Concurrent Image Understanding and Generation: Self-Correcting Coupled Markov Jump Processes

# Concurrent Image Understanding and Generation: Self-Correcting Coupled Markov Jump Processes
Source: [https://arxiv.org/html/2607.13188](https://arxiv.org/html/2607.13188)
\\uselogo\\correspondingauthor

qqdd@google\.com\\reportnumber0001

Armand Comas\\thepaAlexandros Lattas\\thepaStylianos Moschoglou\\thepaPedro VélezGoogle DeepMindAmit RajGoogle DeepMindAaron Germuth\\thepaThabo Beeler\\thepaDimitris SamarasStony Brook UniversityDi Qiu\\thepa

###### Abstract

Human cognition does not separate understanding and generation\. A teacher at a whiteboard speaks and draws*together*, each modality reshapes the other\. In this paper, we bring this coupled loop to artificial systems\. Masked Diffusion Models \(MDMs\) are ideally suited to this task, yet existing samplers either decode text and image interleavedly or independently update them in parallel branches that share only previous\-step history, but not the other modality’s latest decisions*within*the same step; combined with MDMs’ inability to remask, cross\-modal contradictions are neither detected nor repaired\. We introduceSelf\-Correcting Coupled Markov Jump Processes \(SC\-CMJP\), a framework in which one modality’s transition rates are functionals of the other modality’s confidence score, as weighted by cross\-modal attention\. Furthermore, a remasking jump retracts commitments the moment cross\-modal evidence turns against them\. In conjunction with SC\-CMJP, we introduceCO2Jump\(Self\-COrrectingCOupledJump\), a novel training\-free single\-pass sampler for joint multimodal geneneration\. For training and evaluation purposes, we have created and will release three large\-scale joint multimodal generation corpora:JEdit\-1M,JMaze\-200K,JNono\-200K, with matching in\- and out\-of\-distribution benchmarks\.CO2Jumpachieves best joint performance for image understanding and editing as well as visual reasoning \(maze and nonogram solving\)\. The performance of the sampler scales monotonically with the number of denoising steps, evidence that the benefits of cross\-modal coupling*compound*across the trajectory\. Project page:[https://coupled\-jump\.github\.io](https://coupled-jump.github.io/)

![Refer to caption](https://arxiv.org/html/2607.13188v1/x1.png)Figure 1:CO2Jumpin action: text and image co\-author the answer\.Three trajectories ofCO2Jumpon image editing, maze, and nonogram solving, showing the joint state at an intermediate stepttand at the final step\. The image\-editing panel highlights the core mechanism: at stepttthe text branch has already begun committing atarget\-image bounding box in text for the new object*person*; by the final step the image branch has placed the hiker*exactly*inside the finalized box \(we overlay the bounding boxes from generated text on edited image\)\. The text branch*plans*where the edit should land, and the image branch*executes*that plan within the same denoising trajectory — no second forward pass, no external grounder\. Maze and Nonogram show the same coupled\-refinement pattern: partial\-path and partial cell\-fill commitments at stepttconverge with their text\-side answers by the final step\.## 1Introduction

When a teacher explains an idea at a whiteboard, language and drawing take place*together*: each utterance affects the sketch, and each new mark affects the next sentence\. Understanding and generation are not separate stages but a tightly coupled loop, with each modality continuously informing and revising the other as the explanation unfolds\. We aim to operationalize this loop for artificial systems, producing text and image content concurrently rather than sequentially, with one modality shaping the other*as*it is generated\. While our method is modality\-agnostic in principle, in this paper we focus on pairing text for understanding, with images for generation\.

Masked Diffusion Models \(MDMs\)\[austin2021structured,sahoo2024simple,lou2024discrete,shi2024simplified\]are well\-suited for parallel multimodal generation\. Unlike autoregressive sequential pipelines\[deng2025emerging,chen2025janus,xie2026showo\], which first decode the entire textual reasoning trace and then condition image synthesis on it – a unidirectional flow that cannot retract early reasoning errors – MDMs predict all masked tokens simultaneously, admit a clean continuous\-time formulation as Markov Jump Processes\[campbell2022continuous,berghaus2024foundation\], and scale naturally to multiple modalities under a unified vocabulary spanning text and image tokens\[xin2025lumina,yang2025mmada\]\. Their parallel structure makes them, in principle, a natural framework for joint text/image generation in a single decoding loop\.

In practice, existing MDM samplers\[sahoo2024simple,wang2026remasking,tian2026mmadaparallel,chen2026unified,ouyang2026training\]fall short of*true*concurrent joint multimodal generation\. Even samplers that nominally decode both modalities in parallel\[chen2026unified,tian2026mmadaparallel\]factorize each denoising step so that the text and image updates depend only on the previous joint state and not*within*the same step; concurrency reduces to interleaving over a shared history\. The resulting trajectories might drift: text might commit to descriptions the image cannot illustrate, the image might render content the text never described\. Compounding this, standard masked diffusion is unable to remask\[austin2021structured,campbell2022continuous,sahoo2024simple\], once a token is committed it cannot be revised\. Cross\-modal contradictions introduced by an uncoupled parallel decoder persist for the rest of sampling\.

We address both problems jointly\. We introduceSelf\-Correcting Coupled Markov Jump Processes \(SC\-CMJP\), a general framework for concurrent joint multimodal generation in which the two modalities actively cross\-weigh their commitments*within*every denoising step\. One modality’s transition rates become functionals of the other modality’s unmasking confidence score, weighted by cross\-modal attention extracted from the same backbone forward pass\. The unmasking schedule of one modality adapts to the confidence of the other modality\. Combined with a remasking jump in the spirit of ReMDM\[wang2026remasking\], this lifts decoding from one\-way unmasking to a bidirectional birth\-death \(unmask\-remask\) process that can both reveal new tokens and retract earlier ones whenever cross\-modal evidence turns against them\.

Along with SC\-CMJP, we design a single\-pass training\-free samplerCO2Jump\(Self\-COrrectingCOupled Jump\) that highlights two core ideas:*Coupling*the per\-modality transition rates through cross\-modal attention, and*Correcting*earlier commitments via a remasking jump\.CO2Jumpruns on a frozen MDM with no architectural change, no auxiliary evaluator, and a single forward pass per step\. Figure[1](https://arxiv.org/html/2607.13188#S0.F1)illustratesCO2Jumpon all three of our benchmarks; in the image\-editing trajectory, the text branch commits a bounding box for the inserted object and the image branch fills it in\. In the*same*step, the text*plans*where the edit should land, the image branch*executes*that plan\.

To validateCO2Jump, we instantiate it on three concurrent text\-and\-image tasks of increasing semantic difficulty: image editing on an extended ImgEditBench\[ye2025imgedit\]protocol with mAP\-style\[lin2014microsoft\]grounded\-understanding metrics, and two new visual\-reasoning tasks, a maze and a nonogram \(JMazeandJNono\) where text and image are logically interlocked and jointly verifiable against algorithmic ground truth\. To enable training and evaluation, we curate three corpora:JEdit\-1M,JMaze\-200K, andJNono\-200K, all of which we plan to release\. Across all three tasks,CO2Jumpconsistently improves both modalities, beats existing sampling methods\[sahoo2024simple,wang2026remasking,tian2026mmadaparallel\]on concurrent joint image undestanding and generation, espectially joint performance metrics\. In addition,CO2Jumpsampler’s performance scales monotonically with the number of denoising steps\. In summary, our main contributions are:

- •We introduce the first approach to model simultaneous image understanding and generation as a single, unified stochastic process\. To achieve this, we propose Self\-Correcting Coupled Markov Jump Processes, that integrate parallel joint multimodal generation with built\-in self\-correction\.
- •We design a novel coupled samplerCO2Jumpfor joint multimodal sampling, running in a single forward pass per step\.
- •We curated three large\-scale joint\-generation corpora \(JEdit\-1M,JMaze\-200K,JNono\-200K\) along with matching benchmarks that probe both in\-distribution and out\-of\-distribution performance\. We have a plan to release model checkpoints, code, datasets to community\.
- •CO2Jumpimproves the state\-of\-the\-art on concurrent image understanding and editing, as well as visual reasoning tasks against existing sampling methods\. Our sampler’s performance scales monotonically with the number of denoising steps – empirical evidence that the benefits of cross\-modal coupling*compound*across the trajectory\.

## 2Background

### 2\.1Masked Diffusion Models and Remasking

Masked discrete diffusion models\[austin2021structured,sahoo2024simple,lou2024discrete,shi2024simplified\]corrupt a clean sample𝐱∈𝒱L\\mathbf\{x\}\\in\\mathcal\{V\}^\{L\}by gradually replacing tokens with a special absorbing state𝒎\\boldsymbol\{m\}, and learn to invert this corruption\. We adopt the continuous\-time formulation ofcampbell2022continuousas our default view because it admits the cross\-modal coupling and remasking extensions developed in this paper\.

