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Acceleration of Diffusion Language Model through Discrete Average Generator
Yidong Ouyang, Zhengyan Wan, Themis Haris, Tian Tan, Liqian Peng, Henry Li, Ziqian Lin, Jianhang Chen, Maryam Karimzadehgan, Alec Go, George Michailidis
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 95%
Last extracted: 10/1/2026, 7:54:35 AM
Summary
This paper introduces the Discrete Average Generator, a method to accelerate diffusion language models by extending the MeanFlow framework to discrete state spaces governed by Continuous-Time Markov Chains (CTMCs). The authors define an average generator as the normalized increment of the transition kernel, proving a self-consistency identity that serves as a stable training objective. By projecting this identity onto per-coordinate marginals, they achieve a tractable closed-form solution. The method demonstrates significant performance gains, including a 16x acceleration and reduced perplexity on OpenWebText, and comparable results on ImageNet.
Entities (9)
Relation Signals (7)
Discrete Average Generator → evaluatedon → ImageNet
confidence 98% · achieves comparable performance to existing methods on ImageNet.
Discrete Average Generator → evaluatedon → OpenWebText
confidence 98% · On OpenWebText, our method achieves the lowest generative perplexity among the evaluated methods for 8 to 64 sampling steps...
Yidong Ouyang → affiliatedwith → Google
confidence 95% · Affiliation: Google
Yidong Ouyang → affiliatedwith → University of California, Los Angeles
confidence 95% · Yidong Ouyang Affiliation: University of California, Los Angeles
Discrete Average Generator → appliesto → Continuous-Time Markov Chains
confidence 95% · In this work, we introduce the Discrete Average Generator, a principled extension of MeanFlow to Continuous-Time Markov Chains (CTMCs).
Discrete Average Generator → extends → MeanFlow
confidence 95% · In this work, we introduce the Discrete Average Generator, a principled extension of MeanFlow to Continuous-Time Markov Chains (CTMCs).
Discrete Average Generator → reduceserroron → Potts model
confidence 90% · In Potts model simulations, our objective reduces the total variation distance of the K-step sampler by up to 67%.
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Abstract
Abstract:Discrete diffusion models and flow matching have emerged as powerful frameworks for generative modeling over discrete state spaces, yet efficient few-step generation remains a fundamental challenge. In this work, we introduce the Discrete Average Generator, a principled extension of MeanFlow to Continuous-Time Markov Chains (CTMCs). Analogously to how MeanFlow defines an average velocity field over a time interval in continuous spaces, we define an average generator as the normalized increment of the transition kernel over a time interval. We show that this average generator satisfies a self-consistency identity, which provides the foundation for our training objective. We further develop training strategies that align with the standard training paradigm of diffusion language models while keeping the resulting objective tractable. When projected onto per-coordinate marginals, the self-consistency identity admits a closed-form expression, enabling efficient training and inference. In Potts model simulations, our objective reduces the total variation distance of the $K$-step sampler by up to 67%. On OpenWebText, our method achieves the lowest generative perplexity among the evaluated methods for 8 to 64 sampling steps while enabling a $16\times$ acceleration, and achieves comparable performance to existing methods on ImageNet.
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- Source: https://arxiv.org/abs/2609.38364v1
- Canonical: https://arxiv.org/abs/2609.38364v1
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Acceleration of Diffusion Language Model through Discrete Average Generator Yidong Ouyang Affiliation: University of California, Los Angeles Affiliation: Google Affiliation: Work done as a research intern at Google Zhengyan Wan Affiliation: University of California, Los Angeles Themis Haris Affiliation: Google Tian Tan Affiliation: Google Liqian Peng Affiliation: Google Henry Li Affiliation: Google Ziqian Lin Affiliation: Google Jianhang Chen Affiliation: Google Maryam Karimzadehgan Affiliation: Google Alec Go Affiliation: Google George Michailidis Affiliation: University of California, Los Angeles Abstract Discrete diffusion models and flow matching have emerged as powerful frameworks for generative modeling over discrete state spaces, yet efficient few-step generation remains a fundamental challenge. In this work, we introduce the Discrete Average Generator, a principled extension of MeanFlow to Continuous-Time Markov Chains (CTMCs). Analogously to how MeanFlow defines an average velocity field over a time interval in continuous spaces, we define an average generator as the normalized increment of the transition kernel over a time interval. We show that this average generator satisfies a self-consistency identity, which provides the foundation for our training objective. We further develop training strategies that align with the standard training paradigm of diffusion language models while keeping the resulting objective tractable. When projected onto per-coordinate marginals, the self-consistency identity admits a closed-form expression, enabling efficient training and inference. In Potts model simulations, our objective reduces the total variation distance of the K-step sampler by up to 67%. On OpenWebText, our method achieves the lowest generative perplexity among the evaluated methods for 8 to 64 sampling steps while enabling a 16×16× acceleration, and achieves comparable performance to existing methods on ImageNet. keywordsdiscrete diffusion, diffusion language models, MeanFlow, continuous-time Markov chains, few-step generation 1 Introduction Diffusion models and flow matching have achieved remarkable success in continuous domains such as image and audio generation (Lipman et al., 2024). Extending these generative frameworks to discrete spaces, such as natural language, has attracted growing interest, with discrete flow matching and masked diffusion models emerging as promising approaches (Campbell et al., 2024; Gat et al., 2024; Shi et al., 2024; Sahoo et al., 2024). However, a fundamental challenge shared with their continuous counterparts remains: sampling requires multiple sequential network evaluations, making inference computationally expensive. The difficulty is compounded in discrete spaces, where practical samplers update all coordinates independently within a step: with few steps, each coordinate is drawn from a law computed from the instantaneous rate rather than from the finite-step kernel, and the dependence between coordinates updated together is lost. In continuous settings, this bottleneck has been addressed by consistency models (Song et al., 2023), progressive distillation (Salimans and Ho, 2022), and more recently MeanFlow (Geng et al., 2025a), which defines an average velocity field over time intervals and enables high-quality generation in very few steps without multi-step distillation. The key insight of MeanFlow is that the average velocity satisfies a self-consistent identity involving the instantaneous velocity and its material derivative, which can be exploited as a fixed-point training objective. In this work, we propose the Discrete Average Generator, a principled extension of MeanFlow to discrete state spaces governed by Continuous-Time Markov Chains (CTMCs). The central challenge is that the continuous notion of velocity and its spatial gradient has no direct analogue in discrete spaces. We resolve this by defining the CTMC average generator over an interval [t,r][t,r] as the normalized increment of the transition kernel, i.e. the unique object for which a single Euler step of size r−tr-t reproduces the exact transition law. We further derive an average generator identity that forms the basis of our training objective. Inspired by the progress of velocity prediction (u-prediction / v-prediction) to data prediction (x-prediction) made in continuous MeanFlow literature described in Section 2.2, we develop a corresponding progression from transition rate prediction (U-prediction / Q-prediction) to posterior prediction (p~ p-prediction) in Section 3.2. Although the p~ p-prediction is compatible with existing pretrained diffusion language models, it is intractable at the sequence level because of the huge computational cost associated with the multiplication of the generator matrix with dimension ||×|||S|^D×|S|^D. By projecting the identity onto per-coordinate marginals, as described in Section 3.3, we obtain a tractable closed-form involving only ×||D×|S|-dimensional quantities. Our main contributions are as follows: • We introduce the Average Generator for CTMCs and prove the Discrete Average Generator Identity, the discrete analogue of the MeanFlow identity. • We derive a tractable training objective based on p~ p-prediction and per-coordinate marginal projection, which aligns with existing diffusion language model parameterizations and admits a closed-form expression involving only ×||D×|S|-dimensional quantities. • We validate our approach through simulations, text generation on the OpenWebText dataset, and image generation on the ImageNet dataset. On OpenWebText, our method achieves the lowest generative perplexity among the compared methods for 88 to 6464 sampling steps and maintains lower generative perplexity with 16×16× fewer generation steps against the strongest baseline. On ImageNet, our method achieves comparable FID and Inception Score to existing acceleration methods for diffusion language models. 2 Preliminary 2.1 Discrete Flow Matching Consider a Continuous-Time Markov Chain (CTMC) defined on the state space S^D with marginal probability mass functions ptt∈[0,1]\p_t\_t∈[0,1]. Let QtQ_t denote the associated transition rate matrix (i.e., the generator) at time t, which has dimensions ||×|||S|^D×|S|^D. For any distinct states x,z∈x,z ^D (where x≠zx≠ z), the entry Qt(x,z)Q_t(x,z) represents the instantaneous transition intensity from state x to state z at time t. If Qt(x,z)Q_t(x,z) satisfies the following rate-properties: Qt(x,z)≥0, for any z≠x, and ∑z∈Qt(x,z)=0, Q_t(x,z)≥ 0, for any z≠ x, and \ _z ^DQ_t(x,z)=0, and the Kolmogorov forward equation (also known as the continuity equation): p˙t(x)=∑z∈Qt(z,x)pt(z)=∑z≠xQt(z,x)pt(z)−∑z≠xQt(x,z)pt(x), p_t(x)= _z ^DQ_t(z,x)p_t(z)= _z≠ xQ_t(z,x)p_t(z)- _z≠ xQ_t(x,z)p_t(x), (1) the transition rate QtQ_t is also the generator of the CTMC ptt∈[0,1]\p_t\_t∈[0,1]. In particular, pt+h|t(x|z)=δz(x)+Qt(z,x)h+o(h)p_t+h|t(x|z)= _z(x)+Q_t(z,x)h+o(h) for a small time step h. To learn a transition rate QtQ_t that can transport from an initial noise distribution p0p_0 to a target data distribution p1p_1, a well-developed method is to learn the conditional expectation of the conditional transition rate Qt(x,z|x1)Q_t(x,z|x_1) that generates the conditional probability path pt|1(⋅|x1)p_t|1(·|x_1). Then Qt(x,z)=p1|t(x1|x)[Qt(x,z|x1)]Q_t(x,z)=E_p_1|t(x_1|x)[Q_t(x,z|x_1)] can generate the target probability path ptp_t (Campbell et al., 2024, see Proposition 3.1 of), i.e., it satisfies the Kolmogorov forward Equation 1. For completeness, we include the proof in Appendix B.2. We are free to define the conditional probability path pt|1(⋅|x1)p_t|1(·|x_1) and with the corresponding conditional transition rate as needed. A common approach is to construct a coordinate-wise conditional probability