隐空间中的评判：通过语义保持压缩实现高效生成式奖励建模

# Judging in Latent Space: Efficient Generative Reward Modeling via Semantics-Preserving Compression
Source: [https://arxiv.org/html/2610.09788](https://arxiv.org/html/2610.09788)
Xiaobo LiangJunwei YangAffiliation:Soochow University University of CambridgeZiwei ChenZeren ZhangAffiliation:Chalmers University of Technology Peking UniversityHejin WangYubin WangAffiliation:Tsinghua University The Hong Kong University of Science and TechnologyJuntao Li††thanks:Corresponding author\.

###### Abstract

Reward modeling often requires jointly representing and reasoning over multiple evaluation criteria, yet verbalizing this process token by token can incur substantial inference cost\. Recent work on latent reasoning suggests that continuous states may support this computation more compactly\. We introduce LatentGRM, a latent evaluation framework built on semantic chunking, compression, and reconstruction\. By using the structure of rubric\-guided evaluations to guide compression, LatentGRM learns compact continuous trajectories that support autonomous pairwise judgments without generating textual assessments\. A separate interpreter reconstructs evaluation text from these trajectories, providing an offline view of the information retained under compression\. Under matched training data and backbones, LatentGRM achieves competitive aggregate preference accuracy relative to explicit Supervised Fine\-Tuning \(SFT\) judges at both 4B and 8B scales\. Across four benchmark domains, LatentGRM\-8B compresses evaluation trajectories by8\.98\.9–9\.2×9\.2\\timesand reduces total judge inference time by6\.16\.1–7\.0×7\.0\\timesat vote@5\. Controlled rubric interventions show that criterion\-dependent preference information is carried through the latent sequence\. Together, these results demonstrate that continuous latent evaluation can substantially reduce inference cost while preserving competitive judgment quality\.

Figure 1:Accuracy–efficiency scaling of LatentGRM\-8B \(stars\) and Rubric\-RM\-8B \(circles\) from vote@1 to vote@9\. Points run left to right at 1, 3, 5, 7, and 9 votes\.## 1Introduction

Reward modeling often requires integrating evidence across multiple aspects of a response\. A response may be factually correct yet violate an explicit constraint, while a detailed answer may fall short of a request for concision\. These assessments must be combined into a preference that guides decisions in policy optimization and response selection\([Ouyang et al\., 2022](https://arxiv.org/html/2610.09788#bib.bib4);[Zhang et al\., 2025a](https://arxiv.org/html/2610.09788#bib.bib12)\)\. Since reward models are used repeatedly in these settings, both judgment accuracy and inference efficiency are central to their practical value\.

Among existing approaches, Generative Reward Models \(GRMs\) have become a dominant paradigm\. Generative reward models formulate evaluation as language generation, with reasoning\-based judges producing intermediate assessments before issuing a verdict\([Zhang et al\., 2025a](https://arxiv.org/html/2610.09788#bib.bib12);[Mahan et al\., 2024](https://arxiv.org/html/2610.09788#bib.bib13);[Chen et al\., 2026](https://arxiv.org/html/2610.09788#bib.bib16)\)\. Recently, rubric\-based reward modeling has gained increasing attention by organizing these assessments around*rubrics*: explicit sets of task\-specific evaluation criteria, such as correctness, instruction compliance, and clarity\([Liu et al\., 2026](https://arxiv.org/html/2610.09788#bib.bib11);[Yuan et al\., 2026](https://arxiv.org/html/2610.09788#bib.bib18)\)\. While sampling and aggregating multiple evaluations can improve judgment quality, verbalizing judgments and supporting evidence for each criterion can lengthen evaluation trajectories and increase autoregressive decoding costs\. This raises a central question:*can a generative judge retain the information needed for multi\-criterion evaluation while carrying out its intermediate computation more compactly?*

To address this, we turn to continuous latent reasoning, which offers a promising way to represent intermediate computation with fewer autoregressive steps\([Hao et al\., 2025](https://arxiv.org/html/2610.09788#bib.bib7);[Shen et al\., 2025](https://arxiv.org/html/2610.09788#bib.bib8);[Deng et al\., 2025](https://arxiv.org/html/2610.09788#bib.bib9)\)\. Rubric\-based reward modeling provides a particularly structured setting for studying such compact computation\. Its criterion\-level organization and linguistic boundaries can guide where to place compression boundaries, while the criteria themselves provide natural reference points for probing how latent judgments respond to changes in evaluation requirements and what assessment content remains recoverable\.

We introduceLatentGRM, a framework for rubric\-guided latent evaluation\. Its organizing idea is to treat rubric criteria as*intervenable and alignable criterion\-level semantic units*\. Building on Latent\-SFT’s vocabulary\-space distillation framework\([Deng et al\., 2025](https://arxiv.org/html/2610.09788#bib.bib9)\),Semantic Chunkinguses rubric\-aligned blocks to allocate a fixed latent budget, then uses linguistic cues to refine boundaries within each block\. The trained judge generates compact continuous trajectories and predicts pairwise preferences autonomously\. A separateLatent Trace Interpreter\(LTI\) reconstructs evaluation text from saved trajectories outside the online judgment path\. Aligning its output with teacher assessments by criterion provides a fine\-grained measure of reconstruction fidelity on teacher\-derived trajectories\. Rubric structure therefore connects compression, intervention, and interpretation within one evaluation framework\.

Against explicit Supervised Fine\-Tuning \(SFT\) judges trained on matched data and 4B/8B backbones, LatentGRM achieves competitive aggregate preference accuracy across eight benchmark domains\. On four domains, LatentGRM\-8B compresses evaluation trajectories by8\.98\.9–9\.2×9\.2\\timesand reduces total judge inference time by6\.16\.1–7\.0×7\.0\\timesat vote@5\. Controlled rubric edits and latent\-source replacement show that criterion\-dependent preference information is carried through autonomous latent trajectories\. Separately, reconstruction experiments recover criterion\-level states and justifications from teacher\-derived trajectories\. These findings support efficient latent judgment together with the recovery of useful evaluation content\.

Our contributions are:

- •We introduce LatentGRM, to our knowledge the first framework to bring continuous latent reasoning to reward modeling, combining rubric\-guided semantic compression, autonomous latent evaluation, and a separate interpreter for retrospective reconstruction\.
- •Under matched training data and backbones, we demonstrate competitive aggregate preference accuracy at both 4B and 8B scales and substantial reductions in trajectory length and total judge inference time\.
- •We examine the evaluation information retained under compression through controlled rubric interventions and criterion\-aligned reconstruction\. These analyses show that latent trajectories carry criterion\-dependent preference information and that the LTI can recover criterion\-level supporting justifications from teacher\-derived trajectories\.

## 2Related Works

### 2\.1Latent Reasoning

Continuous latent reasoning replaces explicit intermediate tokens with recurrent continuous states, reducing autoregressive decoding while retaining multi\-step computation\. Existing methods recurrently feed hidden states back into the model, distill textual reasoning chains into latent trajectories, or construct continuous thoughts in vocabulary space\([Hao et al\., 2025](https://arxiv.org/html/2610.09788#bib.bib7);[Shen et al\., 2025](https://arxiv.org/html/2610.09788#bib.bib8);[Zhang et al\., 2025b](https://arxiv.org/html/2610.09788#bib.bib17);[Deng et al\., 2025](https://arxiv.org/html/2610.09788#bib.bib9)\)\. These methods have primarily been studied on mathematical and general reasoning tasks\. We adapt vocabulary\-space latent reasoning to reward modeling, where the latent trajectory must represent a multi\-criterion comparison between candidate responses\.

### 2\.2Reward Modeling

Generative reward models formulate evaluation as language generation and can produce critiques or reasoning traces before issuing a judgment\([Mahan et al\., 2024](https://arxiv.org/html/2610.09788#bib.bib13);[Ankner et al\., 2024](https://arxiv.org/html/2610.09788#bib.bib15);[Chen et al\., 2026](https://arxiv.org/html/2610.09788#bib.bib16);[Guo et al\., 2025](https://arxiv.org/html/2610.09788#bib.bib14)\)\. In recent years, rubric\-based reward modeling has rapidly gained attention as a way to decompose holistic judgments into explicit, task\-specific criteria for evaluation, feedback, reward construction, and policy optimization\([Liu et al\., 2026](https://arxiv.org/html/2610.09788#bib.bib11);[Yuan et al\., 2026](https://arxiv.org/html/2610.09788#bib.bib18)\)\. However, such rubric\-based and reasoning\-heavy judges introduce substantial inference costs, since they must generate and apply criteria explicitly at test time\. Our work therefore focuses on the inference cost of reasoning\-based generative judges: LatentGRM compresses the evaluations into continuous trajectories, with matched explicit SFT judges as the primary comparison\. We further repurpose rubrics as anchors for semantic chunking and as an evaluation and analysis tool for latent reasoning\.

