\bf z\}\), where𝐳=\(𝐱,τ\)\{\\bf z\}=\(\{\\bf x\},\\tau\)\. Integrating the coupled dynamics[Equation15](https://arxiv.org/html/2610.42018#S4.E15)fromsstottgives the first identity\.

Fors≤u≤t<1s\\leq u\\leq t<1, bothr↦Fs,r​\(𝐳\)r\\mapsto F\_\{s,r\}\(\{\\bf z\}\)andr↦Fu,r​\(Fs,u​\(𝐳\)\)r\\mapsto F\_\{u,r\}\(F\_\{s,u\}\(\{\\bf z\}\)\)solve the coupled dynamics on\[u,t\]\[u,t\]with the same stateFs,u​\(𝐳\)F\_\{s,u\}\(\{\\bf z\}\)atr=ur=u\. Uniqueness of solutions therefore givesFs,t​\(𝐳\)=Fu,t​\(Fs,u​\(𝐳\)\)F\_\{s,t\}\(\{\\bf z\}\)=F\_\{u,t\}\(F\_\{s,u\}\(\{\\bf z\}\)\)\. ∎

### A\.7Proof of Proposition[6](https://arxiv.org/html/2610.42018#Thmproposition6)

See[6](https://arxiv.org/html/2610.42018#Thmproposition6)

###### Proof\.

Throughout, let𝐳=\(𝐱,τ\)\{\\bf z\}=\(\{\\bf x\},\\tau\)and\(𝐱r,τr\)=Fs,r​\(𝐳\)\(\{\\bf x\}\_\{r\},\\tau\_\{r\}\)=F\_\{s,r\}\(\{\\bf z\}\)forr≥sr\\geq s, so that\(𝐱s,τs\)=\(𝐱,τ\)\(\{\\bf x\}\_\{s\},\\tau\_\{s\}\)=\(\{\\bf x\},\\tau\)\. By[Equation18](https://arxiv.org/html/2610.42018#S4.E18), the averagesvs,tv\_\{s,t\}andηs,t\\eta\_\{s,t\}satisfy

\(t−s\)​vs,t​\(𝐳\)\\displaystyle\(t\-s\)\\,v\_\{s,t\}\(\{\\bf z\}\)=∫stωr​\(𝐱r,τr\)​bτr​\(𝐱r\)​𝑑r=𝐱t−𝐱,\\displaystyle=\\int\_\{s\}^\{t\}\\omega\_\{r\}\(\{\\bf x\}\_\{r\},\\tau\_\{r\}\)\\,b\_\{\\tau\_\{r\}\}\(\{\\bf x\}\_\{r\}\)\\,\{\\rm d\}r=\{\\bf x\}\_\{t\}\-\{\\bf x\},\(t−s\)​ηs,t​\(𝐳\)\\displaystyle\(t\-s\)\\,\\eta\_\{s,t\}\(\{\\bf z\}\)=∫stωr​\(𝐱r,τr\)​𝑑r=τt−τ\.\\displaystyle=\\int\_\{s\}^\{t\}\\omega\_\{r\}\(\{\\bf x\}\_\{r\},\\tau\_\{r\}\)\\,\{\\rm d\}r=\\tau\_\{t\}\-\\tau\.\(31\)Substituting[Equation31](https://arxiv.org/html/2610.42018#A1.E31)into[Equation19](https://arxiv.org/html/2610.42018#S4.E19),

δs,ti​\(𝐳\)=𝐱i\+1−τiηs,ti​\(𝐳\)​vs,ti​\(𝐳\)=𝐱i\+1−τiτti−τi​\(𝐱ti−𝐱i\)\.\\delta\_\{s,t\}^\{i\}\(\{\\bf z\}\)=\{\\bf x\}^\{i\}\+\\frac\{1\-\\tau^\{i\}\}\{\\eta\_\{s,t\}^\{i\}\(\{\\bf z\}\)\}v\_\{s,t\}^\{i\}\(\{\\bf z\}\)=\{\\bf x\}^\{i\}\+\\frac\{1\-\\tau^\{i\}\}\{\\tau^\{i\}\_\{t\}\-\\tau^\{i\}\}\\bigl\(\{\\bf x\}\_\{t\}^\{i\}\-\{\\bf x\}^\{i\}\\bigr\)\.\(32\)Solving[Equation32](https://arxiv.org/html/2610.42018#A1.E32)for𝐱t\{\\bf x\}\_\{t\}and[Equation31](https://arxiv.org/html/2610.42018#A1.E31)forτt\\tau\_\{t\}proves \(i\)\.

Next, we writeδs,t\\delta\_\{s,t\}as a weighted average of the denoiser along the trajectory\. Recall the coupled dynamics,

τ˙ri=ωri​\(𝐱r,τr\),𝐱˙ri=ωri​\(𝐱r,τr\)1−τri​\(Dτri​\(𝐱r\)−𝐱ri\)\.\\dot\{\\tau\}\_\{r\}^\{i\}=\\omega\_\{r\}^\{i\}\(\{\\bf x\}\_\{r\},\\tau\_\{r\}\),\\qquad\\dot\{\\bf x\}\_\{r\}^\{i\}=\\frac\{\\omega\_\{r\}^\{i\}\(\{\\bf x\}\_\{r\},\\tau\_\{r\}\)\}\{1\-\\tau\_\{r\}^\{i\}\}\\bigl\(D\_\{\\tau\_\{r\}\}^\{i\}\(\{\\bf x\}\_\{r\}\)\-\{\\bf x\}\_\{r\}^\{i\}\\bigr\)\.The product rule gives

dd​r​𝐱ri1−τri\\displaystyle\\frac\{\\rm d\}\{\{\\rm d\}r\}\\frac\{\{\\bf x\}\_\{r\}^\{i\}\}\{1\-\\tau\_\{r\}^\{i\}\}=𝐱˙ri1−τri\+ωri​\(𝐱r,τr\)​𝐱ri\(1−τri\)2\\displaystyle=\\frac\{\\dot\{\\bf x\}\_\{r\}^\{i\}\}\{1\-\\tau\_\{r\}^\{i\}\}\+\\frac\{\\omega\_\{r\}^\{i\}\(\{\\bf x\}\_\{r\},\\tau\_\{r\}\)\{\\bf x\}\_\{r\}^\{i\}\}\{\(1\-\\tau\_\{r\}^\{i\}\)^\{2\}\}=ωri​\(𝐱r,τr\)\(1−τri\)2​Dτri​\(𝐱r\)\.\\displaystyle=\\frac\{\\omega\_\{r\}^\{i\}\(\{\\bf x\}\_\{r\},\\tau\_\{r\}\)\}\{\(1\-\\tau\_\{r\}^\{i\}\)^\{2\}\}D\_\{\\tau\_\{r\}\}^\{i\}\(\{\\bf x\}\_\{r\}\)\.\(33\)Integrating[Equation33](https://arxiv.org/html/2610.42018#A1.E33)and using the tokenwise time dynamics,

