Gumbel Straight Flow: Distilling Autoregressive Models into One-step Flow Maps

Yeongmin Kim; Arnaud Doucet; Andrew Campbell; Valentin De Bortoli; Thomas Mensink; David Ruhe

cs.LGSep 30, 2026

Gumbel Straight Flow: Distilling Autoregressive Models into One-step Flow Maps

Authors: Yeongmin Kim, Arnaud Doucet, Andrew Campbell, Valentin De Bortoli, Thomas Mensink, David Ruhe

Organizations: Google DeepMind · Work done as a Student Researcher at Google DeepMind Amsterdam

Abstract

We present Gumbel Straight Flow (GSF), a continuous flow map language model that leverages the noise-data coupling of a pretrained autoregressive language (AR) model. We theoretically demonstrate that the coupling between Gumbel noise and one-hot token sequences induced by an autoregressive model yields non-intersecting linear paths connecting the noise to the sequence representations. To further enhance high-quality few-step path sampling, we use a flow map semigroup objective where the tangent (velocity) condition is guided directly by the AR teacher. Across various benchmarks, including pretraining and downstream tasks, GSF can outperform current few-step language generation baselines.

Figures & tables

Figure 1 : Efficiency-performance trade-offs for GSF, primary distillation baselines (varying step sizes), and AR models (varying model sizes).

Model	Parallel Generation	Deterministic Pairing	Straight Path
Autoregressive Model	✗	✓	✓
Discrete Diffusion Model	✓	✗	✗
Continuous Flow (Diffusion)	✓	✓	✗
Gumbel Straight Flow (GSF)	✓	✓	✓

Table 1 : Comparison of GSF against existing language generation paradigms. This paper shows that the AR model’s coupling forms a non-intersecting straight path and distills the path into GSF.

Panel of Figure 2 — Figure 2 : (Left) Three types of noise–data coupling and their corresponding optimally learned flow paths. (Right) The Gumbel coupling induced by an AR model belongs to the non-intersecting case and yields straight underlying paths.

Figure 3 : Overview of GSF: distilling AR-induced straight paths into a flow map student.

Panel of Figure 4 — Figure 4 : (a) Under the Gumbel-max trick, the selected index tends to have a larger value than non-selected indices. Since only one of the ${\mathcal{V}}$ indices is selected, the marginal distribution follows a standard Gumbel distribution. (b) GSF injects informative noise; consequently, decoding errors occur as the noise-injection signal increases and are distributed more uniformly over time $t$ .

Panel of Figure 5 — Figure 5 : Comparison between FLM with independent coupling and GSF-FM with AR-based coupling on the OpenWebText dataset under a Gaussian prior.

Figure 7 : Pairwise win rate with Gemini 3.6-Flash. Parentheses indicate sampling steps.

Panel of Figure 8 — Figure 8 : (a) Computational cost for single-step sampling on OpenWebText. (b) One-step generative quality across training iterations. (c) Scaling behavior across parameter size and sampling steps

Table 10

Table 11

Appendix figures & tables14 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 9 : Straightness (CE scale) across decoding softmax temperatures for FLM and GSF-FM .

Method	No-auxiliary network	Params.	NFE	GM ( $\downarrow$ )	PPL ( $\downarrow$ )	Ent.
D-MMD	✗	318.1M	1	N/A	N/A	N/A
GSF (ours)	✓	167.8M	1	0.94	114.9	5.07
D-MMD	✗	318.1M	2	N/A	N/A	N/A
GSF (ours)	✓	167.8M	2	0.74	105.6	5.30
D-MMD	✗	318.1M	4	0.72	66.1	5.44
GSF (ours)	✓	167.8M	4	0.73	99.7	5.32

Appendix

Table 9 : Comparison with D-MMD ( Hoogeboom et al., 2026 ) on unconditional OpenWebText.

Initial noise	NFE	GM ( $\downarrow$ )	PPL ( $\downarrow$ )	Ent.
Prior noise	1	0.94	114.9	5.07
Posterior noise	1	0.94	114.3	5.07
Prior noise	2	0.74	105.6	5.30
Posterior noise	2	0.74	105.3	5.30
Prior noise	4	0.73	99.7	5.32
Posterior noise	4	0.74	99.6	5.32

Appendix

Table 10 : Gumbel prior-posterior gap analysis

Figure 10 : Computational requirements of baseline methods to achieve comparable GM performance to one-step GSF.