![Refer to caption](https://arxiv.org/html/2610.09788v1/latentgrm_overview_v1.png)Figure 2:LatentGRM training and inference\. Stage 1 encodes semantic boundaries into vocabulary\-space targets and trains suffix reconstruction; Stage 2 distills autonomous latent judging\. An offline interpreter reconstructs evaluations from saved trajectories\.

## 3Method

### 3\.1Overview

We study pairwise reward modeling with an evaluation inputx=\(q,ℛ,a,b\)x=\(q,\\mathcal\{R\},a,b\), comprising a user request, a task\-specific rubric, and two candidate responses\. Training examples additionally provide an explicit evaluation tracec=\(c1,…,cM\)c=\(c\_\{1\},\\ldots,c\_\{M\}\)and a preference labely∈\{Response A,Response B\}y\\in\\\{\\texttt\{Response A\},\\texttt\{Response B\}\\\}\. An explicit reasoning judge produces:

\[x,<think\>,c1,…,cM,</think\>,y\]\.\[x,\\ \\texttt\{<think\>\},\\ c\_\{1\},\\ldots,c\_\{M\},\\ \\texttt\{</think\>\},\\ y\]\.\(1\)LatentGRM replaces the textual evaluation with fewer continuous steps:

\[x,<think\>,z1,…,zN,</think\>,y\]\.\[x,\\ \\texttt\{<think\>\},\\ z\_\{1\},\\ldots,z\_\{N\},\\ \\texttt\{</think\>\},\\ y\]\.\(2\)We build on Latent\-SFT’s vocabulary\-space representation and two\-stage distillation framework\([Deng et al\., 2025](https://arxiv.org/html/2610.09788#bib.bib9)\), replacing fixed\-rate segmentation with Semantic Chunking\. Stage 1 distills explicit evaluation traces into latent trajectories\. Stage 2 trains the judge to autoregressively generate these trajectories and predict the final preference from the evaluation input at inference\. A separate interpreter reconstructs explicit evaluations from saved trajectories for offline analysis\. Figure[2](https://arxiv.org/html/2610.09788#S2.F2)summarizes the framework\.

### 3\.2Stage 1: Semantic Chunking for Latent Compression

Stage 1 follows an encoder and decoder objective\. At each semantic boundary, the encoder compresses the evaluation input and teacher\-trace prefix into latent states, while the decoder learns to recover the remaining evaluation and final preference from the latent prefix\.

Letc1:Mc\_\{1:M\}denote an evaluation trace containingMMtokens\. Semantic Chunking divides it intoNNspans using ordered boundaries0=b0<b1<⋯<bN=M0=b\_\{0\}<b\_\{1\}<\\cdots<b\_\{N\}=M\. We set:

N=⌈Mρ⌉,N=\\left\\lceil\\frac\{M\}\{\\rho\}\\right\\rceil,\(3\)whereρ\\rhois the fixed compression rate\. Rubric\-guided evaluations naturally separate into compliance check, criterion\-specific analysis, and final synthesis \(Figure[8](https://arxiv.org/html/2610.09788#A7.F8)\)\. We fix the latent budget of each rubric block with an exact\-budget allocator, then refine its internal frontiers using sentence, clause, and dependency evidence\. Semantic Chunking places compression boundaries to preserve semantic integrity as much as possible\.

Figure[3](https://arxiv.org/html/2610.09788#S3.F3)illustrates how the two boundary choices treat the same rubric and sentence\-level passages\.

Example 1: preserving rubric structure and complete claimsSemantic Chunking\- Criterion 4 \[Principle\]: Met\.\|The logic for generating the metaphor is clear\.\|It randomly selects words and combines them into a sentence\.Fixed\-Rate Chunking \(ρ=8\\rho=8\)\- Criterion 4 \[Principle\]:\|Met\. The logic for generating the metaphor\|is clear\. It randomly selects words and\|combines them into a sentence\.

Example 2: retaining sentence\-level evidenceSemantic ChunkingIt does not provide an explanation or a flowchart\.\|Response B correctly explains the code and its logic\.Fixed\-Rate Chunking \(ρ=8\\rho=8\)It does not provide an explanation or\|a flowchart\. Response B correctly explains\|the code and its logic\.

Figure 3:Illustrative frontiers in cropped evaluations\.Green and red mark the semantic and fixed\-rate cuts on identical text\.At boundarybib\_\{i\}, the encoder readsxxand the explicit prefixc1:bic\_\{1:b\_\{i\}\}and produces a hidden statehih\_\{i\}\. To make this state directly consumable by the decoder, we express it through the decoder’s vocabulary interface\. LetWoutW\_\{\\mathrm\{out\}\}be the decoder’s frozen vocabulary output projection andτ\\tauthe projection temperature\. The resulting full\-vocabulary distribution isαi=softmax⁡\(Wout​hi/τ\)\\alpha\_\{i\}=\\operatorname\{softmax\}\(W\_\{\\mathrm\{out\}\}h\_\{i\}/\\tau\)\. We defineSi=TopK⁡\(αi\)S\_\{i\}=\\operatorname\{TopK\}\(\\alpha\_\{i\}\)as the indices of theKKlargest entries ofαi\\alpha\_\{i\}and renormalize their probability mass, soα¯i,v=αi,v/∑u∈Siαi,u\\bar\{\\alpha\}\_\{i,v\}=\\alpha\_\{i,v\}/\\sum\_\{u\\in S\_\{i\}\}\\alpha\_\{i,u\}forv∈Siv\\in S\_\{i\}, andα¯i,v=0\\bar\{\\alpha\}\_\{i,v\}=0otherwise\. Using the decoder’s frozen input embedding tableEinE\_\{\\mathrm\{in\}\}, we form the compressed state:

zi=∑v∈Siα¯i,v​Ein​\[v\]\.z\_\{i\}=\\sum\_\{v\\in S\_\{i\}\}\\bar\{\\alpha\}\_\{i,v\}E\_\{\\mathrm\{in\}\}\[v\]\.\(4\)Thusziz\_\{i\}is a continuous mixture of token embeddings and can be inserted directly into the decoder’s input sequence\. Given\[x,<think\>,z1:i\]\[x,\\texttt\{<think\>\},z\_\{1:i\}\], the decoder predicts the remaining evaluation and final preference:

ℒrec=1N∑i=1NCE\(\[cbi\+1:M,</think\>,y,EOS\]∣\[x,<think\>,z1:i\]\)\.\\mathcal\{L\}\_\{\\mathrm\{rec\}\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\operatorname\{CE\}\\\!\\left\(\[c\_\{b\_\{i\}\+1:M\},\\texttt\{</think\>\},y,\\texttt\{EOS\}\]\\mid\[x,\\texttt\{<think\>\},z\_\{1:i\}\]\\right\)\.\(5\)Here,CE\\operatorname\{CE\}is token\-averaged autoregressive cross\-entropy\. We optimize the encoder, the decoder, and then both jointly\. After training, the encoder exports\(Si,α¯i\)i=1N\(S\_\{i\},\\bar\{\\alpha\}\_\{i\}\)\_\{i=1\}^\{N\}as sparse targets for Stage 2\.

### 3\.3Stage 2: Autonomous Latent Evaluation

Starting from the Stage 1 decoder, Stage 2 trains the judge to predict the exported distributions autoregressively and then output the final preference\. Let𝐠=\(𝐠1,…,𝐠N\)\\mathbf\{g\}=\(\\mathbf\{g\}\_\{1\},\\ldots,\\mathbf\{g\}\_\{N\}\)collect independent Gumbel noise across the trajectory\. For each targetα¯i\\bar\{\\alpha\}\_\{i\}, we add𝐠i\\mathbf\{g\}\_\{i\}to its log\-probabilities and renormalize withinSiS\_\{i\}, obtainingα~i​\(𝐠\)\\widetilde\{\\alpha\}\_\{i\}\(\\mathbf\{g\}\)\. Replacingα¯i\\bar\{\\alpha\}\_\{i\}with these perturbed weights in Eq\.[4](https://arxiv.org/html/2610.09788#S3.E4)gives the corresponding target embeddingz~i​\(𝐠\)\\widetilde\{z\}\_\{i\}\(\\mathbf\{g\}\)\. During training, the prediction at stepiiis conditioned on the preceding perturbed target embeddingsz~1:i−1\(𝐠\)\\widetilde\{z\}\_\{1:i\-1\}\(\\mathbf\{g\}\)\. This stochastic perturbation discourages overfitting to deterministic teacher targets; at inference, the same mechanism produces diverse latent trajectories for voting\. Letxlat=\[x,<think\>\]x\_\{\\mathrm\{lat\}\}=\[x,\\texttt\{<think\>\}\]and letqθq\_\{\\theta\}denote the judge’s predicted full\-vocabulary distribution\. We optimize:

ℒauto=𝔼𝐠\[1N∑i=1NKL\(α~i\(𝐠\)∥qθ\(⋅∣xlat,z~<i\(𝐠\)\)\)\\displaystyle\\mathcal\{L\}\_\{\\mathrm\{auto\}\}=\\mathbb\{E\}\_\{\\mathbf\{g\}\}\\Bigg\[\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\operatorname\{KL\}\\\!\\left\(\\widetilde\{\\alpha\}\_\{i\}\(\\mathbf\{g\}\)\\,\\middle\\\|\\,q\_\{\\theta\}\(\\cdot\\mid x\_\{\\mathrm\{lat\}\},\\widetilde\{z\}\_\{<i\}\(\\mathbf\{g\}\)\)\\right\)\(6\)\+CE\(\[</think\>,y,EOS\]∣xlat,z~1:N\(𝐠\)\)\]\.\\displaystyle\\qquad\+\\operatorname\{CE\}\\\!\\left\(\[\\texttt\{</think\>\},y,\\texttt\{EOS\}\]\\mid x\_\{\\mathrm\{lat\}\},\\widetilde\{z\}\_\{1:N\}\(\\mathbf\{g\}\)\\right\)\\Bigg\]\.The KL term distills each perturbed latent target, while cross\-entropy predicts the end marker, preference label, and EOS from the complete trajectory\. The expectation is estimated with one noise draw per example at each update\.

At inference, the judge feeds back its own predictions\. At stepii, it producesqi=qθ\(⋅∣x,<think\>,z<i\)q\_\{i\}=q\_\{\\theta\}\(\\cdot\\mid x,\\texttt\{<think\>\},z\_\{<i\}\)\. If the unperturbed argmax ofqiq\_\{i\}is</think\>, latent generation terminates and the preference is decoded greedily\. Otherwise, the judge retains and renormalizes the top\-KKprobabilities ofqiq\_\{i\}, applies the same Gumbel perturbation, and substitutes the resulting support and weights into Eq\.[4](https://arxiv.org/html/2610.09788#S3.E4)to formziz\_\{i\}\. We extend vLLM\([Kwon et al\., 2023](https://arxiv.org/html/2610.09788#bib.bib25)\)to support this soft\-embedding feedback while retaining continuous batching, KV caching, and tensor parallelism\.

### 3\.4Retrospective Interpretation of Latent Trajectories

The Latent Trace Interpreter \(LTI\) is a separate readout that reconstructs an explicit evaluation from a completed latent trajectory\. At positionii, it receives saved top\-KKvocabulary indicesSisaveS\_\{i\}^\{\\mathrm\{save\}\}and normalized weightsq¯isave\\bar\{q\}\_\{i\}^\{\\mathrm\{save\}\}\. Using its embedding tableEintE\_\{\\mathrm\{int\}\}, LTI forms:

ziint=∑v∈Sisaveq¯isave​\(v\)​Eint​\[v\],z\_\{i\}^\{\\mathrm\{int\}\}=\\sum\_\{v\\in S\_\{i\}^\{\\mathrm\{save\}\}\}\\bar\{q\}\_\{i\}^\{\\mathrm\{save\}\}\(v\)E\_\{\\mathrm\{int\}\}\[v\],\(7\)the same vocabulary\-mixture construction as Eq\.[4](https://arxiv.org/html/2610.09788#S3.E4)\. Given the full sequencez1:Nintz\_\{1:N\}^\{\\mathrm\{int\}\}, LTI predicts the complete teacher evaluation with the token\-averaged objective:

ℒint=CE\(\[c1:M,</think\>\]∣\[x,<think\>,z1:Nint\]\)\.\\mathcal\{L\}\_\{\\mathrm\{int\}\}=\\operatorname\{CE\}\\\!\\left\(\[c\_\{1:M\},\\texttt\{</think\>\}\]\\mid\[x,\\texttt\{<think\>\},z\_\{1:N\}^\{\\mathrm\{int\}\}\]\\right\)\.\(8\)Unlike Stage 1’s suffix reconstruction, LTI reconstructs the evaluation from its beginning; the preference labelyyis excluded\. Autonomous judging can save its predicted top\-KKindices and weights for the same offline readout, without running LTI in the online decision path\.

## 4Experiments

Our experiments address three questions: \(RQ1\) whether latent evaluation preserves preference accuracy across model scales and responds to controlled changes in rubric criteria; \(RQ2\) how compressing explicit evaluations into latent trajectories affects inference cost; and \(RQ3\) whether criterion\-level assessments can be reconstructed from latent trajectories\.

Table 1:Pairwise preference evaluation \(%\)\.Avg\.\-8 is the mean of the eight benchmark scores\. Shading mark the best score among the Rubric\-RM and LatentGRM configurations\.### 4\.1Experimental Setup

##### Training data\.

We use 35,612 OpenRubrics\([Liu et al\., 2026](https://arxiv.org/html/2610.09788#bib.bib11)\)training records, each containing a request, two responses, a rubric, an evaluation trace, and a binary preference\. OpenRubrics keeps generated rubrics whose judgments match the preference labels\.

##### Evaluation data\.

We evaluate eight benchmark domains: RewardBench Chat and Chat Hard\([Lambert et al\., 2025](https://arxiv.org/html/2610.09788#bib.bib5)\), PPE\-IFEval\([Frick et al\., 2025](https://arxiv.org/html/2610.09788#bib.bib19)\), IFBench\([Pyatkin et al\., 2025](https://arxiv.org/html/2610.09788#bib.bib20)\), RM\-Bench Chat\([Liu et al\., 2025](https://arxiv.org/html/2610.09788#bib.bib21)\), RewardBench 2 Precise IF and Focus\([Malik et al\., 2026](https://arxiv.org/html/2610.09788#bib.bib6)\), and HelpSteer3\([Wang et al\., 2025](https://arxiv.org/html/2610.09788#bib.bib22)\)\.Avg\.\-8is their unweighted mean\.

##### Models and evaluation\.

Our primary baselines are explicit Rubric\-RM trained by SFT on the same OpenRubrics records and Qwen3\-4B or Qwen3\-8B backbones\([Yang et al\., 2025](https://arxiv.org/html/2610.09788#bib.bib24)\)as LatentGRM\. This matched comparison isolates the effect of replacing explicit evaluation traces with compact latent trajectories\. Both methods receive identical task\-specific rubrics, generated once offline by OpenRubrics generator\-8B\. We report a single rollout \(vote@1\) and majority voting over five rollouts \(vote@5\)\. JudgeLRM, RRM, and two RM\-R1 variants serve as additional reference points\([Chen et al\., 2025](https://arxiv.org/html/2610.09788#bib.bib23);[Guo et al\., 2025](https://arxiv.org/html/2610.09788#bib.bib14);[Chen et al\., 2026](https://arxiv.org/html/2610.09788#bib.bib16)\)\. We evaluate every model in both response orders and average the resulting scores\.

### 4\.2RQ1: Effectiveness and Rubric\-Based Process Evaluation

#### 4\.2\.1Preference Accuracy

At vote@5, LatentGRM scores 67\.8 and 69\.0 on Avg\.\-8 at 4B and 8B, compared with 67\.0 and 68\.1 for the matched explicit Rubric\-RM \(Table[1](https://arxiv.org/html/2610.09788#S4.T1)\)\. Its largest advantage is on Focus \(\+6\.0 and \+6\.4 points\); across the other seven domains, Chat Hard and IFBench favor LatentGRM at both scales, RewardBench Chat favors Rubric\-RM, and Precise IF changes direction between 4B and 8B\. PPE, RM\-Bench Chat, and HelpSteer3 differ by at most 0\.4 points\. Thus, compact latent evaluation retains broadly similar preference accuracy across domains, with a concentrated gain on Focus and Chat Hard\. Voting raises LatentGRM’s Avg\.\-8 from 65\.6 to 67\.8 at 4B \(\+2\.2\) and from 66\.7 to 69\.0 at 8B \(\+2\.3\), showing that multiple compact trajectories can improve the final judgment\.

Appendix[E\.4](https://arxiv.org/html/2610.09788#A5.SS4)provides a comparison with a direct judge that predicts only the final preference\.

#### 4\.2\.2Effect of Semantic Chunking

To assess the contribution of Semantic Chunking, we compare it with Fixed\-Rate Chunking on Qwen3\-8B\. Both use the same training data, target compression rateρ=8\\rho=8, latent representation, initialization, and training schedule\.

Table 2:Semantic Chunking ablation on Qwen3\-8B \(%\)\.Semantic Chunking improves Chat Hard, Precise IF, and Focus at both voting budgets \(Table[2](https://arxiv.org/html/2610.09788#S4.T2)\)\. At vote@5, the gains over Fixed\-Rate Chunking are 2\.2, 3\.9, and 1\.7 points, respectively, while Chat decreases by 0\.7 points; the four\-domain mean rises by 1\.8 points\. With the latent budget and training conditions held fixed, this pattern indicates that placing boundaries at semantic units improves compressed judgments most clearly on instruction\-sensitive tasks\.