path and transition rate, pt|1(x|x1)=∏d=1pt|1d(xd|x1d),Qt(x,z|x1)=∑d=1δx (z )Qtd(xd,zd|x1d),p_t|1(x|x_1)= _d=1^Dp^d_t|1(x^d|x_1^d), Q_t(x,z|x_1)= _d=1^D _x d(z d)Q_t^d(x^d,z^d|x_1^d), (2) where we use the notation x d to denote the vector obtained by removing the d-th coordinate of x, i.e., x :=(x1,…,xd−1,xd+1,…,x)x d:=(x^1,…,x^d-1,x^d+1,…,x^D), Qtd(xd,zd|x1d)Q_t^d(x^d,z^d|x_1^d) is the conditional transition rate that generates the conditional probability path pt|1dp_t|1^d, and δx(z) _x(z) the Kronecker delta satisfying δx(z)=1 _x(z)=1 if x=zx=z and δx(z)=0 _x(z)=0 if x≠zx≠ z. A popular choice of probability path and the associated conditional transition rate used in the previous works (Campbell et al., 2024; Gat et al., 2024) is pt|1d(xd|x1d)=(1−κt)p0d(xd)+κtδx1d(xd),Qtd(xd,zd|x1d)=λt(δx1d(zd)−δxd(zd)),p_t|1^d(x^d|x_1^d)=(1- _t)p_0^d(x^d)+ _t _x_1^d(x^d), Q_t^d(x^d,z^d|x_1^d)= _t ( _x_1^d(z^d)- _x^d(z^d) ), (3) where κt:[0,1]→[0,1] _t:[0,1]→[0,1] is a non-decreasing function satisfying κ0=0 _0=0 and κ1=1 _1=1, and λt≜κ˙t1−κt _t κ_t1- _t. Common sources are the uniform distribution p0d=Unif()p_0^d=Unif(S) and the masked source p0d=δp_0^d= _m, which puts all mass on a mask token m. After defining the conditional path and rate, the marginal transition rate is given by Qt(x,z)=∑x1Qt(x,z|x1)p1|t(x1|x)= Q_t(x,z)= _x_1Q_t(x,z|x_1)p_1|t(x_1|x)= ∑d=1δx (z )∑x1dQtd(xd,zd|x1d)p1|td(x1d|x) _d=1^D _x d(z d) _x_1^dQ_t^d(x^d,z^d|x_1^d)p^d_1|t(x_1^d|x) =△ = ∑d=1δx (z )Qtd(x,zd), _d=1^D _x d(z d)Q_t^d(x,z^d), where p1|td(x1d|x)=∑x1 1|t(x1|x)p_1|t^d(x_1^d|x)= _x d_1p_1|t(x_1|x) is the posterior. When the coordinate is clear from the argument we drop the superscript and write p1|t(zd|x)p_1|t(z^d|x) for p1|td(zd|x)p^d_1|t(z^d|x); for the mixture path, Qtd(x,zd)=λtp1|t(zd|x)Q^d_t(x,z^d)= _t\,p_1|t(z^d|x) for zd≠xdz^d≠ x^d. Training objective To learn the transition rate, a common approach is to minimize the Bregman divergence between the conditional generator and the estimated generator (Holderrieth et al., 2025; Lipman et al., 2024; Shaul et al., 2025; Wan et al., 2025): x1,t,xt∼pt|1(xt∣x1)[∑z≠xtDF(Qt(xt,z|x1)∥Qt(xt,z))],E_x_1,t,x_t p_t 1(x_t x_1) [ _z≠ x_tD_F (Q_t(x_t,z|x_1) \|Q_t(x_t,z) ) ], where DFD_F denotes the Bregman divergence induced by F(x)=xlogxF(x)=x x, DF(a∥b)≜F(a)−F(b)−F′(b)(a−b)=alogab−a+bD_F(a\|b) F(a)-F(b)-F (b)(a-b)=a ab-a+b, which is defined for a≥0a≥ 0 and b>0b>0 and vanishes if and only if a=ba=b. After some derivation in Appendix B.1, it is equivalent to learn the posterior network p1|t(x1d|xt)p_1|t(x^d_1|x_t) under the generalized Kullback–Leibler (KL) divergence: x1,t,xt∼pt|1(xt∣x1)[λt∑d=1−(1−δx1d(xtd))logp1|td(x1d|xt)+δx1d(xtd)−p1|td(xtd|xt)], _x_1,t,x_t p_t 1(x_t x_1) [ _t _d=1^D \- (1- _x^d_1(x^d_t) ) p_1|t^d(x^d_1|x_t)+ _x^d_1(x^d_t)-p_1|t^d(x^d_t|x_t) \ ], (4) which is equivalent to Equation 37 of Shaul et al. (2025). If we use the masked source with the mixture path, then Equation 4 recovers the training objective for masked diffusion models (Shi et al., 2024; Sahoo et al., 2024; Ou et al., 2025; Nie et al., 2025). 2.2 MeanFlow In MeanFlow (Geng et al., 2025a), the authors define the average velocity field in continuous space as the displacement from time t to r divided by r−tr-t (0<t<r<10<t<r<1) as u(xt,t,r)≜1r−t∫trv(xs,s)s,u (x_t,t,r ) 1r-t _t^rv (x_s,s )ds, where v(xs,s)v (x_s,s ) denotes the instantaneous velocity, t=0t=0 corresponds to the noise, and t=1t=1 corresponds to the data. Then, the MeanFlow identity can be derived (see Appendix B.3): u(xt,t,r)=v(xt,t)−(t−r)[∂tu(xt,t,r)+v(xt,t)⋅∇xtu(xt,t,r)].u(x_t,t,r)=v(x_t,t)-(t-r)[ _tu(x_t,t,r)+v(x_t,t)· _x_tu(x_t,t,r)]. Therefore, the training objective for MeanFlow can be formulated as the following fixed-point optimization problem: ℒ(θ)=x1,t,r,xt∼pt|1(xt∣x1)‖uθ(xt,t,r)−sg(utgt(xt,t,r))‖22,L(θ)=E_x_1,t,r,x_t p_t 1(x_t x_1) \|u_θ(x_t,t,r)-sg (u_tgt(x_t,t,r) ) \|_2^2, where utgt(xt,t,r)=v(xt,t)−(t−r)(v(xt,t)⋅∇xuθ(xt,t,r)+∂tuθ(xt,t,r))u_tgt(x_t,t,r)=v(x_t,t)-(t-r) (v(x_t,t)· _xu_θ(x_t,t,r)+ _tu_θ(x_t,t,r) ) and sg(⋅)sg(·) denotes the stop-gradient operator. v-prediction parameterization Geng et al. (2025b) identify an underlying limitation of MeanFlow, i.e., the target utgt(xt,t,r)u_tgt(x_t,t,r) depends on the unknown oracle u(xt,t,r)u(x_t,t,r), and is therefore approximated using the network prediction uθ(xt,t,r)u_θ(x_t,t,r). This self-referential construction introduces bias and leads to unstable training. To address this issue, they propose a velocity-based reparameterization derived from the MeanFlow identity: V(xt,t)=uθ(xt,t,r)+(t−r)(v(xt,t)⋅∇xuθ(xt,t,r)+∂tuθ(xt,t,r)).V(x_t,t)=u_θ(x_t,t,r)+(t-r)\, (v(x_t,t)· _xu_θ(x_t,t,r)+ _tu_θ(x_t,t,r) ). (5) Instead of directly regressing u, the model is trained via a standard velocity regression objective: x1,r,t,xt∼pt|1(xt∣x1)‖V(xt,t)−voracle(xt,t)‖22.E_x_1,r,t,x_t p_t 1(x_t x_1) \|V(x_t,t)-v^oracle(x_t,t) \|_2^2. (6) In practice, v(xt,t)v(x_t,t) can be obtained either via the boundary condition v(xt,t)=uθ(xt,t,t)v(x_t,t)=u_θ(x_t,t,t) or via an auxiliary prediction head. This improved reparameterization significantly reduces variance in calculating the Jacobian-vector product, leading to more stable optimization. x-prediction parameterization Beyond velocity prediction, an alternative parameterization is to directly model the clean data. As observed by Lu et al. (2026), predicting x1x_1 instead of the velocity field better preserves the data manifold and improves training stability. Therefore, they reparameterize the MeanFlow as uθ(xt,t,r)=xθ(xt,t,r)−xt1−t.u_θ(x_t,t,r)= x_θ(x_t,t,r)-x_t1-t. Substituting this into Equation 5 and Equation 6, we obtain the training objective for MeanFlow under the x-prediction parameterization. This formulation further improves empirical stability. 3 Proposed Method Table 1: Conceptual and mathematical comparison between MeanFlow and the proposed Discrete Average Generator. Property MeanFlow Discrete Average Generator State Space Continuous domain (ℝdR^d) Discrete space (S^D) via CTMCs Instantaneous Measure Velocity field v(xt,t)v(x_t,t) Transition rate generator Qt(x,z)Q_t(x,z) Average Measure Average Velocity: u(xt,t,r)=1r−t∫trv(xs,s)su(x_t,t,r)= 1r-t _t^rv(x_s,s)ds Average Generator: Ut,r=Pt→r−Ir−tU_t,r= P_t→ r-Ir-t Material Derivative Spatial gradient used: dtu=∂tu+v⋅∇xu ddtu= _tu+v· _xu Dynkin’s formula used: DtU=∂tU+QtU DDtU= _tU+Q_tU Fixed-Point Identity u=v−(t−r)(∂tu+v⋅∇xu)u=v-(t-r)( _tu+v· _xu) Ut,r=Qt−(t−r)(∂tUt,r+QtUt,r)U_t,r=Q_t-(t-r)( _tU_t,r+Q_tU_t,r) Stable Target Parameterization x-prediction (predicting clean data x1x_1) p~ p-prediction (predicting a per-coordinate distribution p~t,r(⋅|xt) p_t,r(·|x_t)) 3.1 Average Generator In continuous MeanFlow, acceleration is enabled by integrating the instantaneous velocity field, where a single Euler step of size r−tr-t reproduces the endpoint of the flow exactly. We define the analogue CTMC generator for discrete spaces. Let Pt→rP_t→ r denote the transition kernel of the CTMC from time t to r, i.e., Pt→r(x,z)=pr|t(z∣x)P_t→ r(x,z)=p_r|t(z x). We define the average generator over [t,r][t,r] as Ut,r≜Pt→r−Ir−t,andUt,t≜Qt,U_t,r P_t→ r-Ir-t, U_t,t Q_t, (7) so that a single Euler step is exact, δx⊤(I+(r−t)Ut,r)=pr|t(⋅|x) _x (I+(r-t)U_t,r )=p_r|t(·|x) for every x. By Proposition 2 in Appendix C.2, Ut,rU_t,r is itself a valid generator (nonnegative off-diagonal entries, zero row sums) and satisfies Ut,r→QtU_t,r→ Q_t as r→tr→ t. We derive the following identity for the transition rate in a discrete state space, with the proof provided in Appendix C.2. Proposition 1 (Average Generator Identity). For a sufficiently smooth generator QtQ_t, the average generator satisfies: Ut,r=Qt−(t−r)(∂tUt,r+QtUt,r).U_t,r=Q_t-(t-r) ( _tU_t,r+Q_tU_t,r ). (8) Equation 8 expresses the average generator as a self-consistent correction of the instantaneous generator. We call QtUt,rQ_tU_t,r the transport term. We compare the proposed framework with its continuous counterpart in Table 1. This identity forms the basis for a fixed-point training objective described in the following section. 3.2 Training Objective of Discrete Average Generator U-prediction objective. Equation 8 motivates directly regressing Ut,rθU_t,r^θ against the stop-gradient target constructed from the right-hand side, where sg(⋅)sg(·) denotes the stop-gradient operator. With U^t,rθ(xt,zd)≜Qt(xt,zd)−(t−r)(∂tUt,rθ(xt,zd)+(QtUt,rθ)(xt,zd)) U^θ_t,r(x_t,z^d) Q_t(x_t,z^d)-(t-r) ( _tU^θ_t,r(x_t,z^d)+(Q_tU^θ_t,r)(x_t,z^d) ), the U-prediction objective is ℒU(θ)=x1,(t,r),xt∼pt|1(xt∣x1)[∑d=1∑zd≠xtdDF(sg(U^t,rθ(xt,zd))∥Ut,rθ(xt,zd))],L_U(θ)=E_x_1,(t,r),x_t p_t 1(x_t x_1) [ _d=1^D _z^d≠ x_t^dD_F\! (sg\! ( U^θ_t,r(x_t,z^d) )\; \|\;U_t,r^θ(x_t,z^d) ) ], (9) with (t,r)(t,r), 0≤t≤r≤10≤ t≤ r≤ 1, drawn from the design of Appendix D.2. However, analogously to the continuous setting (Geng et al., 2025a), the target in Equation 9 depends on Ut,rθU_t,r^θ itself, introducing a self-referential bias that can destabilize training; moreover, unlike Ut,r(xt,zd)U_t,r(x_t,z^d), it need not be nonnegative, as DFD_F requires. Q-prediction objective. To resolve this issue, we follow the reparameterization strategy of Geng et al. (2025b) and express QtQ_t directly in terms of Ut,rθU_t,r^θ via the Average Generator Identity: Qt,rθ(xt,zd)≜Ut,rθ(xt,zd)+sg((t−r)(∂tUt,rθ(xt,zd)+(QtUt,rθ)(xt,zd))).Q_t,r^θ(x_t,z^d) U_t,r^θ(x_t,z^d)+sg\! ((t-r) ( _tU_t,r^θ(x_t,z^d)+ (Q_t\,U_t,r^θ )(x_t,z^d) ) ). (10) Instead of regressing Ut,rθU_t,r^θ directly, we train by regressing the reparameterized instantaneous generator Qt,rθQ_t,r^θ against the conditional rate Qt(xt,zd|x1)Q_t(x_t,z^d|x_1): ℒQ(θ)=x1,(t,r),xt∼pt|1(xt∣x1)[∑d=1∑zd≠xtdDF(Qt(xt,zd|x1)∥Qt,rθ(xt,zd))].L_Q(θ)=E_x_1,(t,r),x_t p_t 1(x_t x_1) [ _d=1^D _z^d≠ x_t^dD_F\! (Q_t(x_t,z^d|x_1)\; \|\;Q_t,r^θ(x_t,z^d) ) ]. (11) This eliminates the self-referential dependence in the target, which in the continuous case yields markedly more stable optimization (Geng et al., 2025b). p~ p-prediction objective. Analogously to the x-prediction parameterization in the continuous setting (Lu et al., 2026), we further reparameterize Ut,rθU^θ_t,r through a distribution p~t,r(⋅|x) p_t,r(·|x) over S for each coordinate, state x and pair (t,r)(t,r), whose dependence on θ we leave implicit. Starting from Qtd(x,zd)=∑x1dp1|td(x1d|x)Qtd(xd,zd|x1d)=λtp1|t(zd|x)Q_t^d(x,z^d)= _x_1^dp_1|t^d(x_1^d|x)\,Q_t^d(x^d,z^d|x_1^d)= _t\,p_1|t(z^d|x) for zd≠xdz^d≠ x^d and modeling Ut,rθU^θ_t,r as a generator with the same single-coordinate structure, we set Ut,rθ(x,z)=μt,rp~t,r(zd|x),x =z ,zd≠xd,μt,r≜κr−κt(r−t)(1−κt),U_t,r^θ(x,z)= _t,r\, p_t,r(z^d|x), x d=z d,\ z^d≠ x^d, _t,r _r- _t(r-t)(1- _t), (12) where κr _r denotes the schedule of Equation 3 evaluated at r. The scale μt,r _t,r makes the one-step kernel of a coordinate, δxd+(r−t)Ut,rθ(x,⋅) _x^d+(r-t)U^θ_t,r(x,·), equal to (1−ωt,r)δxd+ωt,rp~t,rd(⋅|x)(1- _t,r) _x^d+ _t,r p^d_t,r(·|x) with ωt,r≜(r−t)μt,r=κr−κt1−κt∈[0,1] _t,r (r-t) _t,r= _r- _t1- _t∈[0,1], the probability that a coordinate jumps to x1dx_1^d on [t,r][t,r] under the conditional path. It satisfies μt,r→λt _t,r→ _t as r→tr→ t, so the boundary condition Ut,tθ=QtU^θ_t,t=Q_t holds whenever p~t,t=p1|t p_t,t=p_1|t. Directly substituting Equation 12 into Equation 10 formally gives an objective in p~t,r p_t,r, but its transport term (QtUt,rθ)(xt,zd)(Q_tU^θ_t,r)(x_t,z^d) poses several obstacles. In Section 3.3, we first project the transport term onto per-coordinate marginals and adopt a closed-form expression and a one-pass unbiased estimator to reduce the computational costs. 