FlowEdit: Information-Theoretic Control of LLM Reasoning Flows for Ill-posed Problems Involving Conflicts

# FlowEdit: Information-Theoretic Control of LLM Reasoning Flows for Ill-posed Problems Involving Conflicts
Source: [https://arxiv.org/html/2607.20500](https://arxiv.org/html/2607.20500)
Sizhe Tang The George Washington University s\.tang1@gwu\.edu &Guangyu Jiang The George Washington University guangyu\.jiang@gwu\.edu &Yu Li The George Washington University yul@gwu\.edu &Rongqian Chen The George Washington University rongqianc@gwu\.edu &Ioannis G\. Kevrekidis Johns Hopkins University yannisk@jhu\.edu &Tian Lan The George Washington University tlan@gwu\.edu

###### Abstract

Large Language Models \(LLMs\) have demonstrated strong performance on well\-specified reasoning tasks where a feasible answer exists\. However, problems encountered in the open world can become ill\-posed due to inconsistent conditions, conflicting statements, or mutually incompatible requirements\. It leads to ill\-posed problems involving conflicts that admit no valid responses\. We argue that effective reasoning of such ill\-posed problems involving conflicts require novel LLM capabilities to make hidden conflicts explicit, maintain competing hypotheses via multiple reasoning branches when warranted, and generate alternative responses in a single pass, all of which are challenging due to the inherent limitation of next\-token prediction mechanism in LLMs\. To this end, we proposeFlowEdit, a novel framework that leverages information\-theoretic principles to actively quantify and regulate internal reasoning flows of LLMs, for generating a full set of alternative responses under valid hypotheses in a single pass\.FlowEditcan be viewed as enforcing a branch\-aware reasoning process using two dual information\-theoretic objectives on the model’s internal reasoning representations: maximizing the information flow from each selected hypothesis to the branch outcome, while minimizing the overlap and conditional dependence across sibling branches, to provide a diverse, informative set of responses with broad coverage\. We show that this is achieved through tractable variational bounds under boundary embeddings beingϵ\\epsilon\-sufficient, optimizing the underlying conditional mutual information in LLM reasoning process\. Extensive experiments demonstrate thatFlowEditoutperforms leading proprietary models, improving exact\-set\-match accuracy by 68%, while boosting overall response informativeness by 24%\. We further show that flow regulation surfaces in the token stream as a structured redistribution of next\-token entropy that concentrates inside each branch, amplifies at the boundaries between flows, and scales with the number of flows the problem requires\.

![Refer to caption](https://arxiv.org/html/2607.20500v1/x1.png)Figure 1:Overview ofFlowEdit\. On ill\-posed problems involving conflicts, next\-token prediction silently commits to one self\-consistent hypothesis and collapses the answer set onto a single trajectory \(left\)\.FlowEditregulates the model’s internal reasoning flows via two dual information\-theoretic objectives on boundary representations \(right\)\. One example involving conflicts is demonstrated\.## 1Introduction

Large language models \(LLMs\) have shown strong performance across diverse tasks\[[29](https://arxiv.org/html/2607.20500#bib.bib37),[8](https://arxiv.org/html/2607.20500#bib.bib15),[15](https://arxiv.org/html/2607.20500#bib.bib2),[40](https://arxiv.org/html/2607.20500#bib.bib9)\], with mathematical reasoning\[[7](https://arxiv.org/html/2607.20500#bib.bib8)\]standing out as a central capability\. Current approaches primarily focus on well\-specified problems where a feasible answer exists and train models to produce a single reasoning trace, which is supervised either by step\-level rationales through chain\-of\-thought distillation\[[33](https://arxiv.org/html/2607.20500#bib.bib1),[38](https://arxiv.org/html/2607.20500#bib.bib13)\]or by a scalar reward on the final output through reinforcement learning with verifiable rewards\[[27](https://arxiv.org/html/2607.20500#bib.bib17),[12](https://arxiv.org/html/2607.20500#bib.bib18)\]\. This has proven to be effective on well\-defined problems, where a single deterministic line of thought or logic suffices to reach the unique correct answer\. However, reasoning problems encountered in the open world can often become ill\-posed\[[41](https://arxiv.org/html/2607.20500#bib.bib19),[23](https://arxiv.org/html/2607.20500#bib.bib21),[31](https://arxiv.org/html/2607.20500#bib.bib22),[18](https://arxiv.org/html/2607.20500#bib.bib23),[34](https://arxiv.org/html/2607.20500#bib.bib27)\]\. LLMs that lack the ability to analyze the inherent problem structures, examine implicit deficits/conflicts, or regulate how reasoning process unfolds demonstrate limited performance on ill\-posed problems\.

Among various forms of ill\-posed problems, this paper considers ill\-posed problems involving conflicts that admit no valid responses due to inconsistent conditions, conflicting states, or mutually incompatible requirements\. Such type of ill\-posed problems involving conflicts are routinely encountered in areas such as real\-world decision\-making, operations research, and systems engineering\[[42](https://arxiv.org/html/2607.20500#bib.bib6),[24](https://arxiv.org/html/2607.20500#bib.bib4),[3](https://arxiv.org/html/2607.20500#bib.bib3)\]\. While recent work have studied ill\-posed problems with missing conditions\[[41](https://arxiv.org/html/2607.20500#bib.bib19),[31](https://arxiv.org/html/2607.20500#bib.bib22)\]and underspecified reasoning\[[23](https://arxiv.org/html/2607.20500#bib.bib21),[18](https://arxiv.org/html/2607.20500#bib.bib23)\], these are largely cast into a detection problem to flag or reject ill\-posed inputs\[[34](https://arxiv.org/html/2607.20500#bib.bib27),[26](https://arxiv.org/html/2607.20500#bib.bib34)\]or a post\-hoc inference problem about response confidence\[[16](https://arxiv.org/html/2607.20500#bib.bib25),[17](https://arxiv.org/html/2607.20500#bib.bib28),[20](https://arxiv.org/html/2607.20500#bib.bib20)\]\.

We show that for ill\-posed problems involving conflicts, current LLMs either reject the input altogether\[[31](https://arxiv.org/html/2607.20500#bib.bib22)\]or silently commit to one hypothesis \(regarding the conflicts\) among several equally possible alternatives\[[23](https://arxiv.org/html/2607.20500#bib.bib21)\]\. In contrast, our goal is to enable a more proactive reasoning behavior by making hidden conflicts explicit, analyzing and maintaining competing hypotheses via multiple reasoning branches when warranted, and generating alternative responses/recommendations in a single pass\. This is challenging as LLMs’ next\-token prediction mechanism favors sequential elaboration of an initial commitment over the parallel maintenance of several\[[2](https://arxiv.org/html/2607.20500#bib.bib26)\]\. Novel solutions are needed to quantify and regulate multiple internal reasoning flows/branches in LLMs, to generate a full set of alternative responses under valid hypotheses in a single pass\.

We introduceFlowEdit, a training framework that operationalizes this notion of regulating internal reasoning flows and maintaining competing hypotheses through an information\-theoretic control on a language model’s internal representations and information flows\. In particular, we identify the existence of conflicts and treat each reasoning flow as an object summarized by a designated hidden state along the generation trajectory, so that the abstract notion of enforcing multiple reasoning flows/branches becomes a concrete requirement on the geometry of these representations\. A pair of dual objectives–expressed as conditional mutual information–make this requirement precise: each reasoning flow must retain the maximal information needed to maintain its own hypothesis and derive its own conclusion, while distinct flows should have minimal overlap beyond shared common problem analysis once the autoregressive context preceding them has been accounted for, as the reasoning process unfolds in a single pass\.

More precisely,FlowEditoptimizes the max\-min dual objectives by quantifying the conditional mutual information between selected LLM internal states\. By maximizing the information flow from each selected hypothesis to the branch outcome, while minimizing the conditional overlap across sibling branches,FlowEditprovides a diverse, informative set of responses with broad coverage in a single pass\. We show that this optimization can be achieved through tractable variational bounds with boundary embeddings under anϵ\\epsilon\-sufficient condition, provably optimizing the underlying conditional mutual information in LLM reasoning process\. In particular,ϵ\\epsilon\-sufficiency makes the conditioning choice principled rather than heuristic: regulating cross\-flow dependence against the analysis embedding, rather than a prompt\-only embedding, provably isolates the residual redundancy we wish to penalize from the shared structure that distinct flows are entitled to encode\.

FlowEditdemonstrates superior proactive reasoning behavior on ill\-posed problems involving conflicts\. Across three problem domains and ground\-truth branch countsK⋆∈\{1,2,3,4\}K^\{\\star\}\\\!\\in\\\!\\\{1,2,3,4\\\},FlowEdit\-Qwen3\-4B\-Base improves exact\-set\-match accuracy by 68% and information recovery by 24% over the strongest closed\-source baseline, with gains widening asK⋆K^\{\\star\}grows; token\-level analysis further shows these gains coincide with a structured redistribution of next\-token entropy that concentrates inside each branch and amplifies at the boundaries between flows\.

The primary contributions of this paper are as follows:

- •We reframe ill\-posed mathematical reasoning with conflicting conditions as a flow\-regulation problem, training models to enumerate the full set of valid resolutions in a single pass rather than detect and reject\.
- •We derive a flow\-regulation objective from two conditional mutual information requirements, optimized through directionally aligned bounds under anϵ\\epsilon\-sufficiency conditioning that provably isolates residual cross\-flow dependence\.
- •We construct a dataset of problems involving conflicts spanning three domains and ground\-truth branch countsK⋆∈\{1,2,3,4\}K^\{\\star\}\\\!\\in\\\!\\\{1,2,3,4\\\}, with verified hypothesis–answer pairs per branch\.
- •We showFlowEditoutperforms the strongest closed\-source baseline by 68% on exact\-set\-match and 24% on information recovery, and identify a token\-level entropy signature—concentration inside branches, amplification at flow boundaries—that gives a representation\-level account of the gains\.

## 2Related Works and Background

### 2\.1Mathematical Reasoning with Language Models

Research on mathematical reasoning with language models has progressed along three tracks: prompting\-based methods that elicit multi\-step reasoning from frozen models\[[33](https://arxiv.org/html/2607.20500#bib.bib1),[32](https://arxiv.org/html/2607.20500#bib.bib7),[36](https://arxiv.org/html/2607.20500#bib.bib10),[10](https://arxiv.org/html/2607.20500#bib.bib11),[4](https://arxiv.org/html/2607.20500#bib.bib12),[19](https://arxiv.org/html/2607.20500#bib.bib5)\], supervised fine\-tuning that distills step\-level traces into smaller models\[[38](https://arxiv.org/html/2607.20500#bib.bib13),[22](https://arxiv.org/html/2607.20500#bib.bib14),[11](https://arxiv.org/html/2607.20500#bib.bib16)\], and reinforcement learning with verifiable rewards against a single gold answer\[[27](https://arxiv.org/html/2607.20500#bib.bib17),[12](https://arxiv.org/html/2607.20500#bib.bib18),[37](https://arxiv.org/html/2607.20500#bib.bib29)\]\. All three share the same operative assumption: every problem is well\-posed and admits a unique correct answer\. We depart from this point\-valued target and treat the answer to an ill\-posed problem involving conflicts as a set of mutually exclusive resolutions, training the model to enumerate them within a single structured response\.

### 2\.2Ill\-Posed Mathematical Reasoning and Abstention

Recent work examines LLM behavior on problems that violate the well\-posed assumption, through benchmarks of logical traps, unreasonable inputs, and missing or contradictory conditions\[[41](https://arxiv.org/html/2607.20500#bib.bib19),[23](https://arxiv.org/html/2607.20500#bib.bib21),[31](https://arxiv.org/html/2607.20500#bib.bib22),[18](https://arxiv.org/html/2607.20500#bib.bib23)\], alignment strategies that fine\-tune models to outputunsolvable\[[34](https://arxiv.org/html/2607.20500#bib.bib27)\], and parallel efforts in open\-domain QA on unanswerable queries\[[39](https://arxiv.org/html/2607.20500#bib.bib33),[26](https://arxiv.org/html/2607.20500#bib.bib34)\]\. Despite their methodological variety, all of these threads frame ill\-posed input as a detection problem: the model should identify the input as unsolvable and abstain, or solicit clarification to restore well\-posedness\. This framing discards a structure central to mathematical practice, namely that an ill\-posed problem involving conflicts is typically not unsolvable but multiply solvable, with each self\-consistent subset of its conditions defining a well\-posed subproblem with a distinct answer\. We depart from the detection\-and\-rejection paradigm and train LLMs to constructively enumerate the full set of valid resolutions; to our knowledge ours is the first work to formulate ill\-posed mathematical reasoning involving conflicts as an information flow regulation task and to provide both a training objective and a dataset targeting this setting\.