#### 4\.2\.3Rubric Interventions and Latent\-Source Replacement

Figure 4:Rubric\-conditioned preference shifts on 398 pairs\.\(A\) Each trajectory is generated and evaluated under the same rubric\. \(B\) The evaluation prompt is fixed to the baseline rubric while the latent\-trajectory source varies\. Changes are measured against the baseline condition; positive values favor the response targeted by the criterion intervention\. Bands show pointwise 95% question\-cluster bootstrap intervals\.We test whether changing one evaluation criterion shifts the model’s preference and whether the shift is carried by the latent trajectory, motivated by latent\-state intervention studies\([Li et al\., 2026](https://arxiv.org/html/2610.09788#bib.bib3)\)\. With all other criteria fixed, the test measures the direction of the preference shift\.

##### Design and readout\.

For each of 398 RewardBench pairs \(198 Chat and 200 Chat Hard\), we construct three rubrics that differ in one criterion: the*baseline*rubricℛ0\\mathcal\{R\}\_\{0\}is unchanged; the*wording control*ℛp\\mathcal\{R\}\_\{p\}paraphrases the selected criterion without changing its meaning; and the*criterion intervention*ℛc\\mathcal\{R\}\_\{c\}changes that criterion so that it favors a designated responsebib\_\{i\}\. GLM\-5\.2\([GLM\-5 Team et al\., 2026](https://arxiv.org/html/2610.09788#bib.bib30)\)proposes the two edits, and a fresh\-context call verifies thatℛp\\mathcal\{R\}\_\{p\}preserves meaning andℛc\\mathcal\{R\}\_\{c\}changes only the selected criterion, then identifiesbib\_\{i\}\. Neither call receives benchmark labels or judge predictions\. We generate trajectories for all three rubrics in both response orders\.

At 0%, 10%,…\\ldots, 100% of each trajectory, we probe the preference between the two response labels\. For a contextCCcontaining the evaluation prompt and latent prefix, we measure the log\-odds of the labelℓbi\\ell\_\{b\_\{i\}\}against the other labelℓb¯i\\ell\_\{\\bar\{b\}\_\{i\}\}:

Di​\(C\)=log⁡pθ​\(ℓbi∣C\)−log⁡pθ​\(ℓb¯i∣C\)\.D\_\{i\}\(C\)=\\log p\_\{\\theta\}\(\\ell\_\{b\_\{i\}\}\\mid C\)\-\\log p\_\{\\theta\}\(\\ell\_\{\\bar\{b\}\_\{i\}\}\\mid C\)\.\(9\)For either edited conditione∈\{p,c\}e\\in\\\{p,c\\\},Δ​Di=Di​\(Ce\)−Di​\(C0\)\\Delta D\_\{i\}=D\_\{i\}\(C\_\{e\}\)\-D\_\{i\}\(C\_\{0\}\)measures its effect relative to the baseline; positive values indicate a shift towardbib\_\{i\}\. This contrastive readout follows[Wang et al\. \(2023\)](https://arxiv.org/html/2610.09788#bib.bib1);[Heimersheim and Nanda \(2024\)](https://arxiv.org/html/2610.09788#bib.bib2)\.

##### End\-to\-end rubric intervention\.

The criterion intervention shifts preference toward the designated response, reaching a mean log\-odds change of 4\.58 at the trajectory endpoint \(Figure[4](https://arxiv.org/html/2610.09788#S4.F4)A\)\. Its effect becomes substantially larger than that of the wording control in the latter half of the trajectory, indicating sensitivity to the criterion’s meaning rather than its phrasing alone\. The nonzero shift at 0% shows a direct effect of the rubric prompt; the replay control below measures the contribution carried by the latent sequence\.

##### Latent\-source replacement\.

To remove the visible rubric difference, we fix the evaluation prompt toℛ0\\mathcal\{R\}\_\{0\}and replay the latent embeddings generated under each rubric in a fresh baseline context\. The criterion\-intervention source has little effect at early positions, but its influence grows with the supplied trajectory and reaches a mean log\-odds change of 2\.75 at the endpoint \(Figure[4](https://arxiv.org/html/2610.09788#S4.F4)B\); the wording\-control source remains much smaller\. Thus, criterion\-dependent preference information transfers through the latent sequence even when the evaluation prompt remains unchanged\.

### 4\.3RQ2: Practical Efficiency of Latent Evaluation

We evaluate the cost of producing preference judgments\. Retrospective interpretation is performed separately and is excluded from these measurements\.

#### 4\.3\.1Runtime under Natural Generation

We compare Rubric\-RM\-8B and LatentGRM\-8B on four domains at vote@5\. The vLLM measurements use the same eight\-GPU serving configuration and include prefill, scheduling, decoding, and output collection\. For reference, we also report full\-domain time under ordinary Transformers/PyTorch generation\([Wolf et al\., 2020](https://arxiv.org/html/2610.09788#bib.bib28);[Paszke et al\., 2019](https://arxiv.org/html/2610.09788#bib.bib29)\)on the same eight GPUs, with one model replica per GPU and five sequential rollouts per assigned sample\.

Table 3:Inference cost at vote@5\.Steps are mean generated positions per rollout\. Compression and speedup are the Rubric\-RM\-8B/LatentGRM\-8B ratios of generated steps and total inference time\. HF denotes ordinary Transformers/PyTorch generation without vLLM\. Bold marks fewer steps or less time and the corresponding compression/speedup ratios within each domain\.At vote@5, LatentGRM\-8B reduces generated positions by8\.878\.87–9\.23×9\.23\\times, yielding8\.178\.17–8\.82×8\.82\\timesspeedups under ordinary Transformers and6\.116\.11–6\.95×6\.95\\timesunder vLLM \(Table[3](https://arxiv.org/html/2610.09788#S4.T3)\)\. All four domains exceed6×6\\timesvLLM speedup despite mixed accuracy differences\.

Figure[1](https://arxiv.org/html/2610.09788#S0.F1)extends the comparison across vote@1–9\. LatentGRM remains faster and scores higher on Chat Hard and Focus at every voting budget; Chat remains below Rubric\-RM, and Precise IF varies by budget\. More votes generally improve latent judgment quality, with nonmonotonic intermediate results and6\.9×6\.9\\times–7\.8×7\.8\\timesspeedups at vote@9\.

An equal\-length control that forces both judges to generate exactly 50 positions yields only a1\.017×1\.017\\timesruntime difference, showing that latent reasoning does not introduce additional per\-step computational cost\. \(Appendix[F\.2](https://arxiv.org/html/2610.09788#A6.SS2)\)\.

### 4\.4RQ3: Recovering Rubric\-Level Evaluations

Task rubrics provide predefined semantic units for evaluating reconstruction fidelity\. We align decoded and teacher evaluations by response and criterion, measuring whether each assessment’s state and justification are preserved\. This makes changes in individual judgments observable even when most of the reconstructed text remains similar\.

Using cached teacher\-derived trajectories, we compare three independently trained readouts initialized from the LatentGRM\-8B Stage 1 decoder\.Prompt\-onlyreceives the evaluation input without latent states;Replay\-hiddenadditionally reads projected hidden states obtained by replaying the latent sequence;Latent Trace Interpreter \(LTI\)reads its vocabulary\-probability representation \(Section[3\.4](https://arxiv.org/html/2610.09788#S3.SS4)\)\.

The test set contains 607 records with 9,992 criteria, of which 9,768 have explicit states\. State Macro\-F1 scoresMet,Not Met, andPartial; missing or unparseable states count as errors\. Vector Exact requires all states in a fully labeled record to match\. Rubric Semantic measures BERTScore similarity\([Zhang\* et al\., 2020](https://arxiv.org/html/2610.09788#bib.bib26)\)between aligned justifications, assigning zero to missing criteria; whole\-trace ROUGE\-L\([Lin, 2004](https://arxiv.org/html/2610.09788#bib.bib27)\)provides a lexical comparison\.

Table 4:LatentGRM\-8B retrospective reconstruction on the 607\-record criterion\-rich test set \(%\)\.LTI raises State Macro\-F1 from 40\.20 \(Prompt\-only\) and 69\.42 \(Replay\-hidden\) to 98\.93, and Vector Exact from 18\.61 and 33\.03 to 93\.98 \(Table[4](https://arxiv.org/html/2610.09788#S4.T4)\)\. Prompt\-only reaches 63\.36 ROUGE\-L despite its 40\.20 State Macro\-F1, showing that lexical overlap misses criterion\-state errors\. LTI also leads on Rubric Semantic \(74\.65\) and ROUGE\-L \(79\.24\), reflecting stronger criterion\-level recovery on teacher\-derived trajectories\.