3.3 Tractable Training Objective of Discrete Average Generator Under the coordinate-wise construction Equation 2, Qt(x,y)=0Q_t(x,y)=0 unless x and y differ in exactly one coordinate, so QtQ_t is determined by the ×||D×|S| numbers λtp1|t(zd|x) _t\,p_1|t(z^d|x) per state, which we obtain from a frozen pretrained denoiser, and Ut,rθU^θ_t,r has the same structure by Equation 12. We therefore model p~t,r p_t,r with a network HθH^θ that maps (x,t,r)(x,t,r) to D distributions over S, p~t,r(zd|x)=[Hθ(x,t,r)]d,zd p_t,r(z^d|x)=[H^θ(x,t,r)]_d,z^d, so that a single forward pass gives every entry of Ut,rθU^θ_t,r at the current state. Per-coordinate projection. Let xd→sx^d→ s denote the state x with its d-th coordinate replaced by s. For each coordinate d, the per-coordinate projection Πd ^d is the linear map that marginalizes the target state of a matrix onto its d-th coordinate: it takes a matrix M indexed by pairs of states (x,y)∈×(x,y) ^D×S^D and returns the matrix Πd[M] ^d[M] indexed by (x,zd)∈×(x,z^d) ^D×S, Πd[M](x,zd)≜∑y∈:yd=zdM(x,y),zd≠xd, ^d[M](x,z^d)\; \!\! _y ^D:\;y^d=z^d\!\!M(x,y), z^d≠ x^d, (13) i.e. Πd[M]=MEd ^d[M]=ME_d with Ed(y,zd)≜[yd=zd]E_d(y,z^d) 1[y^d=z^d], which sums the row M(x,⋅)M(x,·) over all target states whose d-th coordinate equals zdz^d, whatever the other coordinates are. For a transition kernel, Πd[M](x,⋅) ^d[M](x,·) is the law of the d-th coordinate after the transition. For a generator, Πd[M](x,zd) ^d[M](x,z^d) is the total rate at which coordinate d moves to zdz^d. For a generator that changes only one coordinate at a time, the d-th coordinate transition is given by a single matrix entry, Πd[Qt](x,zd)=Qt(x,xd→zd)=λtp1|t(zd|x) ^d[Q_t](x,z^d)=Q_t(x,x^d→ z^d)= _t\,p_1|t(z^d|x) and Πd[Ut,rθ](x,zd)=μt,rp~t,r(zd|x) ^d[U^θ_t,r](x,z^d)= _t,r p_t,r(z^d|x). For the exact average generator, however, the d-th coordinate transition is given by the d-th marginal of the jump kernel: Πd[Ut,r](x,zd)=pr|td(zd|x)/(r−t) ^d[U_t,r](x,z^d)=p^d_r|t(z^d|x)/(r-t), which is exactly the quantity required by a coordinate-factorized sampler. In contrast, the entry Ut,r(x,xd→zd)U_t,r(x,x^d→ z^d) is proportional to the probability that coordinate d jumps to zdz^d while all other coordinates remain unchanged, with proportionality factor 1/(r−t)1/(r-t). Applying Πd ^d to Equation 8 gives, for every d and zd≠xdz^d≠ x^d, the projected identity underlying Section 3.2, Πd[Ut,r](x,zd) ^d[U_t,r](x,z^d) =Πd[Qt](x,zd)−(t−r)(∂tΠd[Ut,r](x,zd)+Πd[QtUt,r](x,zd)) = ^d[Q_t](x,z^d)-(t-r) ( _t ^d[U_t,r](x,z^d)+ ^d[Q_tU_t,r](x,z^d) ) (14) =Πd[Qt](x,zd)−(t−r)(∂tΠd[Ut,r](x,zd)+∑x′Qt(x,x′)Πd[Ut,r](x′,zd)). = ^d[Q_t](x,z^d)-(t-r) ( _t ^d[U_t,r](x,z^d)+ _x Q_t(x,x )\, ^d[U_t,r](x ,z^d) ). Here we use that Πd ^d is linear and commutes with ∂t _t, and that Πd[QtM](x,zd)=∑x′Qt(x,x′)Πd[M](x′,zd) ^d[Q_tM](x,z^d)= _x Q_t(x,x )\, ^d[M](x ,z^d). The right-hand side involves Ut,rU_t,r only through Πd[Ut,r] ^d[U_t,r] at x and at its single-coordinate neighbors. The projection therefore gives the transport term a well-defined per-coordinate entry and absorbs the two-coordinate moves by summing over them. We call Πd[QtUt,r] ^d[Q_tU_t,r] the projected transport term. Tractable projected transport term. The projected identity Equation 14 is a closed equation for the per-coordinate marginals: the unique solution of Equation 14 is the exact marginal Πd[Ut,r] ^d[U_t,r], which is proved in Proposition 5. A sufficiently expressive model trained with Equation 11 therefore learns the per-coordinate marginals of the finite-step kernel, using only the instantaneous rate and quantities at the current state, without simulating the kernel over [t,r][t,r]. It remains to evaluate the projected transport term for the parameterization Equation 12. We show in Proposition 4 that Πd[QtUt,rθ](xt,zd)=λtμt,r[Cθd(xt,zd)−p1|t(zd∣xt)], ^d [Q_t\,U_t,r^θ ](x_t,z^d)= _t\, _t,r [\,C^d_θ(x_t,z^d)-p_1|t(z^d x_t) ], (15) where Cθd(xt,zd)≜∑e=1(s∼pe1|t(⋅∣xt)[p~t,r(zd∣xte→s)]−p~t,r(zd∣xt))C^d_θ(x_t,z^d)\; \; _e=1^D (E_s p^e_1 t(· x_t) [ p_t,r (z^d x_t^e→ s ) ]- p_t,r (z^d x_t ) ) (16) measures how the prediction at coordinate d moves when a single coordinate of xtx_t jumps according to QtQ_t. Each summand requires evaluating the network at a state that differs from xtx_t in a single coordinate, producing an output of size ×||D×|S|. Thus, evaluating the sum exactly requires one forward pass for each such state, motivating the estimator introduced below. Substituting Equation 15 and ∂tμt,r=μt,r−λtr−t+λtμt,r _t _t,r= _t,r- _tr-t+ _t _t,r into Equation 10, the purely local terms combine and the reparameterized instantaneous rate becomes Qt,rθ(xt,zd)=λtp~t,r(zd∣xt)+sg(t−r)μt,r[∂tp~t,r(zd∣xt)+λtΓθd(xt,zd)],Q_t,r^θ(x_t,z^d)= _t\, p_t,r(z^d x_t)+sg \(t-r)\, _t,r [ _t p_t,r(z^d x_t)+ _t\, ^d_θ(x_t,z^d) ] \, (17) with Γθd(xt,zd)≜Cθd(xt,zd)+p~t,r(zd∣xt)−p1|t(zd∣xt). ^d_θ(x_t,z^d)\; \;C^d_θ(x_t,z^d)+ p_t,r(z^d x_t)-p_1|t(z^d x_t). (18) Writing Equation 17 with λtp~t,r _t p_t,r outside the stop-gradient rescales the gradient at each (t,r)(t,r) by λt/μt,r _t/ _t,r relative to Equation 10, which leaves the fixed point unchanged. For the linear schedule κt=t _t=t the two coincide, since μt,r=λt _t,r= _t. The correction therefore consists of exactly two terms: the partial time derivative of the network at fixed r and λtΓθd _t ^d_θ, which contains the projected transport term: by Equation 15 and Equation 18, λtμt,rΓθd=Πd[QtUt,rθ](xt,zd)+λtμt,rp~t,r(zd∣xt) _t _t,r ^d_θ= ^d[Q_tU^θ_t,r](x_t,z^d)+ _t _t,r p_t,r(z^d x_t), where the last term comes from ∂tμt,r _t _t,r. At r=tr=t both vanish and Equation 17 reduces to the instantaneous rate Qt,tθ(xt,zd)=λtp~t,t(zd∣xt)Q^θ_t,t(x_t,z^d)= _t\, p_t,t(z^d x_t), which equals Qt(xt,zd)Q_t(x_t,z^d) when p~t,t=p1|t p_t,t=p_1|t, so the objective is a strict generalization of Equation 4. CθdC^d_θ is estimated without bias by a single forward pass shared across all coordinates: draw e∼Unif(1,…,)e (\1,…,D\) and s∼p1|te(⋅|xt)s p^e_1|t(·|x_t), evaluate p~t,r(⋅∣xte→s) p_t,r(· x_t^e→ s) once, and set C^θd(xt,zd)=(p~t,r(zd∣xte→s)−p~t,r(zd∣xt)). C^d_θ(x_t,z^d)=D\, ( p_t,r (z^d x_t^e→ s )- p_t,r (z^d x_t ) ). (19) Every summand of CθdC^d_θ, as well as p~t,r−p1|t p_t,r-p_1|t and ∂tp~t,r _t p_t,r sums to zero over zdz^d. Therefore, the shifted probability q^θd(zd|xt)≜p~t,r(zd|xt)+sg(t−r)μt,rλt[∂tp~t,r(zd|xt)+λtΓθd(xt,zd)], q^d_θ(z^d|x_t) p_t,r(z^d|x_t)+sg\ (t-r) _t,r _t[ _t p_t,r(z^d|x_t)+ _t ^d_θ(x_t,z^d)]\, (20) defined for all zd∈z^d , satisfies ∑zd∈q^θd(zd∣xt)=1 _z^d q^d_θ(z^d x_t)=1. Moreover, for zd≠xtdz^d≠ x_t^d, q^θd(zd|xt)=Qt,rθ(xt,zd)/λt q^d_θ(z^d|x_t)=Q^θ_t,r(x_t,z^d)/ _t. Expanding DFD_F in Equation 11 (Appendix C.7), its linear part becomes λt(1−q^θd(xtd|xt)) _t(1- q^d_θ(x_t^d|x_t)), and the objective takes the form of Equation 4 with the posterior replaced by the shifted probability q^θd q^d_θ, ℒp~(θ)=x1,(t,r),xt[λt∑d=1−(1−δx1d(xtd))logq^θd(x1d|xt)−q^θd(xtd|xt)]+const.L_ p(θ)=E_x_1,(t,r),x_t [ _t _d=1^D \- (1- _x_1^d(x_t^d) ) q^d_θ(x_1^d|x_t)- q^d_θ(x_t^d|x_t) \ ]+const. (21) Algorithm 1 Training of Discrete Average Generator 1: Network HθH^θ, schedule κt _t, data distribution p1p_1, frozen denoiser H, boundary fraction 1−ρ1-ρ 2: repeat 3: Sample x1∼p1x_1 p_1, (t,r)(t,r) with t≤rt≤ r from the logit-normal; set r←tr← t with probability 1−ρ1-ρ 4: Sample xt∼pt|1(⋅∣x1)x_t p_t|1(· x_1) ⊳ mixture path 5: Compute p~t,r(⋅∣xt)←Hθ(xt,t,r) p_t,r(· x_t)← H^θ(x_t,t,r), ∂tp~t,r(zd∣xt) _t p_t,r(z^d x_t) at fixed r ⊳ ×||D×|S| 6: Compute p1|t(⋅∣xt)←H(xt,t)p_1|t(· x_t)← H(x_t,t) ⊳ frozen, stop-gradient 7: Sample e∼Unif(1,…,)e (\1,…,D\), s∼p1|te(⋅∣xt)s p_1|t^e(· x_t); evaluate p~t,r(⋅∣xte→s) p_t,r(· x_t^e→ s) 8: Compute C^θd C^d_θ via Equation 19 and Γθd ^d_θ via Equation 18 9: Compute the shifted probability q^θd q^d_θ with stop-gradient via Equation 20 10: Compute ℒp~(θ)L_ p(θ) via Equation 21 11: Update θ←θ−η∇θℒp~(θ)θ←θ-η _θL_ p(θ) 12: until converged Algorithm 1 summarizes training algorithm. The instantaneous posterior p1|tp_1|t is supplied by a frozen pretrained denoiser, which is also the model the network is initialized from; the boundary condition Ut,tθ=QtU^θ_t,t=Q_t gives the same quantity when no pretrained model is available, since a fraction 1−ρ1-ρ of the training pairs are drawn with r=tr=t. Sampling. Sampling takes one Euler step of the learned per-coordinate average generator Πd[Ut,rθ] ^d[U^θ_t,r] independently for every coordinate per interval. On a grid 0=τ0<⋯<τK=10= _0<…< _K=1, each step queries the network once at (xt,t,r)=(xτk,τk,τk+1)(x_t,t,r)=(x_ _k, _k, _k+1) and, independently for every coordinate, replaces it by a draw from p~t,rd(⋅∣xt) p^d_t,r(· x_t) with probability ωt,r=(r−t)μt,r=κr−κt1−κt _t,r=(r-t) _t,r= _r- _t1- _t. The detailed sampling procedure is given in Algorithm 2. Figure 1: Exact total variation between the target and the law of the K-step sampler on the Potts simulation. Values in Table 2. Figure 2: Experimental results on the ImageNet dataset for image generation. Values in Table 4. 4 Experimental Results 4.1 Experimental Setup Datasets. We evaluate our method on simulated data and two standard benchmarks: OpenWebText (Gokaslan and Cohen, 2019) for text generation and ImageNet (Deng et al., 2009) for image generation. For the simulations, we use a Potts target on S^D with ||==4|S|=D=4, a state space small enough to be enumerated exactly. This allows us to evaluate the sampler against the exact ground truth in terms of total variation distance. Details of the data generation procedure are provided in Appendix D.1. For text generation, we use sequences of length 1024, tokenized with the GPT-2 tokenizer (Radford et al., 2019), which has a vocabulary size of 50,257. For image generation, following prior work (Chang et al., 2022; Hayakawa et al., 2025), we tokenize 256×256256× 256 images into 16×1616× 16 discrete latent codes using a pretrained VQGAN (Esser et al., 2020), resulting in sequences of length 256 with a vocabulary size of 1,024. Baselines. We compare with SDTT (Deschenaux and Gulcehre, 2025) and Di4C (Hayakawa et al., 2025), which distill a multi-step teacher and, for Di4C, model the correlations between dimensions, and with ReDi (Yoo et al., 2025), which iteratively rectifies the source–data coupling (ReDik: k rounds; Di4C+ReDi1 combines both). Our method does not distill a student from a teacher trajectory. Implementation. On OpenWebText, we train the model initialized from Di4C+ReDi1 with Equation 11 under the parameterization Equation 34 and the estimator of Equation 35 (Appendix C.6) for 1k iterations. On ImageNet, we train the model initialized from ReDi2 with Equation 31 and the single-draw estimator Equation 30 (Appendix C.4) for 3 epochs. Further details can be found in Appendix D.2. Metrics. For simulated data, we report the total variation distance between the generated distribution and the ground-truth distribution. For text generation, we measure generative perplexity using LLaMA 3.1-8B (Grattafiori et al., 2024) and report entropy to assess diversity and degeneration. For image generation, we generate 50k samples and report Fréchet Inception Distance (FID) (Heusel et al., 2017) and Inception Score (IS) (Salimans et al., 2016). 