𝐱ti1−τti−𝐱i1−τi\\displaystyle\\frac\{\{\\bf x\}\_\{t\}^\{i\}\}\{1\-\\tau\_\{t\}^\{i\}\}\-\\frac\{\{\\bf x\}^\{i\}\}\{1\-\\tau^\{i\}\}=∫stωri​\(𝐱r,τr\)\(1−τri\)2​Dτri​\(𝐱r\)​𝑑r,\\displaystyle=\\int\_\{s\}^\{t\}\\frac\{\\omega\_\{r\}^\{i\}\(\{\\bf x\}\_\{r\},\\tau\_\{r\}\)\}\{\(1\-\\tau\_\{r\}^\{i\}\)^\{2\}\}D\_\{\\tau\_\{r\}\}^\{i\}\(\{\\bf x\}\_\{r\}\)\\,\{\\rm d\}r,\(34\)∫stωri​\(𝐱r,τr\)\(1−τri\)2​𝑑r\\displaystyle\\int\_\{s\}^\{t\}\\frac\{\\omega\_\{r\}^\{i\}\(\{\\bf x\}\_\{r\},\\tau\_\{r\}\)\}\{\(1\-\\tau\_\{r\}^\{i\}\)^\{2\}\}\\,\{\\rm d\}r=11−τti−11−τi=τti−τi\(1−τi\)​\(1−τti\)\.\\displaystyle=\\frac\{1\}\{1\-\\tau\_\{t\}^\{i\}\}\-\\frac\{1\}\{1\-\\tau^\{i\}\}=\\frac\{\\tau\_\{t\}^\{i\}\-\\tau^\{i\}\}\{\(1\-\\tau^\{i\}\)\(1\-\\tau\_\{t\}^\{i\}\)\}\.\(35\)Substituting[Equations34](https://arxiv.org/html/2610.42018#A1.E34)and[35](https://arxiv.org/html/2610.42018#A1.E35)into[Equation32](https://arxiv.org/html/2610.42018#A1.E32),

δs,ti​\(𝐳\)=∫stωri​\(𝐱r,τr\)\(1−τri\)2​Dτri​\(𝐱r\)​𝑑r∫stωri​\(𝐱r,τr\)\(1−τri\)2​𝑑r∈Δ\|V\|−1\.\\delta\_\{s,t\}^\{i\}\(\{\\bf z\}\)=\\frac\{\\displaystyle\\int\_\{s\}^\{t\}\\frac\{\\omega\_\{r\}^\{i\}\(\{\\bf x\}\_\{r\},\\tau\_\{r\}\)\}\{\(1\-\\tau\_\{r\}^\{i\}\)^\{2\}\}D\_\{\\tau\_\{r\}\}^\{i\}\(\{\\bf x\}\_\{r\}\)\\,\{\\rm d\}r\}\{\\displaystyle\\int\_\{s\}^\{t\}\\frac\{\\omega\_\{r\}^\{i\}\(\{\\bf x\}\_\{r\},\\tau\_\{r\}\)\}\{\(1\-\\tau\_\{r\}^\{i\}\)^\{2\}\}\\,\{\\rm d\}r\}\\in\\Delta^\{\|V\|\-1\}\.\(36\)For \(ii\), substituting the diagonal of[Equation18](https://arxiv.org/html/2610.42018#S4.E18)into[Equation19](https://arxiv.org/html/2610.42018#S4.E19)gives

δt,t​\(𝐳\)=𝐱\+𝟏−τωt​\(𝐱,τ\)​ωt​\(𝐱,τ\)​bτ​\(𝐱\)=Dτ​\(𝐱\),ηt,t​\(𝐳\)=ωt​\(𝐱,τ\)\.\\delta\_\{t,t\}\(\{\\bf z\}\)=\{\\bf x\}\+\\frac\{\\mathbf\{1\}\-\\tau\}\{\\omega\_\{t\}\(\{\\bf x\},\\tau\)\}\\,\\omega\_\{t\}\(\{\\bf x\},\\tau\)\\,b\_\{\\tau\}\(\{\\bf x\}\)=D\_\{\\tau\}\(\{\\bf x\}\),\\qquad\\eta\_\{t,t\}\(\{\\bf z\}\)=\\omega\_\{t\}\(\{\\bf x\},\\tau\)\.These diagonal values are also the limits of the off\-diagonal ones\. Since the weights in[Equation36](https://arxiv.org/html/2610.42018#A1.E36)are nonnegative,

‖δs,ti​\(𝐳\)−Dτi​\(𝐱\)‖≤supr∈\[s,t\]‖Dτri​\(𝐱r\)−Dτi​\(𝐱\)‖→t↓s0,\\\|\\delta\_\{s,t\}^\{i\}\(\{\\bf z\}\)\-D\_\{\\tau\}^\{i\}\(\{\\bf x\}\)\\\|\\leq\\sup\_\{r\\in\[s,t\]\}\\\|D\_\{\\tau\_\{r\}\}^\{i\}\(\{\\bf x\}\_\{r\}\)\-D\_\{\\tau\}^\{i\}\(\{\\bf x\}\)\\\|\\xrightarrow\[t\\downarrow s\]\{\}0,and[Equation31](https://arxiv.org/html/2610.42018#A1.E31)gives

limt↓sηs,t​\(𝐳\)=limt↓s1t−s​∫stωr​\(𝐱r,τr\)​𝑑r=ωs​\(𝐱,τ\)\.\\lim\_\{t\\downarrow s\}\\eta\_\{s,t\}\(\{\\bf z\}\)=\\lim\_\{t\\downarrow s\}\\frac\{1\}\{t\-s\}\\int\_\{s\}^\{t\}\\omega\_\{r\}\(\{\\bf x\}\_\{r\},\\tau\_\{r\}\)\\,\{\\rm d\}r=\\omega\_\{s\}\(\{\\bf x\},\\tau\)\.
For \(iii\), fixs<u<ts<u<tand write𝐳u=Fs,u​\(𝐳\)\{\\bf z\}\_\{u\}=F\_\{s,u\}\(\{\\bf z\}\)\. By[Equation31](https://arxiv.org/html/2610.42018#A1.E31), the tokenwise time components of the direct mapFs,t​\(𝐳\)F\_\{s,t\}\(\{\\bf z\}\)and the composed mapFu,t​\(𝐳u\)F\_\{u,t\}\(\{\\bf z\}\_\{u\}\)are

τ\+\(t−s\)​ηs,t​\(𝐳\)andτ\+\(u−s\)​ηs,u​\(𝐳\)\+\(t−u\)​ηu,t​\(𝐳u\),\\tau\+\(t\-s\)\\,\\eta\_\{s,t\}\(\{\\bf z\}\)\\qquad\\text\{and\}\\qquad\\tau\+\(u\-s\)\\,\\eta\_\{s,u\}\(\{\\bf z\}\)\+\(t\-u\)\\,\\eta\_\{u,t\}\(\{\\bf z\}\_\{u\}\),which agree if and only if the identity forη\\etain[Equation22](https://arxiv.org/html/2610.42018#S4.E22)holds\. When this identity holds, letτt\\tau\_\{t\}denote the common tokenwise times, and write

ai=τi,bi=τui,ci=τti\.a\_\{i\}=\\tau^\{i\},\\qquad b\_\{i\}=\\tau\_\{u\}^\{i\},\\qquad c\_\{i\}=\\tau\_\{t\}^\{i\}\.Applying \(i\) to\[s,u\]\[s,u\]and then to\[u,t\]\[u,t\]gives the token state component of the composed map,