##### Downstream utility through data revision\.

We also test whether the recovered evaluation can guide a frozen Qwen3\-8B Base reviser on 240 RewardBench Rubric examples\.Prompt\-onlygives the reviser just the question and candidate response;Rubric\-RM trajectorysupplies explicit textual feedback;Latent onlyinserts the soft top\-10 embedding mixture at each latent step without training the reviser to consume it; andInterpreter\-restored trajectorysupplies LTI’s textual reconstruction of the same latent feedback\. The reviser does not see the rubric\. Appendix[E\.5](https://arxiv.org/html/2610.09788#A5.SS5)gives the input construction and scoring protocol\.

An external evaluator scores each rubric criterion before and after revision\.Repairis the fraction of originally unsatisfied criteria that become satisfied, whereasRegressionis the fraction of originally satisfied criteria that become unsatisfied\.Fullis the percentage of revised answers satisfying every criterion;Criterion satisfactionis the overall percentage of satisfied criterion decisions\.

Table 5:Frozen\-model revision on 240 examples, equally split between Chat and Chat Hard \(percent\)\. Repair is computed over 897 originally unsatisfied criteria and Regression over 1,076 originally satisfied criteria\. Bold marks the best result in each column\.Prompt\-only revision reaches 34\.17% Full\. Explicit Rubric\-RM feedback raises this to 56\.67% and Repair from 54\.07% to 78\.71% \(Table[5](https://arxiv.org/html/2610.09788#S4.T5)\)\. Even without reviser adaptation, the latent\-only interface improves Repair by 9\.25 points and Full by 7\.50 points over Prompt\-only\. Interpreter\-restored feedback yields the best result in every reported column: 80\.49% Repair, 1\.86% Regression, 57\.50% Full, and 90\.12% criterion satisfaction\. Relative to explicit Rubric\-RM feedback, its Full rate is comparable while Regression is lower \(1\.86% versus 4\.65%\)\.

These results show that latent reasoning preserves the data\-synthesis and revision utility of explicit reasoning: after textual restoration, it matches or exceeds the explicit trajectory on every reported metric\. More strikingly, the training\-free Latent\-only interface already improves substantially over Prompt\-only revision, demonstrating that latent feedback is directly usable before any specialized adaptation\.

## 5Conclusion

We introduced LatentGRM, a framework for latent reward modeling built on semantic chunking, compression, and reconstruction\. Rubric structure guides the compression of explicit evaluations into compact continuous trajectories, which the judge uses to predict pairwise preferences without generating evaluation text\. Compared with explicit SFT judges trained on matched data and backbones, LatentGRM achieves competitive aggregate accuracy while substantially reducing trajectory length and inference time\. Controlled rubric interventions show that criterion\-dependent preference information is carried through the latent sequence, while the Latent Trace Interpreter reconstructs criterion\-level assessments for retrospective inspection\. Together, these results show how structured latent trajectories can support efficient judgment and make compressed evaluations inspectable\.

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## Appendix Contents

1. 1\.
2. 2\.
3. 3\.
4. 4\.
5. 5\.[Supplementary RQ1: Effectiveness and Rubric\-Based Process Evaluation](https://arxiv.org/html/2610.09788#A5) - •Data, backbones, and evaluation protocol - •Evaluation\-time parameters and Semantic Chunking training dynamics - •Rubric\-based intervention protocol - •Direct judgment without an intermediate evaluation trace - •Downstream utility through frozen\-model data revision
6. 6\.[Supplementary RQ2: Practical Efficiency of Latent Evaluation](https://arxiv.org/html/2610.09788#A6) - •vLLM implementation for latent reasoning - •Equal\-length runtime control
7. 7\.[Supplementary RQ3: Recovering Rubric\-Level Evaluations](https://arxiv.org/html/2610.09788#A7) - •Readout data, controls, and evaluation protocol - •Metrics, parsing, and reconstruction examples
8. 8\.
9. 9\.

## Appendix ADiscussion

An important challenge in latent reward modeling is preserving essential evaluation evidence under compression\. Fixed\-Rate Chunking and the current heuristic Semantic Chunking approximate evaluation structure through fixed token intervals and linguistic boundary cues, respectively\. Developing compression strategies that better preserve the evidence most relevant to the final judgment remains an important direction\.

A related challenge is recovering the compressed evidence through faithful retrospective interpretation\. Appendix[G\.4\.1](https://arxiv.org/html/2610.09788#A7.SS4.SSS1)illustrates a coherent, nearly complete reconstruction that changes one criterion assessment\. The interpreter receives renormalized top\-KKsoft labels, which preserve only a subset of the vocabulary distribution\. Richer trajectory representations and reconstruction objectives could strengthen the connection between recovered criterion states and their supporting evidence\.

These challenges accompany substantial opportunities\. By shortening generation and reducing inference time, latent reasoning can make repeated reward evaluations more affordable in policy learning, data selection, and related workflows\. The gains from vote@1 to vote@5 also motivate exploring reinforcement learning to improve individual latent trajectories, building on approaches such as Latent\-GRPO\([Deng et al\., 2026](https://arxiv.org/html/2610.09788#bib.bib10)\)\. Together, advances in compression, trajectory optimization, and interpretation could extend the value of latent reward modeling from faster evaluation to more capable and understandable learning systems\.

## Appendix BSemantic Chunking and Exact\-Budget Allocation

Semantic Chunking first divides an evaluation into outer rubric blocks and allocates a fixed global latent budget across them\. It then refines each block’s internal boundaries while holding its allocated count fixed\. The procedure preserves the original token sequence and produces exactlyN=⌈M/8⌉N=\\lceil M/8\\rceillatent positions\. Preprocessing is performed offline and cached\. Figures[5](https://arxiv.org/html/2610.09788#A2.F5)and[6](https://arxiv.org/html/2610.09788#A2.F6)illustrate Semantic Chunking and Fixed\-Rate Chunking\.

### B\.1Rubric Structure and Initial Budget Allocation

We tokenize the evaluation once with character offsets\. The outer blocks comprise the initial compliance and gatekeeper discussion, each response’s individual criterion assessments, and the final synthesis; each response header joins its first criterion\. Structural expressions of at most 10 tokens are protected against internal cuts, including section and response headers, criterion identifiers,\[Hard Rule\],\[Principle\],Justification:,Met\., andNot Met\.We use*boundary*for a selected token position in the partition; each selected boundary serves as a compression frontier in Stage 1\.

Letmsm\_\{s\}be the token length of outer blockss, fors=1,…,Ss=1,\\ldots,S, and letnsn\_\{s\}be its latent budget\. We require

∑s=1Sns=N,⌈ms16⌉≤ns≤⌊ms4⌋,N=⌈∑sms8⌉\.\\sum\_\{s=1\}^\{S\}n\_\{s\}=N,\\qquad\\left\\lceil\\frac\{m\_\{s\}\}\{16\}\\right\\rceil\\leq n\_\{s\}\\leq\\left\\lfloor\\frac\{m\_\{s\}\}\{4\}\\right\\rfloor,\\qquad N=\\left\\lceil\\frac\{\\sum\_\{s\}m\_\{s\}\}\{8\}\\right\\rceil\.\(10\)For a candidate countkk, letds,1:kd\_\{s,1:k\}be the balanced integer partition ofmsm\_\{s\}, with lengths differing by at most one\. Defineδ⁡\(d\)=max⁡\(6−d,0,d−10\)\\delta\(d\)=\\max\(6\-d,0,d\-10\)\. The allocation cost is

As\(k\)=\(∑j𝟏\[δ\(ds,j\)\>0\],∑jδ\(ds,j\),∑j\(ds,j−8\)2\)\.A\_\{s\}\(k\)=\\left\(\\sum\_\{j\}\\mathbf\{1\}\[\\delta\(d\_\{s,j\}\)\>0\],\\quad\\sum\_\{j\}\\delta\(d\_\{s,j\}\),\\quad\\sum\_\{j\}\(d\_\{s,j\}\-8\)^\{2\}\\right\)\.\(11\)We minimize the sum of these costs lexicographically using

D⁡\(s,n\)=minklex​\{D⁡\(s−1,n−k\)\+As​\(k\)\},D⁡\(0,0\)=\(0,0,0\)\.D\(s,n\)=\\min\_\{k\}^\{\\mathrm\{lex\}\}\\\{D\(s\-1,n\-k\)\+A\_\{s\}\(k\)\\\},\\qquad D\(0,0\)=\(0,0,0\)\.\(12\)The three priorities are the number of lengths outside 6–10 tokens, their distance from this interval, and their deviation from the target rate\.