#### Forward process and CTMC equivalence\.

Fort∈\[0,1\]t\\in\[0,1\]and a monotonically decreasing noise scheduleand∈\[0,1\]\\and\\in\[0,1\]withα0≈1\\alpha\_\{0\}\\approx 1,α1≈0\\alpha\_\{1\}\\approx 0, the per\-position marginal of the forward process is

q​\(𝐳t∣𝐱\)=Cat​\(𝐳t;and​𝐱\+\(1−and\)​𝒎\),q\(\\mathbf\{z\}\_\{t\}\\mid\\mathbf\{x\}\)=\\mathrm\{Cat\}\\bigl\(\\mathbf\{z\}\_\{t\};\\ \\and\\,\\mathbf\{x\}\+\(1\-\\and\)\\,\\boldsymbol\{m\}\\bigr\),\(1\)factorized across positions\. The same dynamics admit an equivalent Continuous\-Time Markov Chain \(CTMC\) description: states evolve by stochastic jumps between𝐱\\mathbf\{x\}and𝒎\\boldsymbol\{m\}at the infinitesimal rate

𝐑t=−α˙tand​\(𝐈−𝒎​𝟏⊤\),\\mathbf\{R\}\_\{t\}\\;=\\;\-\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{\\and\}\\bigl\(\\mathbf\{I\}\-\\boldsymbol\{m\}\\mathbf\{1\}^\{\\\!\\top\}\\bigr\),\(2\)under which the integrated transition probabilityq​\(𝐳t=𝐱∣𝐱\)=andq\(\\mathbf\{z\}\_\{t\}=\\mathbf\{x\}\\mid\\mathbf\{x\}\)=\\andrecovers Eq\. \([1](https://arxiv.org/html/2607.13188#S2.E1)\) exactly\.sahoo2024simpleestablish that the discrete\-time absorbing\-state formulation of MDLM and the CTMC formulation ofcampbell2022continuousparameterize the same family of marginals, posteriors, and likelihood bounds; we use the two views interchangeably\.

#### Training objective\.

A denoising network𝐱θ\\mathbf\{x\}\_\{\\theta\}is trained to predict the clean state𝐱\\mathbf\{x\}from𝐳t\\mathbf\{z\}\_\{t\}, and the resulting NELBO collapses to a position\-wise weighted cross\-entropy\[sahoo2024simple,shi2024simplified\]:

ℒNELBO∞=𝔼q,t​∫01α˙t1−and​∑ℓ=1Llog⁡⟨𝐱θℓ​\(𝐳t,t\),𝐱ℓ⟩​d​t\.\{\\mathcal\{L\}^\{\\infty\}\_\{\\text\{NELBO\}\}\}\\;=\\;\\mathbb\{E\}\_\{q,\\,t\}\\\!\\int\_\{0\}^\{1\}\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{1\-\\and\}\\sum\_\{\\ell=1\}^\{L\}\\log\\bigl\\langle\\mathbf\{x\}\_\{\\theta\}^\{\\ell\}\(\\mathbf\{z\}\_\{t\},t\),\\ \\mathbf\{x\}^\{\\ell\}\\bigr\\rangle\\,\\mathrm\{d\}t\.\(3\)The corresponding reverse posterior of standard MDLM has a well\-known inability to remask\[austin2021structured,campbell2022continuous,sahoo2024simple\]: once a token is unmasked, no subsequent reverse step can revisit it, so any error committed at a timestepttpersists for the rest of the trajectory\.

#### Remasking viaσt\\sigma\_\{t\}\.

ReMDM\[wang2026remasking\]repairs this by allowing committed tokens to revert to𝒎\\boldsymbol\{m\}with per\-step probabilityσt∈\[0,1\]\\sigma\_\{t\}\\in\[0,1\], yielding the modified reverse posterior

qσ​\(𝐳s∣𝐳t,𝐱\)=\{Cat​\(𝐳s;\(1−σt\)​𝐱\+σt​𝒎\),𝐳t≠𝒎Cat​\(𝐳s;αs−\(1−σt\)​αt1−αt​𝐱\+1−αs−σt​αt1−αt​𝒎\),𝐳t=𝒎,q\_\{\\sigma\}\(\\mathbf\{z\}\_\{s\}\\mid\\mathbf\{z\}\_\{t\},\\mathbf\{x\}\)=\\begin\{cases\}\\mathrm\{Cat\}\\\!\\bigl\(\\mathbf\{z\}\_\{s\};\\ \(1\-\\sigma\_\{t\}\)\\,\\mathbf\{x\}\+\\sigma\_\{t\}\\,\\boldsymbol\{m\}\\bigr\),&\\mathbf\{z\}\_\{t\}\\neq\\boldsymbol\{m\}\\\\\[2\.0pt\] \\mathrm\{Cat\}\\\!\\left\(\\mathbf\{z\}\_\{s\};\\ \\dfrac\{\\alpha\_\{s\}\-\(1\-\\sigma\_\{t\}\)\\alpha\_\{t\}\}\{1\-\\alpha\_\{t\}\}\\,\\mathbf\{x\}\+\\dfrac\{1\-\\alpha\_\{s\}\-\\sigma\_\{t\}\\alpha\_\{t\}\}\{1\-\\alpha\_\{t\}\}\\,\\boldsymbol\{m\}\\right\),&\\mathbf\{z\}\_\{t\}=\\boldsymbol\{m\},\\end\{cases\}\(4\)which preserves the marginal in Eq\. \([1](https://arxiv.org/html/2607.13188#S2.E1)\) whenσt≤min⁡\(1,\(1−αs\)/αt\)\\sigma\_\{t\}\\leq\\min\(1,\(1\-\\alpha\_\{s\}\)/\\alpha\_\{t\}\)and recovers MDLM atσt=0\\sigma\_\{t\}=0\. The remask jump makes the forward process non\-Markovian, but the reverse stays Markovian and tractable\[wang2026remasking\]\. Existingσt\\sigma\_\{t\}schedules\[wang2026remasking\]are*modality\-agnostic*– they score remasks from intra\-modal likelihoods alone, leaving cross\-modal contradictions undetected in joint generation\.

### 2\.2Markov Jump Processes and Coupled Multimodal Generation

A Markov Jump Process \(MJP\)\[campbell2022continuous,berghaus2024foundation\]is a continuous\-time Markov process on a discrete state space specified by a rate matrixRt​\(z,z′\)R\_\{t\}\(z,z^\{\\prime\}\); the CTMC view\[campbell2022continuous,sahoo2024simple,ouyang2026training\]of MDM is exactly such an MJP over𝒱L\\mathcal\{V\}^\{L\}with rate matrix Eq\. \([2](https://arxiv.org/html/2607.13188#S2.E2)\)\. Exact reverse simulation via Gillespie’s algorithm\[gillespie1976general,gillespie1977exact\]updates one position per jump and is prohibitive at modern sequence lengths, so practical samplers useτ\\tau\-leaping\[gillespie2001approximate\]– a parallel approximation that updates all positions simultaneously, of which the standard MDM reverse step is the first\-order discretization\[campbell2022continuous\]\.

#### Coupled MJPs for Multimodal Generation\.

Coupled jump processes are classical in chemistry\[gillespie1977exact\]and Glauber dynamics\[glauber1963time\], but their coupling structure is hand\-specified and they target physical simulation; standard MJP\-based diffusion likewise treats per\-position rates as conditionally independent given𝐱θ​\(𝐳t\)\\mathbf\{x\}\_\{\\theta\}\(\\mathbf\{z\}\_\{t\}\)even in multimodal settings\. We instead define a*Coupled Markov Jump Process*\(CMJP\) over𝐳t=\(𝐳ttext,𝐳timage\)\\mathbf\{z\}\_\{t\}=\(\\mathbf\{z\}\_\{t\}^\{\\texttt\{text\}\},\\mathbf\{z\}\_\{t\}^\{\\texttt\{image\}\}\)with modality\-specific rate matrices𝐑ta\\mathbf\{R\}\_\{t\}^\{a\}whose intensities depend on the hidden representations and instantaneous confidence of the complementary modality through learned cross\-modal attention\. Combined with a ReMDM\-styleσt\\sigma\_\{t\}jump \(Eq\. \([4](https://arxiv.org/html/2607.13188#S2.E4)\)\), birth \(unmask\) and death \(remask\) rates of one modality are informed by the current commitments of the other, enabling localized cross\-modal self\-correction at sampling time\.

## 3Related Work

#### Discrete Diffusion\.

Absorbing\-state discrete diffusion was introduced in D3PM\[austin2021structured\]and refined into score\-entropy\[lou2024discrete\], simplified\-ELBO\[sahoo2024simple,shi2024simplified\], and any\-order autoregressive\[ou2024your\]formulations\. Scaling to LLMs has been driven by LLaDA\[nie2025large\], Dream\[ye2025dream7d\], and SDAR\[cheng2025sdar\]\. Decoding improvements include block\-wise generation\[arriola2025block\], Top\-KKconfidence selection\[nie2025large,kim2025train\], and length\-adaptive scheduling\[ou2024your\]\. Multimodal extensions tokenize images with VQ\-VAE\[oord2017neural\]and operate over a unified vocabulary spanning text and image tokens,*e\.g\.*Lumina\-DiMOO\[xin2025lumina\]and MMaDA\[yang2025mmada\]\.

#### Self\-Correction via Remasking\.

Three families of method address the failure\-to\-remask problem\.*Predictor\-corrector samplers*\[campbell2022continuous,lezama2023discrete,gat2024discrete,campbell2024generative\]reduceτ\\tau\-leaping error via corrector steps without an explicit remask jump\.*Training\-based remasking*modifies the model: GIDD\[rutte2025generalized\]generalizes forward and reverse processes;zhao2024informedtrain a separate hollow\-transformer evaluator;kim2025fine,huang2025dontfine\-tune the pretrained MDM to estimate per\-token quality\.*Training\-free remasking*keeps the backbone frozen: ReMDM\[wang2026remasking\]adds a heuristicσt\\sigma\_\{t\}schedule,peng2025pathexplore path\-following corrections, andouyang2026traininguse cumulative\-confidence signals\.CO2Jumpsits within the training\-free family, but is the first to couple the remasking signal across modalities and treat cross\-modal contradiction as the self\-correction trigger\.