1−ci1−bi​\[1−bi1−ai​𝐱i\+bi−ai1−ai​δs,ui​\(𝐳\)\]\+ci−bi1−bi​δu,ti​\(𝐳u\)\\displaystyle\\frac\{1\-c\_\{i\}\}\{1\-b\_\{i\}\}\\left\[\\frac\{1\-b\_\{i\}\}\{1\-a\_\{i\}\}\{\\bf x\}^\{i\}\+\\frac\{b\_\{i\}\-a\_\{i\}\}\{1\-a\_\{i\}\}\\delta\_\{s,u\}^\{i\}\(\{\\bf z\}\)\\right\]\+\\frac\{c\_\{i\}\-b\_\{i\}\}\{1\-b\_\{i\}\}\\delta\_\{u,t\}^\{i\}\(\{\\bf z\}\_\{u\}\)=1−ci1−ai​𝐱i\+ci−ai1−ai​\[γi​δs,ui​\(𝐳\)\+\(1−γi\)​δu,ti​\(𝐳u\)\],\\displaystyle=\\frac\{1\-c\_\{i\}\}\{1\-a\_\{i\}\}\{\\bf x\}^\{i\}\+\\frac\{c\_\{i\}\-a\_\{i\}\}\{1\-a\_\{i\}\}\\bigl\[\\gamma\_\{i\}\\delta\_\{s,u\}^\{i\}\(\{\\bf z\}\)\+\(1\-\\gamma\_\{i\}\)\\delta\_\{u,t\}^\{i\}\(\{\\bf z\}\_\{u\}\)\\bigr\],\(37\)where

γi=\(1−ci\)​\(bi−ai\)\(1−bi\)​\(ci−ai\),1−γi=\(1−ai\)​\(ci−bi\)\(1−bi\)​\(ci−ai\)\.\\gamma\_\{i\}=\\frac\{\(1\-c\_\{i\}\)\(b\_\{i\}\-a\_\{i\}\)\}\{\(1\-b\_\{i\}\)\(c\_\{i\}\-a\_\{i\}\)\},\\qquad 1\-\\gamma\_\{i\}=\\frac\{\(1\-a\_\{i\}\)\(c\_\{i\}\-b\_\{i\}\)\}\{\(1\-b\_\{i\}\)\(c\_\{i\}\-a\_\{i\}\)\}\.Both are nonnegative sinceai<bi<ci<1a\_\{i\}<b\_\{i\}<c\_\{i\}<1, soγi∈\[0,1\]\\gamma\_\{i\}\\in\[0,1\]\. By \(i\), the token state component of the direct map is1−ci1−ai​𝐱i\+ci−ai1−ai​δs,ti​\(𝐳\)\\frac\{1\-c\_\{i\}\}\{1\-a\_\{i\}\}\{\\bf x\}^\{i\}\+\\frac\{c\_\{i\}\-a\_\{i\}\}\{1\-a\_\{i\}\}\\delta\_\{s,t\}^\{i\}\(\{\\bf z\}\), so subtracting it from[Equation37](https://arxiv.org/html/2610.42018#A1.E37)leaves

ci−ai1−ai​\[γi​δs,ui​\(𝐳\)\+\(1−γi\)​δu,ti​\(𝐳u\)−δs,ti​\(𝐳\)\]\.\\frac\{c\_\{i\}\-a\_\{i\}\}\{1\-a\_\{i\}\}\\bigl\[\\gamma\_\{i\}\\delta\_\{s,u\}^\{i\}\(\{\\bf z\}\)\+\(1\-\\gamma\_\{i\}\)\\delta\_\{u,t\}^\{i\}\(\{\\bf z\}\_\{u\}\)\-\\delta\_\{s,t\}^\{i\}\(\{\\bf z\}\)\\bigr\]\.Sinceci\>aic\_\{i\}\>a\_\{i\}, the two token state components then agree if and only if the identity forδ\\deltain[Equation22](https://arxiv.org/html/2610.42018#S4.E22)holds\. Hence the semigroup condition[Equation17](https://arxiv.org/html/2610.42018#S4.E17)holds if and only if both identities hold, which proves \(iii\)\. ∎

### A\.8Proof of Proposition[7](https://arxiv.org/html/2610.42018#Thmproposition7)

See[7](https://arxiv.org/html/2610.42018#Thmproposition7)

###### Proof\.

Settingδ^=δ\\hat\{\\delta\}=\\deltaandη^=η\\hat\{\\eta\}=\\eta, the induced mapF^\\hat\{F\}is the exact flow mapFFby[Proposition6](https://arxiv.org/html/2610.42018#Thmproposition6)\(i\), and[Proposition6](https://arxiv.org/html/2610.42018#Thmproposition6)\(ii\) gives

δt,t​\(𝐳\)=Dτ​\(𝐱\),ηt,t​\(𝐳\)=ωt​\(𝐱,τ\),\\delta\_\{t,t\}\(\{\\bf z\}\)=D\_\{\\tau\}\(\{\\bf x\}\),\\qquad\\eta\_\{t,t\}\(\{\\bf z\}\)=\\omega\_\{t\}\(\{\\bf x\},\\tau\),so both diagonal terms vanish\. Moreover,[Proposition6](https://arxiv.org/html/2610.42018#Thmproposition6)\(iii\) gives

δs,t​\(𝐳\)=𝒯s,u,tδ​\(𝐳\),ηs,t​\(𝐳\)=𝒯s,u,tη​\(𝐳\),\\delta\_\{s,t\}\(\{\\bf z\}\)=\\mathcal\{T\}^\{\\delta\}\_\{s,u,t\}\(\{\\bf z\}\),\\qquad\\eta\_\{s,t\}\(\{\\bf z\}\)=\\mathcal\{T\}^\{\\eta\}\_\{s,u,t\}\(\{\\bf z\}\),so both off\-diagonal terms vanish as well, andℒ⁡\(δ,η\)=0\\mathcal\{L\}\(\\delta,\\eta\)=0\.