A parallel tradition in numerical optimization addresses ill\-posed problems involving conflicts at the level of explicit constraints\. The Maximum Feasible Subsystem problem seeks a largest cardinality subset of mutually inconsistent linear constraints that admits a joint solution\[[1](https://arxiv.org/html/2607.20500#bib.bib30),[6](https://arxiv.org/html/2607.20500#bib.bib31)\], and is approached at scale by RANSAC\[[9](https://arxiv.org/html/2607.20500#bib.bib32)\]and randomized Kaczmarz methods that exploit prior knowledge of conflict\-free constraints\[[28](https://arxiv.org/html/2607.20500#bib.bib39),[21](https://arxiv.org/html/2607.20500#bib.bib38)\]\. These methods operate on explicit constraint matrices and return a single resolution, whereas our setting requires the model to surface implicit conflicts from the problem statement and enumerate the full set of resolutions in a single pass\.

## 3Methodology

### 3\.1Problem Formulation and Preliminaries

We formalize the response to an ill\-posed mathematical problem involving conflicts as a task in which a single model output must carry, and keep separate, several reasoning flows in parallel, and we specify the conditions under which a response is correct at the level of information rather than surface tokens\. Well\-posed problems are subsumed as the degenerate caseK⋆​\(x\)=1K^\{\\star\}\(x\)=1, in which the framework reduces to standard single\-trace reasoning; the formulation is therefore a strict generalization of the point\-valued target adopted in prior work, not a separate regime\.

#### Ill\-posed problems and reasoning flows\.

Letxxdenote a problem statement and𝒞​\(x\)=\{c1,…,cm\}\\mathcal\{C\}\(x\)=\\\{c\_\{1\},\\ldots,c\_\{m\}\\\}the conditions stated inxx\. We callxx*ill\-posed*if no assignment of the problem variables satisfies everycic\_\{i\}simultaneously, although strict subsets of𝒞​\(x\)\\mathcal\{C\}\(x\)are jointly satisfiable\. A*hypothesis*z⊆𝒞​\(x\)z\\subseteq\\mathcal\{C\}\(x\)is a self\-consistent subset of conditions, and𝒵⋆​\(x\)\\mathcal\{Z\}^\{\\star\}\(x\)collects those subsets that are maximal under inclusion among self\-consistent ones and induce a well\-posed subproblem with a unique solution\. WithK⋆​\(x\)=\|𝒵⋆​\(x\)\|K^\{\\star\}\(x\)=\|\\mathcal\{Z\}^\{\\star\}\(x\)\|andyk=g​\(x,zk\)y\_\{k\}=g\(x,z\_\{k\}\)the answer under the domain solution mapgg, the ground\-truth target is

𝒴⋆​\(x\)=\{g​\(x,zk\):zk∈𝒵⋆​\(x\)\}=\{y1,…,yK⋆​\(x\)\},\\mathcal\{Y\}^\{\\star\}\(x\)=\\\{g\(x,z\_\{k\}\):z\_\{k\}\\in\\mathcal\{Z\}^\{\\star\}\(x\)\\\}=\\\{y\_\{1\},\\ldots,y\_\{K^\{\\star\}\(x\)\}\\\},\(1\)the well\-posed caseK⋆​\(x\)=1K^\{\\star\}\(x\)=1retained as a degenerate singleton\. Eachzkz\_\{k\}defines a distinct deterministic*reasoning flow*fromxxtoyky\_\{k\}; ill\-posedness is the regime in whichK⋆​\(x\)\>1K^\{\\star\}\(x\)\>1and a correct response must carry several such flows in parallel within a single output\. Our methodological goal is to regulate these flows: to ensure that they form inside the model and remain mutually distinct, rather than collapsing onto whichever flow next\-token prediction commits to first\.

#### Where reasoning flows become observable\.

Regulating reasoning flow at the representational level requires a place to address each flow individually\. We obtain this addressability with the smallest structural intervention sufficient for the purpose: the policyπθ\\pi\_\{\\theta\}is trained to produce an output that interleaves an analysis segmentaawith a sequence of*branch blocks*, one per flow,

o=\[ANA\]​a​\[/ANA\]​\[B\]​z^1​\[SEP\]​y^1​\[/B\]​⋯​\[B\]​z^K^​\[SEP\]​y^K^​\[/B\]​\[END\],o=\\texttt\{\[ANA\]\}\\,a\\,\\texttt\{\[/ANA\]\}\\;\\texttt\{\[B\]\}\\,\\hat\{z\}\_\{1\}\\,\\texttt\{\[SEP\]\}\\,\\hat\{y\}\_\{1\}\\,\\texttt\{\[/B\]\}\\,\\cdots\\,\\texttt\{\[B\]\}\\,\\hat\{z\}\_\{\\hat\{K\}\}\\,\\texttt\{\[SEP\]\}\\,\\hat\{y\}\_\{\\hat\{K\}\}\\,\\texttt\{\[/B\]\}\\;\\texttt\{\[END\]\},\(2\)whereK^\\hat\{K\}is chosen by the model through emission of\[END\]rather than supplied externally, and the structural tokens carry no semantic content of their own\. The role of \([2](https://arxiv.org/html/2607.20500#S3.E2)\) is not to prescribe a format but to designate hidden states at which each flow is summarized: writinght​\(⋅\)h\_\{t\}\(\\cdot\)for the causal\-LM hidden state at tokentt, we read off

ex=h\[ANA\]​\(x\),ea=h\[/ANA\]​\(x,a\),ek=h\[SEP\]k​\(⋯\),eyk=h\[/B\]k​\(⋯\)∈ℝde\_\{x\}=h\_\{\\texttt\{\[ANA\]\}\}\(x\),\\qquad e\_\{a\}=h\_\{\\texttt\{\[/ANA\]\}\}\(x,a\),\\qquad e\_\{k\}=h\_\{\\texttt\{\[SEP\]\}\_\{k\}\}\(\\cdots\),\\qquad e\_\{y\_\{k\}\}=h\_\{\\texttt\{\[/B\]\}\_\{k\}\}\(\\cdots\)\\;\\in\\;\\mathbb\{R\}^\{d\}as the prompt summary, the analysis summary, thekk\-th flow’s hypothesis representation, and its answer representation\[[30](https://arxiv.org/html/2607.20500#bib.bib41),[13](https://arxiv.org/html/2607.20500#bib.bib42)\]\.

### 3\.2Information\-Theoretic Reasoning Flow Regulation

A reasoning flow, as defined in Section[3\.1](https://arxiv.org/html/2607.20500#S3.SS1), is determined by three boundary representations: the analysiseae\_\{a\}, a hypothesiseke\_\{k\}, and an answereyke\_\{y\_\{k\}\}\. Two properties of these representations distinguish a model that genuinely maintains several flows in parallel from one that emits multiple branches at the surface but reasons about them in a degenerate way, and they jointly motivate the developed loss\.

The first property concerns each flow individually\. A flow is well\-formed only if its hypothesis representationeke\_\{k\}carries, given the analysiseae\_\{a\}, the information from which its answereyke\_\{y\_\{k\}\}is determined; otherwise the branch\-block is decorative, with the answer produced fromeae\_\{a\}alone or from cues unrelated to the committed hypothesis\. We call this property*flow sufficiency*, and it imposes on the loss a term that increasesI​\(ek;eyk∣ea\)I\(e\_\{k\};e\_\{y\_\{k\}\}\\mid e\_\{a\}\), pushing the hypothesis representation to carry the residual information that distinguishes its own subproblem from the analysis\-level baseline\.

The second property concerns the relationship between flows\. Several flows coexist in a single output without collapse only if distinct hypothesis representations are not paraphrases of one another at the level of information: the moreeie\_\{i\}andeje\_\{j\}share beyond what is licensed by the shared analysis and the autoregressive context that precedes them, the more the second flow is reducible to the first\. We call this property*flow separation*, and it imposes on the loss a term that reducesI​\(ei;ej∣wi​j\)I\(e\_\{i\};e\_\{j\}\\mid w\_\{ij\}\)as much as the optimization can afford, wherewi​jw\_\{ij\}is a conditioning vector that absorbs the dependence we cannot and should not eliminate\.

The remainder of this section constructs the flow\-regulation lossℒflow\\mathcal\{L\}\_\{\\mathrm\{flow\}\}that enforces these two properties on the boundary embeddings ofπθ\\pi\_\{\\theta\}\. We work withL2L\_\{2\}\-normalized embeddings,e~=e/‖e‖2\\tilde\{e\}=e/\\\|e\\\|\_\{2\}, suppressing the tilde, so that all representations lie on the unit sphere and the estimators below operate on a bounded, comparable scale\. The overview ofFlowEditis demonstrated in Fig\.[1](https://arxiv.org/html/2607.20500#S0.F1)\.

#### Bounds aligned with the optimization directions\.

The two flow properties translate into the formal conditions

maxθ⁡I​\(ek;eyk∣ea\)for each​k,minθ⁡I​\(ei;ej∣wi​j\)for each pair​i<j,\\max\_\{\\theta\}\\ I\(e\_\{k\}\\,;\\,e\_\{y\_\{k\}\}\\mid e\_\{a\}\)\\ \\ \\text\{for each \}k,\\qquad\\min\_\{\\theta\}\\ I\(e\_\{i\}\\,;\\,e\_\{j\}\\mid w\_\{ij\}\)\\ \\ \\text\{for each pair \}i<j,\(3\)which act on the same kind of quantity but ask the optimizer to move it in opposite directions\. This asymmetry is consequential: maximizing a tractable lower bound provably increases the underlying MI, but minimizing the same lower bound does not reduce it—the bound can become loose without the dependence actually decreasing\. Genuine reduction of MI requires an upper bound that majorizes the true quantity from above\. We therefore choose two estimators by their bounding direction, InfoNCE\[[25](https://arxiv.org/html/2607.20500#bib.bib35)\]for sufficiency and CLUB\[[5](https://arxiv.org/html/2607.20500#bib.bib36)\]for separation:

I^NCE​\(ek;eyk∣ea\)=𝔼​\[log⁡exp⁡\(fϕ​\(ek,eyk,ea\)/τ\)1N​∑n=1Nexp⁡\(fϕ​\(ek,eyk\(n\),ea\)/τ\)\],\\widehat\{I\}\_\{\\mathrm\{NCE\}\}\(e\_\{k\}\\,;\\,e\_\{y\_\{k\}\}\\mid e\_\{a\}\)=\\mathbb\{E\}\\\!\\left\[\\log\\frac\{\\exp\\\!\\big\(f\_\{\\phi\}\(e\_\{k\},e\_\{y\_\{k\}\},e\_\{a\}\)/\\tau\\big\)\}\{\\tfrac\{1\}\{N\}\\sum\_\{n=1\}^\{N\}\\exp\\\!\\big\(f\_\{\\phi\}\(e\_\{k\},e\_\{y\_\{k\}\}^\{\(n\)\},e\_\{a\}\)/\\tau\\big\)\}\\right\],\(4\)I^CLUB​\(ei;ej∣wi​j\)=𝔼p​\(ei,ej,wi​j\)​\[log⁡qψ​\(ej∣ei,wi​j\)\]−𝔼p​\(ei,wi​j\)​p​\(ej∣wi​j\)​\[log⁡qψ​\(ej∣ei,wi​j\)\],\\widehat\{I\}\_\{\\mathrm\{CLUB\}\}\(e\_\{i\}\\,;\\,e\_\{j\}\\mid w\_\{ij\}\)=\\mathbb\{E\}\_\{p\(e\_\{i\},e\_\{j\},w\_\{ij\}\)\}\\\!\\big\[\\log q\_\{\\psi\}\(e\_\{j\}\\mid e\_\{i\},w\_\{ij\}\)\\big\]\-\\mathbb\{E\}\_\{p\(e\_\{i\},w\_\{ij\}\)\\,p\(e\_\{j\}\\mid w\_\{ij\}\)\}\\\!\\big\[\\log q\_\{\\psi\}\(e\_\{j\}\\mid e\_\{i\},w\_\{ij\}\)\\big\],\(5\)withfϕf\_\{\\phi\}a learned scoring function,τ=0\.07\\tau=0\.07, negatives\{eyk\(n\)\}\\\{e\_\{y\_\{k\}\}^\{\(n\)\}\\\}drawn from other branches in the minibatch, and the second CLUB expectation taken over shuffled\(ej,wi​j\)\(e\_\{j\},w\_\{ij\}\)pairs\. The variational networkqψ​\(v∣u,w\)q\_\{\\psi\}\(v\\mid u,w\), parameterized as a diagonal\-Gaussian MLP that predicts the mean and log\-variance ofeje\_\{j\}given\(ei,wi​j\)\(e\_\{i\},w\_\{ij\}\), approximates the true conditional distributionp​\(ej∣ei,wi​j\)p\(e\_\{j\}\\mid e\_\{i\},w\_\{ij\}\); the CLUB difference is a valid upper bound onI​\(ei;ej∣wi​j\)I\(e\_\{i\};e\_\{j\}\\mid w\_\{ij\}\)to the extent that this approximation is accurate, andqψq\_\{\\psi\}is therefore updated alongsideθ\\thetaduring training\. CLUB is preferred over geometric penalties such as cosine or orthogonality, which capture only second\-order dependence and cannot register the higher\-order structure two reasoning flows can share\.