#### Concurrent Multimodal Samplers\.

UD\-VLA\[chen2026unified\]factorizes the joint into independent per\-modality terms \(an instance of MDM\[sahoo2024simple\]sampling\); MMaDA\-Parallel\[tian2026mmadaparallel\]interleaves text and image updates across steps but samples them*independently within each step*– both modality updates condition only on the previous joint state, with no cross\-modal feedback inside the step itself, so any coupling is realized only through shared history rather than instantaneous negotiation\. Our experiments compareCO2Jumpagainst three representative samplers covering these regimes: MDM\[sahoo2024simple\], ReMDM\[wang2026remasking\], and MMaDA\-Parallel\[tian2026mmadaparallel\]\. Coupled jump processes in chemistry and statistical physics\[gillespie1977exact,glauber1963time\]use hand\-specified couplings for physical simulation, so neither methods transfer to our setting\.

## 4Self\-Correcting Coupled Markov Jump Processes

![Refer to caption](https://arxiv.org/html/2607.13188v1/x2.png)Figure 2:CO2Jumpsampler\.A single denoising step from𝐳t\\mathbf\{z\}\_\{t\}to𝐳s\\mathbf\{z\}\_\{s\}\. From one forward pass, the model produces per\-token Self\-Confidence for both modalities; cross\-modal attention𝐀timage→text\\mathbf\{A\}^\{\\text\{image\}\\to\\text\{text\}\}\_\{t\}propagates text confidence to image positions, and an entropy\-based gateλ\\lambdamixes self and cross signals into Coupled Confidence\. The*Death*jump remasks the lowest\-confidence committed tokens, and the*Birth*jump reveals the highest\-confidence masked tokens under the noise schedule\.Standard multimodal diffusion samplers often suffer frommodality\-drift, where the generated text and image diverge because their unmasking schedules are independent and irreversible\. We address this by reformulating joint generation as a unified system where modalities actively negotiate their commitment through a shared birth\-death jump process whose intensities are coupled across modalities\. Figure[2](https://arxiv.org/html/2607.13188#S4.F2)illustrates a single denoising step of the resulting sampler\.

### 4\.1Continuous\-Time Likelihood Bounds for Joint Generation

Following MDM derivation\[sahoo2024simple\], fort∈\[0,1\]t\\in\[0,1\]and a monotonically decreasing noise scheduleand∈\[0,1\]\\and\\in\[0,1\], the per\-position marginal of the forward process is

q​\(𝐳t∣𝐱\)=Cat​\(𝐳t;and​𝐱\+\(1−and\)​𝒎\),q\(\\mathbf\{z\}\_\{t\}\\mid\\mathbf\{x\}\)=\\mathrm\{Cat\}\\bigl\(\\mathbf\{z\}\_\{t\};\\ \\and\\,\\mathbf\{x\}\+\(1\-\\and\)\\,\\boldsymbol\{m\}\\bigr\),\(5\)factorized across positions\. We treat text and image as a single sequence over the union vocabulary𝒱=𝒱text∪𝒱image\\mathcal\{V\}=\\mathcal\{V\}\_\{\\texttt\{text\}\}\\cup\\mathcal\{V\}\_\{\\texttt\{image\}\}and share a single absorbing token𝒎\\boldsymbol\{m\}across both modalities, so that the per\-position marginal in Eq\. \([5](https://arxiv.org/html/2607.13188#S4.E5)\) applies uniformly to text and image positions; this is what allows the joint NELBO in Eq\. \([6](https://arxiv.org/html/2607.13188#S4.E6)\) below to factor cleanly without a cross\-modality term\. A joint clean sample is𝐱=\(𝐱text,𝐱image\)∈𝒱L\\mathbf\{x\}=\(\\mathbf\{x\}^\{\\texttt\{text\}\},\\mathbf\{x\}^\{\\texttt\{image\}\}\)\\in\\mathcal\{V\}^\{L\}with total vocabulary sizeL=Ltext\+LimageL=L^\{\\texttt\{text\}\}\+L^\{\\texttt\{image\}\}, and is corrupted byq​\(𝐳t∣𝐱\)q\(\\mathbf\{z\}\_\{t\}\\mid\\mathbf\{x\}\), applied position\-wise across both modalities under a shared scheduleand\\and\.

Because the per\-position marginal factorizes and both modalities use the sameand\\and, the joint NELBO inherits the similar form in MDM\[sahoo2024simple\]with no cross\-modality term in the objective:

ℒNELBO∞=𝔼q,t​∫01α˙t1−and​∑ℓ=1Llog⁡⟨𝐱θℓ​\(𝐳t,t\),𝐱ℓ⟩​d​t\.\{\\mathcal\{L\}^\{\\infty\}\_\{\\text\{NELBO\}\}\}\\;=\\;\\mathbb\{E\}\_\{q,\\,t\}\\\!\\int\_\{0\}^\{1\}\\frac\{\\dot\{\\alpha\}\_\{t\}\}\{1\-\\and\}\\sum\_\{\\ell=1\}^\{L\}\\log\\bigl\\langle\\mathbf\{x\}\_\{\\theta\}^\{\\ell\}\(\\mathbf\{z\}\_\{t\},t\),\\ \\mathbf\{x\}^\{\\ell\}\\bigr\\rangle\\,\\mathrm\{d\}t\.\(6\)A crucial property is implicit in the conditioning: the network output𝐱θℓ​\(𝐳t,t\)\\mathbf\{x\}\_\{\\theta\}^\{\\ell\}\(\\mathbf\{z\}\_\{t\},t\)at any positionℓ\\ellhas access to the*entire*joint state𝐳t=\(𝐳ttext,𝐳timage\)\\mathbf\{z\}\_\{t\}=\(\\mathbf\{z\}\_\{t\}^\{\\texttt\{text\}\},\\mathbf\{z\}\_\{t\}^\{\\texttt\{image\}\}\), including the complementary modality\. Training under Eq\. \([6](https://arxiv.org/html/2607.13188#S4.E6)\) therefore implicitly forces𝐱θ\\mathbf\{x\}\_\{\\theta\}to learn cross\-modal denoising signals – to reconstruct a masked image patch from surrounding pixels*and*concurrent textual clues, and conversely to reconstruct a masked text token from surrounding context*and*the partially decoded image\. The NELBO never references this dependency explicitly, yet at convergence it is encoded in the network’s hidden representations and attention patterns\. Our coupled samplerCO2Jumpextracts these latent cross\-modal signals at inference time via a single forward pass through𝐱θ\\mathbf\{x\}\_\{\\theta\}, with no architectural changes to the trained model and no auxiliary cross\-modal evaluator\.

### 4\.2Asymmetric Sequential Sampling via Chain\-Rule Decomposition

Letccdenote the task conditioning context \(e\.g\., the editing prompt together with the source image for image editing, or the puzzle specification for visual reasoning\)\. A naive sampler for the joint reverse posteriorpθ​\(𝐳stext,𝐳simage∣𝐳t,c\)p\_\{\\theta\}\(\\mathbf\{z\}\_\{s\}^\{\\texttt\{text\}\},\\mathbf\{z\}\_\{s\}^\{\\texttt\{image\}\}\\mid\\mathbf\{z\}\_\{t\},c\)updates both modalities independently from the same𝐳t\\mathbf\{z\}\_\{t\}, ignoring that the latest text decision𝐳stext\\mathbf\{z\}\_\{s\}^\{\\texttt\{text\}\}provides immediate context for inferring the corresponding image state𝐳simage\\mathbf\{z\}\_\{s\}^\{\\texttt\{image\}\}\. We instead exploit the chain\-rule factorization

p​\(𝐳stext,𝐳simage∣𝐳t,c\)=p​\(𝐳stext∣𝐳t,c\)⏟\(I\) text update⋅p​\(𝐳simage∣𝐳t,𝐳stext,c\)⏟\(II\) image update conditioned on text at steps,p\(\\mathbf\{z\}\_\{s\}^\{\\texttt\{text\}\},\\mathbf\{z\}\_\{s\}^\{\\texttt\{image\}\}\\mid\\mathbf\{z\}\_\{t\},c\)\\;=\\;\\underbrace\{p\(\\mathbf\{z\}\_\{s\}^\{\\texttt\{text\}\}\\mid\\mathbf\{z\}\_\{t\},c\)\}\_\{\\text\{\(I\) text update\}\}\\cdot\\underbrace\{p\(\\mathbf\{z\}\_\{s\}^\{\\texttt\{image\}\}\\mid\\mathbf\{z\}\_\{t\},\\mathbf\{z\}\_\{s\}^\{\\texttt\{text\}\},c\)\}\_\{\\text\{\(II\) image update conditioned on text at step $s$\}\},\(7\)which makes explicit that the text update needs no information beyond the current𝐳t\\mathbf\{z\}\_\{t\}, while the image update would ideally condition on the latest sampled𝐳stext\\mathbf\{z\}\_\{s\}^\{\\texttt\{text\}\}\.