4.2 Experimental Results Simulations Figure 2 compares a network trained with the baseline objective Equation 4 with one trained with Equation 21. The correction reduces the total variation for every number of sampling steps K, by 21%21\% at K=4K=4, 43%43\% at K=8K=8 and 67%67\% at K=16K=16, with the same ordering for all three seeds. Text Generation Figure 3 reports generative perplexity and entropy on OpenWebText. From K=8K=8 on, our method improves on the strongest baseline, Di4C+ReDi1: 27.5927.59 vs. 30.9230.92 at K=8K=8, 22.9222.92 vs. 24.8124.81 at K=16K=16, 21.3521.35 vs. 21.8821.88 at K=32K=32, and 16.9716.97 vs. 20.3220.32 at K=64K=64. Most importantly, our method achieves a lower generative perplexity with K=64K=64 than Di4C+ReDi1 with K=1024K=1024, demonstrating a 16×16× acceleration. Qualitative inspection of unconditional samples generated at K=64K=64 (Appendix E) further suggests that our method produces fluent, topically coherent text at this reduced step budget, without the repetitive degeneration that can mask a low generative perplexity. Figure 3: Generative perplexity and entropy on OpenWebText. The dashed horizontal line indicates the PPL of Di4C + ReDi1 at K=1024K=1024 (18.44). Values in Table 3. Image Generation Figure 2 compares the methods on ImageNet 256×256256× 256 with four network evaluations. Our method attains the highest Inception Score, 250250, 4%4\% above the next best (ReDi2, 240240), at a comparable FID (7.817.81 vs. 7.867.86). The ablation of classifier-free guidance can be found in Appendix E.1 and qualitative comparisons are provided in Appendix E.2. 5 Related Work In continuous spaces, sampling is accelerated by higher-order solvers (Song et al., 2020; Lu et al., 2022), caching (Ma et al., 2024), distillation (Salimans and Ho, 2022; Zhou et al., 2024), consistency models (Song et al., 2023), rectified flow (Liu et al., 2022), and methods that learn the solution operator over an interval, including shortcut models (Frans et al., 2025), MeanFlow (Geng et al., 2025a; Geng et al., 2025b; Lu et al., 2026). For discrete diffusion, SDTT (Deschenaux and Gulcehre, 2025) and Di4C (Hayakawa et al., 2025) distill multi-step samplers, ReDi (Yoo et al., 2025) rectifies the coupling, Di[M]O (Zhu et al., 2025) distills a one-step generator by distribution matching, and Fast-dLLM (Wu et al., 2026) and Jacobi Forcing (Hu et al., 2026) accelerate inference. Closest to our work, Categorical Flow Maps (Roos et al., 2026) and Discrete Flow Maps (Potaptchik et al., 2026) learn flow maps of a continuous interpolant on the probability simplex. We instead stay on the discrete state space and learn the average generator of a CTMC, so that the transport term is left multiplication by the generator rather than a Jacobian–vector product along a drift, and the construction applies to any CTMC with coordinate-wise rates. 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Progressive distillation (Salimans and Ho, 2022) iteratively compresses the sampling trajectory, while score identity distillation (Zhou et al., 2024) and consistency models (Song et al., 2023) enforce consistency across different noise levels to enable one- or few-step generation. Rectified flow (Liu et al., 2022) provides another perspective by reformulating the generative process as a straightened flow, which naturally reduces the complexity of sampling. Shortcut models (Frans et al., 2025) and MeanFlow (Geng et al., 2025a) introduce tailored modifications to the flow formulation to achieve faster generation. In particular, MeanFlow defines an average velocity field over time intervals and derives a self-consistent fixed-point identity, which is the direct inspiration for our work. Improved MeanFlow (Geng et al., 2025b) further addresses training instability through a velocity-based reparameterization, and pixel MeanFlow (Lu et al., 2026) proposes an x-prediction parameterization that improves empirical stability by directly supervising the clean data prediction. Acceleration of Diffusion Language Models. Extending acceleration techniques to discrete diffusion models and diffusion language models has attracted increasing attention. SDTT (Deschenaux and Gulcehre, 2025) and Di4C (Hayakawa et al., 2025) accelerate discrete diffusion by distilling multi-step trajectories into fewer steps, where Di4C additionally exploits dimensional correlations to improve distillation quality. ReDi (Yoo et al., 2025) iteratively rectifies the coupling between source and target distributions to reduce the factorization error of discrete flow models, enabling efficient few-step generation. Di[M]O (Zhu et al., 2025) proposes to distill masked diffusion models into a one-step generator via token-level distribution matching in an on-policy framework, and introduces a token initialization strategy to inject stochasticity into the otherwise deterministic initial distribution. On the inference side, Fast-dLLM (Wu et al., 2026) accelerates diffusion LLM inference without additional training by introducing a block-wise approximate KV cache mechanism for bidirectional diffusion models and a confidence-aware parallel decoding strategy. Jacobi Forcing (Hu et al., 2026) takes a complementary approach by converting autoregressive LLMs into causal parallel decoders through progressive distillation on the model’s own Jacobi decoding trajectories, preserving the causal backbone and KV cache compatibility while achieving significant wall-clock speedup. In contrast to these approaches, our method directly learns an average generator for CTMCs without multi-step distillation, and addresses the discrete analogue of the MeanFlow fixed-point identity at the level of the transition rate operator. Appendix B Theoretical Details for Section 2 B.1 Proof of Training Objective for Discrete Flow Matching Proof. We show how the Bregman divergence-based training objective [∑z≠xtDF(Qt(xt,z|x1)∥Qt(xt,z))],F(x)=xlogx,E [ _z≠ x_tD_F (Q_t(x_t,z|x_1)\,\|\,Q_t(x_t,z) ) ], F(x)=x x, reduces to a cross-entropy loss with respect to the posterior p1|tp_1|t when using the coordinate-wise mixture path. Step 0: Off-diagonal restriction. The generator Qt(x,z|x1)Q_t(x,z|x_1) contains a negative diagonal term −δx(z)- _x(z) and is therefore a signed measure. Since the objective only sums over z≠xz≠ x, we restrict both conditional and marginal generators to their off-diagonal parts: Qtoff(x,z|x1):=Qt(x,z|x1)z≠x.Q_t^off(x,z|x_1):=Q_t(x,z|x_1)1_z≠ x. Under this restriction, all quantities are non-negative and the Bregman divergence is well-defined. Step 1: Bregman divergence. For F(x)=xlogxF(x)=x x, the Bregman divergence is DF(a∥b)=alogab−a+b.D_F(a\|b)=a ab-a+b. Step 2: Conditional and oracle rates. From the coordinate-wise mixture path, for a single coordinate d, Qcond(zd)=Qtd(xtd,zd|x1d)zd≠xtd=λtx1d≠xtdδx1d(zd),Q_cond(z^d)=Q_t^d(x_t^d,z^d|x_1^d)1_z^d≠ x_t^d= _t1_\x_1^d≠ x_t^d\\, _x_1^d(z^d), where we used that the original generator contains an additional diagonal term −δxtd(zd)- _x_t^d(z^d), which vanishes after restricting to zd≠xtdz^d≠ x_t^d. The corresponding oracle rate is Qoracle(zd)=Qtd(xtd,zd)=λtp1|td(zd|xt),zd≠xtd.Q_oracle(z^d)=Q_t^d(x_t^d,z^d)= _t\,p_1|t^d(z^d|x_t), z^d≠ x_t^d. Step 3: Support of the conditional rate. The conditional rate has support only at zd=x1dz^d=x_1^d when x1d≠xtdx_1^d≠ x_t^d, and is identically zero otherwise. Hence the analysis splits into two cases. Step 4: KL term at the target jump (x1d≠xtdx_1^d≠ x_t^d). For zd=x1dz^d=x_1^d: Qcond(x1d)logQcond(x1d)Qoracle(x1d)=λtlogλtλtp1|td(x1d|xt)=λt(−logp1|td(x1d|xt)).Q_cond(x_1^d) Q_cond(x_1^d)Q_oracle(x_1^d)= _t _t _t\,p_1|t^d(x_1^d|x_t)= _t\, (- p_1|t^d(x_1^d|x_t) ). Step 5: Linear terms. The remaining terms in DFD_F give −Qcond(x1d)+∑zd≠xtdQoracle(zd)=−λt+λt(1−p1|td(xtd|xt)).-Q_cond(x_1^d)+ _z^d≠ x_t^dQ_oracle(z^d)=- _t+ _t (1-p_1|t^d(x_t^d|x_t) ). Step 6: Case x1d=xtdx_1^d=x_t^d. In this case, Qcond(zd)=0Q_cond(z^d)=0 for all zd≠xtdz^d≠ x_t^d. Hence DF(0∥Qoracle(zd))=Qoracle(zd),D_F(0\|Q_oracle(z^d))=Q_oracle(z^d), and summing over zd≠xtdz^d≠ x_t^d yields λt(1−p1|td(xtd|xt)). _t(1-p_1|t^d(x_t^d|x_t)). Step 7: Combine terms. Combining both cases using the indicator δx1d(xtd) _x_1^d(x_t^d), the Bregman divergence reduces to λt−(1−δx1d(xtd))logp1|td(x1d|xt)+δx1d(xtd)−p1|td(xtd|xt). _t \-(1- _x_1^d(x_t^d)) p_1|t^d(x_1^d|x_t)+ _x_1^d(x_t^d)-p_1|t^d(x_t^d|x_t) \. Summing over all coordinates d=1,…,d=1,…,D, we obtain exactly the training objective [λt∑d=1−(1−δx1d(xtd))logp1|td(x1d|xt)+δx1d(xtd)−p1|td(xtd|xt)].E [ _t _d=1^D \-(1- _x_1^d(x_t^d)) p_1|t^d(x_1^d|x_t)+ _x_1^d(x_t^d)-p_1|t^d(x_t^d|x_t) \ ]. This completes the proof. ∎ B.2 Proof of Proposition 3.1 of Campbell et al. (2024) We include the proof of Proposition 3.1 of Campbell et al. (2024) here for completeness. Let Qt(x,z|x1)Q_t(x,z|x_1) denote the conditional transition rate from state x to z that generates the conditional probability path pt|1(⋅|x1)p_t|1(·|x_1), i.e., it satisfies the Kolmogorov forward equation ∂tpt|1(x|x1)=∑zQt(z,x|x1)pt|1(z|x1). _tp_t|1(x|x_1)= _zQ_t(z,x|x_1)\,p_t|1(z|x_1). Taking expectation over x1∼p1x_1 p_1, we obtain ∂tpt(x) _tp_t(x) =x1∼p1(x1)[∂tpt|1(x|x1)](definition of pt) =E_x_1 p_1(x_1) [ _tp_t|1(x|x_1) ] (definition of $p_t$) =x1∼p1(x1)[∑zQt(z,x|x1)pt|1(z|x1)] =E_x_1 p_1(x_1) [ _zQ_t(z,x|x_1)\,p_t|1(z|x_1) ] =∑z∑x1p1(x1)pt|1(z|x1)Qt(z,x|x1) = _z _x_1p_1(x_1)\,p_t|1(z|x_1)\,Q_t(z,x|x_1) =∑z∑x1pt(z)p1|t(x1|z)Qt(z,x|x1)(Bayes’ rule) = _z _x_1p_t(z)\,p_1|t(x_1|z)\,Q_t(z,x|x_1) (Bayes' rule) =∑zx1∼p1|t(⋅|z)[Qt(z,x|x1)]pt(z). = _zE_x_1 p_1|t(·|z) [Q_t(z,x|x_1) ]\,p_t(z). Comparing with the Kolmogorov forward equation ∂tpt(x)=∑zQt(z,x)pt(z), _tp_t(x)= _zQ_t(z,x)\,p_t(z), we identify the marginal transition rate as Qt(z,x)=x1∼p1|t(⋅|z)[Qt(z,x|x1)].Q_t(z,x)=E_x_1 p_1|t(·|z) [Q_t(z,x|x_1) ]. Equivalently, reparameterizing the indices to express the transition rate from x to z, we obtain Qt(x,z)=x1∼p1|t(⋅|x)[Qt(x,z|x1)].Q_t(x,z)=E_x_1 p_1|t(·|x) [Q_t(x,z|x_1) ]. This shows that Qt(x,z)Q_t(x,z) generates the marginal probability path pt(x)p_t(x). B.3 Proof of the MeanFlow Identity The MeanFlow identity was originally proved by Geng et al. (2025a). We reproduce the proof here for completeness. Let xtx_t follow the flow defined by dtxt=v(xt,t) ddtx_t=v(x_t,t). By definition, the MeanFlow velocity is u(xt,t,r)=1r−t∫trv(xs,s)s.u(x_t,t,r)= 1r-t _t^rv(x_s,s)\,ds. Multiplying both sides by (r−t)(r-t) gives (r−t)u(xt,t,r)=∫trv(xs,s)s.(r-t)\,u(x_t,t,r)= _t^rv(x_s,s)\,ds. Taking the total derivative with respect to t and applying the fundamental theorem of calculus yields dt[(t−r)u(xt,t,r)]=v(xt,t). ddt [(t-r)\,u(x_t,t,r) ]=v(x_t,t). Expanding the left-hand side using the product rule, we obtain (t−r)dtu(xt,t,r)+u(xt,t,r)=v(xt,t).