#### Conditioning that isolates the flow\-level residual\.

The estimators in \([4](https://arxiv.org/html/2607.20500#S3.E4)\)–\([5](https://arxiv.org/html/2607.20500#S3.E5)\) are only as faithful as their conditioning\. The conditioning vectorwi​jw\_\{ij\}in the separation term defines what counts as redundancy: it must remove the dependence that two flows are entitled to share, so that the regularizer acts on residual cross\-flow information rather than on a baseline guaranteed by construction, but it must not remove the dependence that distinguishes them, lest the regularizer be driven to zero on precisely the multi\-flow inputs where it should be active\. The second of these requirements singles out the analysis representationeae\_\{a\}from natural alternatives such as a prompt\-only embeddingex=h\[ANA\]​\(x\)e\_\{x\}=h\_\{\\texttt\{\[ANA\]\}\}\(x\), a preference we now formalize\. DefineIi​j⋆:=I​\(Ei;Ej∣X,A\)I^\{\\star\}\_\{ij\}:=I\(E\_\{i\};E\_\{j\}\\mid X,A\), the flow\-level residual the regularizer is meant to penalize, andΔi​j:=I​\(Ei;Ej∣X\)−Ii​j⋆\\Delta\_\{ij\}:=I\(E\_\{i\};E\_\{j\}\\mid X\)\-I^\{\\star\}\_\{ij\}, the share of cross\-flow dependence contributed byAAbeyondXX\. By constructionΔi​j≥0\\Delta\_\{ij\}\\geq 0on multi\-flow problems—eacha⋆a^\{\\star\}verbalises a partition that determines every branch, contributing dependence beyond whatXXalone supplies, withΔi​j\>0\\Delta\_\{ij\}\>0in the regime whereℒsep\\mathcal\{L\}\_\{\\mathrm\{sep\}\}is intended to act\.

###### Proposition 1\.

SupposeI​\(\(X,A\);\(Ei,Ej\)∣ea\)≤ϵaI\(\(X,A\);\(E\_\{i\},E\_\{j\}\)\\mid e\_\{a\}\)\\leq\\epsilon\_\{a\}andI​\(X;\(Ei,Ej\)∣ex\)≤ϵxI\(X;\(E\_\{i\},E\_\{j\}\)\\mid e\_\{x\}\)\\leq\\epsilon\_\{x\}, i\.e\., the boundary embeddings areϵ\\epsilon\-sufficient summaries of their attention contexts\. Then

\|I\(Ei;Ej∣ea\)−Ii​j⋆\|\\displaystyle\\big\|\\,I\(E\_\{i\};E\_\{j\}\\mid e\_\{a\}\)\-I^\{\\star\}\_\{ij\}\\,\\big\|≤ϵa,\\displaystyle\\;\\leq\\;\\epsilon\_\{a\},\(6\)\|I\(Ei;Ej∣ex\)−\(Ii​j⋆\+Δi​j\)\|\\displaystyle\\big\|\\,I\(E\_\{i\};E\_\{j\}\\mid e\_\{x\}\)\-\(I^\{\\star\}\_\{ij\}\+\\Delta\_\{ij\}\)\\,\\big\|≤ϵx,\\displaystyle\\;\\leq\\;\\epsilon\_\{x\},\(7\)I​\(Ei;Ej∣ex\)−I​\(Ei;Ej∣ea\)\\displaystyle I\(E\_\{i\};E\_\{j\}\\mid e\_\{x\}\)\-I\(E\_\{i\};E\_\{j\}\\mid e\_\{a\}\)≥Δi​j−\(ϵa\+ϵx\)\.\\displaystyle\\;\\geq\\;\\Delta\_\{ij\}\-\(\\epsilon\_\{a\}\+\\epsilon\_\{x\}\)\.\(8\)The same lower bound[8](https://arxiv.org/html/2607.20500#S3.E8)holds whenexe\_\{x\}is replaced by anyσ​\(X\)\\sigma\(X\)\-measurable,ϵx\\epsilon\_\{x\}\-sufficient statistic ofXX\.

The proposition partitions cross\-flow dependence into a residual we wish to penalize and a componentΔi​j\\Delta\_\{ij\}contributed by the analysis\. Conditioning oneae\_\{a\}targets only the former; conditioning onexe\_\{x\}pulls both terms down jointly, which means the gradient delivered toθ\\thetacontains a component that pushes the boundary embeddings toward forgetting the partitionAAverbalised\. On multi\-flow inputs, where the partition is exactly the structure several flows must encode differently, this gradient component drives the very flow collapse the regularizer is meant to prevent\. The proof is given in Appendix[A](https://arxiv.org/html/2607.20500#A1)and the no\-analysis ablation is given in Section Experiment[4\.4](https://arxiv.org/html/2607.20500#S4.SS4)\.

Witheae\_\{a\}fixed as the analysis\-level baseline, the conditioning vector must absorb one further source of dependence specific to the autoregressive setting\. In any causal LM, the receptive field ofeje\_\{j\}forj\>ij\>icontainseie\_\{i\}via attention over the intervening tokens, soI​\(ei;ej∣ea\)I\(e\_\{i\};e\_\{j\}\\mid e\_\{a\}\)has a baseline level fixed by architecture rather than by content; minimizing it without absorbing this baseline penalises the model for an information\-flow it cannot avoid\. We therefore append a preceding\-flow summarye¯<m=1m−1​∑j<mej\\bar\{e\}\_\{<m\}=\\tfrac\{1\}\{m\-1\}\\sum\_\{j<m\}e\_\{j\}withe¯<1:=0\\bar\{e\}\_\{<1\}:=0, giving

wi​j=\[ea;e¯<min⁡\(i,j\)\],w\_\{ij\}\\;=\\;\\big\[\\,e\_\{a\}\\,;\\;\\bar\{e\}\_\{<\\min\(i,j\)\}\\,\\big\],\(9\)under whichI^CLUB\\widehat\{I\}\_\{\\mathrm\{CLUB\}\}measures only the residual cross\-flow redundancy\. The sufficiency term \([3](https://arxiv.org/html/2607.20500#S3.E3)\) requires a strict subset of these absorptions: onlyeae\_\{a\}matters, since the autoregressive baseline is between sibling hypothesis representations and not between a hypothesis and its own answer\. We accordingly setw=eaw=e\_\{a\}inI^NCE\\widehat\{I\}\_\{\\mathrm\{NCE\}\}, recovering \([4](https://arxiv.org/html/2607.20500#S3.E4)\) as written\.

#### End\-to\-End Training Procedure\.

With the directional bounds of \([4](https://arxiv.org/html/2607.20500#S3.E4)\)–\([5](https://arxiv.org/html/2607.20500#S3.E5)\) and the conditioning of \([9](https://arxiv.org/html/2607.20500#S3.E9)\) in place, the per\-property losses

ℒsuf​\(θ\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{suf\}\}\(\\theta\)=−1K​∑k=1KI^NCE​\(ek;eyk∣ea\),\\displaystyle=\-\\frac\{1\}\{K\}\\sum\_\{k=1\}^\{K\}\\widehat\{I\}\_\{\\mathrm\{NCE\}\}\(e\_\{k\}\\,;\\,e\_\{y\_\{k\}\}\\mid e\_\{a\}\),\(10\)ℒsep​\(θ\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{sep\}\}\(\\theta\)=1\|𝒫\|​∑\(i,j\)∈𝒫clip​\(I^CLUB​\(ei;ej∣wi​j\),−1,\+1\),\\displaystyle=\\frac\{1\}\{\|\\mathcal\{P\}\|\}\\sum\_\{\(i,j\)\\in\\mathcal\{P\}\}\\mathrm\{clip\}\\\!\\big\(\\widehat\{I\}\_\{\\mathrm\{CLUB\}\}\(e\_\{i\}\\,;\\,e\_\{j\}\\mid w\_\{ij\}\),\\,\-1,\\,\+1\\big\),\(11\)with𝒫=\{\(i,j\):1≤i<j≤K\}\\mathcal\{P\}=\\\{\(i,j\):1\\leq i<j\\leq K\\\}, combine into the flow\-regulation loss

ℒflow​\(θ\)=ℒsuf​\(θ\)\+ℒsep​\(θ\)\.\\mathcal\{L\}\_\{\\mathrm\{flow\}\}\(\\theta\)\\;=\\;\\mathcal\{L\}\_\{\\mathrm\{suf\}\}\(\\theta\)\\;\+\\;\\mathcal\{L\}\_\{\\mathrm\{sep\}\}\(\\theta\)\.\(12\)The per\-pair clip inℒsep\\mathcal\{L\}\_\{\\mathrm\{sep\}\}bounds the contribution of any single pair\-batch to a fixed nats budget; without it, the variance ofI^CLUB\\widehat\{I\}\_\{\\mathrm\{CLUB\}\}at the small per\-step pair count\|𝒫\|\|\\mathcal\{P\}\|available within one minibatch admits occasional excursions whose magnitude is uncorrelated with the underlying redundancy and whose effect onθ\\thetais therefore noise\. To provide a substrate of structured generation on whichℒflow\\mathcal\{L\}\_\{\\mathrm\{flow\}\}can act, we pair it with the standard token\-level generation loss over the structured output \([2](https://arxiv.org/html/2607.20500#S3.E2)\),

ℒgen​\(θ\)=−log⁡pθ​\(a⋆∣x\)−∑k=1K⋆\[log⁡pθ​\(zk⋆∣x,a⋆,z<k⋆,y<k⋆\)\+log⁡pθ​\(yk⋆∣x,a⋆,z≤k⋆,y<k⋆\)\],\\mathcal\{L\}\_\{\\mathrm\{gen\}\}\(\\theta\)\\;=\\;\-\\log p\_\{\\theta\}\(a^\{\\star\}\\mid x\)\-\\sum\_\{k=1\}^\{K^\{\\star\}\}\\big\[\\log p\_\{\\theta\}\(z\_\{k\}^\{\\star\}\\mid x,a^\{\\star\},z\_\{<k\}^\{\\star\},y\_\{<k\}^\{\\star\}\)\+\\log p\_\{\\theta\}\(y\_\{k\}^\{\\star\}\\mid x,a^\{\\star\},z\_\{\\leq k\}^\{\\star\},y\_\{<k\}^\{\\star\}\)\\big\],\(13\)and the policy is trained onℒgen\+λ​ℒflow\\mathcal\{L\}\_\{\\mathrm\{gen\}\}\+\\lambda\\,\\mathcal\{L\}\_\{\\mathrm\{flow\}\}whereλ\\lambdacontrols the relative weight\.

## 4Experiments

### 4\.1Experiment Setup

Dataset\.We construct a dataset of5,0005\{,\}000ill\-posed reasoning problems involving conflicts\. Each instance is a tuple\(x,a,\{\(zk,yk\)\}k=1K⋆\)\(x,a,\\\{\(z\_\{k\},y\_\{k\}\)\\\}\_\{k=1\}^\{K^\{\\star\}\}\): a problemxx, an analysisaaidentifying its underspecified conditions, andK⋆∈\{1,2,3,4\}K^\{\\star\}\\\!\\in\\\!\\\{1,2,3,4\\\}valid resolutions, each pairing a hypothesiszkz\_\{k\}with the answeryky\_\{k\}that follows under it\. We explicitly include well\-posed instances \(K⋆=1K^\{\\star\}\\\!=\\\!1\) so the model cannot exploit a fixed enumeration prior\. Problems span three domain categories \(Pure Math, Daily, Application\) and three difficulty tiers\. Each instance is rendered into a single causal sequence\[ANA\]​a​\[/ANA\]​\[B\]​z1​\[SEP\]​y1​\[/B\]​⋯​\[END\]\\texttt\{\[ANA\]\}\\,a\\,\\texttt\{\[/ANA\]\}\\,\\texttt\{\[B\]\}\\,z\_\{1\}\\,\\texttt\{\[SEP\]\}\\,y\_\{1\}\\,\\texttt\{\[/B\]\}\\cdots\\texttt\{\[END\]\}, exposing one boundary token per hypothesis and answer on whichℒflow\\mathcal\{L\}\_\{\\mathrm\{flow\}\}is computed\. Further details are in Appendix[D\.1](https://arxiv.org/html/2607.20500#A4.SS1.SSS0.Px5)\.