Evaluating Term \(II\) exactly demands a second forward pass through𝐱θ\\mathbf\{x\}\_\{\\theta\}at every diffusion step – prohibitive at inference time\. We avoid this cost by approximating𝐳stext\\mathbf\{z\}\_\{s\}^\{\\texttt\{text\}\}with quantities already produced by the single forward pass at𝐳t\\mathbf\{z\}\_\{t\}: the image side reads the model’s self\-belief over𝐳ttext\\mathbf\{z\}\_\{t\}^\{\\texttt\{text\}\}, propagates it through cross\-modal attention, and uses the result as a low\-cost surrogate for the unavailable𝐳stext\\mathbf\{z\}\_\{s\}^\{\\texttt\{text\}\}\. This yields an*asymmetric scoring rule*: text positions are scored by their pure self\-confidence, while image positions are scored by an entropy\-gated mixture of self\-confidence and a cross\-modal signal – theCoupled Confidence\. Both scores then drive the death\-and\-birth dynamics\.

### 4\.3Self\-Confidence and Coupled Confidence

Self\-Belief via Gumbel\-Max\.At each timestep, the network produces beliefspθ​\(𝐱ℓ∣𝐳t,c\)p\_\{\\theta\}\(\\mathbf\{x\}^\{\\ell\}\\mid\\mathbf\{z\}\_\{t\},c\)over each position’s clean token\. To escape local minima, we apply Gumbel\-max sampling,𝐱^ℓ=arg⁡maxv∈𝒱⁡\(log⁡pθ​\(v∣𝐳t,c\)\+γv\)\\hat\{\\mathbf\{x\}\}^\{\\ell\}=\\arg\\max\_\{v\\in\\mathcal\{V\}\}\\bigl\(\\log p\_\{\\theta\}\(v\\mid\\mathbf\{z\}\_\{t\},c\)\+\\gamma\_\{v\}\\bigr\)with i\.i\.d\. Gumbel noiseγv\\gamma\_\{v\}\. The*Self\-Confidence*of positionℓ\\ellis the model’s probability for its sampled choice:SelfConfℓ=pθ​\(𝐱^ℓ∣𝐳t,c\)\\texttt\{SelfConf\}\_\{\\ell\}=p\_\{\\theta\}\(\\hat\{\\mathbf\{x\}\}^\{\\ell\}\\mid\\mathbf\{z\}\_\{t\},c\)\.

#### Text Scoring\.

Because Term \(I\) of Eq\. \([7](https://arxiv.org/html/2607.13188#S4.E7)\) has no cross\-modal dependency, the per\-position text score is the self\-confidence directly:

Scoretext,ℓ=SelfConftext,ℓ\.\\texttt\{Score\}\_\{\\texttt\{text\},\\ell\}\\;=\\;\\texttt\{SelfConf\}\_\{\\texttt\{text\},\\ell\}\.\(8\)

#### Cross\-Modal Negotiation\.

For Term \(II\), we approximate the conditioning on𝐳stext\\mathbf\{z\}\_\{s\}^\{\\texttt\{text\}\}using𝐳ttext\\mathbf\{z\}\_\{t\}^\{\\texttt\{text\}\}via cross\-attention\. Let𝐇timage∈ℝLimage×D\\mathbf\{H\}\_\{t\}^\{\\texttt\{image\}\}\\in\\mathbb\{R\}^\{L^\{\\texttt\{image\}\}\\times D\}and𝐇ttext∈ℝLtext×D\\mathbf\{H\}\_\{t\}^\{\\texttt\{text\}\}\\in\\mathbb\{R\}^\{L^\{\\texttt\{text\}\}\\times D\}be the hidden representations of the two modalities, extracted post\-hoc from the model’s final layer during the same forward pass that producedSelfConf– no architectural modification or auxiliary attention head is introduced\. We compute the image→\\totext cross\-attention

𝐀timage→text=Softmax​\(𝐇timage​\(𝐇ttext\)⊤D\+𝐁t\),\\mathbf\{A\}\_\{t\}^\{\\texttt\{image\}\\to\\texttt\{text\}\}\\;=\\;\\texttt\{Softmax\}\\\!\\left\(\\frac\{\\mathbf\{H\}\_\{t\}^\{\\texttt\{image\}\}\(\\mathbf\{H\}\_\{t\}^\{\\texttt\{text\}\}\)^\{\\top\}\}\{\\sqrt\{D\}\}\+\\mathbf\{B\}\_\{t\}\\right\),\(9\)where𝐁t\\mathbf\{B\}\_\{t\}is a mask\-aware bias that downweights attention toward currently masked positions\. The cross\-modal negotiation is implemented as the cross signal for image positionℓ\\ell, which is the attention\-weighted text self\-confidence:

CrossSignalℓ=∑j=1Ltext𝐀t,ℓ​jimage→text⋅SelfConftext,j\.\\texttt\{CrossSignal\}\_\{\\ell\}\\;=\\;\\sum\_\{j=1\}^\{L^\{\\texttt\{text\}\}\}\\mathbf\{A\}\_\{t,\\ell j\}^\{\\texttt\{image\}\\to\\texttt\{text\}\}\\cdot\\texttt\{SelfConf\}\_\{\\texttt\{text\},j\}\.\(10\)

#### Dynamic Trust through Entropy\-Based Gating\.

The weight given to the cross signal versus the self signal on the image side is a*single scalar gate*λ∈\[0,1\]\\lambda\\in\[0,1\], computed once per denoising step and applied uniformly to every image position\. We deriveλ\\lambdafrom per\-modality predictive uncertainty: lettingℋ¯a=1La​∑ℓ=1La\[−∑vpθ​\(v∣𝐳t,c\)a,ℓ​log⁡pθ​\(v∣𝐳t,c\)a,ℓ\]\\bar\{\\mathcal\{H\}\}\_\{a\}=\\tfrac\{1\}\{L^\{a\}\}\\sum\_\{\\ell=1\}^\{L^\{a\}\}\\bigl\[\-\\sum\_\{v\}p\_\{\\theta\}\(v\\mid\\mathbf\{z\}\_\{t\},c\)\_\{a,\\ell\}\\log p\_\{\\theta\}\(v\\mid\\mathbf\{z\}\_\{t\},c\)\_\{a,\\ell\}\\bigr\]denote the mean token\-level Shannon entropy in modalityaa, averaged over*all*positions of that modality,

λ=ℋ¯imageℋ¯image\+ℋ¯text\+ϵ\.\\lambda\\;=\\;\\frac\{\\bar\{\\mathcal\{H\}\}\_\{\\texttt\{image\}\}\}\{\\bar\{\\mathcal\{H\}\}\_\{\\texttt\{image\}\}\+\\bar\{\\mathcal\{H\}\}\_\{\\texttt\{text\}\}\+\\epsilon\}\.\(11\)When the image is locally chaotic \(highℋ¯image\\bar\{\\mathcal\{H\}\}\_\{\\texttt\{image\}\}\),λ→1\\lambda\\to 1and the image modality defers to the text for evidentiary support; when the text is the uncertain side,λ→0\\lambda\\to 0and the image relies on its own self\-belief\. The single\-gate formulation follows directly from Eq\. \([7](https://arxiv.org/html/2607.13188#S4.E7)\) – only the image term carries a cross\-modal dependency to be resolved\.

#### Shared Percentile Rank Space\.

Because the text and image vocabularies differ in size \(\|𝒱text\|≫\|𝒱image\|\|\\mathcal\{V\}\_\{\\texttt\{text\}\}\|\\gg\|\\mathcal\{V\}\_\{\\texttt\{image\}\}\|\), absolute confidence values are not directly comparable across modalities\. We therefore project each scalar score into its empirical percentile rank within its own modality, ensuring the image\-side mixture below is well\-conditioned regardless of vocabulary scale\.

#### Coupled Confidence for Image Scoring\.

Combining the rank\-normalized self and cross signals:

Scoreimage,ℓ=CoupledConfℓ=\(1−λ\)⋅Rank​\(SelfConfimage,ℓ\)\+λ⋅Rank​\(CrossSignalℓ\)\.\\texttt\{Score\}\_\{\\texttt\{image\},\\ell\}\\;=\\;\\texttt\{CoupledConf\}\_\{\\ell\}\\;=\\;\(1\-\\lambda\)\\cdot\\text\{Rank\}\(\\texttt\{SelfConf\}\_\{\\texttt\{image\},\\ell\}\)\+\\lambda\\cdot\\text\{Rank\}\(\\texttt\{CrossSignal\}\_\{\\ell\}\)\.\(12\)

### 4\.4TheCO2JumpSampler

We interpret the joint reverse generation as a modality\-specific*birth\-death*jump process\. Standard masked diffusion only “births” tokens \(unmask\) and never updates them\.CO2Jumpadditionally allows tokens of either modality to “die” \(remask\) when their scores fall below the schedule\-driven death rate, allowing both intra\-modal errors and cross\-modal contradictions to be retracted\.