(t-r)\, ddtu(x_t,t,r)+u(x_t,t,r)=v(x_t,t). Rearranging terms leads to the MeanFlow identity u(xt,t,r)=v(xt,t)−(t−r)dtu(xt,t,r).u(x_t,t,r)=v(x_t,t)-(t-r)\, ddtu(x_t,t,r). Here, the total derivative is given by dtu(xt,t,r)=∂tu(xt,t,r)+v(xt,t)⋅∇xtu(xt,t,r). ddtu(x_t,t,r)= _tu(x_t,t,r)+v(x_t,t)· _x_tu(x_t,t,r). Appendix C Theoretical Details C.1 Notation and the material derivative of a CTMC Throughout this appendix, xd→sx^d→ s denotes the state obtained from x by replacing its d-th coordinate with s, matrices are indexed by states and act on the right on row-vectors of probabilities, and Pt→rP_t→ r denotes the propagator of the CTMC, Pt→r(x,y)=pr|t(y|x)P_t→ r(x,y)=p_r|t(y|x), which is the unique solution of ∂rPt→r=Pt→rQr⏟forward,∂tPt→r=−QtPt→r⏟backward,Pt→t=I. _rP_t→ r=P_t→ r\,Q_r_forward, _tP_t→ r=-\,Q_t\,P_t→ r_backward, P_t→ t=I. (22) For each coordinate e define the row-stochastic matrix PteP^e_t that resamples coordinate e from the posterior, Pte(x,xe→s)≜p1|te(s|x)P^e_t(x,x^e→ s) p^e_1|t(s|x) for all s∈s (the diagonal s=xes=x^e included) and Pte(x,y)=0P^e_t(x,y)=0 otherwise. The coordinate-wise marginal generator of Section 3 is then exactly Qt=λt∑e=1(Pte−I),Ut,rθ=μt,r∑d=1(P~t,rd,θ−I),Q_t= _t _e=1^D (P^e_t-I ), U^θ_t,r= _t,r _d=1^D ( P^d,θ_t,r-I ), (23) with P~t,rd,θ(x,xd→s)=p~t,rd(s|x) P^d,θ_t,r(x,x^d→ s)= p^d_t,r(s|x). Indeed, for zd≠xdz^d≠ x^d the (x,xd→zd)(x,x^d→ z^d) entry of Equation 23 is λtp1|t(zd|x) _tp_1|t(z^d|x), the diagonal is λt∑e(p1|te(xe|x)−1) _t _e(p^e_1|t(x^e|x)-1), and all entries at Hamming distance ≥2≥ 2 vanish; in particular QtQ_t is single-site supported. MeanFlow’s identity is obtained by differentiating (t−r)u(xt,t,r)(t-r)u(x_t,t,r) along the flow, which produces the material derivative dt=∂t+v⋅∇x ddt= _t+v· _x. The correct discrete counterpart is supplied by Dynkin’s formula. Let (Xt)(X_t) be the CTMC with generator QtQ_t and let ftf_t be a time-dependent function on S^D, viewed as a column vector, which is C1C^1 in t. Then Mt≜ft(Xt)−∫0t(∂sfs+Qsfs)(Xs)sM_t\; \;f_t(X_t)- _0^t ( _sf_s+Q_sf_s )(X_s)\,ds is a martingale, and consequently dt[ft(Xt)]=[(∂tft+Qtft)(Xt)] ddtE[f_t(X_t)]=E[( _tf_t+Q_tf_t)(X_t)]. We therefore define the material derivative along the CTMC as Dt≜∂t+Qt⋅, DDt\; \; _t+Q_t\,·, (24) where Qt⋅Q_t\,· denotes left multiplication (applied columnwise for matrix-valued fields). This is the discrete analogue of v⋅∇xv· _x: the generator is the directional derivative along the process. C.2 Proof of Proposition 1 We first record the basic properties of Ut,r=Pt→r−Ir−tU_t,r= P_t→ r-Ir-t defined in Equation 7. Proposition 2 (Basic properties of the average generator). For 0≤t<r≤10≤ t<r≤ 1: (a) Ut,r(x,z)≥0U_t,r(x,z)≥ 0 for all z≠xz≠ x and ∑zUt,r(x,z)=0 _zU_t,r(x,z)=0, i.e. Ut,rU_t,r is itself a valid generator; (b) δx⊤(I+(r−t)Ut,r)=pr|t(⋅|x) _x (I+(r-t)U_t,r)=p_r|t(·|x) for every x, i.e. a single Euler step of size r−tr-t reproduces the exact transition law; (c) limr→tUt,r=Qt _r→ tU_t,r=Q_t. Proof. (a) Pt→rP_t→ r is row-stochastic, so its off-diagonal entries are nonnegative and its rows sum to one; subtracting I and dividing by r−t>0r-t>0 preserves nonnegativity off the diagonal and yields zero row sums. (b) Immediate from δx⊤Pt→r=pr|t(⋅|x) _x P_t→ r=p_r|t(·|x). (c) limr→tPt→r−Ir−t=∂rPt→r|r=t=Qt _r→ t P_t→ r-Ir-t= _rP_t→ r|_r=t=Q_t by Equation 22. ∎ Proof of Proposition 1. Let Ωt,r≜(t−r)Ut,r=I−Pt→r _t,r (t-r)\,U_t,r=I-P_t→ r. By the backward equation in Equation 22, ∂tΩt,r=−∂tPt→r=QtPt→r=Qt(I−Ωt,r)=Qt−QtΩt,r, _t _t,r=- _tP_t→ r=Q_t\,P_t→ r=Q_t (I- _t,r )=Q_t-Q_t\, _t,r, that is ∂tΩt,r+QtΩt,r=Qt _t _t,r+Q_t _t,r=Q_t, which is exactly Dt[(t−r)Ut,r]=Qt DDt[(t-r)U_t,r]=Q_t with the material derivative Equation 24. Expanding the left-hand side with the product rule, ∂tΩt,r=Ut,r+(t−r)∂tUt,r,QtΩt,r=(t−r)QtUt,r, _t _t,r=U_t,r+(t-r)\, _tU_t,r, Q_t _t,r=(t-r)\,Q_tU_t,r, and substituting gives Ut,r+(t−r)(∂tUt,r+QtUt,r)=QtU_t,r+(t-r)( _tU_t,r+Q_tU_t,r)=Q_t, i.e. Equation 8. The boundary case r=tr=t reduces to Ut,t=QtU_t,t=Q_t, consistent with Proposition 2(c). ∎ C.3 Marginal projection and the tractable projected transport term The exact Ut,rU_t,r of Equation 7 is not single-site supported (a finite-time propagator moves many coordinates at once), whereas the model Equation 23 is. We therefore enforce Equation 8 after projecting onto per-coordinate marginals with Πd ^d of Equation 13, which is precisely the information a coordinate-factorized sampler consumes. Proposition 3 (Properties of Πd ^d). Let zd≠xdz^d≠ x^d. Then (a) Πd ^d is linear and commutes with ∂t _t; (b) if M is single-site supported then Πd[M](x,zd)=M(x,xd→zd) ^d[M](x,z^d)=M(x,x^d→ z^d), in particular Πd[Qt](x,zd)=λtp1|t(zd|x) ^d[Q_t](x,z^d)= _tp_1|t(z^d|x) and Πd[Ut,rθ](x,zd)=μt,rp~t,r(zd|x) ^d[U^θ_t,r](x,z^d)= _t,r p_t,r(z^d|x); (c) Πd[I](x,zd)=0 ^d[I](x,z^d)=0 and Πd[Pte](x,zd)=δedp1|t(zd|x) ^d[P^e_t](x,z^d)= _ed\,p_1|t(z^d|x); (d) Πd[Ut,r](x,zd)=pr|td(zd|x)/(r−t) ^d[U_t,r](x,z^d)=p^d_r|t(z^d|x)/(r-t), so that matching Πd[Ut,rθ]=Πd[Ut,r] ^d[U^θ_t,r]= ^d[U_t,r] is exactly the condition that the coordinate-factorized one-step sampler reproduce the correct per-coordinate marginals of the true jump kernel. Proof. (a) is immediate. (b) The only y within Hamming distance one of x with yd=zd≠xdy^d=z^d≠ x^d is y=xd→zdy=x^d→ z^d. (c) Apply (b) to I (supported on y=xy=x, which has yd=xd≠zdy^d=x^d≠ z^d) and to PteP^e_t (supported on y=xe→sy=x^e→ s, which requires e=de=d and s=zds=z^d). (d) Follows from Equation 7 and Equation 13. ∎ Applying Πd ^d to Equation 8 and using Proposition 3(a)–(b) gives the projected identity we train against: for every d and every zd≠xdz^d≠ x^d, Πd[Ut,r](x,zd)=λtp1|t(zd|x)−(t−r)(∂tΠd[Ut,r](x,zd)+Πd[QtUt,r](x,zd)). ^d [U_t,r ](x,z^d)= _t\,p_1|t(z^d|x)-(t-r) ( _t\, ^d [U_t,r ](x,z^d)+ ^d [Q_tU_t,r ](x,z^d) ). (25) Only the projected transport term couples different coordinates. This is not because Πd ^d fails to commute with QtQ_t: since Πd[M]=MEd ^d[M]=ME_d acts on the target state, it commutes with left multiplication, Πd[QtM]=QtΠd[M] ^d[Q_tM]=Q_t\, ^d[M], as made explicit in Equation 26. Rather, QtQ_t acts on the source state, so QtΠd[Ut,r]Q_t\, ^d[U_t,r] evaluates Πd[Ut,r] ^d[U_t,r] at the states xe→sx^e→ s reached by a jump of any coordinate e, not only of d. The next proposition evaluates it in closed-form for the single-site model, which is what makes the objective tractable and proves Equation 15. Proposition 4 (Tractable projected transport term). Let QtQ_t and Ut,rθU^θ_t,r be as in Equation 23. Then for every d and every zd≠xdz^d≠ x^d, Πd[QtUt,rθ](x,zd)=λtμt,r[Cθd(x,zd)−p1|t(zd∣x)], ^d [Q_tU^θ_t,r ](x,z^d)= _t\, _t,r [\,C^d_θ(x,z^d)-p_1|t(z^d x) ], with CθdC^d_θ as in Equation 16. Proof. Expanding the product of the two expressions in Equation 23, QtUt,rθ=λtμt,r∑e=1∑d′=1(PeP~d′−Pe−P~d′+I),Q_tU^θ_t,r= _t _t,r _e=1^D _d =1^D (P^e P^d -P^e- P^d +I ), where we drop the sub/superscripts t,r,θt,r,θ for readability. Fix d and zd≠xdz^d≠ x^d and apply Πd ^d term by term. Step 1. By Proposition 3(c), Πd[I]=0 ^d[I]=0, ∑e,d′Πd[Pe](x,zd)=p1|t(zd|x) _e,d ^d[P^e](x,z^d)=D\,p_1|t(z^d|x) and ∑e,d′Πd[P~d′](x,zd)=p~t,r(zd|x) _e,d ^d[ P^d ](x,z^d)=D\, p_t,r(z^d|x), each sum contributing a factor D from the free index. Step 2. For the products, write (PeP~d′)(x,y)=∑sPe(x,xe→s)P~d′(xe→s,y)(P^e P^d )(x,y)= _sP^e(x,x^e→ s)\, P^d (x^e→ s,y), so that y must be of the form y=xe→s,d′→s′y=x^e→ s,\,d → s . Requiring yd=zd≠xdy^d=z^d≠ x^d forces d∈e,d′d∈\e,d \ with the corresponding new symbol equal to zdz^d. Two cases: (i) d′=d =d: then s′=zds =z^d and the remaining index s is free and summed over by Πd ^d, giving, for every e (including e=de=d), ∑sp1|te(s|x)p~t,r(zd|xe→s)=s∼pe1|t(⋅|x)[p~t,r(zd|xe→s)] _sp^e_1|t(s|x) p_t,r(z^d|x^e→ s)=E_s p^e_1|t(·|x)[ p_t,r(z^d|x^e→ s)]. (i) d′≠d ≠ d: then coordinate d can only be set by the first factor, so e=de=d and s=zds=z^d; the index s′s is free and summed by Πd ^d, and since P~d′ P^d is row-stochastic, ∑s′p~t,r(s′|xd→zd)=1 _s p_t,r(s |x^d→ z^d)=1. Each of the −1D-1 values of d′d contributes p1|t(zd|x)p_1|t(z^d|x). Summing, ∑e,d′Πd[PeP~d′](x,zd)=∑es∼pe1|t(⋅|x)[p~t,r(zd|xe→s)]+(−1)p1|t(zd|x). _e,d ^d [P^e P^d ](x,z^d)= _eE_s p^e_1|t(·|x) [ p_t,r(z^d|x^e→ s) ]+(D-1)\,p_1|t(z^d|x). Step 3. Combining Steps 1 and 2, Πd[QtUθ](x,zd)=λtμt,r[∑es∼pe1|t(⋅|x)[p~t,r(zd|xe→s)]−p~t,r(zd|x)−p1|t(zd|x)], ^d [Q_tU^θ ](x,z^d)= _t _t,r [ _eE_s p^e_1|t(·|x) [ p_t,r(z^d|x^e→ s) ]-D\, p_t,r(z^d|x)-p_1|t(z^d|x) ], which is the claim after grouping the D copies of p~t,r(zd|x) p_t,r(z^d|x) into the D summands of Equation 16. ∎ What the objective learns. The projected identity Equation 25 is closed in the per-coordinate marginals. For a matrix M with zero row sums, extend Πd[M](x,⋅) ^d[M](x,·) to zd=xdz^d=x^d by Πd[M](x,xd)≜∑y:yd=xdM(x,y)=−∑s≠xdΠd[M](x,s) ^d[M](x,x^d) _y:\,y^d=x^dM(x,y)=- _s≠ x^d ^d[M](x,s). Exchanging the sums in Equation 13 gives Πd[QtM](x,zd)=∑y:yd=zd∑x′Qt(x,x′)M(x′,y)=∑x′Qt(x,x′)Πd[M](x′,zd), ^d [Q_tM ](x,z^d)= _y:\,y^d=z^d _x Q_t(x,x )\,M(x ,y)= _x Q_t(x,x )\, ^d[M](x ,z^d), (26) so the projected transport term depends on M only through its off-diagonal per-coordinate marginals at the states x′x reachable from x in one jump; Equation 35 is Equation 26 written out for a single-site M. In particular Πd[QtUt,rθ]=Πd[QtUt,r] ^d[Q_tU^θ_t,r]= ^d[Q_tU_t,r] whenever Πd[Ut,rθ]=Πd[Ut,r] ^d[U^θ_t,r]= ^d[U_t,r] for all d and all states, so replacing Ut,rU_t,r by the single-site model in the projected transport term is not an approximation. Proposition 5 (Fixed point of the projected identity). Fix r and assume that t↦Qtt Q_t is continuous on [t0,r][t_0,r]. Let Vt(x,zd)V_t(x,z^d), zd≠xdz^d≠ x^d, be per-coordinate rates that are C1C^1 in t and bounded on [t0,r)[t_0,r), extended to zd=xdz^d=x^d by zero row sums, and suppose that for all t∈[t0,r)t∈[t_0,r), x, d and zd≠xdz^d≠ x^d, Vt(x,zd)=Πd[Qt](x,zd)−(t−r)(∂tVt(x,zd)+∑x′Qt(x,x′)Vt(x′,zd)).V_t(x,z^d)= ^d[Q_t](x,z^d)-(t-r) ( _tV_t(x,z^d)+ _x Q_t(x,x )\,V_t(x ,z^d) ). (27) Then Vt=Πd[Ut,r]V_t= ^d[U_t,r], i.e. Vt(x,zd)=pr|td(zd|x)/(r−t)V_t(x,z^d)=p^d_r|t(z^d|x)/(r-t). Proof. By Equation 26, Equation 25 states that Πd[Ut,r] ^d[U_t,r], which is bounded by Proposition 2(c), satisfies Equation 27. For uniqueness, let Wt≜(t−r)VtW_t (t-r)V_t. Since ∂tWt=Vt+(t−r)∂tVt _tW_t=V_t+(t-r) _tV_t, Equation 27 is equivalent to the linear system ∂tWt=Πd[Qt]−tWt _tW_t= ^d[Q_t]-Q_tW_t with (tW)(x,zd)≜∑x′Qt(x,x′)W(x′,zd)(Q_tW)(x,z^d) _x Q_t(x,x )W(x ,z^d), in the finitely many off-diagonal entries of WtW_t (the diagonal ones being fixed by zero row sums). Boundedness of VtV_t gives Wt→0W_t→ 0 as t→rt→ r. A linear ODE with continuous coefficients has a unique solution with a prescribed terminal value, so WtW_t, and hence VtV_t, is unique. ∎ At the population optimum of Equation 11, a model that can match the marginal rate satisfies Qt,rθ(xt,zd)=Πd[Qt](xt,zd)Q^θ_t,r(x_t,z^d)= ^d[Q_t](x_t,z^d), the minimizer of the Bregman divergence to the conditional rate. By Equation 10 and Equation 26 this is Equation 27 for Vt=Πd[Ut,rθ]V_t= ^d[U^θ_t,r], so Πd[Ut,rθ]=Πd[Ut,r] ^d[U^θ_t,r]= ^d[U_t,r]: the objective learns the per-coordinate marginals of the exact finite-step kernel. Regressing Πd[Ut,rθ] ^d[U^θ_t,r] directly onto Πd[Ut,r] ^d[U_t,r] has the same solution, but requires samples of the kernel over [t,r][t,r], i.e. multi-step trajectories of a teacher; the projected identity only requires the instantaneous rate and single-coordinate perturbations of the current state. C.4 The masked path For the masked source p0d=δp_0^d= _m, with m the mask token and ℳ(x)M(x) the set of masked coordinates of x, the posterior and the network satisfy the carry-over property p1|te(⋅|x)=p~t,re(⋅|x)=δxep^e_1|t(·|x)= p^e_t,r(·|x)= _x^e for e∉ℳ(x)e (x), and neither predicts m. If e∉ℳ(x)e (x) then xe→s=x^e→ s=x almost surely and the e-th summand of Equation 16 vanishes; if e=d∈ℳ(x)e=d (x), carry-over gives p~t,rd(⋅|xd→s)=δs p^d_t,r(·|x^d→ s)= _s and s[δs(zd)]=p1|t(zd|x)E_s[ _s(z^d)]=p_1|t(z^d|x). Hence Cθd(xt,zd)=(p1|t(zd|xt)−p~t,r(zd|xt))+Ccrossd(xt,zd),C^d_θ(x_t,z^d)= (p_1|t(z^d|x_t)- p_t,r(z^d|x_t) )+C^d_cross(x_t,z^d), (28) with Ccrossd(xt,zd)≜∑e∈ℳ(xt)∖d(s∼pe1|t(⋅|xt)[p~t,r(zd|xte→s)]−p~t,r(zd|xt))C^d_cross(x_t,z^d) _e (x_t) \d\ (E_s p^e_1|t(·|x_t)[ p_t,r(z^d|x_t^e→ s)]- p_t,r(z^d|x_t) ), so that Γθd ^d_θ of Equation 18, which contains the projected transport term, reduces to CcrossdC^d_cross and Equation 17 becomes Qt,rθ(xt,zd)=λtp~t,r(zd∣xt)+sg(t−r)μt,r[∂tp~t,r(zd∣xt)+λtCcrossd(xt,zd)].Q_t,r^θ(x_t,z^d)= _t\, p_t,r(z^d x_t)+sg \(t-r)\, _t,r [ _t p_t,r(z^d x_t)+ _t\,C^d_cross(x_t,z^d) ] \. (29) Since unmasked coordinates contribute nothing, CcrossdC^d_cross is estimated with a lower variance than Equation 19 by drawing e∼Unif(ℳ(xt))e (M(x_t)) and s∼p1|te(⋅|xt)s p^e_1|t(·|x_t) and setting C^crossd(xt,zd)=|ℳ(xt)|⋅[e≠d]⋅(p~t,r(zd∣xte→s)−p~t,r(zd∣xt)). C^d_cross(x_t,z^d)= |M(x_t) |· 1[e≠ d]· ( p_t,r (z^d x_t^e→ s )- p_t,r (z^d x_t ) ). (30) For an unmasked d the shifted probability is q^θd=δxtd q^d_θ= _x_t^d, and for a masked d it satisfies q^θd(|xt)=0 q^d_θ(m|x_t)=0, so Equation 21 reduces to a weighted cross-entropy over the masked coordinates, ℒp~(θ)=x1,(t,r),xt[λt∑d∈ℳ(xt)−log(p~t,r(x1d∣xt)+sg(t−r)μt,rλt[∂tp~t,r(x1d∣xt)+λtCcrossd(xt,x1d)])]+const.L_ p(θ)=E_x_1,(t,r),x_t [ _t\!