Withnsn\_\{s\}fixed, a local dynamic program constructs feasible reference boundariesb0s,1:nsb^\{0\}\_\{s,1:n\_\{s\}\}using lengths in\[4,16\]\[4,16\]and avoiding protected expressions\. It applies the same three priorities to the realized segment lengths, then uses total displacement from the evenly spaced boundariesround⁡\(j​ms/ns\)\\operatorname\{round\}\(jm\_\{s\}/n\_\{s\}\)as the final tie\-breaker\.

### B\.2Linguistic Boundary Costs

We combine rule\-based sentence detection with a spaCy dependency parser to identify natural sentence and clause boundaries\. Before parsing, structural templates, code, mathematics, and markup are replaced with equal\-length whitespace\. This prevents non\-prose spans from interfering with the syntactic analysis while preserving character alignment, so linguistic evidence can be mapped back to the original token boundaries\.

Each candidate boundary receives a base costβ⁡\(b\)\\beta\(b\)from local linguistic evidence\. We prefer boundaries at the ends of sentences or complete clauses, and discourage boundaries that split structural labels, attached punctuation, compact code or mathematical expressions, or lexical units\. Sentence endings are detected with deterministic rules that avoid common false positives such as decimals, ellipses, abbreviations, and protected expressions\. Table[6](https://arxiv.org/html/2610.09788#A2.T6)summarizes these preferences\.

The dependency parse additionally assigns a local risk levelr⁡\(b\)∈\{0,1,2\}r\(b\)\\in\\\{0,1,2\\\}, where higher values indicate a more severe disruption of a short local dependency\. We consider only relations spanning at most eight tokens so that long parser arcs do not penalize large portions of a sentence\. The combined boundary cost is

C⁡\(b\)=β⁡\(b\)\+3​r​\(b\)\.C\(b\)=\\beta\(b\)\+3r\(b\)\.\(13\)Cuts inside words, numbers, or identifiers receive a high base cost and are also marked separately, allowing lexical integrity to be optimized before the remaining soft preferences\. We additionally use a rule\-based fragment penaltyF⁡\(a,b\)∈\{0,4,8\}F\(a,b\)\\in\\\{0,4,8\\\}to discourage short, incomplete rubric fragments, while assigning zero cost to complete headers and status statements\.

Table 6:Base costs for candidate internal boundaries\. Lower costs favor a cut; protected expressions are excluded before scoring\.
### B\.3Constrained Local Refinement

Consider one outer block of lengthmmwith fixed countnnand reference boundariesb01:nb^\{0\}\_\{1:n\}\. A candidate partitionb0:nb\_\{0:n\}hasb0=0b\_\{0\}=0,bn=mb\_\{n\}=m, and lengthsdj=bj−bj−1d\_\{j\}=b\_\{j\}\-b\_\{j\-1\}\. We require

4≤dj≤16,bj∉ℱ\(j<n\),\|bj−bj0\|≤w,∑j<n𝟏\[r\(bj\)=2\]≤∑j<n𝟏\[r\(bj0\)=2\],\\begin\{split\}&4\\leq d\_\{j\}\\leq 16,\\qquad b\_\{j\}\\notin\\mathcal\{F\}\\quad\(j<n\),\\\\ &\|b\_\{j\}\-b\_\{j\}^\{0\}\|\\leq w,\\qquad\\sum\_\{j<n\}\\mathbf\{1\}\[r\(b\_\{j\}\)=2\]\\leq\\sum\_\{j<n\}\\mathbf\{1\}\[r\(b\_\{j\}^\{0\}\)=2\],\\end\{split\}\(14\)whereℱ\\mathcal\{F\}contains the protected boundaries and the default search radius isw=4w=4tokens\.

LetW⁡\(𝐛\)W\(\\mathbf\{b\}\)count the internal boundaries that cut a lexical unit\. Among feasible partitions, we first minimizeWW, then minimize

J⁡\(𝐛\)=∑j<nC⁡\(bj\)\+∑j=1nF⁡\(bj−1,bj\)\+0\.50​∑j=1nδ​\(dj\)2\+0\.02∑j=1n\(dj−m/n\)2\+0\.10∑j<n\|bj−bj0\|\.\\begin\{split\}J\(\\mathbf\{b\}\)=\{\}&\\sum\_\{j<n\}C\(b\_\{j\}\)\+\\sum\_\{j=1\}^\{n\}F\(b\_\{j\-1\},b\_\{j\}\)\+0\.50\\sum\_\{j=1\}^\{n\}\\delta\(d\_\{j\}\)^\{2\}\\\\ &\+0\.02\\sum\_\{j=1\}^\{n\}\(d\_\{j\}\-m/n\)^\{2\}\+0\.10\\sum\_\{j<n\}\|b\_\{j\}\-b\_\{j\}^\{0\}\|\.\\end\{split\}\(15\)The first two terms favor linguistic boundaries and complete fragments; the remaining terms control preferred length, length variance, and displacement from the reference partition\.

A dynamic\-programming state records the chunk count, current boundary, and risk\-2 count\. Transitions add feasible chunks and minimize\(W,J\)\(W,J\)lexicographically, breaking ties by the number of moved boundaries and then their coordinates\.

We adopt the candidate if it reducesWW\. WhenWWis unchanged, we requireJJnot to increase and the boundary\-plus\-fragment costH⁡\(𝐛\)=∑j<nC⁡\(bj\)\+∑jF⁡\(bj−1,bj\)H\(\\mathbf\{b\}\)=\\sum\_\{j<n\}C\(b\_\{j\}\)\+\\sum\_\{j\}F\(b\_\{j\-1\},b\_\{j\}\)to decrease by at least 0\.5; otherwise we retain the reference\.

Algorithm 1Syntax\-Aware Semantic Chunking with an Exact Latent BudgetRequire :teacher evaluationcc, tokenizer𝒯\\mathcal\{T\}, rateρ=8\\rho=8

1

\(𝐯,𝐨\)←TokenizeOnce​\(c,𝒯\)\(\\mathbf\{v\},\\mathbf\{o\}\)\\leftarrow\\textsc\{TokenizeOnce\}\(c,\\mathcal\{T\}\)
2

𝒪←OuterRubricBlocks​\(c,𝐨\)\\mathcal\{O\}\\leftarrow\\textsc\{OuterRubricBlocks\}\(c,\\mathbf\{o\}\)
3

N←⌈\|𝐯\|/8⌉N\\leftarrow\\lceil\|\\mathbf\{v\}\|/8\\rceil
4

\(n1,…,nS\)←AllocateBudget​\(𝒪,N\)\(n\_\{1\},\\ldots,n\_\{S\}\)\\leftarrow\\textsc\{AllocateBudget\}\(\\mathcal\{O\},N\)
5

ℱ←ProtectedStructuralCuts​\(c,𝐨\)\\mathcal\{F\}\\leftarrow\\textsc\{ProtectedStructuralCuts\}\(c,\\mathbf\{o\}\)
6

\(C,r,W,F\)←BoundaryAndFragmentEvidence​\(c,𝐨\)\(C,r,W,F\)\\leftarrow\\textsc\{BoundaryAndFragmentEvidence\}\(c,\\mathbf\{o\}\)
7foreach*outer blockss*do

8

𝐛s0←ProtectedReferencePartition​\(ms,ns,ℱs\)\\mathbf\{b\}\_\{s\}^\{0\}\\leftarrow\\textsc\{ProtectedReferencePartition\}\(m\_\{s\},n\_\{s\},\\mathcal\{F\}\_\{s\}\)
9

𝐛s←RefineAndSelect​\(𝐛s0,C,r,W,F,w=4\)\\mathbf\{b\}\_\{s\}\\leftarrow\\textsc\{RefineAndSelect\}\(\\mathbf\{b\}\_\{s\}^\{0\},C,r,W,F,w=4\)
10if*W⁡\(𝐛s\)\>0W\(\\mathbf\{b\}\_\{s\}\)\>0*then

11

𝐛s′←RefineAndSelect​\(𝐛s0,C,r,W,F,w=5\)\\mathbf\{b\}\_\{s\}^\{\\prime\}\\leftarrow\\textsc\{RefineAndSelect\}\(\\mathbf\{b\}\_\{s\}^\{0\},C,r,W,F,w=5\)
12if*W⁡\(𝐛s′\)<W⁡\(𝐛s\)W\(\\mathbf\{b\}\_\{s\}^\{\\prime\}\)<W\(\\mathbf\{b\}\_\{s\}\)*then

13

𝐛s←𝐛s′\\mathbf\{b\}\_\{s\}\\leftarrow\\mathbf\{b\}\_\{s\}^\{\\prime\}
Return :concatenated block\-local boundaries, shifted to global token offsets

Refinement leaves each outer block’s span and latent count unchanged without increasing lexical or risk\-2 cuts\. Stage 1 uses the resulting compression frontiers with the supervision objective in Appendix[C](https://arxiv.org/html/2610.09788#A3)\.