#### Death Step \(Remasking\)\.

At each discretization step\(t→s\)\(t\\to s\)we set the death rateσt\\sigma\_\{t\}to satisfy the valid\-posterior constraintσt≤min⁡\(η,\(1−αs\)/αt\)\\sigma\_\{t\}\\leq\\min\(\\eta,\(1\-\\alpha\_\{s\}\)/\\alpha\_\{t\}\)\. For each modalityaa, the idealized unmasked count isUta=αt​LaU\_\{t\}^\{a\}=\\alpha\_\{t\}L^\{a\}and the idealized masked count isMta=\(1−αt\)​LaM\_\{t\}^\{a\}=\(1\-\\alpha\_\{t\}\)L^\{a\}\(we use⌊⋅⌋\\lfloor\\cdot\\rfloorto obtain integer token counts\)\. The death quota isNremaska=⌊Uta⋅σt⌋N\_\{\\text\{remask\}\}^\{a\}=\\lfloor U\_\{t\}^\{a\}\\cdot\\sigma\_\{t\}\\rfloor, and theNremaskaN\_\{\\text\{remask\}\}^\{a\}unmasked tokens with the*lowest*Scorea\\texttt\{Score\}\_\{a\}are reverted to absorbing token𝒎\\boldsymbol\{m\}\. Per the asymmetric rule \(Eq\. \([7](https://arxiv.org/html/2607.13188#S4.E7)\)\), remasking usesSelfConffor text \(Eq\. \([8](https://arxiv.org/html/2607.13188#S4.E8)\)\) andCoupledConffor image \(Eq\. \([12](https://arxiv.org/html/2607.13188#S4.E12)\)\)\.

#### Birth Step \(Unmasking\)\.

To keep the global signal\-to\-noise progression matched to the schedule, the birth quotaNunmaska=⌊αs−αt1−αt⋅Mta⌋\+Δ​NaN\_\{\\text\{unmask\}\}^\{a\}=\\lfloor\\tfrac\{\\alpha\_\{s\}\-\\alpha\_\{t\}\}\{1\-\\alpha\_\{t\}\}\\cdot M\_\{t\}^\{a\}\\rfloor\+\\Delta N^\{a\}combines the base unmask count with an extra\-unmask compensationΔ​Na=⌊σt⋅αt1−αt⋅Mta⌋\\Delta N^\{a\}=\\lfloor\\sigma\_\{t\}\\cdot\\tfrac\{\\alpha\_\{t\}\}\{1\-\\alpha\_\{t\}\}\\cdot M\_\{t\}^\{a\}\\rfloorderived from the ReMDM\[wang2026remasking\]posterior\. Among the post\-remask masked tokens of modalityaa, we reveal theNunmaskaN\_\{\\text\{unmask\}\}^\{a\}with the*highest*Scorea\\texttt\{Score\}\_\{a\}, again with the asymmetric definition\. Algorithm[1](https://arxiv.org/html/2607.13188#alg1)gives the full pseudocode\.

Algorithm 1CO2JumpSampler \(Asymmetric Scoring\)1:Input:Pretrained

𝐱θ\\mathbf\{x\}\_\{\\theta\}, schedule

αt\\alpha\_\{t\}, remasking ratio

η\\eta, steps

TT, modality lengths

Ltext,LimageL^\{\\texttt\{text\}\},L^\{\\texttt\{image\}\}, conditioning

cc\.

2:Initialize

𝐳1←\{𝒎\}Ltext\+Limage\\mathbf\{z\}\_\{1\}\\leftarrow\\\{\\boldsymbol\{m\}\\\}^\{L^\{\\texttt\{text\}\}\+L^\{\\texttt\{image\}\}\}\.

3:for

i=Ti=Tdown to

11do

4:

t←i/T,s←\(i−1\)/Tt\\leftarrow i/T,\\ \\ s\\leftarrow\(i\-1\)/T
5:

𝐳s←𝐳t\\mathbf\{z\}\_\{s\}\\leftarrow\\mathbf\{z\}\_\{t\}// initialize next\-step state; death/birth jumps below write into𝐳s\\mathbf\{z\}\_\{s\}

6:// Single forward pass

7:Run

𝐱θ​\(𝐳t,c\)\\mathbf\{x\}\_\{\\theta\}\(\\mathbf\{z\}\_\{t\},c\); extract

𝐇ttext,𝐇timage\\mathbf\{H\}\_\{t\}^\{\\texttt\{text\}\},\\mathbf\{H\}\_\{t\}^\{\\texttt\{image\}\}and mean entropies

ℋ¯text,ℋ¯image\\bar\{\\mathcal\{H\}\}\_\{\\texttt\{text\}\},\\bar\{\\mathcal\{H\}\}\_\{\\texttt\{image\}\}over all positions

8:Sample

𝐱^ℓ\\hat\{\\mathbf\{x\}\}^\{\\ell\}via Gumbel noise; record

SelfConfℓ\\texttt\{SelfConf\}\_\{\\ell\}for both modalities

9:// Asymmetric scoring \(Eq\. \([7](https://arxiv.org/html/2607.13188#S4.E7)\)\)

10:*Text*

Scoretext,ℓ←SelfConftext,ℓ\\texttt\{Score\}\_\{\\texttt\{text\},\\ell\}\\leftarrow\\texttt\{SelfConf\}\_\{\\texttt\{text\},\\ell\}// Term \(I\): no cross\-modal dependence

11:*Image*compute

𝐀timage→text\\mathbf\{A\}\_\{t\}^\{\\texttt\{image\}\\to\\texttt\{text\}\},

CrossSignalℓ\\texttt\{CrossSignal\}\_\{\\ell\}\(Eqs\.[9](https://arxiv.org/html/2607.13188#S4.E9)and[10](https://arxiv.org/html/2607.13188#S4.E10)\), scalar gate

λ\\lambda\(Eq\.[11](https://arxiv.org/html/2607.13188#S4.E11)\)

12:

Scoreimage,ℓ←CoupledConfℓ\\texttt\{Score\}\_\{\\texttt\{image\},\\ell\}\\leftarrow\\texttt\{CoupledConf\}\_\{\\ell\}via Eq\. \([12](https://arxiv.org/html/2607.13188#S4.E12)\)// Term \(II\): approx\.𝐳stext\\mathbf\{z\}\_\{s\}^\{\\texttt\{text\}\}via attention

13:// Death & Birth Jumps

14:

σt←min⁡\(η,\(1−αs\)/αt\)\\sigma\_\{t\}\\leftarrow\\min\\\!\\bigl\(\\eta,\\ \(1\-\\alpha\_\{s\}\)/\\alpha\_\{t\}\\bigr\)// valid\-posterior constraint

15:for

a∈\{text,image\}a\\in\\\{\\texttt\{text\},\\texttt\{image\}\\\}do

16:

Uta←⌊αt​La⌋U\_\{t\}^\{a\}\\leftarrow\\lfloor\\alpha\_\{t\}L^\{a\}\\rfloor,

Mta←⌊\(1−αt\)​La⌋M\_\{t\}^\{a\}\\leftarrow\\lfloor\(1\-\\alpha\_\{t\}\)L^\{a\}\\rfloor// idealized counts, floored at runtime

17:

Nremaska←⌊Uta​σt⌋N\_\{\\text\{remask\}\}^\{a\}\\leftarrow\\lfloor U\_\{t\}^\{a\}\\sigma\_\{t\}\\rfloor,

Nunmaska←⌊αs−αt1−αt​Mta⌋\+⌊σt​αt1−αt​Mta⌋N\_\{\\text\{unmask\}\}^\{a\}\\leftarrow\\bigl\\lfloor\\tfrac\{\\alpha\_\{s\}\-\\alpha\_\{t\}\}\{1\-\\alpha\_\{t\}\}\\,M\_\{t\}^\{a\}\\bigr\\rfloor\+\\bigl\\lfloor\\sigma\_\{t\}\\,\\tfrac\{\\alpha\_\{t\}\}\{1\-\\alpha\_\{t\}\}\\,M\_\{t\}^\{a\}\\bigr\\rfloor
18:*Death:*in

𝐳s\\mathbf\{z\}\_\{s\}, remask the

NremaskaN\_\{\\text\{remask\}\}^\{a\}tokens of modality

aawith*lowest*

Scorea\\texttt\{Score\}\_\{a\}to

𝒎\\boldsymbol\{m\}
19:*Birth:*in

𝐳s\\mathbf\{z\}\_\{s\}, reveal the

NunmaskaN\_\{\\text\{unmask\}\}^\{a\}masked tokens of modality

aawith*highest*

Scorea\\texttt\{Score\}\_\{a\}
20:endfor

21:endfor

22:Output:Unified multimodal sample

𝐳0\\mathbf\{z\}\_\{0\}

## 5Datasets and Benchmarks for Joint Multimodal Generation

To train and evaluateCO2Jumpacross both photographic and logical multimodal tasks, we curate three joint\-generation datasets:JEdit\-1Mfor image editing,JMaze\-200Kfor maze solving, andJNono\-200Kfor nonogram solving, each paired with a held\-out benchmark\. All three corpora share the same record schema \(prompt, source image, target image, structured understanding, thinking trace\); Figure[3](https://arxiv.org/html/2607.13188#S5.F3)shows the curation pipeline\.