Conversely, all four terms are nonnegative andλ\>0\\lambda\>0, soℒ⁡\(δ^,η^\)=0\\mathcal\{L\}\(\\hat\{\\delta\},\\hat\{\\eta\}\)=0if and only if each term vanishes almost surely under the training distribution\. Since KL divergence and the squared norm vanish exactly when their arguments agree, and stop\-gradient leaves the target values unchanged, this gives

δ^t,t​\(𝐳\)=Dτ​\(𝐱\),η^t,t​\(𝐳\)=ωt​\(𝐱,τ\),δ^s,t​\(𝐳\)=𝒯s,u,tδ​\(𝐳\),η^s,t​\(𝐳\)=𝒯s,u,tη​\(𝐳\)\.\\hat\{\\delta\}\_\{t,t\}\(\{\\bf z\}\)=D\_\{\\tau\}\(\{\\bf x\}\),\\qquad\\hat\{\\eta\}\_\{t,t\}\(\{\\bf z\}\)=\\omega\_\{t\}\(\{\\bf x\},\\tau\),\\qquad\\hat\{\\delta\}\_\{s,t\}\(\{\\bf z\}\)=\\mathcal\{T\}^\{\\delta\}\_\{s,u,t\}\(\{\\bf z\}\),\\qquad\\hat\{\\eta\}\_\{s,t\}\(\{\\bf z\}\)=\\mathcal\{T\}^\{\\eta\}\_\{s,u,t\}\(\{\\bf z\}\)\.The first two are[Equation21](https://arxiv.org/html/2610.42018#S4.E21), and expanding the targets in the last two gives[Equation22](https://arxiv.org/html/2610.42018#S4.E22), which hold forδ^\\hat\{\\delta\}andη^\\hat\{\\eta\}\. ∎

### A\.9Proof of Corollary[8](https://arxiv.org/html/2610.42018#Thmproposition8)

See[8](https://arxiv.org/html/2610.42018#Thmproposition8)

###### Proof\.

We first show that the tokenwise time terms of[Equation24](https://arxiv.org/html/2610.42018#S4.E24)vanish\. By assumption,η^\\hat\{\\eta\}is fixed to the exactη\\eta, and sinceτt=τ⁡\(t\)\\tau\_\{t\}=\\tau\(t\)for every state,[Equation18](https://arxiv.org/html/2610.42018#S4.E18)gives

ηs,t=τ⁡\(t\)−τ⁡\(s\)t−s,ηt,t=τ˙​\(t\)=ωt\.\\eta\_\{s,t\}=\\frac\{\\tau\(t\)\-\\tau\(s\)\}\{t\-s\},\\qquad\\eta\_\{t,t\}=\\dot\{\\tau\}\(t\)=\\omega\_\{t\}\.The diagonal tokenwise time term therefore vanishes\. Substitutingη\\etainto[Equation23](https://arxiv.org/html/2610.42018#S4.E23)gives

𝒯s,u,tη=\(u−s\)​ηs,u\+\(t−u\)​ηu,tt−s=\(τ⁡\(u\)−τ⁡\(s\)\)\+\(τ⁡\(t\)−τ⁡\(u\)\)t−s=ηs,t,\\mathcal\{T\}^\{\\eta\}\_\{s,u,t\}=\\frac\{\(u\-s\)\\,\\eta\_\{s,u\}\+\(t\-u\)\\,\\eta\_\{u,t\}\}\{t\-s\}=\\frac\{\(\\tau\(u\)\-\\tau\(s\)\)\+\(\\tau\(t\)\-\\tau\(u\)\)\}\{t\-s\}=\\eta\_\{s,t\},so the off\-diagonal tokenwise time term vanishes as well\.

We next rewrite the token state terms\. Since𝐱t∼pτ⁡\(t\)\{\\bf x\}\_\{t\}\\sim p\_\{\\tau\(t\)\}by[Corollary4](https://arxiv.org/html/2610.42018#Thmproposition4), the states in[Equation24](https://arxiv.org/html/2610.42018#S4.E24)are𝐳=\(Iτ⁡\(t\),τ⁡\(t\)\)\{\\bf z\}=\(I\_\{\\tau\(t\)\},\\tau\(t\)\), andF^s,u\\hat\{F\}\_\{s,u\}andγ\\gammain𝒯s,u,tδ\\mathcal\{T\}^\{\\delta\}\_\{s,u,t\}are evaluated atτ⁡\(s\)\\tau\(s\),τ⁡\(u\)\\tau\(u\), andτ⁡\(t\)\\tau\(t\)\. The off\-diagonal token state term is thus the first term of[Equation25](https://arxiv.org/html/2610.42018#S4.E25)\. For the diagonal term,𝐱𝟏i\{\\bf x\}\_\{\\bf 1\}^\{i\}is one\-hot and has zero entropy, so

𝖪𝖫\(𝐱𝟏∥δ^t,t\(𝐳\)\)=∑i=1L\[−logδ^t,ti\(𝐳\)\]𝐲i,\\mathsf\{KL\}\(\{\\bf x\}\_\{\\bf 1\}\\,\\\|\\,\\hat\{\\delta\}\_\{t,t\}\(\{\\bf z\}\)\)=\\sum\_\{i=1\}^\{L\}\\left\[\-\\log\\hat\{\\delta\}^\{i\}\_\{t,t\}\(\{\\bf z\}\)\\right\]\_\{\{\\bf y\}^\{i\}\},which is the tokenwise cross\-entropy of[Equation13](https://arxiv.org/html/2610.42018#S4.E13)withD^τ⁡\(t\)=δ^t,t\\hat\{D\}\_\{\\tau\(t\)\}=\\hat\{\\delta\}\_\{t,t\}\. Applying the decomposition[Equation29](https://arxiv.org/html/2610.42018#A1.E29)withτ=τ⁡\(t\)\\tau=\\tau\(t\)gives

𝔼t,𝐳,𝐱𝟏\[𝖪𝖫\(𝐱𝟏∥δ^t,t\(𝐳\)\)\]\\displaystyle\\mathbb\{E\}\_\{t,\{\\bf z\},\{\\bf x\}\_\{\\bf 1\}\}\\,\\big\[\\mathsf\{KL\}\(\{\\bf x\}\_\{\\bf 1\}\\,\\\|\\,\\hat\{\\delta\}\_\{t,t\}\(\{\\bf z\}\)\)\\big\]=C\+𝔼t,𝐳\[𝖪𝖫\(Dτ⁡\(t\)\(Iτ⁡\(t\)\)∥δ^t,t\(𝐳\)\)\],\\displaystyle=C\+\\mathbb\{E\}\_\{t,\{\\bf z\}\}\\,\\big\[\\mathsf\{KL\}\(D\_\{\\tau\(t\)\}\(I\_\{\\tau\(t\)\}\)\\,\\\|\\,\\hat\{\\delta\}\_\{t,t\}\(\{\\bf z\}\)\)\\big\],C\\displaystyle C≔𝔼t,𝐳​\[∑i=1LH⁡\(Dτ⁡\(t\)i​\(Iτ⁡\(t\)\)\)\],\\displaystyle\\coloneqq\\mathbb\{E\}\_\{t,\{\\bf z\}\}\\left\[\\sum\_\{i=1\}^\{L\}H\(D\_\{\\tau\(t\)\}^\{i\}\(I\_\{\\tau\(t\)\}\)\)\\right\],where KL sums over token positions andCCdoes not depend onδ^\\hat\{\\delta\}\. Hence[Equation25](https://arxiv.org/html/2610.42018#S4.E25)equals[Equation24](https://arxiv.org/html/2610.42018#S4.E24)plusCC\. Since[Equation24](https://arxiv.org/html/2610.42018#S4.E24)is nonnegative and vanishes at the exactδ\\deltaby[Proposition7](https://arxiv.org/html/2610.42018#Thmproposition7), the minimum of[Equation25](https://arxiv.org/html/2610.42018#S4.E25)isCC, and its minimizers are exactly the zero\-loss solutions of[Proposition7](https://arxiv.org/html/2610.42018#Thmproposition7)\. ∎