GSF (Ours) (1-Step, 1024 tokens)	FMLM (1-Step, 1024 tokens)
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Gemini 3.6-Flash Pairwise Judge ( Overall : Ours Win \| Coherence : Ours Win \| Grammaticality : Ours Win \| Fluency : Ours Win \| Non-Repetition : Ours Win ) Both texts are limited in terms of meaning and structure at a single sampling step. However, Ours (left) is preferred over FMLM (right) , which contains more artifacts and broken punctuation. Ours , while still largely incoherent, is closer to natural English phrasing.

Appendix

Table 11 : Unconditional 1024-token generation comparison on OpenWebText at 1-Step (NFE=1) along with the Gemini 3.6-Flash pairwise evaluation result.

GSF (Ours) (2-Step, 1024 tokens)	FMLM (2-Step, 1024 tokens)
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Appendix

Table 12 : Unconditional 1024-token generation comparison on OpenWebText at 2-Step (NFE=2) along with the Gemini 3.6-Flash pairwise evaluation result.

GSF (Ours) (4-Step, 1024 tokens)	FMLM (4-Step, 1024 tokens)
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Gemini 3.6-Flash Pairwise Judge ( Overall : Ours Win \| Coherence : Ours Win \| Grammaticality : Ours Win \| Fluency : Ours Win \| Non-Repetition : Ours Win ) Both texts show limited long-range logical flow and occasional syntactic issues. However, Ours (left) is preferred over FMLM (right) . Ours maintains a more consistent thematic focus (politics, elections, and personal reflections) resembling a speech transcript, whereas FMLM changes more abruptly across unrelated topics (e.g., McDonald’s, UAVs, Orwell, Coca-Cola) with more frequent grammatical errors and artifacts.

Appendix

Table 13 : Unconditional 1024-token generation comparison on OpenWebText at 4-Step (NFE=4) along with the Gemini 3.6-Flash pairwise evaluation result.

GSF (Ours) (8-Step, 1024 tokens)

an election that serves the people’s vote and, which in most of those cases, helped and changed the voters. But that did happen. The end of the story that, for now, that the hope of this telly is to, without reference to, an election you would, and the electorate, this kind of election you will and whoever towards this idea needs an election that reflects almost completely the other parties that are mentioned in the past, politics and something that’s not on the mind that that sometimes where the CPP is important. This would fall out—in these elections that were successful, it seemed like the election. Which, what then would surprise us, we did never come into. But, it took good that, the people who value most everyone thing. The folks were right then because of this call about a single party in the country and, at the beginning where he wrote that article he could only meet all young voters and—he could then put this work on all over anti-corruption and racist voters as a factor in helping him. (That’s them leaning on him a little.) Which might make his one well starting that a third party. He said everyone would in advance. He certainly told to eventually he was, when I was really putting it into that thinking he would think maybe him lower, and we have a lot to try these guys out of touch with our country. Now, in case we want to make it, he has been in his well, turned an old fashioned post beyond politics. There’s not everything I needed except to be the best at the past a number of times made no one. The very recent original, which, in particular, is basically, a national newspaper isn’t often as a pretty successful success where you might think as much as a story. And if they work big without a chance. (The authors say) of these guys may find out, two weeks that yes, everybody was quite involved in the good times the state and race for the network.)ism that you were telling them, and that had something just that they did in any respect. The real, was the quote, followed, by my father, which he makes: The father was a family from—and. So, that family was born, and what bought him the first, so I used that to pay political parties do. By all three years ago, I mean who, say, ask anyhow I can change that’s already my year. Because of that our companies are very different. Some people in our industry are big,, and who say change is going to happen. It’s dead-wise. We could have to close one plant. It should be opening up up somewhere for us, but that is life, like middle TV or perhapses or whatever, poor or whatever to choose from. There’s exactly how those folks know you the events around them are talk, maybe, but there’s one for. And, so here we shall know our fellow people who are still happy and emonese, because we say what they are going to stop about. You’ get this with these things of action; they are most take it or go down the political road?, what I’m gonna trying to say, here. Now three and cut so they are making all sense and the real of party officials. And I still hold this in the public, perhaps that time and politics comes later.. a lot of the same way things: politics open up until it’s no longer policy and people always talk. But who’s elected Chavez and the president of the people? It is too soon to me to get back into people say, they created the best chance to break him and other candidates seem to be a political for themselves the next in 2012 That also message which that guy is now going to make a longer story telling the same thing for this. From this comes, of stories that are still the core base of their people all around states that the way Bernie Sanders does in you would be quite a later, in many more ways ways. It was as well as mine. My words told me yesterday that I saw the two. The first was a second term debate one week; you asked, “It is something.") A week I won two elections better than pretty much both in. And I needed to be over. Which, for now, needs to worry about what you also want to be and because that was not his job or hangman. And he was content with your money. Not mind, but mostly. Well he had his life so we kept at the beginning point away the very beginning (or maybe and heart, etc?) before the beginning of every new campaign and adisme. So the middle down the road all right answer to that the stuff is, but just points of value. I was D