Baselines and training\.We compare against closed\-source prompted models \(Claude Haiku 4\.5, GPT\-5, Gemini 2\.5 Pro\) and the same open\-source backbones under prompting only, each evaluated under a*blind*prompt \(no mention of ill\-posedness\) and an*ill\-posed*prompt \(explicit enumeration instruction\); the ill\-posed prompt is held identical across baselines to isolate the training contribution from prompt engineering\. We instantiateFlowEditon Qwen3\-4B\-Base and Qwen2\.5\-3B\-Instruct\[[35](https://arxiv.org/html/2607.20500#bib.bib43)\], training both underℒgen\+λ​ℒflow\\mathcal\{L\}\_\{\\mathrm\{gen\}\}\+\\lambda\\ \\mathcal\{L\}\_\{\\mathrm\{flow\}\}for 2,000 optimizer steps withλ=0\.1\\lambda=0\.1, using the*ill\-posed*prompt\. The parameter sensitivity experiments are illustrated in Appendix[B\.1](https://arxiv.org/html/2607.20500#A2.SS1)We report exact\-set\-match \(EM\) and information recovery \(IR\), broken down by domain and byK⋆∈\{1,2,3,4\}K^\{\\star\}\\in\\\{1,2,3,4\\\}\.

### 4\.2Main Results

![Refer to caption](https://arxiv.org/html/2607.20500v1/x2.png)

![Refer to caption](https://arxiv.org/html/2607.20500v1/x3.png)

Figure 2:Training dynamics ofFlowEditbased on Qwen\-3\-4B Base and Qwen\-2\.5\-3B\-Instruct with metrics of EM and IR\. It is illustrated that the training ofFlowEditcan converge stably with high performance\. Shaded regions indicate standard deviation across three seeds\.*Exact\-set\-match*\(EM\) credits a response only when𝒴^​\(x\)=𝒴⋆​\(x\)\\hat\{\\mathcal\{Y\}\}\(x\)=\\mathcal\{Y\}^\{\\star\}\(x\), treating every deviation as a uniform failure and remaining silent on the share of valid resolutions a partially correct response recovers\. We complement it with*information recovery*\(IR\),

IR​\(x\)=\|𝒴^​\(x\)∩𝒴⋆​\(x\)\|max⁡\(\|𝒴^​\(x\)\|,\|𝒴⋆​\(x\)\|\),\\mathrm\{IR\}\(x\)\\;=\\;\\frac\{\|\\hat\{\\mathcal\{Y\}\}\(x\)\\cap\\mathcal\{Y\}^\{\\star\}\(x\)\|\}\{\\max\\\!\\big\(\|\\hat\{\\mathcal\{Y\}\}\(x\)\|,\\,\|\\mathcal\{Y\}^\{\\star\}\(x\)\|\\big\)\},\(14\)which penalises missing flows through the numerator \(a silent commitment onK⋆=4K^\{\\star\}\{=\}4yieldsIR≤1/4\\mathrm\{IR\}\\\!\\leq\\\!1/4\) and spurious flows through the denominator, ruling out the trivial strategy of over\-emitting to maximise coverage\. IR reduces to EM onK⋆=1K^\{\\star\}\{=\}1inputs\.

#### Training Dynamics\.

Figure[2](https://arxiv.org/html/2607.20500#S4.F2)tracks EM and IR across training\. The two metrics target the failure modesℒflow\\mathcal\{L\}\_\{\\mathrm\{flow\}\}prevents: missing flows depress IR, while flow collapse onto duplicate hypotheses depresses EM without lowering IR\. At both scales the two curves ascend jointly and peak in the same region \(EM=0\.508/IR=0\.538\\mathrm\{EM\}\{=\}0\.508/\\mathrm\{IR\}\{=\}0\.538at 4B, step∼1075\{\\sim\}1075;EM=0\.444/IR=0\.462\\mathrm\{EM\}\{=\}0\.444/\\mathrm\{IR\}\{=\}0\.462at 3B, step∼1525\{\\sim\}1525\)\. Joint ascent is the trajectory\-level counterpart of the dual\-objective design:ℒsuf\\mathcal\{L\}\_\{\\mathrm\{suf\}\}raises IR by anchoring each branch to its answer, whileℒsep\\mathcal\{L\}\_\{\\mathrm\{sep\}\}raises EM by enforcing set\-distinctness\. Decoupling—IR rising without EM—does not appear at either scale, indicating that the two terms are jointly active rather than competing for capacity\. We adopt single\-criterion selection on validation EM for all results in Table[1](https://arxiv.org/html/2607.20500#S4.T1)\.

#### Scaling behaviour\.

The 4B and 3B trajectories share their qualitative shape, indicating thatℒflow\\mathcal\{L\}\_\{\\mathrm\{flow\}\}transfers across scale without re\-tuning\. Scaling from 3B to 4B improves IR by0\.0760\.076and EM by0\.0640\.064at comparable rates, consistent with the dual objective being jointly capacity\-bound rather than one term saturating first\. The persistent gap of∼0\.03\{\\sim\}0\.03between IR and EM at both scales reflects the structural asymmetry of the metrics: set\-equality is strictly stronger than coverage at every capacity\.

#### Performance and informativeness\.

FlowEdit\-Qwen3\-4B\-Base attainsEM=0\.47\\mathrm\{EM\}\{=\}0\.47andIR=0\.51\\mathrm\{IR\}\{=\}0\.51, exceeding the strongest closed\-source baseline \(Haiku\-4\.5 with the ill\-posed prompt,0\.28/0\.410\.28/0\.41\) by\+68%\+68\\%on EM and\+24%\+24\\%on IR, consistently across all three domains and at both scales\. The smaller relative gain on IR is diagnostic: baselines that silently commit onK⋆\>1K^\{\\star\}\{\>\}1inputs already accrue1/K⋆1/K^\{\\star\}partial credit, so IR headroom is structurally narrower than EM\. Against open\-source prompted baselines the gap widens to\+88%\+88\\%EM and\+24%\+24\\%IR \(over Qwen3\-4B with the ill\-posed prompt\), confirming the gain stems from training rather than base\-model capacity\. The advantage is most pronounced where silent commitment is most costly: onK⋆=2K^\{\\star\}\{=\}2inputs,FlowEditreachesEM=0\.40,0\.54,0\.65\\mathrm\{EM\}\{=\}0\.40,0\.54,0\.65across domains versus0\.20,0\.27,0\.520\.20,0\.27,0\.52for the strongest closed\-source baseline—the regime where parallel hypothesis maintenance pays off most against the1/K⋆1/K^\{\\star\}ceiling of single\-hypothesis prompting\.

Table 1:Branch\-level performance and informativeness on three problem domains \(Pure Math,Daily,Application\), broken down by ground\-truth branch countK⋆∈\{1,2,3,4\}K^\{\\star\}\\\!\\in\\\!\\\{1,2,3,4\\\}, with per\-domain and overall weighted averages\. Best result in each column is inbold\.FlowEditconsistently outperforms across all baselines\.

### 4\.3Token\-Level Entropy Signatures of Flow Regulation

![Refer to caption](https://arxiv.org/html/2607.20500v1/x4.png)Figure 3:Per\-sample teacher\-forced next\-token entropy, one example perK⋆∈\{1,2,3,4\}K^\{\\star\}\\in\\\{1,2,3,4\\\}\. The trained trace suppresses entropy inside answer spans and amplifies it at branch\-opening\[B\]tokens\.![Refer to caption](https://arxiv.org/html/2607.20500v1/x5.png)Figure 4:Window\-aggregated entropy \(mean±\\pm95% CI\) at\[B\]\(D1\\mathrm\{D\}\_\{1\}\) and\[SEP\]\(D2\\mathrm\{D\}\_\{2\}\)\. TheD1\\mathrm\{D\}\_\{1\}gap vanishes atK⋆=1K^\{\\star\}\\\!=\\\!1and grows withK⋆K^\{\\star\}\.We probe whetherℒflow\\mathcal\{L\}\_\{\\mathrm\{flow\}\}leaves observable signatures in token\-level entropy through two diagnostic windows\.D1\\mathrm\{D\}\_\{1\}spans the 10 tokens after each\[B\]\(hypothesis opening\);D2\\mathrm\{D\}\_\{2\}spans the 10 tokens after each\[SEP\]\(answer span\)\. The two windows isolate the two loss terms:ℒsuf\\mathcal\{L\}\_\{\\mathrm\{suf\}\}predicts trainedD2\\mathrm\{D\}\_\{2\}collapses toward zero, whileℒsep\\mathcal\{L\}\_\{\\mathrm\{sep\}\}predicts trainedD1\\mathrm\{D\}\_\{1\}retains nontrivial mass over𝒵⋆​\(x\)∖\{z^<k\}\\mathcal\{Z\}^\{\\star\}\(x\)\\setminus\\\{\\hat\{z\}\_\{<k\}\\\}\.

Figure[3](https://arxiv.org/html/2607.20500#S4.F3)illustrates both effects on representative samples; Figure[4](https://arxiv.org/html/2607.20500#S4.F4)confirms them in expectation\. On everyK⋆\>1K^\{\\star\}\\\!\>\\\!1panel, trainedD2\\mathrm\{D\}\_\{2\}lies strictly below the base with disjoint confidence intervals across all 10 positions, while trainedD1\\mathrm\{D\}\_\{1\}retains boundary mass that decays only gradually\. Two regularities sharpen the reading\. \(i\) OnK⋆=1K^\{\\star\}\\\!=\\\!1, trained and baseD1\\mathrm\{D\}\_\{1\}coincide:ℒsep\\mathcal\{L\}\_\{\\mathrm\{sep\}\}admits no active pair, so the absence of a boundary spike reflects the objective by design\. \(ii\) TheD1\\mathrm\{D\}\_\{1\}gap grows monotonically fromK⋆=2K^\{\\star\}\\\!=\\\!2toK⋆=4K^\{\\star\}\\\!=\\\!4, contrary to collapse—under separation, each new element of𝒵⋆​\(x\)\\mathcal\{Z\}^\{\\star\}\(x\)must be held distinct from those already opened, registering its uncertainty at the\[B\]that opens it\.

Read with Table[1](https://arxiv.org/html/2607.20500#S4.T1), this dissociation localizes the effect ofℒflow\\mathcal\{L\}\_\{\\mathrm\{flow\}\}to the token positions our derivation predicts—evidence thatFlowEditshapes the reasoning trajectory at the boundaries where flows are addressed, rather than producing diffuse gains across the sequence\.

### 4\.4Ablation Study

Removing either component ofℒflow\\mathcal\{L\}\_\{\\mathrm\{flow\}\}degrades both metrics, but droppingℒsuf\\mathcal\{L\}\_\{\\mathrm\{suf\}\}produces a markedly larger drop on EM \(−0\.07\-0\.07\) and IR \(−0\.05\-0\.05\) than droppingℒsep\\mathcal\{L\}\_\{\\mathrm\{sep\}\}\(−0\.04\-0\.04and−0\.03\-0\.03\), identifying sufficiency as the dominant term\. This matches the geometric roles in §3\.2:ℒsuf\\mathcal\{L\}\_\{\\mathrm\{suf\}\}anchors eacheke\_\{k\}toeyke\_\{y\_\{k\}\}and fixes what each flow encodes, whileℒsep\\mathcal\{L\}\_\{\\mathrm\{sep\}\}only requires distinctei,eje\_\{i\},e\_\{j\}\. Without the anchor, separation spreads representations apart in directions that need not correspond to distinct answers, explaining whyℒgen\\mathcal\{L\}\_\{\\mathrm\{gen\}\}alone matchesℒgen\+ℒsep\\mathcal\{L\}\_\{\\mathrm\{gen\}\}\+\\mathcal\{L\}\_\{\\mathrm\{sep\}\}to within seed variance \(0\.3370\.337vs0\.3300\.330EM\): the dual objective is coupled rather than additive\.