## Appendix BScheduler Parameterization

### B\.1Schedule construction

Figure 6:Power schedulesτj​\(t\)=tαj\\tau^\{j\}\(t\)=t^\{\\alpha\_\{j\}\}withβ=log⁡3\\beta=\\log 3\(left\) and their speedsktjk\_\{t\}^\{j\}\(right\)\. Darker curves denote higher\-priority positions, with exponents ranging fromαj=1/3\\alpha\_\{j\}=1/3\(darkest\) toαj=3\\alpha\_\{j\}=3\(lightest\), and the dashed line denotes the synchronous schedule\.All schedulers in our experiments take the common form of[Equation27](https://arxiv.org/html/2610.42018#S4.E27),

ωti​\(𝐱,τ\)=1−τi1−t​kti​\(𝐱,τ\),t<1\.\\omega\_\{t\}^\{i\}\(\{\\bf x\},\\tau\)=\\frac\{1\-\\tau^\{i\}\}\{1\-t\}\\,k\_\{t\}^\{i\}\(\{\\bf x\},\\tau\),\\qquad t<1\.This form sendsτt→𝟏\\tau\_\{t\}\\to\\mathbf\{1\}ast↑1t\\uparrow 1wheneverkik^\{i\}is bounded away from zero neart=1t=1\. Writingkrik^\{i\}\_\{r\}forkri​\(𝐱r,τr\)k^\{i\}\_\{r\}\(\{\\bf x\}\_\{r\},\\tau\_\{r\}\)along the trajectory, each tokenwise time satisfies

dd​r​log⁡\(1−τri\)=−τ˙ri1−τri=−kri1−r,\\frac\{\\rm d\}\{\{\\rm d\}r\}\\log\(1\-\\tau^\{i\}\_\{r\}\)=\-\\frac\{\\dot\{\\tau\}^\{i\}\_\{r\}\}\{1\-\\tau^\{i\}\_\{r\}\}=\-\\frac\{k^\{i\}\_\{r\}\}\{1\-r\},and integrating over\[s,t\]\[s,t\]gives

1−τti=\(1−τsi\)exp\(−∫stkri1−rdr\)\.1\-\\tau^\{i\}\_\{t\}=\(1\-\\tau^\{i\}\_\{s\}\)\\exp\\\!\\left\(\-\\int\_\{s\}^\{t\}\\frac\{k^\{i\}\_\{r\}\}\{1\-r\}\\,\{\\rm d\}r\\right\)\.\(38\)Sincekri≥0k^\{i\}\_\{r\}\\geq 0andτ0i=0\\tau^\{i\}\_\{0\}=0, the tokenwise timeτti\\tau^\{i\}\_\{t\}never decreases and stays in\[0,1\)\[0,1\)for everyt<1t<1\. Moreover, ifkri≥ck^\{i\}\_\{r\}\\geq cfor somec\>0c\>0and allr∈\[s,1\)r\\in\[s,1\), then∫st𝑑r/\(1−r\)=log⁡1−s1−t\\int\_\{s\}^\{t\}\{\\rm d\}r/\(1\-r\)=\\log\\frac\{1\-s\}\{1\-t\}gives

1−τti≤\(1−τsi\)​\(1−t1−s\)c→0as​t↑1\.1\-\\tau^\{i\}\_\{t\}\\leq\(1\-\\tau^\{i\}\_\{s\}\)\\left\(\\frac\{1\-t\}\{1\-s\}\\right\)^\{c\}\\to 0\\quad\\text\{as \}t\\uparrow 1\.All schedulers in our experiments satisfy this condition, since the top\-KKconfidence and entropy\-bounded schedulers useki≥kmin\>0k^\{i\}\\geq k\_\{\\min\}\>0, and the power schedules havektj→1k^\{j\}\_\{t\}\\to 1ast↑1t\\uparrow 1\.

The schedulers differ in whatkik^\{i\}depends on\. Let

pi=Dτi​\(𝐱\),ci=‖pi‖∞,hi=H⁡\(pi\)p\_\{i\}=D\_\{\\tau\}^\{i\}\(\{\\bf x\}\),\\qquad c\_\{i\}=\\\|p\_\{i\}\\\|\_\{\\infty\},\\qquad h\_\{i\}=H\(p\_\{i\}\)\(39\)denote the predicted clean\-token distribution, its confidence, and its entropy, respectively\. All position indices and statistics below refer to generated tokens, and fixed context tokens remain unchanged\.

#### Power schedules\.

Several of the schedulers below use the power schedule

τj​\(t\)=tαj\\tau^\{j\}\(t\)=t^\{\\alpha\_\{j\}\}\(40\)\([Figure6](https://arxiv.org/html/2610.42018#A2.F6)\)\. Its rateωtj=αj​tαj−1\\omega\_\{t\}^\{j\}=\\alpha\_\{j\}t^\{\\alpha\_\{j\}\-1\}takes the common form, withkkdepending only ontt,

ktj=ωtj​1−t1−τj​\(t\)=αj​tαj−1​1−t1−tαj,k\_\{t\}^\{j\}=\\omega\_\{t\}^\{j\}\\,\\frac\{1\-t\}\{1\-\\tau^\{j\}\(t\)\}=\\alpha\_\{j\}t^\{\\alpha\_\{j\}\-1\}\\,\\frac\{1\-t\}\{1\-t^\{\\alpha\_\{j\}\}\},\(41\)which tends to one ast↑1t\\uparrow 1\. Given an order over the generated positions, letρj∈\[0,1\]\\rho\_\{j\}\\in\[0,1\]denote the normalized rank of positionjj, with zero denoting the highest priority, and set

αj=exp⁡\[β⁡\(2​ρj−1\)\],\\alpha\_\{j\}=\\exp\\bigl\[\\beta\\,\(2\\rho\_\{j\}\-1\)\\bigr\],\(42\)so that the exponents range frome−βe^\{\-\\beta\}toeβe^\{\\beta\}and positions with higher priority are denoised faster\. Except for the synchronous schedule, we useβ=log⁡3\\beta=\\log 3, so that the exponents range from1/31/3to33\. We use the following power schedules\.