![Refer to caption](https://arxiv.org/html/2607.13188v1/x3.png)Figure 3:Dataset curation pipeline\.ForJEdit\-1M, raw editing pairs are augmented by an oracle Qwen3\-VL\-235B that produces both the per\-image scene\-graph understanding and the thinking trace\. ForJMaze\-200KandJNono\-200K, source/target images and the structured understanding are produced algorithmically\.### 5\.1JEdit\-1M: Joint Image Editing and Understanding

#### Training corpus\.

JEdit\-1Mcombines a 724k\-pair subset of ImgEdit\[ye2025imgedit\]with a 368k\-pair subset of OmniEdit\[wei2025omniedit\], yielding 1M \(prompt, source, target\) tuples\. Because raw editing data lacks the joint understanding and thinking supervision that joint multimodal generation requires, we augment each pair with two fields synthesized by Qwen3\-VL\-235B\[bai2025qwen3vltechnicalreport\]: \(i\) a pixel\-aligned source/target scene\-graph extraction with per\-category indices preserved across panels, and \(ii\) a logic\-based thinking trace conditioned on both images, the prompt, and the scene graph from \(i\)\. Figure[4](https://arxiv.org/html/2607.13188#S5.F4)provides a concrete example of a training sample of theJEdit\-1Mdataset\.

![Refer to caption](https://arxiv.org/html/2607.13188v1/x4.png)Figure 4:JEdit\-1Mtraining sample\.A natural\-language edit prompt \(bottom\) paired with the source and target images \(right\), the oracle Qwen3\-VL\-235B thinking trace, and the per\-panel source/target scene\-graph understanding\. The bounding boxes and labels from MLLM’s grounding solution are overlaid on the corresponding images\.
#### Benchmark and metrics\.

We extend ImgEditBench\[ye2025imgedit\]with two metric families that score joint understanding \+ generation behavior\.Text qualityis generative perplexity under a frozen GPT\-2 Large\[radford2019language\]plus token\-distribution entropy as a mode\-collapse safeguard, following MDLM\[sahoo2024simple\]and ReMDM\[wang2026remasking\]\.Image\-edit fidelityis the standard ImgEditBench oracle score from Gemini 3 Flash\[gemini3flash\]\.Image\-grounded understandingis COCO\-style mAP@0\.5:0\.95\[lin2014microsoft\]against a*pseudo*scene graph constructed per\-sample by Gemini on the model’s own \(source, generated\-target\) pair: pseudo grounding is necessary because each model produces a different target image\. We report mAP separately for source and target sections plus their average\.

### 5\.2JMaze\-200KandJNono\-200K: Visual Reasoning

To stress\-test joint generation on tasks where text and image are*logically interlocked*, we add two synthetic visual\-reasoning corpora\. In both, the input fully specifies a unique solution and the two modalities are independently verifiable against algorithmic ground truth\.

#### Corpora\.

We use themaze\-dataset\[ivanitskiy2023mazedataset\]library to generateJMaze\-200Kcomprising 200k DFS\-perfect mazes with grid sizes uniformly sampled from\{6,…,20\}\\\{6,\\ldots,20\\\}; the input shows walls plus a green start and red end, the output overlays the solution path in blue, and the text answer is the path as an\(r,c\)\(r,c\)sequence\.JNono\-200Kis 200k nonogram puzzles with grid sizes\{5,…,25\}\\\{5,\\ldots,25\\\}, generated by a mixture of pattern synthesizers biased toward multi\-run row/column clues; the input renders the emptyN×NN\\\!\\times\\\!Ngrid with marginal clue numbers, and the output fills cells black to satisfy every clue\. For both corpora, the thinking trace is synthesized by an oracle Gemini 3 Flash conditioned on the source/target images and the algorithmic ground\-truth answer\.

#### Parallel\-form supervision\.

Both the structured understanding and the thinking trace are written in a form native to parallel decoding rather than autoregressive ordering: maze solutions use global\(r,c\)\(r,c\)coordinates instead of relative moves \(*Left/Right/Up/Down*\), and nonogram supervision emphasizes bidirectional row/column constraint propagation instead of row\-by\-row solving\. A relative direction is only well\-defined once the previous coordinate is committed, while a global coordinate keeps its meaning regardless of which other tokens have been unmasked, aligning the supervision with the parallel denoising the sampler actually performs\.

#### Benchmarks and metrics\.

For each task we hold out a 500\-sample test set whose grid sizes*exceed*the training range on both ends:JMaze\-Test500spans\{3,…,22\}\\\{3,\\ldots,22\\\}\(in\-dist\{6,…,20\}\\\{6,\\ldots,20\\\}, OOD\{3,4,5,21,22\}\\\{3,4,5,21,22\\\}\);JNono\-Test500spans\{3,…,27\}\\\{3,\\ldots,27\\\}\(in\-dist\{5,…,25\}\\\{5,\\ldots,25\\\}, OOD\{3,4,26,27\}\\\{3,4,26,27\\\}\)\. Our headline metric isjoint accuracy, a sample counts correct only when both the text answer \(judged by a tolerant Gemini extractor against the algorithmic ground truth\) and the generated image \(compared side\-by\-side with the ground\-truth image, by Gemini 3 Flash\) are correct\.Text accuracyandimage accuracyare reported separately as diagnostics; on in\-distribution and OOD splits\.

## 6Experiments

We evaluateCO2Jumpagainst three sampler families across the three tasks of Section[5](https://arxiv.org/html/2607.13188#S5)\. To ensure a strictly controlled comparison: we initialize from Lumina\-DiMOO\[xin2025lumina\], fine\-tune on the corresponding corpus, and swap only the inference\-time sampler\. For each task, we fine\-tune the model on64×64\\timesH100 80 GB GPUs with a total batch size of512512and learning rate2×10−52\\\!\\times\\\!10^\{\-5\}\. Baselines includes:MDM\[sahoo2024simple\], independent modalities with no remasking;ReMDM\[wang2026remasking\], independent modalities with remasking; andMMaDA\-Parallel\[tian2026mmadaparallel\], modalities interleave and are independent with no remasking\.CO2Jumpis the only entry that generates joint modalities concurrently with cross\-modal coupling and self\-correction\. All samplers use a cosineαt\\alpha\_\{t\}schedule; for the remasking variants \(ReMDM andCO2Jump\), we use a remasking schedule withσt=0\.01\\sigma\_\{t\}=0\.01ont∈\[0\.25,0\.75\]t\\in\[0\.25,0\.75\]and0elsewhere\.

### 6\.1Joint Image Editing and Understanding

Table 1:Image\-editing results on the extended ImgEditBench protocol\.Bottom block: apple\-to\-apple sampler comparison; above the rule: pre\-trained Lumina\-DiMOO and Qwen3\-VL\-8B references\.Bold/underline= best/second best across the four fine\-tuned samplers\.![Refer to caption](https://arxiv.org/html/2607.13188v1/x5.png)Figure 5:Scaling sampling steps\.ImgEditBench \(left\) and overall mAP \(right\) vs\. NFE\.CO2Jumpis the only sampler with monotonically rising curves; the gap to baselines widens with NFE\.Table[1](https://arxiv.org/html/2607.13188#S6.T1)reports results on extended ImgEditBench\. Our headline metric is per\-section mAP@0\.5:0\.95 from the pseudo\-grounding pipeline; target\-mAP is the cleanest test of cross\-modal coupling because it requires the image branch to place each edited object inside the bounding boxes that the text branch declares*and*the text branch to describe what the image branch actually produces\.

#### Coupled generation beats a strong sequential grounder\.

Qwen3\-VL\-8B is a stringent reference: same scale as our backbone, and the supervision used to fine\-tune all four samplers comes from the Qwen3\-VL family\[bai2025qwen3vltechnicalreport\]\. Because Qwen3\-VL\-8B itself cannot generate images, we evaluate its grounding on the same \(source, generated\-target\) image pairs produced by ourCO2Jumpsampler, effectively giving it the generated target image as input for free\. YetCO2Jumpproduces grounded text*concurrently*with the image and still exceeds Qwen3\-VL\-8B on the same images, on both target mAP \(0\.3460\.346vs\.0\.3300\.330\) and overall mAP \(0\.3690\.369vs\.0\.3600\.360\)\.

#### Better understanding leads to better generation\.

Pre\-trained Lumina\-DiMOO has near\-zero understanding mAP \(0\.0190\.019\); fine\-tuning unlocks it across all samplers, and the sampler with the strongest understanding \(CO2Jump,0\.3690\.369overall\) also achieves the best ImgEditBench score \(1\.931\.93\), while the weakest \(MMaDA\-Parallel,0\.3350\.335\) is worst on ImgEditBench \(1\.441\.44\)\. This is the empirical signature of the chain\-rule decomposition \(Section[4\.2](https://arxiv.org/html/2607.13188#S4.SS2)\): once text decisions inform image decisions within a single step, sharper text grounding tightens the image\-side score distribution\.

#### Scaling sampling steps\.

Sweeping NFE from 8 to 512 \(Figure[5](https://arxiv.org/html/2607.13188#S6.F5)\),CO2Jumpis the only sampler that improves*monotonically*on both metrics: ImgEditBench rises from1\.721\.72to1\.931\.93and overall mAP from0\.0740\.074to0\.3690\.369\. Single\-modality baselines plateau and regress at high NFE, and MMaDA\-Parallel*degrades*from1\.521\.52to1\.441\.44, its uncoupled schedule cannot productively use additional steps\. The gap also widens: at 8 NFE the four methods sit within0\.0090\.009mAP, but at 512 NFECO2Jumpis\+0\.015/\+0\.016/\+0\.034\+0\.015/\+0\.016/\+0\.034ahead of MDM / ReMDM / MMaDA\-Parallel\. Coupling*compounds*across steps rather than saturating\.