Replacingeae\_\{a\}withexe\_\{x\}in Eq\.[9](https://arxiv.org/html/2607.20500#S3.E9)produces an EM degradation of−0\.07\-0\.07, comparable to removingℒsuf\\mathcal\{L\}\_\{\\mathrm\{suf\}\}outright\. This is the empirical counterpart of Proposition 1: underexe\_\{x\}the regularizer also penalises the partition\-encoding dependenceΔi​j\\Delta\_\{ij\}thatAAcontributes beyondXX, forcing boundary embeddings to forget precisely the structure distinct flows must encode differently\. That a mis\-conditioned separation term is as costly as undermining sufficiency justifies treatingϵ\\epsilon\-sufficiency as load\-bearing rather than heuristic\.FlowEditoutperforms every ablation on both metrics across all three domains\.

Table 2:Ablation on Qwen2\.5\-3B\-Instruct\.

## 5Conclusion

We presentedFlowEdit, a framework that recasts reasoning under conflicting conditions from a detection problem into a flow\-regulation problem: rather than asking whether an ill\-posed input should be rejected, we ask how a model should keep several valid resolutions distinct as a single autoregressive trace unfolds\. The two requirements this entails, namely that each flow be sufficient for its own answer and that distinct flows not collapse into paraphrases of one another, admit a clean expression as conditional mutual information objectives on boundary representations\. Across three domains and four branch counts,FlowEditwith Qwen3\-4B\-Base improves exact\-set\-match accuracy by68%68\\%and information recovery by24%24\\%over the strongest closed\-source baseline, with gains widening asK⋆K^\{\\star\}grows\. Token\-level analysis further reveals that these gains coincide with a structured redistribution of next\-token entropy: uncertainty concentrates inside each branch and amplifies at the boundaries between flows, with the boundary signature scaling monotonically with the number of resolutions the problem requires\. These results give a representation\-level account of why parallel hypothesis maintenance succeeds where sequential elaboration does not\. More broadly, this work suggests that the limits of next\-token prediction on open\-world inputs may be addressable by shaping the internal information geometry of the trace itself\. A limitation of the present scope is thatFlowEditregulates flow geometry once branching has begun; the upstream decision of whether to branch remains implicit in the policy and are not the objects of the loss, which could be considered for the next step of studying internal reasoning flows\.

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## Appendix AProof of Proposition 1

We restate the setting before the proof for self\-containedness\. LetXXdenote the random problem statement andAAthe analysis text generated fromXX, so thatea=ha​\(X,A\)e\_\{a\}=h\_\{a\}\(X,A\)andex=hx​\(X\)e\_\{x\}=h\_\{x\}\(X\)are deterministic functions of their attention contexts\. The boundary embeddingsE1,…,EKE\_\{1\},\\dots,E\_\{K\}are produced autoregressively afterAA\. We assume theϵ\\epsilon\-sufficiency bounds

I​\(\(X,A\);\(Ei,Ej\)\|ea\)≤ϵa,I​\(X;\(Ei,Ej\)\|ex\)≤ϵx,I\\\!\\bigl\(\(X,A\);\\,\(E\_\{i\},E\_\{j\}\)\\,\\big\|\\,e\_\{a\}\\bigr\)\\;\\leq\\;\\epsilon\_\{a\},\\qquad I\\\!\\bigl\(X;\\,\(E\_\{i\},E\_\{j\}\)\\,\\big\|\\,e\_\{x\}\\bigr\)\\;\\leq\\;\\epsilon\_\{x\},\(15\)and writeIi​j⋆:=I​\(Ei;Ej∣X,A\)I^\{\\star\}\_\{ij\}:=I\(E\_\{i\};E\_\{j\}\\mid X,A\)andΔi​j:=I​\(Ei;Ej∣X\)−Ii​j⋆\\Delta\_\{ij\}:=I\(E\_\{i\};E\_\{j\}\\mid X\)\-I^\{\\star\}\_\{ij\}\.

### Two elementary facts

\(F1\) Determinism\.IfC=g​\(U\)C=g\(U\)thenI\(⋅;⋅∣U,C\)=I\(⋅;⋅∣U\)I\(\\,\\cdot\\,;\\,\\cdot\\mid U,C\)=I\(\\,\\cdot\\,;\\,\\cdot\\mid U\)\.

\(F2\) Chain rule\.I​\(U;V,W∣C\)=I​\(U;V∣C\)\+I​\(U;W∣C,V\)I\(U;V,W\\mid C\)=I\(U;V\\mid C\)\+I\(U;W\\mid C,V\), all three terms non\-negative; this holds for arbitrary conditioningCC\.

### Step 1: bound\|I\(Ei;Ej∣ea\)−Ii​j⋆\|\|I\(E\_\{i\};E\_\{j\}\\mid e\_\{a\}\)\-I^\{\\star\}\_\{ij\}\|

Apply \(F2\) toI​\(\(X,A\);\(Ei,Ej\)∣ea\)I\(\(X,A\);\(E\_\{i\},E\_\{j\}\)\\mid e\_\{a\}\)in two orderings:

I​\(\(X,A\);\(Ei,Ej\)∣ea\)\\displaystyle I\\bigl\(\(X,A\);\(E\_\{i\},E\_\{j\}\)\\mid e\_\{a\}\\bigr\)=I​\(\(X,A\);Ei∣ea\)\+I​\(\(X,A\);Ej∣ea,Ei\)\\displaystyle=I\\bigl\(\(X,A\);E\_\{i\}\\mid e\_\{a\}\\bigr\)\+I\\bigl\(\(X,A\);E\_\{j\}\\mid e\_\{a\},E\_\{i\}\\bigr\)=I​\(\(X,A\);Ej∣ea\)\+I​\(\(X,A\);Ei∣ea,Ej\)\.\\displaystyle=I\\bigl\(\(X,A\);E\_\{j\}\\mid e\_\{a\}\\bigr\)\+I\\bigl\(\(X,A\);E\_\{i\}\\mid e\_\{a\},E\_\{j\}\\bigr\)\.\(16\)All four summands are non\-negative; \([15](https://arxiv.org/html/2607.20500#A1.E15)\) bounds the LHS byϵa\\epsilon\_\{a\}, so each summand is≤ϵa\\leq\\epsilon\_\{a\}\. In particular,

I​\(\(X,A\);Ei∣ea\)≤ϵa,I​\(\(X,A\);Ei∣ea,Ej\)≤ϵa\.I\\bigl\(\(X,A\);E\_\{i\}\\mid e\_\{a\}\\bigr\)\\leq\\epsilon\_\{a\},\\qquad I\\bigl\(\(X,A\);E\_\{i\}\\mid e\_\{a\},E\_\{j\}\\bigr\)\\leq\\epsilon\_\{a\}\.\(17\)
Apply \(F2\) toI​\(Ei;\(X,A\),Ej∣ea\)I\(E\_\{i\};\(X,A\),E\_\{j\}\\mid e\_\{a\}\)in two orderings:

I​\(Ei;\(X,A\),Ej∣ea\)\\displaystyle I\(E\_\{i\};\(X,A\),E\_\{j\}\\mid e\_\{a\}\)=I​\(Ei;Ej∣ea\)\+I​\(Ei;\(X,A\)∣ea,Ej\)\\displaystyle=I\(E\_\{i\};E\_\{j\}\\mid e\_\{a\}\)\+I\\bigl\(E\_\{i\};\(X,A\)\\mid e\_\{a\},E\_\{j\}\\bigr\)=I​\(Ei;\(X,A\)∣ea\)\+I​\(Ei;Ej∣ea,X,A\)\.\\displaystyle=I\\bigl\(E\_\{i\};\(X,A\)\\mid e\_\{a\}\\bigr\)\+I\\bigl\(E\_\{i\};E\_\{j\}\\mid e\_\{a\},X,A\\bigr\)\.\(18\)Equating the two expansions and rearranging,

I​\(Ei;Ej∣ea\)−I​\(Ei;Ej∣ea,X,A\)=I​\(Ei;\(X,A\)∣ea\)−I​\(Ei;\(X,A\)∣ea,Ej\)\.I\(E\_\{i\};E\_\{j\}\\mid e\_\{a\}\)\-I\(E\_\{i\};E\_\{j\}\\mid e\_\{a\},X,A\)=I\\bigl\(E\_\{i\};\(X,A\)\\mid e\_\{a\}\\bigr\)\-I\\bigl\(E\_\{i\};\(X,A\)\\mid e\_\{a\},E\_\{j\}\\bigr\)\.\(19\)By symmetry of mutual information in its two arguments, the two RHS terms in \([19](https://arxiv.org/html/2607.20500#A1.E19)\) equal the two quantities bounded in \([17](https://arxiv.org/html/2607.20500#A1.E17)\) respectively, so each lies in\[0,ϵa\]\[0,\\epsilon\_\{a\}\]\. The difference of two values in\[0,ϵa\]\[0,\\epsilon\_\{a\}\]lies in\[−ϵa,ϵa\]\[\-\\epsilon\_\{a\},\\epsilon\_\{a\}\], hence

\|I\(Ei;Ej∣ea\)−I\(Ei;Ej∣ea,X,A\)\|≤ϵa\.\\bigl\|\\,I\(E\_\{i\};E\_\{j\}\\mid e\_\{a\}\)\-I\(E\_\{i\};E\_\{j\}\\mid e\_\{a\},X,A\)\\,\\bigr\|\\;\\leq\\;\\epsilon\_\{a\}\.Sinceea=ha​\(X,A\)e\_\{a\}=h\_\{a\}\(X,A\), applying \(F1\) withU=\(X,A\)U=\(X,A\)givesI​\(Ei;Ej∣ea,X,A\)=I​\(Ei;Ej∣X,A\)=Ii​j⋆I\(E\_\{i\};E\_\{j\}\\mid e\_\{a\},X,A\)=I\(E\_\{i\};E\_\{j\}\\mid X,A\)=I^\{\\star\}\_\{ij\}, so

\|I\(Ei;Ej∣ea\)−Ii​j⋆\|≤ϵa\.\\bigl\|\\,I\(E\_\{i\};E\_\{j\}\\mid e\_\{a\}\)\-I^\{\\star\}\_\{ij\}\\,\\bigr\|\\;\\leq\\;\\epsilon\_\{a\}\.\(20\)

### Step 2: bound\|I\(Ei;Ej∣ex\)−\(Ii​j⋆\+Δi​j\)\|\|I\(E\_\{i\};E\_\{j\}\\mid e\_\{x\}\)\-\(I^\{\\star\}\_\{ij\}\+\\Delta\_\{ij\}\)\|

We repeat Step 1 with the substitution\(X,A\)↦X\(X,A\)\\\!\\mapsto\\\!X,ea↦exe\_\{a\}\\\!\\mapsto\\\!e\_\{x\},ϵa↦ϵx\\epsilon\_\{a\}\\\!\\mapsto\\\!\\epsilon\_\{x\}\. The logical structure is identical, so we state the substituted intermediate results\.