\! _d (x_t)\!\!- ( p_t,r(x_1^d x_t)+sg \ (t-r) _t,r _t [ _t p_t,r(x_1^d x_t)+ _t\,C^d_cross(x_t,x_1^d) ] \ ) ]+const. (31) This is the objective we use on ImageNet. C.5 Sampling with the Discrete Average Generator Algorithm 2 Sampling with the Discrete Average Generator 1: Network HθH^θ, schedule κt _t, step count K, grid 0=τ0<τ1<⋯<τK=10= _0< _1<·s< _K=1 2: xτ0∼p0x_ _0 p_0 ⊳ uniform noise; all-mask state for the masked source 3: for k=0,…,K−1k=0,…,K-1 do 4: t←τkt← _k, r←τk+1r← _k+1 ⊳ a single jump of size r−tr-t 5: p~t,r(⋅∣xt)←Hθ(xt,t,r) p_t,r(· x_t)← H^θ(x_t,t,r) ⊳ one NFE, ×||D×|S| 6: ωt,r←(r−t)μt,r=κr−κt1−κt _t,r←(r-t)\, _t,r= _r- _t1- _t ⊳ ωt,r∈[0,1] _t,r∈[0,1] by monotonicity of κ 7: for each coordinate d=1,…,d=1,…,D independently do 8: xrd←zd∼p~t,rd(⋅∣xt)x_r^d← z^d p^d_t,r(· x_t) with probability ωt,r _t,r, else xrd←xtdx_r^d← x_t^d ⊳ masked source: unmasked xtdx_t^d carried over 9: end for 10: end for 11: return xτKx_ _K Sampling. Algorithm 2 takes, independently for every coordinate, one Euler step of the learned per-coordinate average generator; its one-step kernel is the product Kt,rθ(xt,y)=∏d=1[δxtd(yd)+(r−t)Πd[Ut,rθ](xt,yd)],K^θ_t,r(x_t,y)\;=\; _d=1^D [ _x_t^d(y^d)+(r-t)\, ^d [U^θ_t,r ](x_t,y^d) ], (32) whose d-th factor is a distribution over S, its entry at yd=xtdy^d=x_t^d being fixed by the zero row sum of Πd[Ut,rθ] ^d[U^θ_t,r]. This is not the full Euler step δxt⊤(I+(r−t)Ut,rθ) _x_t (I+(r-t)U^θ_t,r): since Ut,rθU^θ_t,r is single-site supported, the latter moves at most one coordinate and, for large ωt,r _t,r, has a negative diagonal entry. In Equation 32 the per-coordinate jump probability is (r−t)Πd[Ut,rθ](xt,zd)=(r−t)μt,rp~t,r(zd|xt)=ωt,rp~t,r(zd|xt)(r-t)\, ^d[U^θ_t,r](x_t,z^d)=(r-t)\, _t,r\, p_t,r(z^d|x_t)= _t,r\, p_t,r(z^d|x_t). Three properties follow directly from Section 3 and are worth making explicit. (i) Exact marginals. By Proposition 3(d), if Πd[Ut,rθ]=Πd[Ut,r] ^d[U^θ_t,r]= ^d[U_t,r] for every d, then the d-th factor of Equation 32 is exactly the d-th marginal of pr|t(⋅|xt)p_r|t(·|x_t), for any step size: no discretisation error is incurred by the per-coordinate jump. The remaining error of a single step is the dependence between the coordinates under pr|t(⋅|xt)p_r|t(·|x_t), which a product kernel cannot represent. Proposition 5 shows that these exact marginals are the population target of our objective. (i) Train/test consistency. The factor μt,r _t,r appearing in the sampler is the same one used to parameterise Ut,rθU^θ_t,r during training, so the jump probability ωt,r _t,r is by construction the quantity the training objective was written for. Had we instead kept the instantaneous scale, Uθ=λtp~1|tU^θ= _t p_1|t, the sampler would use (r−t)λt(r-t) _t while the conditional path jumps with probability κr−κt1−κt _r- _t1- _t; these agree only for the linear schedule κt=t _t=t, and diverge as O((r−t)2κ¨t)O((r-t)^2 κ_t) otherwise. Carrying μt,r _t,r explicitly removes this mismatch for any schedule. (i) Where the learning shows up. The product in Equation 32 is over coordinates, so a single step cannot reproduce the correlations of the true kernel Pt→rP_t→ r. What the objective can improve is therefore the per-coordinate marginals. We do not claim a guarantee on how much exact marginals reduce the error of a K-step sampler; we measure this exactly in Section 4.2. Setting r=tr=t makes ωt,r→0 _t,r→ 0 and recovers the instantaneous rate, so Algorithm 2 degenerates gracefully to the standard ancestral sampler as K grows. Finally, the network must be queried with the same (t,r)(t,r) conditioning it saw in training: a K-step run with a uniform grid only ever presents step sizes r−t=1/Kr-t=1/K, so the training distribution over (t,r)(t,r) has to place mass on the step sizes that will actually be used at test time. C.6 The uniform-state (DUO) process Everything above is stated for the mixture path of Equation 2, whose conditional rate Qtd(xd,zd|x1d)=λt(δx1d(zd)−δxd(zd))Q_t^d(x^d,z^d|x_1^d)= _t( _x_1^d(z^d)- _x^d(z^d)) does not depend on the source p0p_0. For a uniform source this is the CTMC in which a single noise sample x0x_0 is drawn once and each coordinate either still shows x0x_0 or has already switched to x1x_1. Uniform-state models in the DUO family (Sahoo et al., 2025) use a different CTMC with the same marginals pt|1p_t|1: the noise is re-randomised at every step, as in D3PM. This appendix records the parameterization we use on top of the uniform-state checkpoints of our text experiments. Throughout this subsection we follow the convention of DUO, in which αt _t decreases with t, t=0t=0 is data, a step goes from t to r<tr<t, and c′≜(1−αt)/||c (1- _t)/|S|. Write αt|s=αt/αs _t|s= _t/ _s. Conditional and marginal rates. The conditional rate is obtained by differentiating the bridge q(xs|xt,x1)∝[αt|sδxs(xt)+1−αt|s||][αsδx1(xs)+1−αs||]q(x_s|x_t,x_1) [ _t|s _x_s(x_t)+ 1- _t|s|S| ] [ _s _x_1(x_s)+ 1- _s|S| ] at s=ts=t: Qt(xt,z|x1)=−α˙tαt||⋅αt[z=x1]+c′αt[x1=xt]+c′,z≠xt.Q_t(x_t,z\,|\,x_1)= - α_t _t|S|· _t 1[z=x_1]+c _t 1[x_1=x_t]+c , z≠ x_t. (33) Averaging Equation 33 over x1∼p1|t(⋅|xt)x_1 p_1|t(·|x_t) — which by Proposition 3.1 of Campbell et al. (2024) is the marginal generator — splits on x1∈z,xt,otherx_1∈\z,x_t,other\ and yields the closed-form −α˙tαt||[pzαt+c′c′+pxtc′αt+c′+(1−pz−pxt)] - α_t _t|S| [p_z _t+c c +p_x_t c _t+c +(1-p_z-p_x_t) ]. Unlike on the mixture path, this rate is not proportional to the posterior mass pzp_z of z, so the parameterization Ut,rθ=μt,rp~t,rU^θ_t,r= _t,r p_t,r does not carry over. Parameterisation. We take Ut,rθU^θ_t,r from the exact marginalization of the bridge, Πd[Ut,rθ](xt,zd)=1t−rv∼p~dt,r(⋅|xt)[q(xrd=zd|xtd,x1d=v)],zd≠xtd, ^d [U^θ_t,r ](x_t,z^d)= 1t-r\,E_v p^d_t,r(·|x_t) [q(x_r^d=z^d\,|\,x_t^d,\,x_1^d=v) ], z^d≠ x_t^d, (34) which, because the bridge is a ratio of two functions affine in the one-hot encoding of v whose denominator takes only two values, has a closed-form costing O(||)O(|S|) per coordinate. The sampler draws each coordinate from the same kernel, so a single per-coordinate Euler step of Equation 34 is exactly the sampler step. The projected transport term in general form. Proposition 4 is a special case of an identity that holds for any pair of single-site matrices. Let qe(x,⋅)q^e(x,·) and ud(x,⋅)u^d(x,·) denote the coordinate-e row of QtQ_t and the coordinate-d row of Ut,rθU^θ_t,r, both with zero row sum. Then Πd[QtUt,rθ](x,zd) ^d [Q_tU^θ_t,r ](x,z^d) =∑e=1∑sqe(x,s)ud(xe→s,zd) = _e=1^D _sq^e(x,s)\,u^d (x^e→ s,z^d ) =∑e=1Λe(x)(s∼νe[ud(xe→s,zd)]−ud(x,zd)), = _e=1^D ^e(x) (E_s ν^e [u^d(x^e→ s,z^d) ]-u^d(x,z^d) ), (35) with Λe(x)=∑s≠xeqe(x,s) ^e(x)= _s≠ x^eq^e(x,s) the total outflow rate at coordinate e and νe=qe/Λeν^e=q^e/ ^e the normalised jump law. The terms with d′≠d ≠ d in the expansion vanish because ud′u^d has zero row sum. Substituting the mixture-path rows recovers λtμt,r(Cθd−p1|t) _t _t,r(C^d_θ-p_1|t) of Equation 15; with the uniform-state rows, one draw e∼Unif(1,…,)e (\1,…,D\), s∼νes ν^e gives the unbiased estimate Λe(x)(ud(xe→s,zd)−ud(x,zd))D\, ^e(x) (u^d(x^e→ s,z^d)-u^d(x,z^d) ) at the cost of a single forward pass. The objective is the Bregman divergence Equation 11 between Equation 33 and the reparameterized rate, summed over all zd≠xtdz^d≠ x_t^d. Sampling. The sampler draws each coordinate from the kernel Equation 34, i.e. it takes one per-coordinate Euler step of the learned average generator. Two conditioning details matter for a model trained this way. First, the final noise-removal step sets αr=1 _r=1, i.e. r=0r=0, so its effective step size is the whole of t rather than the grid spacing; passing no step size at all there queries a conditioning path the network never sees in training. Second, a K-step run on a uniform grid only ever presents t−r=1/Kt-r=1/K, so the training distribution over (t,r)(t,r) must place mass there; the logit-normal design over (t,r)(t,r) used in our experiments places mass on the step sizes 1/K1/K of all budgets K≤64K≤ 64 that we evaluate. Finally, nucleus filtering and softmax temperature must be left at their neutral settings: both reshape the kernel after the fact and would break the calibration that Equation 34 relies on. C.7 Proof of Training Objective for Discrete Average Generator Proof. We show how the objective ℒ(θ)=[∑d=1∑zd≠xtdDF(Qt(xt,zd|x1)∥Πd[Ut,rθ](xt,zd)+sg((t−r)(∂tΠd[Ut,rθ](xt,zd)+Πd[QtUt,rθ](xt,zd))))],L(θ)=E [ _d=1^D _z^d≠ x_t^dD_F\! (Q_t(x_t,z^d|x_1)\; \|\; ^d [U_t,r^θ ](x_t,z^d)+sg\! ((t-r) ( _t ^d [U_t,r^θ ](x_t,z^d)+ ^d [Q_t\,U_t,r^θ ](x_t,z^d) ) ) ) ], with Πd[Ut,rθ](xt,zd)=μt,rp~t,r(zd|xt) ^d[U_t,r^θ](x_t,z^d)= _t,r\, p_t,r(z^d|x_t), reduces to Equation 36 and then to Equation 21. Step 1: Bregman divergence. For F(x)=xlogxF(x)=x x, the Bregman divergence is DF(a∥b)=alogab−a+b.D_F(a\|b)=a ab-a+b. Step 2: Conditional and target rates. For a single coordinate d, restricting to off-diagonal entries zd≠xtdz^d≠ x_t^d, the conditional rate is Qcond(zd)=Qtd(xtd,zd|x1d) 1zd≠xtd=λt 1x1d≠xtdδx1d(zd).Q_cond(z^d)=Q_t^d(x_t^d,z^d|x_1^d)\,1_z^d≠ x_t^d= _t\,1_\x_1^d≠ x_t^d\\, _x_1^d(z^d). The target rate is Qθ(zd)=Πd[Ut,rθ](xt,zd)+At(xt,zd)=μt,rp~t,r(zd|xt)+At(xt,zd),Q_θ(z^d)= ^d [U_t,r^θ ](x_t,z^d)+A_t(x_t,z^d)= _t,r\, p_t,r(z^d|x_t)+A_t(x_t,z^d), where At(xt,zd)≜sg((t−r)(∂tΠd[Ut,rθ](xt,zd)+Πd[QtUt,rθ](xt,zd))).A_t(x_t,z^d) \! ((t-r) ( _t ^d [U_t,r^θ ](x_t,z^d)+ ^d [Q_t\,U_t,r^θ ](x_t,z^d) ) ). Step 3: Support of the conditional rate. The conditional rate has support only at zd=x1dz^d=x_1^d when x1d≠xtdx_1^d≠ x_t^d, and is identically zero otherwise. The analysis splits into two cases. Step 4: Log term (x1d≠xtdx_1^d≠ x_t^d). At zd=x1dz^d=x_1^d: Qcond(x1d)logQcond(x1d)Qθ(x1d)=λtlogλt−λtlogQθ(x1d),Q_cond(x_1^d) Q_cond(x_1^d)Q_θ(x_1^d)= _t _t- _t Q_θ(x_1^d), whose first term does not depend on θ. Step 5: Linear terms (x1d≠xtdx_1^d≠ x_t^d). The linear terms from DFD_F at zd=x1dz^d=x_1^d are −Qcond(x1d)+Qθ(x1d)=−λt+Qθ(x1d).-Q_cond(x_1^d)+Q_θ(x_1^d)=- _t+Q_θ(x_1^d). For all zd≠x1dz^d≠ x_1^d (and zd≠xtdz^d≠ x_t^d), Qcond(zd)=0Q_cond(z^d)=0, so DF(0∥Qθ(zd))=Qθ(zd)D_F(0\|Q_θ(z^d))=Q_θ(z^d). Summing over all zd≠xtdz^d≠ x_t^d gives ∑zd≠xtdQθ(zd) _z^d≠ x_t^dQ_θ(z^d) =μt,r(1−p~t,r(xtd|xt))+∑zd≠xtdAt(xt,zd). = _t,r (1- p_t,r(x_t^d|x_t) )+ _z^d≠ x_t^dA_t(x_t,z^d). The last term cannot in general be rewritten as −At(xt,xtd)-A_t(x_t,x_t^d): that step would require ∑zdAt(xt,zd)=0 _z^dA_t(x_t,z^d)=0, whereas ∑zd∂tΠd[Ut,rθ](xt,zd)=∂tμt,r≠0 _z^d _t ^d[U^θ_t,r](x_t,z^d)= _t _t,r≠ 0. We therefore keep the linear contribution as ∑zd≠xtdQθ(zd) _z^d≠ x_t^dQ_θ(z^d). For the rate Equation 17 with a normalized p~t,r p_t,r it takes a simple form: since ∑zdp~t,r(zd|xt)=1 _z^d p_t,r(z^d|x_t)=1 for all t implies ∑zd∂tp~t,r(zd|xt)=0 _z^d _t p_t,r(z^d|x_t)=0, and each summand of CθdC^d_θ, as well as p~t,r−p1|t p_t,r-p_1|t, is a difference of two normalized distributions, the correction sums to zero over zd∈z^d and ∑zd≠xtdQθ(zd)=λt∑zd≠xtdq^θd(zd|xt)=λt(1−q^θd(xtd|xt)), _z^d≠ x_t^dQ_θ(z^d)= _t _z^d≠ x_t^d q^d_θ(z^d|x_t)= _t (1- q^d_θ(x_t^d|x_t) ), with q^θd q^d_θ the shifted probability of Section 3.3. On the masked path, q^θd(|xt)=0 q^d_θ(m|x_t)=0 for masked d, so this is the constant λt _t. Combining the linear contributions in this case gives −Qcond(x1d)+∑zd≠xtdQθ(zd)=−λt+∑zd≠xtdQθ(zd),-Q_cond(x_1^d)+ _z^d≠ x_t^dQ_θ(z^d)=- _t+ _z^d≠ x_t^dQ_θ(z^d), where the leading −λt- _t is independent of θ and is absorbed into the additive constant of the loss. Step 6: Case x1d=xtdx_1^d=x_t^d. In this case Qcond(zd)=0Q_cond(z^d)=0 for all zd≠xtdz^d≠ x_t^d, so DF(0∥Qθ(zd))=Qθ(zd)D_F(0\|Q_θ(z^d))=Q_θ(z^d) and the total contribution is ∑zd≠xtdQθ(zd) _z^d≠ x_t^dQ_θ(z^d), i.e. the same linear expression as in Step 5 with the logarithmic term absent and without the −λt- _t offset. Step 7: Combine terms. Combining both cases using the indicator δx1d(xtd) _x_1^d(x_t^d), dropping the θ-independent terms (the aloga a part of DFD_F and the offset −λt(1−δx1d(xtd))- _t(1- _x_1^d(x_t^d))), and summing over all coordinates d=1,…,d=1,…,D, we obtain ℒQ(θ)=x1,(t,r),xt[∑d=1−(1−δx1d(xtd))λtlogQt,rθ(xt,x1d)+∑zd≠xtdQt,rθ(xt,zd)]+const,L_Q(θ)=E_x_1,(t,r),x_t [ _d=1^D \- (1- _x_1^d(x_t^d) )\, _t\, Q_t,r^θ(x_t,x_1^d)+ _z^d≠ x_t^dQ_t,r^θ(x_t,z^d) \ ]+const, (36) where Qt,rθ(xt,zd)=Qθ(zd)Q_t,r^θ(x_t,z^d)=Q_θ(z^d) is the reparameterized instantaneous rate of Equation 10. Substituting Equation 17, writing logQt,rθ(xt,x1d)=logλt+logq^θd(x1d|xt) Q^θ_t,r(x_t,x_1^d)= _t+ q^d_θ(x_1^d|x_t) and using the linear term above yields Equation 21, and on the masked path Equation 31. This completes the proof. ∎ Appendix D Experimental Details D.1 Experimental Setup of Simulations Data distribution We consider a data distribution in which every pair of coordinates is coupled, i.e. we take the Potts law q(x)∝exp(β∑1≤i<j≤[xi=xj]+εx),εx∼0.3(0,1),q(x)\; \; (β\!