GSF (Ours) (16-Step, 1024 tokens)

an election that serves the people’s vote and, which in most of those cases, helped and changed the voters. But that did happen. The end of the story that, for now, that the hope of this telly is to, without regard to, an election you would, and the Democrats, this type of election you will and whoever perhaps this idea needs an election that reflects almost completely the other ideas that are mentioned in the past, politics and something that’s not on the mind that that sometimes where the CPP is important. This would fall out—in these cases that were successful, it seemed like the times. Well, what then would affect us, we did never come into. But, it took good that, the people who value most common thing. The leaders were right then because of this call about a single party in the country and, at the beginning where he wrote that race he could only nominate all young democr voters—he could then put this work on all over anti-Obama and environmental voters as a leader in those him. (That’s them calling on him a little.) Which might make his one well starting that a third party. He said everyone would in Congress. He certainly told to eventually he was, when President Obama, said it into that race he would think described him as, and we have a lot to try these guys out of touch with our country. Now, in case we want to make it, he has been in his well, turned an old Republican post beyond politics. There’s not everything I ever heard to be the best at the past a number of times made no one. The very recent original, which, in particular, is word, a national newspaper isn’t often as a pretty successful result where you might think as much as a story. And if they work big without a staff. (The authors say) of these guys may find out, two weeks that yes, etc was quite involved in the good of the state and vote for the networkestism that you were telling them, and that had something just that they did in any other. The real, was the son, father, and my father, which he makes: The father was a family member—and. So, that family was born, and what bought him the first, so I used that to pay political dividends do. Now all three of this, I mean who, say, ask anyhow I can change that’s already my year. Because of that our companies are very different. These people in our industry are big,, and who say change is going to happen. It’s dead-wise. We could have to close one door. It should be opened up to everybody for us, but that is life, like townons or foxes or whatever, poor or whatever to choose from. There’s exactly how those folks know you the events around them are changing, maybe, but there’s one side. And, so here we shall know our fellow people who are still alive and Republicans are talking, because we say what they are going to stop about. You’ get this with these things of action; they are most take it or go down the political road? This what I’m gonna trying to say, here. Now three and cut so they are making all sense and the real of party decisions. And I still hold this in the public, perhaps that time and politics comes later up a lot of the same way things: politics open up until it’s no longer policy and people always stay. But who’s Mitt staff and the president of the Republic? It is too soon to me to get back into people say, they created the best chance to break him and other candidates hope to be a political for themselves the next in 2012 That also message which that guy is now going to make a longer story.. the two thing for this. Before this comes, please remember that I still lost the trust of their people all around states that the way Mitt Romney does in you would be quite a later, in many more ways ways. It was as well as mine. My words told me yesterday that I saw the two. The first was a second term till one week; you asked, “It is something.") A week I won two elections better than pretty much both in. And I needed to be over. Because, for now he needs to hold about what you also want to be and because that was not his job or layman. And he was content with your money. Not mind, but mostly. Well he had his life so we kept at the beginning point away the very beginning (or over and heart, etc) before the beginning of every new campaign and adisme. So the middle down the road all right back to that the stuff is, but just points of value. I was D

Appendix

Table 14 : Unconditional 1024-token generation trajectories of GSF (Ours) at 8-Step (top) and 16-Step (bottom) on OpenWebText.