- •Synchronous\.Settingβ=0\\beta=0giveskj=1k^\{j\}=1andτj​\(t\)=t\\tau^\{j\}\(t\)=t, which recovers the synchronous schedule of[Lee et al\. \(2026\)](https://arxiv.org/html/2610.42018#bib.bib1)\.
- •Left\-to\-right\.ρj\\rho\_\{j\}is the normalized position of tokenjjamong the generated positions\.
- •Algorithmic order for Sudoku\.ρj=r𝐜​\(j\)/R𝐜\\rho\_\{j\}=r\_\{\\bf c\}\(j\)/R\_\{\\bf c\}, wherer𝐜​\(j\)r\_\{\\bf c\}\(j\)is the round in which celljjis resolved andR𝐜R\_\{\\bf c\}is the last round \([SectionC\.3](https://arxiv.org/html/2610.42018#A3.SS3)\)\. Cells resolved in the same round share the same exponent\.
- •Initial entropy\.Following[Bansal and Sanghavi \(2026\)](https://arxiv.org/html/2610.42018#bib.bib8),ρj\\rho\_\{j\}is the rank of positionjjby increasing entropy of the initial predictionD𝟎​\(𝐱0\)D\_\{\\bf 0\}\(\{\\bf x\}\_\{0\}\), kept fixed during generation, so that positions with lower entropy are denoised earlier\.kkthus depends on the token states only through the initial state𝐱0\{\\bf x\}\_\{0\}\.

The following two schedulers instead setkik^\{i\}from the current predictions at every step\.

#### Top\-KKconfidence scheduler\.

Following MaskGIT\([Chang et al\., 2022](https://arxiv.org/html/2610.42018#bib.bib26)\), we advance theKKmost confident positions faster, whereKKis a fractionf⁡\(t\)=1−cos⁡\(π​t/2\)f\(t\)=1\-\\cos\(\\pi t/2\)of the generated positions and thus increases over time\. Letc~i\\widetilde\{c\}\_\{i\}denote the confidencecic\_\{i\}standardized over the generated positions, and letq⁡\(t\)q\(t\)be its quantile at level1−f⁡\(t\)1\-f\(t\), so that a fractionf⁡\(t\)f\(t\)of positions havec~i\>q⁡\(t\)\\widetilde\{c\}\_\{i\}\>q\(t\)\. We set

ki=kmin\+A​sigmoid​\(c~i−q⁡\(t\)T\),k^\{i\}=k\_\{\\min\}\+A\\,\\mathrm\{sigmoid\}\\\!\\left\(\\frac\{\\widetilde\{c\}\_\{i\}\-q\(t\)\}\{T\}\\right\),\(43\)wherekmin\>0k\_\{\\min\}\>0is the speed of the remaining positions,A\>0A\>0is the additional speed for the selected ones, andT\>0T\>0is a temperature that controls how sharplykkchanges around the threshold\.

#### Entropy\-bounded scheduler\.

Following EB\-Sampler\([Ben\-Hamu et al\., 2025](https://arxiv.org/html/2610.42018#bib.bib16)\), we order token positions by decreasing confidence or increasing entropy\. Letπj\\pi\_\{j\}denote thejj\-th position in this order\. For the firstjjpositions, we sum their entropies and subtract the largest:

Bj=∑ℓ=1jhπℓ−max1≤ℓ≤j⁡hπℓ\.B\_\{j\}=\\sum\_\{\\ell=1\}^\{j\}h\_\{\\pi\_\{\\ell\}\}\-\\max\_\{1\\leq\\ell\\leq j\}h\_\{\\pi\_\{\\ell\}\}\.\(44\)We assign largerkkto positions beforeBjB\_\{j\}exceeds a thresholdϵ\\epsilon, using a sigmoid to decreasekkgradually:

kπj=kmin\+A​sigmoid​\(ϵ−BjT\)\.k^\{\\pi\_\{j\}\}=k\_\{\\min\}\+A\\,\\mathrm\{sigmoid\}\\\!\\left\(\\frac\{\\epsilon\-B\_\{j\}\}\{T\}\\right\)\.\(45\)HereA\>0A\>0controls the additional speed, andT\>0T\>0controls how sharply it decreases aroundϵ≥0\\epsilon\\geq 0\.

### B\.2Tokenwise time rate reparameterization

Taking the logarithm of[Equation38](https://arxiv.org/html/2610.42018#A2.E38)at each position gives

log⁡𝟏−τs𝟏−τt=∫stkr​\(𝐱r,τr\)1−r​𝑑r,log⁡1−s1−t=∫st11−r​𝑑r\.\\log\\frac\{\\mathbf\{1\}\-\\tau\_\{s\}\}\{\\mathbf\{1\}\-\\tau\_\{t\}\}=\\int\_\{s\}^\{t\}\\frac\{k\_\{r\}\(\{\\bf x\}\_\{r\},\\tau\_\{r\}\)\}\{1\-r\}\\,\{\\rm d\}r,\\qquad\\log\\frac\{1\-s\}\{1\-t\}=\\int\_\{s\}^\{t\}\\frac\{1\}\{1\-r\}\\,\{\\rm d\}r\.Dividing the first by the second shows thatκs,t\\kappa\_\{s,t\}defined in[Section4\.3](https://arxiv.org/html/2610.42018#S4.SS3)is the average ofkrk\_\{r\}along the trajectory weighted by1/\(1−r\)1/\(1\-r\),

κs,t=∫stkr​\(𝐱r,τr\)/\(1−r\)​𝑑r∫st1/\(1−r\)​𝑑r\.\\kappa\_\{s,t\}=\\frac\{\\int\_\{s\}^\{t\}k\_\{r\}\(\{\\bf x\}\_\{r\},\\tau\_\{r\}\)/\(1\-r\)\\,\{\\rm d\}r\}\{\\int\_\{s\}^\{t\}1/\(1\-r\)\\,\{\\rm d\}r\}\.We now derive the relation betweenκ^s,s\\hat\{\\kappa\}\_\{s,s\}and the diagonal rateη^s,s\\hat\{\\eta\}\_\{s,s\}\. By[Proposition6](https://arxiv.org/html/2610.42018#Thmproposition6)\(i\), the left side of[Equation28](https://arxiv.org/html/2610.42018#S4.E28)is the tokenwise time componentτs\+\(t−s\)​η^s,t​\(𝐳s\)\\tau\_\{s\}\+\(t\-s\)\\,\\hat\{\\eta\}\_\{s,t\}\(\{\\bf z\}\_\{s\}\)of the flow map\. Its derivative inttatt=st=sis thereforeη^s,s​\(𝐳s\)\\hat\{\\eta\}\_\{s,s\}\(\{\\bf z\}\_\{s\}\)\.