\! _1≤ i<j \!\! 1 [x^i=x^j ]\;+\; _x ), _x 0.3\,N(0,1), where β≥0β≥ 0 controls how strongly the coordinates are coupled, and the perturbations εx _x are drawn once and then held fixed so as to break the permutation symmetry of the energy. We use β=1.5β=1.5 and dimension =4D=4. The vocabulary size is 44. Experimental setup and implementation details. The state space has ||=256|S|^D=256 elements, so the target q, the posterior p1|tp_1|t, the projected transport term Equation 15 and the law of the K-step sampler are all obtained by enumeration, i.e., no quantity we report is estimated from samples. We parameterize the network as MLPs with SiLU activation functions, consisting of four hidden layers with hidden dimension 256, conditioned on Fourier features of t, r, r−tr-t and −log(1−t)- (1-t). We first train the base denoiser with Equation 4 using the Adam optimizer with a learning rate of 2e-3 for 1.2e4 steps, and then train only the correction head with Equation 21 for 3e3 steps with the base frozen, so that every objective we compare shares an identical base denoiser. The pairs (t,r)(t,r) are drawn from the evaluation grids, with 25%25\% of the draws at r=tr=t; the instantaneous posterior p1|tep^e_1|t is supplied by the frozen base denoiser. In the sampling stage, we evaluate the trained network at every state and propagate the law of the sampler exactly, for a number of steps K∈2,3,4,6,8,12,16,24,32K∈\2,3,4,6,8,12,16,24,32\, and record the total variation between q and that law. We run each setting with 3 seeds and report the mean. D.2 Implementation Details of OpenWebText and ImageNet The network is conditioned on t and on the step size r−tr-t. On OpenWebText, τt _t and τr _r are the larger and the smaller of two logit-normal draws σ((Pmean,Pstd2))σ(N(P_mean,P^2_std)) with (Pmean,Pstd)=(0,1.5)(P_mean,P_std)=(0,1.5) and (−1.0,1.5)(-1.0,1.5); on ImageNet, τt _t is the arccos mask ratio and τr _r is a logit-normal draw with (−3.0,2.0)(-3.0,2.0) clipped to τr≤τt _r≤ _t, which places more mass on the large steps of the 11–44 step regime. A fraction 1−ρ=0.251-ρ=0.25 of the pairs is drawn with r=tr=t, where the correction vanishes and the objective reduces to Equation 4; this is what keeps the boundary calibrated. On ImageNet the sampling hyperparameters of every method, ours and the baselines, are selected by a sweep and we report the best configuration of each method; on OpenWebText the baseline numbers are taken from Yoo et al. (2025). D.3 Detailed values for Section 4 Detailed values for simulations, OpenWebText, and ImageNet can be found in Tables 2, 3, and 4, respectively. Table 2: Total variation Potts model simulations. Mean over three seeds. These are the values plotted in Figure 2. K 22 33 44 66 88 1212 1616 2424 3232 Baseline 0.60200.6020 0.37070.3707 0.24560.2456 0.13510.1351 0.09070.0907 0.05080.0508 0.03560.0356 0.02250.0225 0.01360.0136 Ours 0.53810.5381 0.31700.3170 0.19340.1934 0.09570.0957 0.05170.0517 0.01810.0181 0.01160.0116 0.00800.0080 0.00530.0053 Table 3: OpenWebText results for Figure 3: generative perplexity measured with LLaMA 3.1-8B and entropy, as a function of the number of sampling steps K. Baseline numbers are taken from Yoo et al. (2025). Best perplexity per column in bold. Sampling steps K Model 44 88 1616 3232 6464 Generative perplexity DUO+DCD 97.2297.22 72.1872.18 54.8254.82 46.0546.05 42.3842.38 ReDi1 79.3579.35 48.5348.53 37.1337.13 31.2131.21 29.1229.12 ReDi2 69.0169.01 45.1145.11 36.4236.42 31.5631.56 29.8529.85 ReDi3 53.2453.24 36.3336.33 30.3430.34 27.7527.75 26.7826.78 Di4C 55.4255.42 36.5236.52 27.6627.66 23.0423.04 21.5421.54 Di4C + ReDi1 47.5047.50 30.9230.92 24.8124.81 21.8821.88 20.3220.32 Ours 56.0856.08 27.5927.59 22.9222.92 21.3521.35 16.9716.97 Entropy DUO+DCD 4.824.82 5.255.25 5.355.35 5.365.36 5.355.35 ReDi1 5.435.43 5.495.49 5.485.48 5.445.44 5.425.42 ReDi2 5.435.43 5.445.44 5.445.44 5.415.41 5.395.39 ReDi3 5.335.33 5.305.30 5.285.28 5.265.26 5.255.25 Di4C 4.814.81 5.025.02 5.075.07 5.055.05 5.035.03 Di4C + ReDi1 4.964.96 5.075.07 5.085.08 5.095.09 5.075.07 Ours 4.964.96 5.065.06 5.095.09 5.185.18 5.025.02 Table 4: ImageNet 256×256256× 256 with four network evaluations. Every method is reported at the best sampling configuration found by a sweep. Best value per column in bold. Model FID ↓ IS ↑ MaskGIT (Chang et al., 2022) 10.90 184 SDTT (Deschenaux and Gulcehre, 2025) 8.97 205 Di4C (Hayakawa et al., 2025) 6.20 216 ReDi1 (Yoo et al., 2025) 7.58 228 ReDi2 (Yoo et al., 2025) 7.86 240 Ours 7.81 250 Appendix E Unconditional Text Generation Samples (K=64K=64) Below we provide unconditional 1,0241,024-token text samples generated by our OpenWebText model using the exact-marginalization sampler of Equation 34–Equation 32 at K=64K=64 sampling steps and temperature T=1.0T=1.0. Across the evaluation batch, this configuration achieves the generative perplexity and entropy reported for “Ours” at K=64K=64 in Table 3 (LLaMA-3.1-8B generative perplexity 16.9716.97, entropy 5.025.02). Sample 1: Energy Policy & Regulatory Hearing (K=64K=64, T=1.0T=1.0) <|endoftext|> statement that he was "deeply disappointed" that the company’s actions were "not in line with our public policy" during the regulatory process. "Everyone have a right to have concerns about energy security, and we do not believe that actions like this are in the vital national interest," Price said in the statement. Pamper E. Price said that the firm paid roughly $14.1 million to lobbying groups in 2013, but a lawsuit filed by the Institute for Energy Policy Studies, the Washington-based Ceres’ think group, that it won’t be any more attention from the government. The company is expected to go before the House Committee on Commerce and Transportation before joining the House of Representatives this fall and hold a hearing. The White House said in a statement on Friday that it would continue to get the company’s representatives to investigate its activities. "We are very disappointed this firm played an important role in efforts to implement critical energy security in our region and look forward to continuing to improve the decision being made to the American people of our region," the company said in the statement. The firms ties to politics have raised questions about the effectiveness of the Obama administration’s political attacks on the industry, including the closure of a variety of plants in the United States and the hiring of about 1,200 employees from 35 individuals and nonprofit groups, including fossil fuel firms and environmental groups. The industry is heavily backed by special interests. One August, the firm’s actions were part of the efforts to block the construction of the Dakota Access pipeline through Minnesota, the project being planned up in North Dakota carrying coal and crude oil. The Trump administration said the actions were related a March 2015 rule, which temporarily halted efforts to save large sections of the proposed pipeline. Protesters repeatedly blocked a portion of the proposed pipeline. “We look forward to hearing before the House today whether a few of these actions are in line with our national policy,” Price said. “There is no evidence, in fact, that these actions involving Goldstone Perkins are politically motivated and are not the policies we should be guided by.”<|endoftext|> Sample 2: Public Health Study & International News (K=64K=64, T=1.0T=1.0) <|endoftext|> by reducing their daily intake of nutrients, such as fruits, vegetables, and nuts, could contribute to a large decrease in the number of displaced in China. The report, on the issue of food insecurity in China during the growing season, was authored by researchers from the Federal Institute of Public Affairs and the Department of Public Health of the University of Shanghai. The team, led by professors Zhou, U.S. Department of Agriculture, and Liu, also focused on the issue of the National Food Standards, which are mostly designed to combat the problem of hunger and food insecurity in China. A Professor of Public Health at the University, Wei Mao, who not involved as a senior author of the report, said that the administration decided not to enforce the National Standards "due to the misinformation or distortion of information that is currently available. Our results that excessively high-fat foods such as fruits, vegetables, can lead to high levels of blood sugar. These foods have increase the risk of heart disease and increase the risk of diabetes." The consumption of sugary drinks contributes to a high risk of food insecurity. Tens of thousand of the people currently without food in China is associated with the risk of diabetes mellitus which has nearly doubled in the past 25 years. And according to a UN monitor, the majority of the citizens of the world, Southeast Asia and East Asia, live on poor quality food conditions. Earlier this month, Shanghai Municipal Commission for Food Safety and Integrity, maker of the Standards, said that the government was carrying out an effort to improve the use of fresh vegetables in Beijing during the growing season. It’s critical to understand different varieties of vegetables offer, such as fresh varieties, to encourage people to eat more vegetables, according to the report. The authors also explained that the global problem of food insecurity is that the government’s largely neglected for decades, and which has hampered the efforts to combat the situation in China. As a result, they said, the need to reduce over-consumption and use of fresh food is a top priority in China.<|endoftext|> By James Mayr London - World News staff British military and intelligence agencies on Friday called on the international community to prosecute the extremist organisations and anti-government forces to defeat in the name of peace and stability. The calls were made by at least two thirds of the leading members of British military, for the restoration of peace talks between Turkey and the rebels in northern Syria, and for an end to the war in Syria, which has left 250,000 people fleeing the country from abroad.