GSF (Ours) (32-Step, 1024 tokens)

an election that serves the people’s lives and, which in most of those cases, helps and supports the family. But that did happen. The end of the story that, for now, that the hope of this telly is to, without regard to, an election you would, and the audience, this kind of election you will and whoever perhaps this idea needs an image that reflects almost completely the other parties that are mentioned in the past, while keeping something that’s not on the ground that that sometimes where the CPP is important. This would fall out—in these cases that were successful, it seemed like the times. Well, what then would affect us, we did never come into. But, it took good that, the people who value most common thing. The leaders were right then because of this call about a single party in the country and, at the conference where he wrote that race he could only invite all young democr voters—he could then put this work on all over anti-Obama and gay material as a candidate in those states. (That’s them calling on him a little differently) might make his one well field that a third party. He said everyone would in Congress. He certainly told to eventually he was, when President Obama, said it into that race he would think described him as, and we have a lot to try these people out of touch with our country. Now, in case we want to make it try to reach out in his well, turn an old fashioned post beyond politics. There’s not everything I ever want to be the best at the past a number of times made no one. The very recent original, which, in particular, is word, a national newspaper isn’t often as a pretty successful result where you might think as much as a story. And if they work big without a staff. (The authors say) of these guys may find out, two weeks that yes, etc was quite involved in the good of the state and vote for the networkestism that you were telling them, and that had something just that they did in any other. The real, was the quote, followed, by my father, which he makes: The father was a family member—and…. So, that family was born, and what bought him the first, so I used that to pay political dividends do. These all three of whom were a couple who, say, ask anyhow I can change that’s already my year. Because of that our companies are very different. These people in our industry are bigots, and who say change is going to happen. It’s dead-wise. We could have to close one door. It should be opened up to everybody for us, but that is life, like kidsons or Georges or whatever, poor or whatever to learn from. There’s gonna love those folks know you the events around them are gone, yes, but there’s one side. And, so here we’ve seen people who are still happy and nobody are talking, because we say what they are going to stop about. You’ get this with these things of action; they are most take it or go down the political road? This what I’m gonna trying to say, here. Now three and cut so they are making all effort and the real election party trying. And I still hold this in the public, thinking that time and politics comes later up a lot of the same way things: stay open up until it’s no longer policy and I always watch. But who’s Borges and the president of the Philippines? It is too soon to me to get back into people say, they were fighting with promises to break him and other candidates (to be a political leader); the site in 2012 That also happened which that guy is now going to make a longer story): the two thing for this. Where this comes, reality is that I still lost the trust of their people all around states that the way Mitt Romney does in you would be quite a later, in many more ways ways. It was as well as mine. My words told me yesterday that I saw the two. The first was a second term; one week; you asked, “It is something— A week I won two elections better than pretty much both in. And I needed to be over. Because, for now he needs to hold about what you also want to be and because that was not his job or layman. And he was content with your money. We failed, but we failed, he had his life so we kept at the beginning point, the very beginning (or over and gone, yes, before the beginning of every new campaign and ad.)e. So the stay down the road all right back to that the stuff is, but just points of value. I was D

GSF (Ours) (64-Step, 1024 tokens)