Applying \(F2\) toI​\(X;\(Ei,Ej\)∣ex\)I\(X;\(E\_\{i\},E\_\{j\}\)\\mid e\_\{x\}\)in two orderings and using \([15](https://arxiv.org/html/2607.20500#A1.E15)\) yields, in analogy with \([17](https://arxiv.org/html/2607.20500#A1.E17)\),

I​\(X;Ei∣ex\)≤ϵx,I​\(X;Ei∣ex,Ej\)≤ϵx\.I\(X;E\_\{i\}\\mid e\_\{x\}\)\\leq\\epsilon\_\{x\},\\qquad I\(X;E\_\{i\}\\mid e\_\{x\},E\_\{j\}\)\\leq\\epsilon\_\{x\}\.\(21\)Applying \(F2\) toI​\(Ei;X,Ej∣ex\)I\(E\_\{i\};X,E\_\{j\}\\mid e\_\{x\}\)in two orderings and rearranging,

I​\(Ei;Ej∣ex\)−I​\(Ei;Ej∣ex,X\)=I​\(Ei;X∣ex\)−I​\(Ei;X∣ex,Ej\),I\(E\_\{i\};E\_\{j\}\\mid e\_\{x\}\)\-I\(E\_\{i\};E\_\{j\}\\mid e\_\{x\},X\)=I\(E\_\{i\};X\\mid e\_\{x\}\)\-I\(E\_\{i\};X\\mid e\_\{x\},E\_\{j\}\),whose RHS lies in\[−ϵx,ϵx\]\[\-\\epsilon\_\{x\},\\epsilon\_\{x\}\]by symmetry of MI and \([21](https://arxiv.org/html/2607.20500#A1.E21)\)\. Sinceex=hx​\(X\)e\_\{x\}=h\_\{x\}\(X\), \(F1\) withU=XU=XgivesI​\(Ei;Ej∣ex,X\)=I​\(Ei;Ej∣X\)I\(E\_\{i\};E\_\{j\}\\mid e\_\{x\},X\)=I\(E\_\{i\};E\_\{j\}\\mid X\), so

\|I\(Ei;Ej∣ex\)−I\(Ei;Ej∣X\)\|≤ϵx\.\\bigl\|\\,I\(E\_\{i\};E\_\{j\}\\mid e\_\{x\}\)\-I\(E\_\{i\};E\_\{j\}\\mid X\)\\,\\bigr\|\\;\\leq\\;\\epsilon\_\{x\}\.By definitionI​\(Ei;Ej∣X\)=Ii​j⋆\+Δi​jI\(E\_\{i\};E\_\{j\}\\mid X\)=I^\{\\star\}\_\{ij\}\+\\Delta\_\{ij\}, giving

\|I\(Ei;Ej∣ex\)−\(Ii​j⋆\+Δi​j\)\|≤ϵx\.\\bigl\|\\,I\(E\_\{i\};E\_\{j\}\\mid e\_\{x\}\)\-\(I^\{\\star\}\_\{ij\}\+\\Delta\_\{ij\}\)\\,\\bigr\|\\;\\leq\\;\\epsilon\_\{x\}\.\(22\)

### Step 3: combine

From \([20](https://arxiv.org/html/2607.20500#A1.E20)\) and \([22](https://arxiv.org/html/2607.20500#A1.E22)\),I​\(Ei;Ej∣ea\)≤Ii​j⋆\+ϵaI\(E\_\{i\};E\_\{j\}\\mid e\_\{a\}\)\\leq I^\{\\star\}\_\{ij\}\+\\epsilon\_\{a\}andI​\(Ei;Ej∣ex\)≥Ii​j⋆\+Δi​j−ϵxI\(E\_\{i\};E\_\{j\}\\mid e\_\{x\}\)\\geq I^\{\\star\}\_\{ij\}\+\\Delta\_\{ij\}\-\\epsilon\_\{x\}\. Subtracting,

I​\(Ei;Ej∣ex\)−I​\(Ei;Ej∣ea\)≥Δi​j−\(ϵa\+ϵx\)\.I\(E\_\{i\};E\_\{j\}\\mid e\_\{x\}\)\-I\(E\_\{i\};E\_\{j\}\\mid e\_\{a\}\)\\;\\geq\\;\\Delta\_\{ij\}\-\(\\epsilon\_\{a\}\+\\epsilon\_\{x\}\)\.\(23\)

### Extension to arbitraryσ​\(X\)\\sigma\(X\)\-measurableCC

For anyC=g​\(X\)C=g\(X\)satisfyingI​\(X;\(Ei,Ej\)∣C\)≤ϵxI\(X;\(E\_\{i\},E\_\{j\}\)\\mid C\)\\leq\\epsilon\_\{x\}, Step 2 used only \(i\) the determinism property \(F1\), which applies becauseCCis a deterministic function ofXX, and \(ii\) theϵx\\epsilon\_\{x\}\-sufficiency bound withCCin place ofexe\_\{x\}\. Both conditions hold for any suchCC, so\|I\(Ei;Ej∣C\)−I\(Ei;Ej∣X\)\|≤ϵx\|I\(E\_\{i\};E\_\{j\}\\mid C\)\-I\(E\_\{i\};E\_\{j\}\\mid X\)\|\\leq\\epsilon\_\{x\}and the lower bound \([23](https://arxiv.org/html/2607.20500#A1.E23)\) carries over withexe\_\{x\}replaced byCC\.□\\square

### On the sign ofΔi​j\\Delta\_\{ij\}

Proposition 1 is stated for arbitraryΔi​j\\Delta\_\{ij\}; the claim that conditioning oneae\_\{a\}rather thanexe\_\{x\}*shrinks*cross\-flow MI requires in addition thatΔi​j≥0\\Delta\_\{ij\}\\geq 0\. We treat this as a property of the data construction rather than as a consequence of the graphical structure alone\. The pipeline of Section 4\.1 generates eacha⋆a^\{\\star\}as a verbalisation of the latent decomposition that already determines\{\(zk⋆,yk⋆\)\}k=1K⋆\\\{\(z^\{\\star\}\_\{k\},y^\{\\star\}\_\{k\}\)\\\}\_\{k=1\}^\{K^\{\\star\}\}fromxx, soAAcontributes a refinement of partition information already implicit inXXrather than an independent source of dependence between branches\. Strict positivity,Δi​j\>0\\Delta\_\{ij\}\>0, then corresponds to the multi\-flow regime in whichAAidentifies whichzk⋆z^\{\\star\}\_\{k\}each branch will encode and is therefore the regime in whichℒsep\\mathcal\{L\}\_\{\\mathrm\{sep\}\}is intended to act; the empirical counterpart is the no\-analysis ablation reported in Section 4\.4, where replacingeae\_\{a\}withexe\_\{x\}inwi​jw\_\{ij\}degrades EM by0\.070\.07, comparable to removingℒsuf\\mathcal\{L\}\_\{\\mathrm\{suf\}\}outright\.

## Appendix BFurther Analysis

### B\.1Sensitivity to the Flow\-Regulation Weightλ\\lambda

![Refer to caption](https://arxiv.org/html/2607.20500v1/x6.png)Figure 5:Training dynamics under varying flow\-regulation weightλ∈\{0\.05,0\.10,0\.15,0\.20\}\\lambda\\in\\\{0\.05,0\.10,0\.15,0\.20\\\}on Qwen3\-4B\-Base\. EM \(left\) and IR \(right\) are reported every100100optimizer steps\. Trajectories atλ∈\{0\.05,0\.10\}\\lambda\\in\\\{0\.05,0\.10\\\}remain confined to a narrow band after step∼500\\sim\\\!500, whereasλ=0\.15\\lambda\{=\}0\.15andλ=0\.20\\lambda\{=\}0\.20exhibit visible excursions, indicating that stability degrades withλ\\lambda\.![Refer to caption](https://arxiv.org/html/2607.20500v1/x7.png)Figure 6:End\-of\-training EM and IR as a function of the flow\-regulation weightλ\\lambda, selected by validation EM\. Both metrics peak jointly atλ=0\.10\\lambda\{=\}0\.10\(EM=0\.508\{=\}0\.508, IR=0\.542\{=\}0\.542\) and decline monotonically on either side, with steeper degradation above the optimum than below\.We sweepλ∈\{0\.05,0\.10,0\.15,0\.20\}\\lambda\\in\\\{0\.05,0\.10,0\.15,0\.20\\\}on Qwen3\-4B\-Base with all other settings fixed\. Figure[5](https://arxiv.org/html/2607.20500#A2.F5)reports per\-step trajectories; Figure[6](https://arxiv.org/html/2607.20500#A2.F6)reports end\-of\-training values selected by validation EM\.

Concave response with interior optimum\.EM and IR peak jointly atλ=0\.10\\lambda\{=\}0\.10\(EM=0\.508\{=\}0\.508, IR=0\.542\{=\}0\.542\) and decline monotonically on either side, with the EM–IR gap staying within\[0\.030,0\.038\]\[0\.030,0\.038\]\. The two metrics co\-vary because a single weight scales bothℒsuf\\mathcal\{L\}\_\{\\mathrm\{suf\}\}andℒsep\\mathcal\{L\}\_\{\\mathrm\{sep\}\}, which act on disjoint representational targets and are jointly capacity\-bound rather than antagonistic\.

Asymmetric degradation\.Raisingλ\\lambdafrom0\.100\.10to0\.200\.20costs0\.0340\.034EM and0\.0380\.038IR; lowering it to0\.050\.05costs only0\.0080\.008EM and0\.0050\.005IR\. At largeλ\\lambda, the geometric constraints on boundary embeddings overwhelm the token\-level gradient, trading branch correctness for representational separation; at smallλ\\lambda,ℒgen\\mathcal\{L\}\_\{\\mathrm\{gen\}\}remains the dominant signal and a directionally correct nudge fromℒflow\\mathcal\{L\}\_\{\\mathrm\{flow\}\}recovers most of the gain\.

Stability degrades withλ\\lambda\.All runs reach their plateau by step∼500\\sim\\\!500\. Beyond this,λ∈\{0\.05,0\.10\}\\lambda\\in\\\{0\.05,0\.10\\\}remain in a narrow band;λ=0\.15\\lambda\{=\}0\.15shows excursions of∼0\.10\\sim\\\!0\.10EM near step800800; andλ=0\.20\\lambda\{=\}0\.20drops from∼0\.52\\sim\\\!0\.52to∼0\.32\\sim\\\!0\.32near step14001400before partial recovery\. The variance ofI^CLUB\\hat\{I\}\_\{\\mathrm\{CLUB\}\}scales withλ\\lambda, and at large weight becomes commensurate with the token\-level signal despite the\[−1,\+1\]\[\-1,\+1\]clip;λ=0\.10\\lambda\{=\}0\.10keeps the two losses in stable equilibrium\.

Recommendation\.Best\-step EM spans\[0\.474,0\.508\]\[0\.474,0\.508\]and IR spans\[0\.504,0\.542\]\[0\.504,0\.542\]across the sweep, so FlowEdit is not sensitive to fine tuning ofλ\\lambda\. We adoptλ=0\.10\\lambda\{=\}0\.10throughout\.

### B\.2Further Reading of the Token\-Level Entropy Signature

Section[4](https://arxiv.org/html/2607.20500#S4)establishes thatℒflow\\mathcal\{L\}\_\{\\mathrm\{flow\}\}leaves dissociated marks on the two diagnostic windows\. We add one observation that strengthens the representation\-level reading without introducing additional measurements\.

The redistribution induced byℒflow\\mathcal\{L\}\_\{\\mathrm\{flow\}\}is not a uniform damping of next\-token uncertainty\. Across allK⋆K^\{\\star\}in Figure[4](https://arxiv.org/html/2607.20500#S4.F4), the trainedD2\\mathrm\{D\}\_\{2\}curve sits at a near\-zero plateau with disjoint confidence bands from base, indicating that the answer span has become almost deterministic once a hypothesis is committed\. The same trained model retains visibly higherD1\\mathrm\{D\}\_\{1\}mass at the branch\-opening anchor, so the residual uncertainty the policy preserves is concentrated at the tokens that designate which hypothesis a new flow will encode rather than at the tokens that elaborate it\. The two windows therefore describe a redistribution of next\-token uncertainty rather than its global suppression: the trace becomes more decisive within each branch precisely at the positions where commitment to a particular flow has already been made\.

## Appendix CImplementation Details

This appendix consolidates the training and evaluation details required to reproduce the results in Section[4](https://arxiv.org/html/2607.20500#S4): dataset splits, the joint training procedure, optimizer configuration, decoding settings for evaluation, and the compute used to obtain experiment data\.

### C\.1Data Splits

The 5,000 ill\-posed problems described in Section[4](https://arxiv.org/html/2607.20500#S4)are partitioned once into a training and a held\-out validation split, fixed across all runs\. We use a 90/10 split stratified jointly over branch countK⋆∈\{1,2,3,4\}K^\{\\star\}\\\!\\in\\\!\\\{1,2,3,4\\\}and the three domain categories \(Pure Math, Daily, Application\), so the marginal distribution ofK⋆K^\{\\star\}and of domain in validation matches that in training; this prevents differences inK⋆K^\{\\star\}frequency from being absorbed into the metric\.

### C\.2Backbones and Adapter Configuration

FlowEditis instantiated onQwen3\-4B\-BaseandQwen2\.5\-3B\-Instruct\[[35](https://arxiv.org/html/2607.20500#bib.bib43)\]\. We attach a LoRA\[[14](https://arxiv.org/html/2607.20500#bib.bib44)\]adapter to the four attention projections\{q,k,v,o\}\\\{q,k,v,o\\\}of every transformer block \(rankr=16r\{=\}16,α=32\\alpha\{=\}32, dropout 0\.05\); MLP projections are kept frozen\. The six structural tokens\{\\\{\[ANA\],\[/ANA\],\[B\],\[SEP\],\[/B\],\[END\]\}\\\}extend the tokenizer vocabulary; their input and output embedding rows are added to the trainable parameter set so the policy can both consume and emit them\. ForQwen2\.5\-3B\-Instruct, which uses tied input/output embeddings, this means a single shared row per special token is updated\. All training is inbfloat16withflash\_attention\_2; activations are recomputed via gradient checkpointing\.