<|endoftext|> Sample 3: International Diplomacy & Economic Aid (K=64K=64, T=1.0T=1.0) <|endoftext|>United States has spoken out against what critics say is a poverty-ridden, expansionary and precarious economic situation in Egypt. Deputy Foreign Minister Rafa Mansour-Hahallah told reporters on Saturday the United States was “aware of a serious crisis with the country’s dire situation.” “The U.S. is actively in the process to overcome this humanitarian crisis,” she said, after releasing a report saying that the Egyptian economy is in a “grave state of stagnation in the country.” The report said that the focus was on strengthening the security of the people and distributing programs to boost growth to end financial hardships for those living below the poverty level. She added that the situation of country is in a difficult position that any growth in activity could jeopardizing its security. Total oil production was cut by 5.2 million people in 2009 to just 3.1 million, according to the report. To mitigate the effects of the cuts, the federal subsidy is to freeze until the end of March. The proposed cuts, which will be needed to remove 3.5 million from the budget, where the central government has reserves, are only temporary. However, according to the report, the important thing is to improve the quality of life in the United States and “to restore job security and life changing services through the government’s ambitious infrastructure plan,” administration officials said on Saturday. The plan is focused on creating jobs by 2011 and is meant to provide relief to the tens of millions of women, children and people living past the retirement age by bringing in full-time jobs. The State Department spokesman, Jay Carney, said that the federal government was focusing mainly to poverty-era programs and measures to help the affected people, most of whom are still facing job insecurity. “The government working hard on words, speaking by themselves.” Carney said, referring to comments published in The Washington Post. The U.S. government is determined to play a leading role in addressing this humanitarian situation, the official said. In addition to the call on the central government to take no further restrictions on oil and gas production, the United States however made it clear that the government would not place further restrictions. “The United States continues engaging with the people, and encouraging businesses to do as much as they can help restore the security of the people, as soon as now.” the United States Department said in a statement.<|endoftext|> Sample 4: Internet Openness, Privacy Law & Climate Study (K=64K=64, T=1.0T=1.0) <|endoftext|> on June 7, 2011, in the city of Seattle, the US and Russia agreed to collaborate on three projects proposed by Russia, with the goal of protecting personal information and then they stole this information from the United States for the past 15 years. The most pressing thing they have to do is put pressure on the US government to pay for these projects because their practices have changed, and the peasants affected by this pipeline project are now before the eyes of the world," he said<|endoftext|> Turkey In January 2015, a citizen of Iran is six months pregnant in Turkey allegedly find themselves attempting to collect personal information of the Turkish government during a meeting of parliament of Turkey March 7, 2015. The Iranian was attempting to collect information about patient behavior, health and safety, and national security, breaking Turkish privacy laws and attempting to co-operate with Turkish authorities. No criminal charge was committed. The fine imposed on the person was determined by international law. Germany The United States, the United Kingdom, and the Netherlands support a variety of scientific, technological and social programs which benefit the people and environment of the world, and serve to be the engine of the global economy. This support is supported by the National Institutes of Health Sciences, the National Institutes of Health, the National Institute of Science and Technology, the Medical School of the University of Maryland, the School of Medicine and Public Health, the Massachusetts Institute of Technology, Energy and Environment, and Harvard University’s School of Science, Space and Public Policy, and the Center for Environmental Science and Data Analysis at the University of Maryland. United States The United Nations Security Council reached an agreement on the openness of the Internet on July 11, 2011. One of the goals was to protecting national security and the openness of the Internet. On a global level, the Internet is probably one of the largest ecosystems in the world today. For example, the size of the web today is about 2.5 billion, the nature of the content has slightly through a variety of events, including melting Arctic sea ice. But over the course of the year, the surface extent of Arctic sea ice, increased steadily worldwide. A new study, shows glaciers now for about 55 percent of the world’s surface ice, and about one-to-third of it is sea ice. (Changes to address climate change adaptation were made by three international experts, from the Institute for Media, Policy and Media Research, and from the World Economic Forum-WHO.) In June 2011, under Barack Obama, the United States and Russia reached a series of agreement which is designed to promote greater online openness on the Internet. The elements of this agreement: an international commitment to Internet openness, every online form of communication, full Web-wide Internet access, formal reviews, and certifying Internet providers for computers, mobile phones, and other forms of information.<|endoftext|> Sample 5: Sports Journalism: NHL & Preseason Coverage (K=64K=64, T=1.0T=1.0) <|endoftext|>Yes. Muthyomun is an important member of the Chicago side which went through the very well and ended up in a third position in the W-L playoffs. The club had a favorable relationship with the Philadelphia Eagles and took over under the direction of Muthyomun in January. At the same time, the club now looking to reach at least 10th in W-L playoffs. Chicago head coach Antoine Bouchard is expected to visit the Chicago Blackhawks and make his preseason debut Wednesday, March 14 at the Chicago Blackhawks. In the past few months, he has been able to play as a key part of the lineup for the Chicago Blackhawks, according to the English-language website. Antoine Brouwer is a key part of the lineup for the Blackhawks and has already signed with the club most recently from the Anaheim Ducks. Anaheim had a great team last season to make the Stanley Cup Playoffs, and he is expected to be a part of the lineup for the first time since his injury in the Eastern Conference Playoffs. The Chicago Blackhawks had a great 2016 season and have a real chance at making another playoff run in the Eastern Conference. The club finished 23-16 in the regular season and was able to earn a third place spot in the WHLs first bid for the Stanley Cup by making a powerful shot that passed of his hand and his right foot bounced to the back of the net. Following hot start to the season he then scored 2 beautiful third and 2 goals in the third period against the Dallas Stars (4-2-1) and he was able to capitalize on it. He is considered one of the best players in the league and he has developed a good amount of confidence in his game. His right hand gives him good opportunities to make plays and boost his acceleration, for him to be able to hold on to his feet. While playing with his left foot down, he will have to make sure to find some movement at the end of the season with the Chicago Blackhawks. Brouwer returned to the Blackhawks after playing well in the American Hockey League. He was named the American Hockey League Most Outstanding Player of the Year and played a key part in the Chicago Blackhawks’ victory over the Los Angeles Kings in the playoffs.<|endoftext|> Sample 6: US Politics & Election Coverage (K=64K=64, T=1.0T=1.0) <|endoftext|> the most dangerous people in the federal government in 2016. The FBI confirmed they only interviewed Flynn on January 20. Washington (CNN) Matt Flynn reportedly not been interviewed by the FBI after the call was held, Director Comey reportedly chose not to answer all questions about it. Senate Democratic leaders said the action was the effective result of the official action and viewed the actions of President Trump as "a coup." "In order to keep Russia out of our country, we must continue to rely on the resolve of all members of Congress and support the American people," said Baucus, D-Mont., chairman of the council of The Church of Jesus Christ of Latter-day Saints. Vice President Joe Biden, at the center of hacking, said in a State of the Union address that he didn’t immediately follow the department’s emergency proclamation but it was an "appropriate decision." The accusations made their way through White House channels Saturday, said Frank Reich, a senior fellow at the liberal Center for American Progress and a member of the White House under President Donald Trump. "We have repeatedly repeated in the last days about the Trump campaign’s contradictory statements about the use of chemical weapons against civilians," Reich told The New York Times. "Now, Mr. Biden’s e-mails are a very damaging line of attack for the incoming administration." Senate Majority Whip Max Baucus, D-Mont., declined to comment about the allegations. On Saturday, he said, "We just have to see the results of the investigation conducted by prosecutors in our community." Former Attorney General Scott Baquet, who served under Robert M. Mueller as attorney general of the White House, has condemned the accusations. "It’s difficult to say whether there’s nothing else, but they certainly are there, and it’s highly subjectively concluded that there’s no merit in this case," he said.<|endoftext|> E.1 CFG Ablation on ImageNet Ablation Studies Sweeping classifier-free guidance (CFG) weight w∈1.5,2.0,3.0,4.0,5.0w∈\1.5,2.0,3.0,4.0,5.0\ at 44 and 88 sampling steps (Figure 4), we find that precision increases and recall decreases monotonically with w at both step budgets, that the best 44-step FID (7.817.81) is attained at w=3.0w=3.0, and that the optimal guidance weight shifts down to w=1.5w=1.5 (FID 6.906.90) once step budget increases to 88, where iterative refinement already improves fidelity so a lower guidance weight suffices to preserve diversity. Figure 5 shows the corresponding 44-step samples across w, with the random seed held fixed per class. Figure 4: Ablation on the classifier-free guidance weight w used to sample the training pairs, evaluated at 44- and 88-step generation on ImageNet 256×256256× 256 (50k samples). (a) Precision and recall vs. w. (b) FID vs. w, with the best configuration at each step budget annotated. (c) Inception score and density vs. w. Figure 5: Qualitative effect of the CFG weight w used to sample the training pairs, at 44-step generation on ImageNet 256×256256× 256, for nine classes: goldfish, ostrich, hourglass, racer, volcano, goldfinch, sea anemone, cheeseburger and spotted salamander. The random seed is held fixed per class across w∈1.0,1.5,2.0,3.0,4.0,5.0w∈\1.0,1.5,2.0,3.0,4.0,5.0\, so only the effect of w varies; the optimal 44-step setting (w=3.0w=3.0, FID 7.817.81) is highlighted. E.2 Additional Qualitative Samples on ImageNet Figure 6 demonstrates the 44-step qualitative comparison to four ImageNet classes (goldfinch, sea anemone, cheeseburger, spotted salamander) following Yoo et al. (2025). Figure 6: 44-step generation results on ImageNet (classes goldfinch, sea anemone, cheeseburger, spotted salamander), two samples per class and model. Our method (bottom row, FID 7.817.81) is appended below the baselines.