Writing the power in[Equation28](https://arxiv.org/html/2610.42018#S4.E28)as an exponential,

\(1−t1−s\)κ^s,t=exp⁡\[κ^s,t​log⁡1−t1−s\],\\left\(\\frac\{1\-t\}\{1\-s\}\\right\)^\{\\hat\{\\kappa\}\_\{s,t\}\}=\\exp\\\!\\left\[\\hat\{\\kappa\}\_\{s,t\}\\log\\frac\{1\-t\}\{1\-s\}\\right\],and differentiating with respect tottby the chain and product rules gives

∂t\(1−t1−s\)κ^s,t=\(1−t1−s\)κ^s,t⊙\[∂tκ^s,t​log⁡1−t1−s−κ^s,t1−t\]\.\\partial\_\{t\}\\left\(\\frac\{1\-t\}\{1\-s\}\\right\)^\{\\hat\{\\kappa\}\_\{s,t\}\}=\\left\(\\frac\{1\-t\}\{1\-s\}\\right\)^\{\\hat\{\\kappa\}\_\{s,t\}\}\\odot\\left\[\\partial\_\{t\}\\hat\{\\kappa\}\_\{s,t\}\\log\\frac\{1\-t\}\{1\-s\}\-\\frac\{\\hat\{\\kappa\}\_\{s,t\}\}\{1\-t\}\\right\]\.Att=st=s, we have1−t1−s=1\\frac\{1\-t\}\{1\-s\}=1andlog⁡1−t1−s=0\\log\\frac\{1\-t\}\{1\-s\}=0, so the term involving∂tκ^s,t\\partial\_\{t\}\\hat\{\\kappa\}\_\{s,t\}vanishes and

∂t\(1−t1−s\)κ^s,t\|t=s=−κ^s,s1−s\.\\left\.\\partial\_\{t\}\\left\(\\frac\{1\-t\}\{1\-s\}\\right\)^\{\\hat\{\\kappa\}\_\{s,t\}\}\\right\|\_\{t=s\}=\-\\frac\{\\hat\{\\kappa\}\_\{s,s\}\}\{1\-s\}\.Differentiating both sides of[Equation28](https://arxiv.org/html/2610.42018#S4.E28)att=st=stherefore gives

η^s,s\(𝐳s\)=−\(𝟏−τs\)⊙\(−κ^s,s​\(𝐳s\)1−s\)=𝟏−τs1−s⊙κ^s,s\(𝐳s\)\.\\hat\{\\eta\}\_\{s,s\}\(\{\\bf z\}\_\{s\}\)=\-\(\\mathbf\{1\}\-\\tau\_\{s\}\)\\odot\\left\(\-\\frac\{\\hat\{\\kappa\}\_\{s,s\}\(\{\\bf z\}\_\{s\}\)\}\{1\-s\}\\right\)=\\frac\{\\mathbf\{1\}\-\\tau\_\{s\}\}\{1\-s\}\\odot\\hat\{\\kappa\}\_\{s,s\}\(\{\\bf z\}\_\{s\}\)\.On the other hand, the scheduler of[Equation27](https://arxiv.org/html/2610.42018#S4.E27)at the same state is

ωs​\(𝐱s,τs\)=𝟏−τs1−s⊙ks​\(𝐱s,τs\)\.\\omega\_\{s\}\(\{\\bf x\}\_\{s\},\\tau\_\{s\}\)=\\frac\{\\mathbf\{1\}\-\\tau\_\{s\}\}\{1\-s\}\\odot k\_\{s\}\(\{\\bf x\}\_\{s\},\\tau\_\{s\}\)\.Subtracting the two expressions gives

η^s,s​\(𝐳s\)−ωs​\(𝐱s,τs\)=𝟏−τs1−s⊙\(κ^s,s​\(𝐳s\)−ks​\(𝐱s,τs\)\)\.\\hat\{\\eta\}\_\{s,s\}\(\{\\bf z\}\_\{s\}\)\-\\omega\_\{s\}\(\{\\bf x\}\_\{s\},\\tau\_\{s\}\)=\\frac\{\\mathbf\{1\}\-\\tau\_\{s\}\}\{1\-s\}\\odot\\bigl\(\\hat\{\\kappa\}\_\{s,s\}\(\{\\bf z\}\_\{s\}\)\-k\_\{s\}\(\{\\bf x\}\_\{s\},\\tau\_\{s\}\)\\bigr\)\.\(46\)Since every entry of\(𝟏−τs\)/\(1−s\)\(\\mathbf\{1\}\-\\tau\_\{s\}\)/\(1\-s\)is positive forτs<𝟏\\tau\_\{s\}<\\mathbf\{1\}ands<1s<1, the diagonal identityη^s,s=ωs\\hat\{\\eta\}\_\{s,s\}=\\omega\_\{s\}holds if and only ifκ^s,s=ks\\hat\{\\kappa\}\_\{s,s\}=k\_\{s\}\. Moreover, taking the squared norm of[Equation46](https://arxiv.org/html/2610.42018#A2.E46)gives

‖η^s,s​\(𝐳s\)−ωs​\(𝐱s,τs\)‖22=∑i=1L\(1−τsi1−s\)2​\(κ^s,si​\(𝐳s\)−ksi​\(𝐱s,τs\)\)2,\\\|\\hat\{\\eta\}\_\{s,s\}\(\{\\bf z\}\_\{s\}\)\-\\omega\_\{s\}\(\{\\bf x\}\_\{s\},\\tau\_\{s\}\)\\\|\_\{2\}^\{2\}=\\sum\_\{i=1\}^\{L\}\\left\(\\frac\{1\-\\tau\_\{s\}^\{i\}\}\{1\-s\}\\right\)^\{2\}\\bigl\(\\hat\{\\kappa\}^\{i\}\_\{s,s\}\(\{\\bf z\}\_\{s\}\)\-k^\{i\}\_\{s\}\(\{\\bf x\}\_\{s\},\\tau\_\{s\}\)\\bigr\)^\{2\},so the implemented termλ​‖ks−κ^s,s‖22\\lambda\\\|k\_\{s\}\-\\hat\{\\kappa\}\_\{s,s\}\\\|\_\{2\}^\{2\}only removes these positive weights, and both terms vanish exactly whenκ^s,s=ks\\hat\{\\kappa\}\_\{s,s\}=k\_\{s\}\.

## Appendix CExperiment Details

Table 4:Per\-task hyperparameters\.All models on the same task share the architecture and optimizer, and differ only in the training and inference schedules\.### C\.1Training details

#### Architecture and hyperparameters\.

All models on the same task share the architecture, and the tokenwise timeτ∈\[0,1\]L\\tau\\in\[0,1\]^\{L\}is embedded per position for the*a*FLM teacher andaFMLM\. Per\-task hyperparameters are listed in[Table4](https://arxiv.org/html/2610.42018#A3.T4)\. We train all models with AdamW\([Loshchilov and Hutter, 2017](https://arxiv.org/html/2610.42018#bib.bib41)\)withβ1=0\.9\\beta\_\{1\}=0\.9,β2=0\.999\\beta\_\{2\}=0\.999,ϵ=10−8\\epsilon=10^\{\-8\}, and no weight decay\. The learning rate is linearly warmed up to3×10−43\\times 10^\{\-4\}over 2,500 steps and kept constant afterwards, and gradients are clipped to norm 1\.0\. We use bfloat16 mixed precision and keep an exponential moving average of the weights with decay 0\.9999, which is used for evaluation\. All models are trained on NVIDIA A100 and H200 GPUs\.