### C\.3Training Procedure

Both backbones are trained directly from the base checkpoint by jointly optimizingℒgen\+λ​ℒflow\\mathcal\{L\}\_\{\\mathrm\{gen\}\}\+\\lambda\\,\\mathcal\{L\}\_\{\\mathrm\{flow\}\}for 2,000 optimizer steps, withλ\\lambdafixed at0\.100\.10throughout training\. The token\-level termℒgen\\mathcal\{L\}\_\{\\mathrm\{gen\}\}in Eq\.[13](https://arxiv.org/html/2607.20500#S3.E13)and the flow\-regulation termℒflow=ℒsuf\+ℒsep\\mathcal\{L\}\_\{\\mathrm\{flow\}\}=\\mathcal\{L\}\_\{\\mathrm\{suf\}\}\+\\mathcal\{L\}\_\{\\mathrm\{sep\}\}in Eq\.[12](https://arxiv.org/html/2607.20500#S3.E12)are computed on every mini\-batch and back\-propagated through the same set of trainable parameters \(LoRA adapter\+\+special\-token embedding rows\)\. No warm\-start, distillation, or staged ramp onλ\\lambdais used; the joint objective is active from the first optimizer step\.

The CLUB networkqψq\_\{\\psi\}is a two\-layer diagonal\-Gaussian MLP with hidden size256256and dropout 0\.1, predicting the mean and log\-variance ofeje\_\{j\}given the conditioning\(ei,wi​j\)\(e\_\{i\},w\_\{ij\}\)in Eq\.[9](https://arxiv.org/html/2607.20500#S3.E9)\. To keep the upper\-bound property ofI^CLUB\\widehat\{I\}\_\{\\mathrm\{CLUB\}\}approximately tight,qψq\_\{\\psi\}is updated three times per outer optimizer step using its own AdamW optimizer \(see Table[3](https://arxiv.org/html/2607.20500#A3.T3)\) before the policy step that consumes its score\. The InfoNCE scoring headfϕf\_\{\\phi\}has the same hidden size and dropout, is updated jointly with the policy, and uses temperatureτ=0\.07\\tau\{=\}0\.07\. The per\-pair CLUB term is clipped to\[−1,\+1\]\[\-1,\+1\]before averaging, as stated in Eq\.[11](https://arxiv.org/html/2607.20500#S3.E11), to bound the contribution of any single mini\-batch pair to a fixed nats budget\.

All entries in Table[3](https://arxiv.org/html/2607.20500#A3.T3)were either fixed at their HuggingFace Trainer defaults or adopted unchanged from the LoRA fine\-tuning recipe shipped with the Qwen 2\.5 / 3 release\[[35](https://arxiv.org/html/2607.20500#bib.bib43)\], and reused across both backbones and every ablation arm without per\-arm tuning\. We use random seed4242for primary runs and report mean±\\pmone standard deviation across three independent seeds in Fig\.[2](https://arxiv.org/html/2607.20500#S4.F2)and Tab\.[1](https://arxiv.org/html/2607.20500#S4.T1)\.

GroupHyperparameterValuePolicy optimizerOptimizerAdamWLearning rate5×10−55\\\!\\times\\\!10^\{\-5\}LR scheduleLinear warm\-up \+ cosine decayWarm\-up ratio0\.05Weight decay0\.01Adam\(β1,β2,ϵ\)\(\\beta\_\{1\},\\beta\_\{2\},\\epsilon\)\(0\.9,0\.999,1×10−8\)\(0\.9,\\,0\.999,\\,1\\\!\\times\\\!10^\{\-8\}\)Gradient clip \(ℓ2\\ell\_\{2\}norm\)1\.0CLUB optimizer \(qψq\_\{\\psi\}\)OptimizerAdamWLearning rate5×10−45\\\!\\times\\\!10^\{\-4\}Inner updates per outer step3Adam\(β1,β2,ϵ\)\(\\beta\_\{1\},\\beta\_\{2\},\\epsilon\)\(0\.9,0\.999,1×10−8\)\(0\.9,\\,0\.999,\\,1\\\!\\times\\\!10^\{\-8\}\)BatchPer\-device batch size2Gradient accumulation4 \(4B, 2\-GPU\) / 8 \(3B, 1\-GPU\)Effective batch \(problems / step\)16Maximum sequence length2048DurationTotal optimizer steps2,000Mixed precisionbfloat16Table 3:Optimizer and step\-size settings used to trainFlowEdit\. The same configuration is used for bothQwen3\-4B\-BaseandQwen2\.5\-3B\-Instructexcept for the gradient\-accumulation factor, which is adjusted to keep the effective batch size constant across the two\-GPU \(4B\) and one\-GPU \(3B\) configurations\.
### C\.4Decoding and Evaluation Settings

For every evaluation point reported in the paper, including the training dynamics curves in Fig\.[2](https://arxiv.org/html/2607.20500#S4.F2), we decode the validation split with nucleus sampling atT=0\.7T\{=\}0\.7,p=0\.9p\{=\}0\.9, with a hard cap of10241024new tokens forQwen3\-4B\-Baseand15001500forQwen2\.5\-3B\-Instruct, and\[END\]as the stop string\. Decoding is done with the LoRA adapter merged into the base model\. The branchesY^​\(x\)\\hat\{Y\}\(x\)are recovered from the decoded string by parsing the\[B\]⋅\\cdot\[SEP\]⋅\\cdot\[/B\]blocks; matches againstY⋆​\(x\)Y^\{\\star\}\(x\)use numeric tolerance10−310^\{\-3\}for floating\-point answers and a SymPy\-based canonicalization for symbolic ones, identically acrossFlowEditand the prompted baselines\. We average each\(K⋆,domain\)\(K^\{\\star\},\\text\{domain\}\)cell over all examples that fall in it, and then average those cell values to form the per\-domain and overall columns of Table[1](https://arxiv.org/html/2607.20500#S4.T1)\. Variance across the three independent seeds is reported as the shaded band in Fig\.[2](https://arxiv.org/html/2607.20500#S4.F2)\(one standard deviation\)\.

### C\.5Compute and Wall\-Clock Time

All experiments were run on NVIDIA H100 80 GB GPUs through PyTorch≥\\geq2\.3 with HuggingFace Transformers≥\\geq4\.46 and PEFT≥\\geq0\.14\.Qwen3\-4B\-Basewas trained on two GPUs with DDP and mixed\-precisionbfloat16;Qwen2\.5\-3B\-Instructfits on a single H100 and was trained without DDP\. Joint training of theℒgen\+λ​ℒflow\\mathcal\{L\}\_\{\\mathrm\{gen\}\}\+\\lambda\\,\\mathcal\{L\}\_\{\\mathrm\{flow\}\}objective for 2,000 optimizer steps takes approximately 15 hours of wall\-clock time per backbone per seed\.

Including the three independent seeds reported in Fig\.[2](https://arxiv.org/html/2607.20500#S4.F2)and Tab\.[1](https://arxiv.org/html/2607.20500#S4.T1), the four ablation arms in Tab\.[2](https://arxiv.org/html/2607.20500#S4.T2)\( w/oℒflow\\mathcal\{L\}\_\{\\mathrm\{flow\}\}, w/oℒsep\\mathcal\{L\}\_\{\\mathrm\{sep\}\}, w/oℒsuf\\mathcal\{L\}\_\{\\mathrm\{suf\}\}, prompt\-only conditioningexe\_\{x\}\), and the four\-pointλ\\lambdasweep in Appendix[B\.1](https://arxiv.org/html/2607.20500#A2.SS1), the total compute spent on the reported numbers is on the order of several hundred H100\-GPU\-hours\. Closed\-source baselines \(Claude Haiku 4\.5, GPT\-5, Gemini 2\.5 Pro\) were evaluated through their public APIs and contributed only inference cost \(no training\)\.

## Appendix DDataset Details

### D\.1Dataset Construction Pipeline

The dataset is produced by a four\-stage pipeline that fixes the conflict structureK⋆K^\{\\star\}before any natural language is generated, then realizes a problem statement around it\. All four stages are implemented as independent calls to a single frontier LLM; no other model is involved, and the conflict skeleton produced in Stage 1 is preserved verbatim through subsequent stages so thatK⋆K^\{\\star\}is determined by construction rather than recovered post hoc\.

#### Stage 1: Conflict\-structure design\.

For each instance we sampleK⋆K^\{\\star\}from a target distribution over\{1,2,3,4\}\\\{1,2,3,4\\\}, a sub\-domain from a fixed pool that groups into the three high\-level domains \(Pure Math, Daily, Application\), and a difficulty tier from\{\\\{middle school, high school, early college\}\\\}\. The model is prompted to design the conflict skeleton before any prose: it nominates the conflicting quantity, lists itsK⋆K^\{\\star\}candidate values, and specifies a multi\-step computation chain that propagates the conflicting quantity to the final answer\. The chain is required to contain enough computation steps that branches are distinguished by genuine derivation rather than by surface substitution\. A self\-check at the end of the stage asserts that allK⋆K^\{\\star\}branch answers are distinct and that no condition is entangled across branches; failures are discarded\.

#### Stage 2: Surface realization\.

A second call rewrites the Stage\-1 skeleton into the user\-facing problemxx, identifies the contradicting conditions, writes the shared analysis proseaa, and emits a list ofK⋆K^\{\\star\}branch records, each specifying the trusted and discarded condition subset and the well\-posed sub\-problem the branch solves\. No numeric answer is produced at this stage\.

#### Stage 3: Per\-branch solving\.

Each branch’s sub\-problem is dispatched as an independent solver call that returns a derivation ending in a boxed final value\. The narrative may use only the trusted conditions of that branch, and every arithmetic step is shown explicitly to support the verification in Stage 4\.

#### Stage 4: Verification, filtering, and spot\-check\.

A fourth call verifies each branch against five criteria: assumption coherence with the trusted condition subset, step\-by\-step logical validity, arithmetic correctness recomputed by an independent symbolic evaluator, answer–narrative agreement, and overall narrative quality\. The arithmetic check is performed outside the LLM loop so that verification of numeric correctness does not depend on the same model that produced the derivation\. A branch is retained only if all five criteria pass\. We then apply exact and near\-duplicate removal on problem text, drop instances whose branches collide on the boxed value \(which would collapse\|𝒴⋆​\(x\)\|\|\\mathcal\{Y\}^\{\\star\}\(x\)\|belowK⋆K^\{\\star\}\), and discard samples with empty or non\-informative final answers\. A subsequent human spot\-check on a stratified random sample is conducted to confirm mathematical correctness, conflict structure, and analysis fidelity, with failed instances returned to Stage 2 for re\-realization rather than silently dropped, to avoid biasing the difficulty distribution\.

#### Final composition\.

The released dataset contains 5,000 instances spanning the three domains and the four values ofK⋆K^\{\\star\}\. The marginal distribution ofK⋆K^\{\\star\}is skewed toward smaller branch counts, reflecting thatK⋆=4K^\{\\star\}\{=\}4instances require two independent binary conflicts to co\-occur in a single coherent problem statement and are correspondingly rarer; we explicitly includeK⋆=1K^\{\\star\}\{=\}1instances so that the model cannot exploit a fixed enumeration prior\. Because problems are synthesized from conflict\-structure templates rather than collected from existing sources, contamination with public math benchmarks is not expected by construction\.

To illustrate the structure described in Section[4](https://arxiv.org/html/2607.20500#S4), we include one representative training instance for each value ofK⋆∈\{1,2,3,4\}K^\{\\star\}\\\!\\in\\\!\\\{1,2,3,4\\\}\. Every instance is stored as a JSON object with the same schema—problem,analysis, and a list ofbranches, each carrying ahypothesisand the correspondinganswer—together with the bookkeeping fieldsK,id,domain, anddifficulty\. Listings[1](https://arxiv.org/html/2607.20500#LST1)–[4](https://arxiv.org/html/2607.20500#LST4)step through the four cases: the degenerateK⋆=1K^\{\\star\}=1instance where multi\-source data verifies as consistent and the framework reduces to standard reasoning; a binaryK⋆=2K^\{\\star\}=2conflict between two authoritative specifications; aK⋆=3K^\{\\star\}=3instance with three mutually exclusive candidate values for a single quantity; and aK⋆=4K^\{\\star\}=4instance whose two independent binary disagreements yield four maximal self\-consistent combinations\. Longanalysisandanswerbodies are abbreviated with “\[\.\.\.\]” for readability; the underlying records are stored verbatim and used unmodified during training\.