an election that serves the people’s lives and, which in most of those cases, helps and supports the family. But that did happen. The end of the story that, for now, that the hope of this telly is to, without regard to, an election you would, and the culture, this kind of society you will and you realize this idea needs an image that represents almost completely the other ideas that are mentioned in the past, while there something that can become being focused on the ground that that sometimes where the CIO is important. This would fall out—in these cases that were successful, it seemed like the times. And, what then would that tell me we did never come into? But, it took good that, the people who value most common thing. The folks were right then because of this book about a single town in the country south, at the beginning where he wrote that race he could only write all young adult industry—he could then put this work on all over anti-war and racial material as a leader in those states. (That kind of push them around on him a little bit) might make his one well field that a third party. He said everyone would in Congress. He was told to eventually he was, when President Obama, said it into that race he would think described him as, and we have a lot to try these people out of touch with our country. Now, in case we want to make it try to write out in his well, turn an old fashioned post beyond risk. There’s not everything I think except to be the best at the past a number of times made no one. The very recent original, which, in particular, is word, a national newspaper isn’t often as a pretty successful result where you might think as much as a story. And if they work big without a staff. (The authors say) of these guys may find out, two check that out, everybody was quite involved in the good of the state and vote for the networkestism that you were telling them, and that had something just that they did in any other. The real, was the quote, followed, by my father, which he makes: The father was a family member—and…. So, that family was born, and what bought him the first, so everybody used that to pay political dividends do. And all three of this were a big who, oh, man any of us can change that’s already my year. Because of that our companies are very different. These people in our industry are big,, and who say change is going to happen. It’s total-wise. We could have to close one school. It should be opened up to everybody for us, but that is life, like kidsons or Georges or whatever, poor or whatever to learn from. There’s gonna love those folks know you the events around them are gone, yes, but there’s one side. And, so here we’ve seen people who are still happy and nobody are talking, because we say what they are going to stop about. You’ get this with these things– action; they are most of it or go down the political road. This what I’m gonna trying to say, here. Now three and cut so they are making all decisions and the real ones will try. And I still hold this in the public, thinking that time and politics comes later up a lot of the same way things: stay open up until it’s no longer policy and I always watch. But who’s Borges and the president of the press? It is too soon to me to get back into race stuff, they were going with Hillary to break him and other candidates (howviously a political network was the site in 2012 That also happened which that guy is now going to make a longer story): the two thing for this. Where this comes, remember is that I still leave the base of their story all around states that the way Mitt Romney does in you would be quite a later, in many more ways ways. It was as well as mine. My words told me yesterday that I saw the two. The first was a second term; one week; you asked, “It is something— A week I won two elections better than pretty much both in. And I needed to be over. Then, yes; he needs to hold about what you also want to be and because that was not his job or layman. And he was content with your money. We failed, but we failed, he had his life so we kept at the beginning point, the very beginning (or over and heart, etc) before the beginning of every new campaign and adisme. So the stay down the road all right back to that the stuff is, but just points of value. I was D

Appendix

Table 15 : Unconditional 1024-token generation trajectories of GSF (Ours) at 32-Step (top) and 64-Step (bottom) on OpenWebText.