### 6\.2Visual Reasoning: Maze and Nonogram Solving

Table[2](https://arxiv.org/html/2607.13188#S6.T2)reports joint accuracy on visual reasoning benchmarks on in\-distribution and OOD grid sizes\.

#### CO2Jumpoutperforms other samplers in all six columns\.

Improvements over the best baseline range from\+0\.008\+0\.008\(Maze, In\-Dist\) to\+0\.062\+0\.062\(Nonogram, OOD\)\. No baseline is the runner\-up everywhere — MDM is second on Maze but fourth on Nonogram, and MMaDA\-Parallel is the reverse — so consistent joint correctness across both task structures is itself evidence that the coupling mechanism is generic rather than benchmark\-specific\.

Table 2:Joint accuracy onJMaze\-Test500andJNono\-Test500\.A sample is correct only when text and image both match the algorithmic ground truth\.Bold/underline= best/second best\.![Refer to caption](https://arxiv.org/html/2607.13188v1/x6.png)Figure 6:Qualitative comparison\.Top left: Nonogram \(red✗marks clue violations\); Bottom left: Maze; Right: per\-step MDM vs\.CO2Jumptrajectory\. OnlyCO2Jumpsatisfies all clues and paths, and its joint trajectory stays consistent across both modalities\.
#### Out\-of\-distribution generalization\.

CO2Jumpalso holds up best on grid sizes outside the training range\. On Maze it leads on the OOD split \(0\.3200\.320vs\.0\.3120\.312for the runner\-up\), and on Nonogram it widens its lead substantially \(0\.1750\.175vs\.0\.1130\.113\)\. MDM drops55%55\\%from In\-Dist to OOD on Nonogram and MMaDA\-Parallel21%21\\%, whileCO2Jumpstays roughly flat\. It shows that cross\-modal coupling transfers solution\-style information across grid sizes more robustly than uncoupled samplers\.

#### Qualitative trajectories expose the uncoupling failure mode\.

In Figure[6](https://arxiv.org/html/2607.13188#S6.F6), every baseline violates at least one Nonogram clue \(red✗\), whileCO2Jumpsatisfies all clues; on the maze, baselines wander into wrong corridors or cut through walls andCO2Jumpreproduces the ground\-truth path\. The right column exposes*why*: by step 160 of MDM’s trajectory the text answer is correct but the image has drifted onto a different path, because under MDM’s independence factorization the image branch never sees the latest committed text\.CO2Jump’s chain\-rule decomposition propagates each text commitment into image scoring at the same step, so both modalities converge on the same solution\.

### 6\.3Ablation Study and Analysis

#### Ablation\.

Table[3](https://arxiv.org/html/2607.13188#S6.T3)removes each mechanism in isolation\. Removing Shared Percentile Rank causes the largest drop in image\-edit fidelity \(1\.93→1\.871\.93\\to 1\.87\): without a common scale, one of Self\-Confidence and Cross Signal arbitrarily dominates the blend\. Removing Entropy\-Based Gating leaves source\-mAP unchanged but cuts target\-mAP by0\.0300\.030and overall mAP by0\.0150\.015, the gate’s main role is in late\-stage sampling where the target image is committed under freshly grounded text\. Removing Self\-Correction is the most damaging change for understanding \(−3\.2\-3\.2on overall mAP\) and drops editing by0\.010\.01: without remasking, late\-discovered cross\-modal contradictions cannot be repaired\.

Table 3:Ablation onCO2Jump\.Each row removes one mechanism: Shared Percentile Rank, Entropy\-Based Gating \(λimage\\lambda\_\{\\text\{image\}\}\), or Self\-Correction\.Bold/underline= best/second best\.![Refer to caption](https://arxiv.org/html/2607.13188v1/x7.png)Figure 7:\(a\)Per\-step text/image entropy and the gateλimage\\lambda\_\{\\text\{image\}\}\. The image starts confident \(source prior\), text starts uncertain;λimage\\lambda\_\{\\text\{image\}\}rises from≈0\.05\\approx 0\.05to≈0\.85\\approx 0\.85as text commits\.\(b\)Image\-side remask events in \(Self\-Confidence, Cross Signal\) space: most fire in the lower\-left region; a smaller hotspot at high\-Self / low\-Cross corresponds to coupling\-driven revocations\.
#### Dynamic trust via entropy\-based gating\.

Figure[7](https://arxiv.org/html/2607.13188#S6.F7)\(a\) traces per\-step entropies andλimage\\lambda\_\{\\text\{image\}\}for a representative editing sample\. The image branch starts confident \(source\-image prior\) while text starts uncertain \(only the editing prompt as context\), soλimage≈0\.05\\lambda\_\{\\text\{image\}\}\\approx 0\.05and the image relies more on self\-confidence\. As text commits, its entropy falls and the image moves into regions outside its prior;λimage\\lambda\_\{\\text\{image\}\}rises to≈0\.85\\approx 0\.85, and the image branch increasingly leans on the text\-grounded cross signal — the empirical signature of the chain\-rule decomposition\.

#### Coupling quadrant\.

Figure[7](https://arxiv.org/html/2607.13188#S6.F7)\(b\) shows every image\-side remask event in \(Self\-Confidence, Cross Signal\) space\. Most events concentrate in the lower\-left region where both signals are low, so the coupled confidence\(1−λ\)⋅Self\+λ⋅Cross\(1\-\\lambda\)\\cdot\\text\{Self\}\+\\lambda\\cdot\\text\{Cross\}is small and the death jump is triggered\. A smaller hotspot in the high\-Self / low\-Cross quadrant corresponds to coupling\-driven unmasking — the image was locally confident but the text disagreed — and these account for the gap between the fullCO2Jumpand the “−\-Self\-Correction” ablation row in Table[3](https://arxiv.org/html/2607.13188#S6.T3)\.

## 7Conclusion

We presentedSelf\-Correcting Coupled Markov Jump Processes \(SC\-CMJP\), a framework in which the two modalities of a unified MDM negotiate their commitments*within*every denoising step, andCO2Jump, a training\-free single\-pass sampler that instantiates it on a frozen backbone\. By coupling one modality’s transition rates to the other’s emerging confidence via cross\-modal attention, and by allowing committed tokens to be retracted through a remasking jump triggered by cross\-modal contradictions,CO2Jumpcloses the concurrent\-generation loop that prior parallel samplers leave open\.CO2Jumpdelivers best performance across joint image understanding and editing, as well as visual reasoning tasks and it scales monotonically with denoising steps, evidence that cross\-modal coupling compounds across the trajectory\. Our experiments instantiate the framework on the canonical text\-and\-image pair; the SC\-CMJP recipe is modality\-agnostic, and extending it to audio, video, or structured outputs is left to future work\.

## Appendix AQualitative Example fromCO2Jump: Joint Image Editing and Understanding

Figure[8](https://arxiv.org/html/2607.13188#A1.F8)shows a completeCO2Jumpsample fromImgEditBench: prompt, source image, generated target image, the per\-panel scene\-graph understanding, and the thinking trace, all produced concurrently in a single denoising loop\. Bounding boxes overlaid on each image are exactly those emitted by the model’s own text branch \(no external grounder\), normalized to a1001×10011001\\\!\\times\\\!1001canvas\. The example illustrates the planning\-then\-execution pattern that drives the gain on extended ImgEditBench: the text branch isolatesshellas the only dynamic object and keeps every other entity \(tortoise,arm,eye,ground,rock\) at identical bounding boxes between source and target, while changing only the shell’s attributes from\(scaly\)to\(smooth\); the image branch realizes precisely that delta, removing the high\-frequency scaly pattern on the shell while leaving body, limbs, and surroundings untouched\.

![Refer to caption](https://arxiv.org/html/2607.13188v1/x8.png)Figure 8:Joint image editing with concurrent understanding onJEdit\-1M\.Prompt, model\-generated thinking trace, scene\-graph source/target analyses, and the source/generated images with predicted bounding boxes overlaid \(boxes are the model’s own text\-side predictions, not ground truth\)\. The text branch declares only theshellas a dynamic object — every other entity has identical bounding boxes in the source and target analyses — and the image branch realizes exactly this localized texture edit: scaly→\\tosmooth on the shell, with body, limbs, ground and rock preserved\.
## Appendix BQualitative Example fromCO2Jump: Maze Solving

Figure[9](https://arxiv.org/html/2607.13188#A2.F9)shows aCO2Jumpsample fromJMaze: the input maze \(with green start and red end\), the model\-generated thinking trace and coordinate path, and the solved maze with the blue path overlaid\. Both the textual coordinate sequence and the rendered solution path are produced concurrently in a single denoising loop\. The thinking trace reasons globally over the maze — identifying the upper half as a dead\-end region, the lower corridors as the main artery, and the central zig\-zag through the bottom row as the southern bypass — before committing the explicit\(r,c\)\(r,c\)sequence in<answer\>, and the image branch traces exactly that sequence, demonstrating that the path drawn on the grid agrees position\-by\-position with the coordinates the text branch declares\.