#### *a*FLM training\.

For each task, we train one*a*FLM teacher with the diffusion forcing loss of[Equation13](https://arxiv.org/html/2610.42018#S4.E13)for the number of steps in[Table4](https://arxiv.org/html/2610.42018#A3.T4)\. The teacher is initialized from MDLM \([Section4\.3](https://arxiv.org/html/2610.42018#S4.SS3)\) unless stated otherwise\. We use the*a*FLM checkpoint at 80k steps, and[Figure4](https://arxiv.org/html/2610.42018#S5.F4)continues training up to 240k steps only to compare the two initializations\. For Alg\. order†in[Figure2](https://arxiv.org/html/2610.42018#S5.F2), we instead train the denoiser only along the algorithmic order of[SectionC\.3](https://arxiv.org/html/2610.42018#A3.SS3)\.

#### Distillation\.

aFMLM \(EB\),aFMLM \(Top\-KK\), andaFMLM \(Sync\.\) are initialized from the*a*FLM teacher and distilled with[Equation24](https://arxiv.org/html/2610.42018#S4.E24), whileaFMLM \(Alg\. order\) is trained from scratch with[Equation25](https://arxiv.org/html/2610.42018#S4.E25)without a teacher\. Following[Lee et al\. \(2026\)](https://arxiv.org/html/2610.42018#bib.bib1), we sample three timess<u<ts<u<tby first samplingd=t−s∼𝖴⁡\[0,1\]d=t\-s\\sim\\mathsf\{U\}\[0,1\], then samplings∼𝖴⁡\[0,1−d\]s\\sim\\mathsf\{U\}\[0,1\-d\], and settingu=\(s\+t\)/2u=\(s\+t\)/2\. With learned schedulers, the state at timessis obtained by rolling out the dynamics from time00toss\([Section4\.3](https://arxiv.org/html/2610.42018#S4.SS3)\)\. We run this rollout curriculum for five rounds, where the first round rolls out the teacher with 32 steps and each later round rolls out the current flow map with half as many steps, down to 2 steps\. The final round with 2 rollout steps continues for the remaining training steps\. We set the tokenwise time loss weight in[Equation24](https://arxiv.org/html/2610.42018#S4.E24)toλ=1\.0\\lambda=1\.0\. For noise downscaling \([Section4\.3](https://arxiv.org/html/2610.42018#S4.SS3)\), we draw the initial noise withσ=0\.1\\sigma=0\.1on GSM8K both in distillation and at inference, and keepσ=1\.0\\sigma=1\.0on Sudoku\.

#### Scheduler hyperparameters\.

For the top\-KKconfidence scheduler, we use\(T,A\)=\(0\.2,12\)\(T,A\)=\(0\.2,12\)on all Sudoku difficulty levels and\(0\.5,48\)\(0\.5,48\)on GSM8K, withkmin=0\.1k\_\{\\min\}=0\.1\. For the entropy\-bounded scheduler, we use\(ϵ,T,A\)=\(0\.01,0\.001,12\)\(\\epsilon,T,A\)=\(0\.01,0\.001,12\)on all Sudoku difficulty levels and\(3,0\.3,24\)\(3,0\.3,24\)on GSM8K, withkmin=0\.1k\_\{\\min\}=0\.1throughout\. For the prescribed power schedules reported here, we useβ=log⁡3\\beta=\\log 3, giving exponents in\[1/3,3\]\[1/3,3\], except for the synchronous schedule, which usesβ=0\\beta=0\. For the initial entropy scheduler, we rank the positions by the entropy of the initial prediction and assign the power schedule withβ=log⁡3\\beta=\\log 3according to this rank\.

#### Evaluation\.

All models are sampled on a uniform grid of the base timettwith the given number of steps\. On this grid, the*a*FLM teachers integrate[Equation15](https://arxiv.org/html/2610.42018#S4.E15),aFMLM jumps between adjacent grid points withF^\\hat\{F\}, and the tokenwise times evolve according to the scheduler\. We evaluate on the full 1,319 GSM8K test questions and 2,000 Sudoku puzzles for each difficulty\. For Sudoku, we report exact solving accuracy, and for GSM8K, we execute the generated code and report the accuracy of its output\. We measure the training cost in[Table2](https://arxiv.org/html/2610.42018#S5.T2)on a single H200 GPU\.

### C\.2Baseline details

For[Table1](https://arxiv.org/html/2610.42018#S5.T1), we take the one\-step results reported in each paper and otherwise evaluate the checkpoints released by[Deschenaux and Gulcehre \(2026\)](https://arxiv.org/html/2610.42018#bib.bib30)and[Agarwal et al\. \(2026\)](https://arxiv.org/html/2610.42018#bib.bib3)\. For FMLM\+, we report the GSM8K result with MDLM initialization\. For DBTM, we report its only one\-step GSM8K result, which uses the refinement\-in\-loop objective\. For AlphaDLM, we report its best one\-step result, which the sequence\-level objective with its hyperparameterk=16k=16achieves\. Since DBTM and AlphaDLM report no one\-step results on Sudoku and release no checkpoints, we mark them with –\.

Figure 7:Algorithmic ordering for a hard Sudoku puzzle\. Plain digits are the given clues, and labelsr1–r3indicate the round in which each empty cell is resolved\. Cells resolved in the current round are shaded in darker blue, while previously assigned rounds remain visible\. Cells still blank after round 3 are resolved in later rounds \(not shown\)\.
### C\.3Algorithmic ordering for Sudoku

Following[Shah et al\. \(2024\)](https://arxiv.org/html/2610.42018#bib.bib18), we start from the given clues𝐜\{\\bf c\}and compute the valid digits of each empty cell under the row, column, and box constraints\. At each round, we fill every cell that has only one valid digit or is the only valid location of a digit in its row, column, or box, and we repeat until no cell can be resolved\.[Figure7](https://arxiv.org/html/2610.42018#A3.F7)illustrates the first three rounds\. We denote byr𝐜​\(j\)r\_\{\\bf c\}\(j\)the round in which celljjis resolved and assign the last roundR𝐜R\_\{\\bf c\}to unresolved cells, withR𝐜=1R\_\{\\bf c\}=1if no cell is resolved\.

### C\.4Details of the analysis in[Section5\.3](https://arxiv.org/html/2610.42018#S5.SS3)

We apply the logit lens after the attention and MLP sublayers of each transformer block, decoding each hidden state with the final layer norm and output head of the model\. For each token, we take the earliest sublayer from which its decoded prediction matches the final output and remains so\. For the*a*FLM teacher, we run 128\-step EB sampling from the same initial noise and regard a token as decided at the first step where its tokenwise time exceeds 0\.95\. The four columns of[Figure5](https://arxiv.org/html/2610.42018#S5.F5)show the first 10%, 50%, 85%, and 100% of the tokens in this order, and[Table5](https://arxiv.org/html/2610.42018#A3.T5)lists the block of eachaFMLM column\.

Table 5:Transformer block \(out of 12\) at which each column of[Figure5](https://arxiv.org/html/2610.42018#S5.F5)is taken\.