GSF (Ours) (1-Step, 1024 tokens)	Gumbel-Distill (1-Step, 1024 tokens)
team was trying to put what I think, of his same. I would have got that kind of thing is was in good one. So reports of course ( because the man who was part of most of his job I was like at each other partly because we never getting a back because the news is to on. As that happens the my time I’m my story and don’t But the change doesn for me, and I want to become they me there before we made my job, because probably our only thing we would do. So I get a long, of thinking I was in no doubt. What I know, to happen at the highest L of number this was the results that any of us did so at. You like the want. If it does team, but they of play their heads up for whatever reason with each other. Every team want just take that in our heads to share what we’re gonna have a lot for things. So what we will tell with that if you trust them or you would a to, sorry for what you have, the chances of to your in place big was of course you can give because the world with something no one always up and taken them up. But this seems don’t just show any of the support that you see to getting real. You said then thinking that our other people makes have a little case for the I and my man — this?" which can be about, every important I, why there is not something being, in this look something I we do give in the right thing. It’s this with me and try — since we know again we simply to give me, noDo you think I have a lot. I think as to why I and these my here video come have. I started to spend a short trying to take my that, I like to continue myself Some other people by whom I will never know that people won’t with which they were themselves up years and so I don’t believe, whoever is on day from coast to possible. many things, to say, "you has no" when I’m and my friends do a few up for your in. I know though I just change my own work and some ideas or bad wrongs, which helps us was better. The team is to’m so sure that I think as far as the " problem is at the" which we have. It’s very easy to think and you should before, you’d say something we just had our own mind at, just you to this process and think much of your own model. is the reason which we hope what you might do, we’ll do is them to go. That’s what the team are and this can do from it, it is enough to happen. And we’ve got a common amount to have in the past, before up this kind of time making and take we take something like going to do one work we have to pretty doing a little one that is it easy to do it and we just do the starting to ensure that this doesn’t happen. What if a line not what it takes to exercise out our right to be a person? I. be all to in him, and make sure we that all the good, are not and so we’s come at the top. We did this kind of The work here is easy and we’ll take the first act of ourselves while. Then, last time, we saw how that it’s really done. We agree with a it of all, now at the end that you and our all on it and then, we have gone through our lives. What city guys that are some very important in the National? What has been that year, we are back and what I can like for is because they came out — from my we to find a has to be and the next day every do in "just take 8" does not really I other want and to take that, looking And having all of our lives, friends, our job games, start. your and family issues is a better idea? Trust them. for if to do that already of this site, you can feel we have gone this whole 90 because of time each individual. Such an opportunity for not only is, where more or less have take the past. of And I think that is’re for those that is biased against When we Well I only see an example of you or not that refers to "what’s a those of my really." Apart from me, to be honest as I don’t find myself all the time. What do you like me? The fact that our biggest we are all that we know so have so a lot around that use and some a to every a team. for to make that people are being happy or frustrated, since the other just can’t get one of at best and generally to good work very.". you get the list isn’t just because pretty is just when they are I still, they’re no longer in no good works at the moment. 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Gemini 3.6-Flash Pairwise Judge ( Overall : Ours Win \| Coherence : Ours Win \| Grammaticality : Ours Win \| Fluency : Ours Win \| Non-Repetition : Ours Win ) GD (1-Step) (right) consists largely of disconnected words, fragmented numbers, and broken punctuation. Ours (1-Step) (left) , while containing some disjointed phrasing and weak logical flow, produces valid English words organized into readable sentences and clauses, and is therefore preferred across all evaluated dimensions.

Appendix

Table 16 : Unconditional 1024-token generation comparison on OpenWebText between GSF (Ours) (1-Step) and Gumbel-Distill ( Zhang et al., 2026 ) (1-Step) along with the Gemini 3.6-Flash pairwise evaluation result.

GSF (Ours) (1-Step, 1024 tokens)	Gumbel-Distill (2-Step, 1024 tokens)
team was trying to put what I think, of his same. I would have got that kind of thing is was in good one. So reports of course ( because the man who was part of most of his job I was like at each other partly because we never getting a back because the news is to on. As that happens the my time I’m my story and don’t But the change doesn for me, and I want to become they me there before we made my job, because probably our only thing we would do. So I get a long, of thinking I was in no doubt. What I know, to happen at the highest L of number this was the results that any of us did so at. You like the want. If it does team, but they of play their heads up for whatever reason with each other. Every team want just take that in our heads to share what we’re gonna have a lot for things. So what we will tell with that if you trust them or you would a to, sorry for what you have, the chances of to your in place big was of course you can give because the world with something no one always up and taken them up. But this seems don’t just show any of the support that you see to getting real. You said then thinking that our other people makes have a little case for the I and my man — this?" which can be about, every important I, why there is not something being, in this look something I we do give in the right thing. It’s this with me and try — since we know again we simply to give me, noDo you think I have a lot. I think as to why I and these my here video come have. I started to spend a short trying to take my that, I like to continue myself Some other people by whom I will never know that people won’t with which they were themselves up years and so I don’t believe, whoever is on day from coast to possible. many things, to say, "you has no" when I’m and my friends do a few up for your in. I know though I just change my own work and some ideas or bad wrongs, which helps us was better. The team is to’m so sure that I think as far as the " problem is at the" which we have. It’s very easy to think and you should before, you’d say something we just had our own mind at, just you to this process and think much of your own model. is the reason which we hope what you might do, we’ll do is them to go. That’s what the team are and this can do from it, it is enough to happen. And we’ve got a common amount to have in the past, before up this kind of time making and take we take something like going to do one work we have to pretty doing a little one that is it easy to do it and we just do the starting to ensure that this doesn’t happen. What if a line not what it takes to exercise out our right to be a person? I. be all to in him, and make sure we that all the good, are not and so we’s come at the top. We did this kind of The work here is easy and we’ll take the first act of ourselves while. Then, last time, we saw how that it’s really done. We agree with a it of all, now at the end that you and our all on it and then, we have gone through our lives. What city guys that are some very important in the National? What has been that year, we are back and what I can like for is because they came out — from my we to find a has to be and the next day every do in "just take 8" does not really I other want and to take that, looking And having all of our lives, friends, our job games, start. your and family issues is a better idea? Trust them. for if to do that already of this site, you can feel we have gone this whole 90 because of time each individual. Such an opportunity for not only is, where more or less have take the past. of And I think that is’re for those that is biased against When we Well I only see an example of you or not that refers to "what’s a those of my really." Apart from me, to be honest as I don’t find myself all the time. What do you like me? 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Gemini 3.6-Flash Pairwise Judge ( Overall : Ours Win \| Coherence : Ours Win \| Grammaticality : Ours Win \| Fluency : Ours Win \| Non-Repetition : Ours Win ) GD (2-Step) (right) consists largely of disconnected words, symbols, punctuation marks, and incomplete sentences. Ours (1-Step) (left) , while rambling and grammatically imperfect in places, contains readable English sentences and clauses that resemble natural speech, making it the chosen option.