### B\.4Realized Allocation and a Complete Example

The training cache contains 35,612 evaluations, 27,541,067 explicit tokens, and 3,458,245 latent positions: 97\.11 latents per evaluation and 7\.96 explicit tokens per latent\. The per\-example latent counts match Fixed\-Rate Chunking\. As Table[7](https://arxiv.org/html/2610.09788#A2.T7)shows, 86\.18% of chunks contain 6–10 tokens\. Refinement moves 1,836,211 internal boundaries, reducing lexical cuts from 142,295 to 151 and risk\-2 cuts from 694,857 to 946\.

Table 7:Chunk lengths in the Qwen3\-8B semantic training cache\. All 35,612 evaluations retain their exact rate\-8 latent budgets\.Figures[5](https://arxiv.org/html/2610.09788#A2.F5)and[6](https://arxiv.org/html/2610.09788#A2.F6)compare the boundary locations produced by Semantic Chunking and Fixed\-Rate Chunking on the same cached evaluation, containing 219 explicit tokens and 28 latent positions\.

\-\-\- Compliance Check \-\-\-
Identified Gatekeeper Criterion:\|Criterion 1: The response directly addresses\|the question posed by providing\|a clear and explicit answer\.
\-\-\- Analysis \-\-\-\|
Global assessment /
gatekeeper discussion\*\*Response A:\*\*\|
\- Criterion 1 \[Hard Rule\]: Met\.\|The response provides the explicit answer "7"\.\|
\- Criterion 2 \[Principle\]: Met\.\|The answer "7"\|is self\-contained and clear\.\|
\- Criterion 3 \[Principle\]: Met\.\|The response uses conventional notation\|\(a numeral\) to communicate the answer\.\|
\*\*Response B:\*\*\|
\- Criterion 1 \[Hard Rule\]: Met\.\|The response provides the explicit answer "7"\.\|
\- Criterion 2 \[Principle\]: Met\.\|The answer is presented in a complete sentence,\|making it self\-contained and clear\.\|
\- Criterion 3 \[Principle\]: Met\.\|The response uses appropriate and conventional\|language to communicate the answer effectively\.\|
Response\-wise
rubric analysis\-\-\- Final Judgment \-\-\-
Justification:\|Both responses successfully meet all criteria\.\|Response B is slightly better as it provides\|a more complete and conversational answer,\|directly addressing the question in a full sentence\|while still being concise and clear\.\|
Preference label \(outside the chunked trace\): Response BFinal synthesis
and judgmentFigure 5:A complete evaluation from the semantic training cache, with the syntax\-aware frontiers used for training\. All 219 explicit tokens and all 28 latent frontiers are shown inside one frame\. Contiguous background regions follow the coarse semantic hierarchy, their identities are written at the right, and each uniform dark\|marks one latent frontier\.\-\-\- Compliance Check \-\-\-
Identified Gatekeeper\|Criterion: Criterion 1: The response\|directly addresses the question posed by providing a\|clear and explicit answer\.
\-\-\- Analysis \-\-\-\|
\*\*Response A:\*\*
\- Criterion\|1 \[Hard Rule\]: Met\. The\|response provides the explicit answer "7"\.\|
\- Criterion 2 \[Principle\]:\|Met\. The answer "7" is\|self\-contained and clear\.
\- Criterion\|3 \[Principle\]: Met\. The\|response uses conventional notation \(a numeral\)\|to communicate the answer\.
\*\*Response B\|:\*\*
\- Criterion 1 \[Hard\|Rule\]: Met\. The response provides the\|explicit answer "7"\.
\- Criterion\|2 \[Principle\]: Met\. The\|answer is presented in a complete sentence,\|making it self\-contained and clear\.
\-\|Criterion 3 \[Principle\]: Met\|\. The response uses appropriate and conventional language\|to communicate the answer effectively\.
\-\-\- Final\|Judgment \-\-\-
Justification: Both responses successfully\|meet all criteria\. Response B is slightly\|better as it provides a more complete and\|conversational answer, directly addressing the question\|in a full sentence while still being concise\|and clear\.\|
Preference label \(outside the chunked trace\): Response BFixed\-Rate
Chunking \(ρ=8\\rho=8\)Figure 6:Fixed\-Rate Chunking of exactly the same evaluation trace as Figure[5](https://arxiv.org/html/2610.09788#A2.F5)\. The 219 explicit tokens are divided into 27 complete groups of eight and one final group of three, yielding the same 28 latent tokens\. Only the red frontier locations change: unlike Semantic Chunking, they may split a phrase, a rubric criterion, or a transition between coarse semantic regions\.Figure 7:Stage 2 loss for the LatentGRM\-8B Semantic Chunking ablation\.Semantic Chunking maintains a lower Stage 2 loss after the initial decline\.

## Appendix CStage 1 Latent Distillation and Stage 2 Autonomous Evaluation

We now give the full objectives for an evaluation inputxx, teacher evaluationcc, and preference labelyy\. Letϕ\\phiandψ\\psidenote the trainable Stage 1 encoder and decoder parameters, and letθ\\thetadenote the trainable Stage 2 judge parameters\. Semantic Chunking specifiesSeg\(c\)=\{cbi−1\+1:bi\}i=1N\\mathrm\{Seg\}\(c\)=\\\{c\_\{b\_\{i\-1\}\+1:b\_\{i\}\}\\\}\_\{i=1\}^\{N\}, adapting the compression frontiers to rubric structure\. Throughout this section,c1:Mc\_\{1:M\}denotes the tokenized evaluation retained under the training sequence\-length limit\.

### C\.1Encoder Sequence and Vocabulary\-Space State

Let0=b0<b1<⋯<bN=M0=b\_\{0\}<b\_\{1\}<\\cdots<b\_\{N\}=Mbe the compression frontiers and letκi\\kappa\_\{i\}denote the compression placeholder followingcbi−1\+1:bic\_\{b\_\{i\-1\}\+1:b\_\{i\}\}\. The encoder sequence is

ξenc=\[x,<think\>,c1:b1,κ1,cb1\+1:b2,κ2,…,cbN−1\+1:bN,κN,</think\>\]\.\\begin\{split\}\\xi^\{\\mathrm\{enc\}\}=\[x,\\texttt\{<think\>\},c\_\{1:b\_\{1\}\},\\kappa\_\{1\},c\_\{b\_\{1\}\+1:b\_\{2\}\},\\kappa\_\{2\},\\ldots,c\_\{b\_\{N\-1\}\+1:b\_\{N\}\},\\kappa\_\{N\},\\texttt\{</think\>\}\]\.\\end\{split\}\(16\)The encoder attention mask is causal and additionally prevents each query from attending to earlier compression placeholders\. Ifuuandvvare non\-padding query and key positions, respectively, then

Au​venc=1⟺v≤u∧¬\(v∈\{κj\}j=1N∧v<u\)\.A^\{\\mathrm\{enc\}\}\_\{uv\}=1\\quad\\Longleftrightarrow\\quad v\\leq u\\ \\land\\ \\neg\(v\\in\\\{\\kappa\_\{j\}\\\}\_\{j=1\}^\{N\}\\land v<u\)\.\(17\)Hencehi=Encϕ⁡\(ξenc,Aenc\)κih\_\{i\}=\\operatorname\{Enc\}\_\{\\phi\}\(\\xi^\{\\mathrm\{enc\}\};A^\{\\mathrm\{enc\}\}\)\_\{\\kappa\_\{i\}\}can usexxandc1:bic\_\{1:b\_\{i\}\}, but neither future evaluation tokens nor earlier compression placeholders\.