Listing 1:Dataset example withK⋆=1K^\{\\star\}=1\(well\-posed degenerate case: the multi\-source data*appears*to conflict, but arithmetic verification shows the conditions are jointly consistent, so the framework collapses to standard single\-trace reasoning\)\. Domain:*combinatorics / counting*; difficulty: middle school\.\{

"id":"sample\_2368",

"K":1,

"domain":"combinatorics/counting",

"difficulty":"middleschool",

"problem":"Ajarcontainsredmarblesandbluemarbles\.Thereare5bluemarblesinthejar\.Theredmarbleswereaddedinthreeseparatebatches:thefirstbatchhad8redmarbles,thesecondbatchhad4additionalredmarbles\(making12totalredmarbles\),andthethirdbatchhad3moreredmarbles\(making15totalredmarbles\)\.Ifyourandomlyselect3marblesfromthejar,howmanywayscanyouselectthemsuchthatatleastonemarbleisblue?",

"analysis":"Ineedtofirstestablishthetotalcompositionofmarblesinthejar\.Theproblemstatesthereare5bluemarbles,anddescribestheredmarblesbeingaddedinthreebatches\.Letmeverifythearithmetic:thefirstbatchcontributes8redmarbles,thesecondbatchadds4morebringingthetotalto12,andthethirdbatchadds3morebringingthefinaltotalto15\.Checkingthissequence:8\+4=12\\\\checkmark,and12\+3=15\\\\checkmark\.Thearithmeticisconsistentthroughout,sothejarcontains5blueand15redmarblesforatotalof20\.Theproblemiswell\-posedandreducestoastandardcombinatoricscalculation\.",

"branches":\[

\{

"hypothesis":"Allgivenconditionsarecorrectandconsistent\.Thejarcontains5bluemarblesand15redmarbles\(20totalmarbles\)\.",

"answer":"Usingcomplementarycounting,thenumberofwaystopick3marbleswithatleastoneblueequalsC\(20,3\)\-C\(15,3\)=1140\-455=685\.Theansweris\\\\boxed\{685\}\."

\}

\]

\}

Listing 2:Dataset example withK⋆=2K^\{\\star\}=2\(binary conflict between architectural and regulatory specifications\)\. Domain:*construction / area & materials*; difficulty: early college\.\{

"id":"sample\_4101",

"K":2,

"domain":"construction/areaandmaterials",

"difficulty":"earlycollege",

"problem":"Aconstructioncompanyisbuildingarectangularfoundationforawarehouse\.Thefoundationhasalengthof20metersandrequiresconcretetobepouredtoadepthof0\.6meters\.Theconcretecosts$300percubicmeter\.Accordingtothearchitecturalplans,thefoundationwidthshouldbe60%ofthelength\.However,thesitesupervisornotesthatlocalbuildingcodesrequirethewidthtobeatleast15metersforthistypeofstructure\.Whatisthetotalcostoftheconcreteneededforthisfoundation?",

"analysis":"Ineedtocalculatethevolumeofconcreterequiredandmultiplybythecostpercubicmeter\.\[\.\.\.\]Thearchitecturalplanscallforawidthof0\.6\\\\times20=12meters\.However,localbuildingcodesrequireaminimumwidthof15meters\.Thiscreatesadirectconflict:thearchitecturalwidthof12mislessthanthelegallyrequiredminimumof15m\.Bothrequirementscannotbesatisfiedsimultaneously,soImustdeterminewhichspecificationtakesprecedencetoproceedwiththecostcalculation\.",

"branches":\[

\{

"hypothesis":"Thearchitecturalplansareauthoritativeandthebuildingcoderequirementiseitheroutdated,misapplied,orwillbeaddressedthroughavarianceprocess",

"answer":"Width=0\.6\\\\times20=12m\.Volume=20\\\\times12\\\\times0\.6=144m^3\.Cost=144\\\\times300=$43,200\.Theansweris\\\\boxed\{$43,200\}\."

\},

\{

"hypothesis":"Thebuildingcoderequirementislegallybindingandtakesprecedenceoverthearchitecturalspecification,requiringadesignmodification",

"answer":"Width=15m\(codeminimum\)\.Volume=20\\\\times15\\\\times0\.6=180m^3\.Cost=180\\\\times300=$54,000\.Theansweris\\\\boxed\{$54,000\}\."

\}

\]

\}

Listing 3:Dataset example withK⋆=3K^\{\\star\}=3\(three mutually exclusive choices among candidate principals\)\. Domain:*finance / interest & investment*; difficulty: middle school\.\{

"id":"sample\_0898",

"K":3,

"domain":"finance/interestandinvestment",

"difficulty":"middleschool",

"problem":"Sarahwantstoinvestmoneyinasavingsaccountthatpays8%simpleinterestperyear\.After2years,shewillreceivea$20bonusfromthebankforbeingaloyalcustomer\.However,therewillbea$20processingfeewhenshewithdrawshermoney\.Sarahmentionsthreedifferentamountsshemightinvest:shecouldusemoneyfromherbirthdaygiftstotaling$200,orshecouldcombinethatwithherallowancesavingsof$100foratotalof$300,orshecoulduseherpart\-timejobearningsof$250\.HowmuchmoneywillSarahhaveafterthe2\-yearinvestmentperiod,includingthebonusandafterpayingthefee?",

"analysis":"IneedtodetermineSarah’sfinalamountaftera2\-yearinvestmentwith8%simpleinterest,plusa$20bonus,minusa$20processingfee\.\[\.\.\.\]Theproblempresentsthreedifferentpossibleinvestmentamounts:$200,$300,or$250\.Thequestionasksfor’howmuchmoneywillSarahhave’asifexpectingasingleanswer,butthefinalamountdependsentirelyonwhichinitialinvestmentamountshechooses,yetnosinglechoiceisspecified\[\.\.\.\]IcannotproceedwithacalculationuntilIdeterminewhichinvestmentscenariotouse\.",

"branches":\[

\{

"hypothesis":"Sarahchoosestoinvestthe$200fromherbirthdaygifts,treatingtheotheramountsasmerelyconsideredalternatives",

"answer":"Interest=200\\\\times0\.08\\\\times2=$32\.Balance=200\+32\+20\-20=$232\.Theansweris\\\\boxed\{$232\}\."

\},

\{

"hypothesis":"Sarahchoosestoinvestthecombined$300frombirthdaygiftsandallowancesavings,treatingtheotheramountsasmerelyconsideredalternatives",

"answer":"Interest=300\\\\times0\.08\\\\times2=$48\.Balance=300\+48\+20\-20=$348\.Theansweris\\\\boxed\{$348\}\."

\},

\{

"hypothesis":"Sarahchoosestoinvestthe$250fromherpart\-timejobearnings,treatingtheotheramountsasmerelyconsideredalternatives",

"answer":"Interest=250\\\\times0\.08\\\\times2=$40\.Balance=250\+40\+20\-20=$290\.Theansweris\\\\boxed\{$290\}\."

\}

\]

\}

Listing 4:Dataset example withK⋆=4K^\{\\star\}=4\(two independent binary conflicts—price and discount—producing four maximal self\-consistent combinations, each with a distinct final amount\)\. Domain:*shopping / pricing & discounts*; difficulty: middle school\.\{

"id":"sample\_3941",

"K":4,

"domain":"shopping/pricinganddiscounts",

"difficulty":"middleschool",

"problem":"Sarahisbuyingabackpackandawaterbottle\.Thewaterbottlecosts$12\.Shehasconflictinginformationaboutthebackpack:onetagshows$40whileanothershows$50\.Thestoreclerkmentionsthere’sadiscountonbackpacks,butSarahhearddifferentamounts\-either20%offor30%off\.Afterthediscountisappliedtothebackpack,sheneedstopay8%salestaxonhertotalpurchase\(backpack\+waterbottle\)\.Finally,shecanusea$5storecreditsheearnedfrompreviouspurchases\.HowmuchwillSarahpayintotal?",

"analysis":"IneedtocalculateSarah’stotalcostbydeterminingthediscountedbackpackprice,addingthewaterbottlecost,applyingsalestax,andsubtractingthestorecredit\.\[\.\.\.\]Thebackpackhastwodifferentpricetagsshowing$40and$50,whichcannotbothbecorrectsimultaneously,andthediscountrateisreportedasboth20%and30%fromdifferentsources\.Thepricedisagreementandthediscountdisagreementareindependent:eachyieldsitsownpairofself\-consistentvalues,socombiningthemgivesfourmaximalself\-consistenthypotheses,eachproducingadifferentfinalamount\.",

"branches":\[

\{

"hypothesis":"The$40pricetagiscorrectandthe20%discountapplies,whilethe$50tagand30%discountinformationareerrors",

"answer":"Discountedbackpack:40\\\\times0\.80=$32\.Subtotal:32\+12=$44\.With8%tax:44\\\\times1\.08=$47\.52\.After$5credit:$42\.52\.Theansweris\\\\boxed\{$42\.52\}\."

\},

\{

"hypothesis":"The$40pricetagiscorrectandthe30%discountapplies,whilethe$50tagand20%discountinformationareerrors",

"answer":"Discountedbackpack:40\\\\times0\.70=$28\.Subtotal:28\+12=$40\.With8%tax:40\\\\times1\.08=$43\.20\.After$5credit:$38\.20\.Theansweris\\\\boxed\{$38\.20\}\."

\},

\{

"hypothesis":"The$50pricetagiscorrectandthe20%discountapplies,whilethe$40tagand30%discountinformationareerrors",

"answer":"Discountedbackpack:50\\\\times0\.80=$40\.Subtotal:40\+12=$52\.With8%tax:52\\\\times1\.08=$56\.16\.After$5credit:$51\.16\.Theansweris\\\\boxed\{$51\.16\}\."

\},

\{

"hypothesis":"The$50pricetagiscorrectandthe30%discountapplies,whilethe$40tagand20%discountinformationareerrors",

"answer":"Discountedbackpack:50\\\\times0\.70=$35\.Subtotal:35\+12=$47\.With8%tax:47\\\\times1\.08=$50\.76\.After$5credit:$45\.76\.Theansweris\\\\boxed\{$45\.76\}\."

\}

\]

\}

## Appendix EPrompts

We evaluate models with two prompt strategies that bracket the spectrum of ill\-posedness disclosure: a*blind*prompt that does not mention multiple interpretations \([8](https://arxiv.org/html/2607.20500#A5.F8)\), and an*ill\-posed–aware*prompt that explicitly tells the model the problem admits multiple valid solutions \([7](https://arxiv.org/html/2607.20500#A5.F7)\)\.

Prompt 1: \(ill\-posed–aware\)System message:```
You are a careful problem solver. The problems you receive
are ILL-POSED -- they typically contain conflicting or
ambiguous conditions that admit multiple valid
interpretations. Each problem has K valid solutions, where K
depends on how many independent contradictions exist. Your
job is to identify each valid interpretation and solve the
problem under each.
```

User message:```
Solve the following ill-posed problem. The problem has
multiple valid solutions depending on which subset of
conditions you trust. Identify ALL valid interpretations and
provide the answer under each.

Respond with a single JSON object and nothing else:
{
  "analysis": "Identify the specific conflicts or
               ambiguities, then list all possible
               interpretations.",
  "branches": [
    {"hypothesis": "Which conditions are trusted in this
                    interpretation (and which are discarded)",
     "answer": "Final answer under this interpretation
                (number or expression)"}
  ]
}

Critical rules:
- The number of branches MUST equal the number of distinct
  valid interpretations that genuinely arise from
  contradictions in the problem.
- If the problem is actually well-posed (no real
  contradictions), output exactly one branch.
- Do NOT invent speculative branches just to enumerate; only
  include an interpretation if it follows from a real
  conflict between conditions.
- Each branch’s answer must be the result computed under
  that branch’s hypothesis.

Problem:
{problem}
```

Figure 7:Ill\-posed–aware prompt\. The model is told the input is ill\-posed and asked to enumerate all valid interpretations\.Prompt 2: \(blind\)System message:```
You are a careful problem solver. Read the problem, think step
by step, then give your final answer.
```

User message:```
Solve the following problem.

Respond with a single JSON object and nothing else. The JSON
must have exactly these fields:
- "reasoning": your step-by-step reasoning as a string
- "answer": your final answer as a string

Problem:
{problem}
```

Figure 8:Blind prompt\. The model is not told that problems may be ill\-posed\.