Appendix

Table 17 : Unconditional 1024-token generation comparison on OpenWebText between GSF (Ours) (1-Step) and Gumbel-Distill ( Zhang et al., 2026 ) (2-Step) along with the Gemini 3.6-Flash pairwise evaluation result.

GSF (Ours) (1-Step, 1024 tokens)	Gumbel-Distill (4-Step, 1024 tokens)
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Gemini 3.6-Flash Pairwise Judge ( Overall : Ours Win \| Coherence : Ours Win \| Grammaticality : Ours Win \| Fluency : Ours Win \| Non-Repetition : Ours Win ) GD (4-Step) (right) contains frequent special tokens, fragmented subwords, irregular punctuation, and broken formatting. Ours (1-Step) (left) , while still rambling and globally incoherent, maintains valid English words and basic sentence-like structures throughout.

Appendix

Table 18 : Unconditional 1024-token generation comparison on OpenWebText between GSF (Ours) (1-Step) and Gumbel-Distill ( Zhang et al., 2026 ) (4-Step) along with the Gemini 3.6-Flash pairwise evaluation result.

[Prompt #1:] was my mainstay. I despise apple candies, and though grape was usually on an even level with orange, grape juice certainly wasn’t. Later, I found a new flavor, or at least a flavor that was new to [Continuation #1:] me with my friends! It was just a taste of me, it was a sweet sweet flavor. Seeing many of them at the side like the sweet fruit, and the flavors changed my eyes on where they hung out up, too, with orange. [Continuation #2:] this era when my change began growing, I still found problems, like apples to eat in the kitchen. Having somewhere within the recipes I released before I would arrive at my office, I was thinking of finding things for an actual taste. The flavor of [Continuation #3:] control virtually all day. I assured her it was a decade before. That I needed to pass me year, in order to enjoy and respond to all tastes. The flavors, which is definitely on your doorstep now! I highly recommend them. If a

[Prompt #2:] train with Stu Hart for the latter’s Stampede Wrestling promotion.[6] He trained under Hart in Calgary before debuting on January 16, 1970 in a match with Dan Kroffat.[14] After wrestling briefly under his real name, [Continuation #1:] he wore a wrestling name with Hart behind his name; not 1-10 but he won titles and retired on April decision. In 1971, Smelton was the champion of WWE Wrestling before the official event of the Diva.[15] The match [Continuation #2:] he claimed to have started a career in professional wrestling Wrestling in the 1980 industry.[15] After losingring to a hospital from a St injury in Montreal, five in 2011 for the remainder of his career, after he was announced in 1970 as Hart’s [Continuation #3:] Hart was officiallyized and his replacement by the Calgary family in Canada. He was at Stampede School of age 18; at least in Canada to a degree pro in general.[15] Despite an average score of 2 (19) in his debut