![Refer to caption](https://arxiv.org/html/2607.13188v1/x9.png)Figure 9:Joint maze solving onJMaze\.CO2Jump’s thinking trace and answer \(left\) alongside the input maze \(top right; green start, red end\) and the solved maze with the blue path overlaid \(bottom right\)\. The text branch first reasons globally about which corridors are dead\-ends and which form the main artery, then commits the path as an explicit\(r,c\)\(r,c\)coordinate sequence; the image branch traces precisely that sequence on the grid in the same denoising trajectory\.
## Appendix CQualitative Example fromCO2Jump: Nonogram Solving

Figure[10](https://arxiv.org/html/2607.13188#A3.F10)shows aCO2Jumpsample fromJNono: the input nonogram \(an empty7×77\\\!\\times\\\!7grid with row and column clue numbers\), the model\-generated thinking trace and per\-row filled\-cell answer, and the solved grid with the corresponding cells filled black\. The thinking trace performs bidirectional constraint propagation rather than row\-by\-row solving: it identifies the most globally\-constraining lines first \(rows whose largest clue forces a 4\-cell block, columns whose clue sums match the grid width\), explains how the row and column constraints tighten each other concurrently, and only then commits the row\-by\-row filled\-cell ranges in<answer\>; the image branch fills precisely those ranges on the grid, satisfying every clue\.

![Refer to caption](https://arxiv.org/html/2607.13188v1/x10.png)Figure 10:Joint nonogram solving onJNono\.CO2Jump’s thinking trace and answer \(left\) alongside the input puzzle \(top right; row clues at left margin, column clues at top margin\) and the solved grid with cells filled black \(bottom right\)\. The text branch reasons in a bidirectional, constraint\-propagation style — starting from the most globally\-constrained rows and columns and using each forced fill to tighten the other dimension — before committing the per\-row filled\-cell ranges; the image branch fills exactly those ranges on the grid in the same denoising trajectory, satisfying every clue\.
## Appendix DSystem Prompts for Dataset Curation

### D\.1Understanding Prompt: Pixel\-Aligned Scene Graph Extraction

The following system prompt was used to extract the structured understanding \(bounding boxes and attributes\) from the source and target images\.

Task: Pixel\-Aligned Scene Graph Extraction for Image Editing Pairs\.

\#\#\# Objective: Analyze the transition from the Source Image to the Target Image\. You must extract a high\-fidelity grounding dataset that maps how visual tokens change in response to the Editing Prompt\. Focus on maintaining spatial "anchors" for static regions while precisely detailing the "delta" in edited regions\.

\#\#\# Inputs: 1\. Source Image: \(The original state\) 2\. Target Image: \(The modified state\) 3\. Editing Prompt: "\{editing\_prompt\_var\}"

\#\#\# Strict Constraints: THINKING BEFORE ANSWERING:In your <think\> tags, you MUST explicitly compare the two images\. List: 1\. Static entities \(Background/Context\)\. 2\. Transformed entities \(Attribute/State changes\)\. 3\. New or Deleted entities \(Structural changes\)\.

FIDELITY OVER INTENT:Describe what is visually rendered in the Target Image\. If the prompt asks for a "red car" but the Target Image contains a "pink car," you MUST label it as "pink car\."

COMPLETENESS:Annotate the main subject, all objects involved in the edit, and key background anchors \(e\.g\., floor, sky, walls\) to provide global context\.

COORDINATES:Use normalized integer bounding boxes \[x1, y1, x2, y2\] on a \[0\-1000\] scale\. Bounding boxes must be tight\.

IDENTITY CONSISTENCY:For any entity present in both images, the ’category\_name’ and ’\[index\]’ MUST be identical\.

PER\-CATEGORY INDEXING:Every new object category MUST start its index at \[0\]\. For example, the first ’car’ is ’car \[0\]’, and the first ’person’ is ’person \[0\]’\. Do NOT use a single global incrementing index for the whole scene\.

STRUCTURAL LOGIC:If an object is deleted in the Target, do not list it in the TARGET IMAGE ANALYSIS\. If an object is added, list it only in the TARGET section\.

\#\#\# Prohibited Actions: DO NOTuse generic placeholder strings \(e\.g\., "category\_name", "attribute\_list"\)\. DO NOTinclude the brackets around the category name or attributes in the final answer; only the index should remain in brackets as per format\.

\#\#\# Output Format: <answer\> SOURCE IMAGE ANALYSIS: category\_name \(attribute\_list\) \[index\]: \[x1, y1, x2, y2\]

TARGET IMAGE ANALYSIS: category\_name \(attribute\_list\) \[index\]: \[x1, y1, x2, y2\] </answer\>

\#\#\# Reference Example: <answer\> SOURCE IMAGE ANALYSIS: mountain \(snowy, jagged\) \[0\]: \[0, 50, 1000, 450\] person \(black hoodie, walking\) \[0\]: \[420, 550, 580, 910\] person \(blue jeans, standing\) \[1\]: \[600, 550, 700, 910\]

TARGET IMAGE ANALYSIS: mountain \(snowy, jagged\) \[0\]: \[0, 50, 1000, 450\] person \(yellow raincoat, walking\) \[0\]: \[420, 550, 580, 910\] person \(blue jeans, standing\) \[1\]: \[600, 550, 700, 910\] </answer\>

### D\.2JEdit\-1MThinking Prompt: Reasoning Trace Synthesis

The following system prompt was used with Qwen3\-VL\-235B to synthesize the logic\-based reasoning trace for theJEdit\-1Mdataset\.

Task: Synthesize a Logic\-Based Reasoning Trace for Image Editing\.

\#\#\# Inputs: 1\. Source and Target Images: Visual evidence\. 2\. Editing Prompt \(The intended transformation\): \{editing\_prompt\_var\} 3\. Image Understanding \(Bounding box coordinates and object labels for both states\): \{answer\}

\#\#\# Objective: Synthesize a logic\-based reasoning trace that analyzes the visual transition relative to the Editing Prompt\. Critically evaluate whether the target state successfully fulfills the prompt’s instructions or deviates from the intended logic\.

\#\#\# Strict Reasoning Guidelines: Categorize Objects:Group objects into: Static \(persistent coordinates/attributes\), Dynamic \(modified coordinates/attributes\), and Deleted/Added \(present in only one state\)\.

Quantify Change:Reference bounding box shifts or attribute updates as direct evidence of the transition\.

Critical Alignment Check:Explicitly compare the visual delta against the Editing Prompt\. Identify hallucinations or instruction mismatches\.

Connect to Intent:Explain how the detections fulfill—or fail to fulfill—the specific requirements of the Editing Prompt\.

Constraint:Keep the response dense and information\-rich, within a max of 350 words\. Avoid generic phrases\. Start with the direct logical flow\.

\#\#\# Output Format: <think\> \[Your concise logical trace here\] </think\>

### D\.3Thinking Prompt: Maze Solving Parallel Reasoning

The following system prompt was used to synthesize the parallel\-reasoning trace for the maze\-solving problem in theJMaze\-200Kdataset\.

Task: Synthesize a parallel\-reasoning trace for a maze\-solving problem\.

\#\#\# Inputs: 1\. Input maze image: walls, a green start cell, and a red end cell\. No path drawn\. 2\. Output maze image: the same maze with the correct solution path drawn in blue\. 3\. Maze structure as text \(adjacency list \+ start \+ end\): \{prompt\} 4\. Ground\-truth solution path: \{answer\}

\#\#\# Objective: Write a natural reasoning trace that reflects a global\-to\-local refinement process, illustrating how the maze is solved simultaneously rather than purely sequentially\. Briefly describe the maze boundaries \(grid size, start position, end position\)\. Identify the global structure first: note major bottlenecks, large dead\-end regions to mask out, or key anchor points/corridors between the start and end\. Describe the path formation as a concurrent process—e\.g\., establishing a bridge through a central corridor while simultaneously connecting the start and end regions to that main artery, verifying local wall\-openings via the adjacency list\. Write in natural prose, as a holistic chain of thought\. No bullet lists\. Keep it under 350 words\.

\#\#\# Output format \(strict — respond with exactly this tag and nothing else\): <think\> \{\}\[Your reasoning trace\] </think\>

### D\.4Thinking Prompt: Nonogram Parallel Reasoning

The following system prompt was used to synthesize the parallel\-reasoning trace for the nonogram puzzle in theJNono\-200Kdataset\.

Task: Synthesize a parallel\-reasoning trace for a nonogram \(picross\) puzzle\.

\#\#\# Inputs: 1\. Source image: empty NxN grid with row clue numbers along the left margin and column clue numbers along the top margin\. 2\. Target image: the same grid with cells filled to satisfy every clue, revealing a pixel\-art picture\. 3\. Puzzle structure as text \(size \+ row clues \+ column clues\): \{prompt\} 4\. Ground\-truth solution \(filled cell ranges per row\): \{answer\}

\#\#\# Objective: Write a holistic reasoning trace that reflects parallel constraint propagation, NOT row\-by\-row sequential solving\. Briefly describe the puzzle \(grid size, rough shape of the revealed picture\)\. Identify the most globally\-constraining lines first: rows or columns where the sum of clues plus required gaps equals N \(full\-line forces\), or lines whose single largest clue produces "definite\-fill" cells via the left\-most / right\-most overlap argument\. Describe how column constraints and row constraints tighten each other concurrently: a forced fill produced by a row clue confirms a position in some column’s clue list, which can in turn unlock more rows\. Avoid sequential "first solve row 0, then row 1" prose\. Emphasize the bidirectional, global interplay between row and column constraints\. Write in natural prose, no bullet lists\. Keep it under 250 words\.

\#\#\# Output format \(strict — respond with exactly this tag and nothing else\): <think\> \{\}\[Your reasoning trace\] </think\>

## References