[Prompt #3:] take no position on the race. Corrado said it’s uncharted territory to have an incumbent who is not a Democrat and who sits far to the left of the Democratic Party. He said the entire membership of the 37th District Democrats [Continuation #1:] were all supporters of the other Democratic and rank-and-file group “would be successful.” Critics say the Democratic Party and the growing would reflect the weight of the pressure that is on board members the Democratic Party or as Democrats [Continuation #2:] will be split from January 2015, to replace former Democrats. “Of course the Democrats could be at all with those people but up what the party did was and the Democrats’ fault.” “It’s all [Continuation #3:] are Democrats, the number nine from Republican Democrats, and are more suited from the left for the District of members and Democratic Democratic Committee (CCC) and the Democratic Party. This is a mistake. “He’s what we have

Appendix

Table 19 : Conditional text generation samples on OpenWebText ( prefix_len = 50 , seq_len = 100 ). Given the ground-truth conditioning prefix ( [Prompt] ), GSF generates diverse and coherent continuations ( [Continuation] ) from different initial noise seeds. (4-step)

[Doc:] The number eight was capped 15 times at international level, touring Australia and New Zealand and skippering the side during the 1965 Five Nations. Perry featu … and scored 115 points for Bedford. A Bedford statement said: "The club extends its deepest sympathy to David’s family." [GT:] Former England rugby union captain David Perry has died at the age of 79. [1-Step:] Ex- and Cambridge captain David Perry has died at the age. [2-Step:] Former Bedford and Cambridge captain David Perry has died at the age of 44.

[Doc:] "It’s a question of reason," he said on French TV. His decision prompted immediate thanks from Mr Macron, who is neck-and-neck for the first round of the electi … that she did little work for the hundreds of thousands of euros she received in public funds as parliamentary assistant. [GT:] France’s ex-Prime Minister Manuel Valls has thrown his weight behind the centrist candidate for the presidency, Emmanuel Macron, and not his own Socialist party’s candidate. [1-Step:] French Minister Manuel Valls has won the backing government’s president to become right-right as Emmanuel Macron this president. [2-Step:] French Minister Manuel Valls has won the right government’s candidate to a front-right to be Macron as president of Irish 2016 after.

[Doc:] Colombian leader Juan Manuel Santos and the Farc rebel commander known as Timochenko signed the deal in an emotional ceremony on Monday evening. "I would like t … defeats in its worst year 2012: Start of peace talks in Havana 2016: Definitive ceasefire Full timeline of Farc conflict [GT:] Colombia’s government has signed a historic peace deal with the Farc rebel group, ending more than 50 years of armed conflict. [1-Step:] The largest rebels in Farc has signed an agreement the government of an historic that deal deal for the that country an treaty on a peace deal against the country and the government.. in Cart. [2-Step:] The historic rebels in F Colombia has signed an agreement the signing of an historic that deal deal for the the Farc rebels on a peace deal against the country and the government rebel place in 1965.

[Doc:] But his term in office has been dogged by the deep schism between his own Fatah movement in the West Bank and the militant Hamas movement, which controls the Ga … me minister from May 2003 until his resignation almost four months later after a power struggle with Arafat over control [GT:] Mahmoud Abbas, also known as Abu Mazen, has led the Palestinian Authority since he was elected as its president in January 2005. [1-Step:] Mahm Mahmoud Abbas has spent a first pro at a top figure in the Palestinian of Mahmoud issues over [2-Step:] Mah’s Mahmoud Abbas has won a first figure- a top figure in the Palestinian in Mahmoud issues that has ended a en between over the decade.

[Doc:] SABC says the move, which has been hailed by local musicians after years of campaigning, will promote South African culture and heritage. There will be a specia … ofit from a boost in sales of their music. And for many listeners too, the decision seems to have struck the right note. [GT:] South Africa’s national broadcaster SABC has brought in a new quota system, requiring 90% of the music played on its 18 radio stations to be homegrown. [1-Step:] South Africa’s country’s first station is to new music play production play, with the stations in the past stations. [2-Step:] South Africa’s arm has launched a new music play production passes, with the release of the pastowing towards its launchmost.

Appendix

Table 20 : Extreme summarization (XSum) samples generated by GSF (Ours) at 1-Step and 2-Step . Each block shows 4 lines: the source news document ( [Doc] ), the ground-truth summary ( [GT] ), and our [1-Step] and [2-Step] generations.

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