LetVVbe the vocabulary size,ddthe hidden dimension, andτ\\tauthe projection temperature\. The decoder’s frozen output projectionWout∈ℝV×dW\_\{\\mathrm\{out\}\}\\in\\mathbb\{R\}^\{V\\times d\}mapshih\_\{i\}to vocabulary scores, while its frozen input embedding tableEin∈ℝV×dE\_\{\\mathrm\{in\}\}\\in\\mathbb\{R\}^\{V\\times d\}maps a vocabulary distribution back to the decoder’s input space\. The full\-vocabulary construction isαi=softmax⁡\(Wout​hi/τ\)\\alpha\_\{i\}=\\operatorname\{softmax\}\(W\_\{\\mathrm\{out\}\}h\_\{i\}/\\tau\)andzifull=Ein⊤​αiz\_\{i\}^\{\\mathrm\{full\}\}=E\_\{\\mathrm\{in\}\}^\{\\top\}\\alpha\_\{i\}\. We retain the indices of theKKlargest entries inSi=TopK⁡\(αi\)S\_\{i\}=\\operatorname\{TopK\}\(\\alpha\_\{i\}\)and compute

si,v\\displaystyle s\_\{i,v\}=Wout​\[v\]⊤​hi/τ,\\displaystyle=W\_\{\\mathrm\{out\}\}\[v\]^\{\\top\}h\_\{i\}/\\tau,\(18\)α¯i,v\\displaystyle\\bar\{\\alpha\}\_\{i,v\}=exp⁡si,v∑u∈Siexp⁡si,u,v∈Si,\\displaystyle=\\frac\{\\exp s\_\{i,v\}\}\{\\sum\_\{u\\in S\_\{i\}\}\\exp s\_\{i,u\}\},\\qquad v\\in S\_\{i\},\(19\)zi\\displaystyle z\_\{i\}=∑v∈Siα¯i,v​Ein​\[v\]\.\\displaystyle=\\sum\_\{v\\in S\_\{i\}\}\\bar\{\\alpha\}\_\{i,v\}E\_\{\\mathrm\{in\}\}\[v\]\.\(20\)Equivalently,α¯i,v=αi,v/∑u∈Siαi,u\\bar\{\\alpha\}\_\{i,v\}=\\alpha\_\{i,v\}/\\sum\_\{u\\in S\_\{i\}\}\\alpha\_\{i,u\}onSiS\_\{i\}, with zero probability outsideSiS\_\{i\}\. The full\-vocabulary formula is recovered whenK=VK=V\. Gradients pass through the mixture weights to the encoder statehih\_\{i\}but not intoWoutW\_\{\\mathrm\{out\}\}orEinE\_\{\\mathrm\{in\}\}\.

### C\.2Suffix Reconstruction Loss

The suffix\-reconstruction objective trains each latent prefix to support the subsequent explicit evaluation and final preference\. The remaining spans concatenate tocbi\+1:Mc\_\{b\_\{i\}\+1:M\}, followed by the preference suffix\. Our implementation uniformly samples one frontier and materializes its corresponding causal suffix view\.

For frontierii, define the decoder prefix and target

ui\\displaystyle u\_\{i\}=\[x,<think\>,z1:i\],\\displaystyle=\[x,\\texttt\{<think\>\},z\_\{1:i\}\],\(21\)ti\\displaystyle t\_\{i\}=\[cbi\+1:M,</think\>,y,EOS\]\.\\displaystyle=\[c\_\{b\_\{i\}\+1:M\},\\texttt\{</think\>\},y,\\texttt\{EOS\}\]\.\(22\)IfLiL\_\{i\}is the tokenized length oftit\_\{i\}, the per\-frontier loss is

ℓi\(ϕ,ψ\)=−1Li∑j=1Lilogpψ\(ti,j∣ui,ti,<j\),\\ell\_\{i\}\(\\phi,\\psi\)=\-\\frac\{1\}\{L\_\{i\}\}\\sum\_\{j=1\}^\{L\_\{i\}\}\\log p\_\{\\psi\}\\\!\\left\(t\_\{i,j\}\\mid u\_\{i\},t\_\{i,<j\}\\right\),\(23\)and the complete Stage 1 objective is

ℒrec​\(ϕ,ψ\)=1N​∑i=1Nℓi​\(ϕ,ψ\)\.\\mathcal\{L\}\_\{\\mathrm\{rec\}\}\(\\phi,\\psi\)=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\ell\_\{i\}\(\\phi,\\psi\)\.\(24\)The implementation samplesI∼Uniform⁡\{1,…,N\}I\\sim\\operatorname\{Uniform\}\\\{1,\\ldots,N\\\}for each example and evaluates onlyℓI\\ell\_\{I\}\. This estimator is unbiased because

𝔼I​\[ℓI\]=∑i=1N1N​ℓi=ℒrec\.\\mathbb\{E\}\_\{I\}\[\\ell\_\{I\}\]=\\sum\_\{i=1\}^\{N\}\\frac\{1\}\{N\}\\ell\_\{i\}=\\mathcal\{L\}\_\{\\mathrm\{rec\}\}\.\(25\)For the sampled frontier,\[uI,tI\]\[u\_\{I\},t\_\{I\}\]uses consecutive position indices and a causal decoder mask\. The compressed explicit prefix and future latent positions are omitted, while earlier target tokens remain visible through teacher forcing\. Loss is applied only totIt\_\{I\}\. Each sampled causal view provides one estimate of the full\-suffix objective\.

### C\.3The Three Stage 1 Optimization Phases

Letϕ0\\phi\_\{0\}andψ0\\psi\_\{0\}denote the initial encoder and decoder parameters\. The three phases optimize the same reconstruction objective with different parameter blocks:

ϕ^\\displaystyle\\widehat\{\\phi\}=arg⁡minϕ​ℒrec​\(ϕ,ψ0\),\\displaystyle=\\arg\\min\_\{\\phi\}\\mathcal\{L\}\_\{\\mathrm\{rec\}\}\(\\phi,\\psi\_\{0\}\),encoder phase: decoder frozen,\\displaystyle\\text\{encoder phase: decoder frozen\},\(26\)ψ^\\displaystyle\\widehat\{\\psi\}=arg⁡minψ​ℒrec​\(ϕ^,ψ\),\\displaystyle=\\arg\\min\_\{\\psi\}\\mathcal\{L\}\_\{\\mathrm\{rec\}\}\(\\widehat\{\\phi\},\\psi\),decoder phase: encoder frozen,\\displaystyle\\text\{decoder phase: encoder frozen\},\(27\)\(ϕ∗,ψ∗\)\\displaystyle\(\\phi^\{\*\},\\psi^\{\*\}\)=argminϕ,ψℒrec\(ϕ,ψ\)initialized at\(ϕ^,ψ^\),\\displaystyle=\\arg\\min\_\{\\phi,\\psi\}\\mathcal\{L\}\_\{\\mathrm\{rec\}\}\(\\phi,\\psi\)\\quad\\text\{initialized at \}\(\\widehat\{\\phi\},\\widehat\{\\psi\}\),joint phase\.\\displaystyle\\text\{joint phase\}\.\(28\)After joint training, the encoder runs once over every full teacher trace and exports\(Si,α¯i\)\(S\_\{i\},\\bar\{\\alpha\}\_\{i\}\)for all frontiers\. These sparse targets are stored by dataset index\. Stage 2 initializes from the jointly trained decoder adapter, and the encoder’s role ends after target export\.

### C\.4Stage 2 Soft\-Target Training and Inference

Letqθq\_\{\\theta\}denote the Stage 2 judge’s predicted distribution over the full vocabulary\. For every stored top\-KKtarget, training resamples independentgi,v∼Gumbel⁡\(0,1\)g\_\{i,v\}\\sim\\operatorname\{Gumbel\}\(0,1\)at each update and forms

α~i,v​\(𝐠\)=exp⁡\(\(log⁡α¯i,v\+λg​gi,v\)/τg\)∑u∈Siexp⁡\(\(log⁡α¯i,u\+λg​gi,u\)/τg\),v∈Si\.\\widetilde\{\\alpha\}\_\{i,v\}\(\\mathbf\{g\}\)=\\frac\{\\exp\\left\(\(\\log\\bar\{\\alpha\}\_\{i,v\}\+\\lambda\_\{g\}g\_\{i,v\}\)/\\tau\_\{g\}\\right\)\}\{\\sum\_\{u\\in S\_\{i\}\}\\exp\\left\(\(\\log\\bar\{\\alpha\}\_\{i,u\}\+\\lambda\_\{g\}g\_\{i,u\}\)/\\tau\_\{g\}\\right\)\},\\quad v\\in S\_\{i\}\.\(29\)whereλg\\lambda\_\{g\}controls the noise scale,τg\\tau\_\{g\}is the perturbation temperature, andα~i,v​\(𝐠\)=0\\widetilde\{\\alpha\}\_\{i,v\}\(\\mathbf\{g\}\)=0forv∉Siv\\notin S\_\{i\}\. This perturbation discourages overfitting to deterministic teacher targets and provides diverse latent trajectories at inference\. Teacher forcing insertsz~i​\(𝐠\)=∑v∈Siα~i,v​\(𝐠\)​Ein​\[v\]\\widetilde\{z\}\_\{i\}\(\\mathbf\{g\}\)=\\sum\_\{v\\in S\_\{i\}\}\\widetilde\{\\alpha\}\_\{i,v\}\(\\mathbf\{g\}\)E\_\{\\mathrm\{in\}\}\[v\]at theii\-th latent input position\. The prediction immediately preceding this position is matched to the same perturbed distribution, so latent inputs and KL targets share each noise draw\. The student distribution remains normalized over the full vocabulary\. For one draw, the sparse distillation loss is

ℒKL​\(𝐠\)=1N​∑i=1N∑v∈Siα~i,v​\(𝐠\)​log⁡α~i,v​\(𝐠\)qθ​\(v∣x,<think\>,z~<i​\(𝐠\)\)\.\\mathcal\{L\}\_\{\\mathrm\{KL\}\}\(\\mathbf\{g\}\)=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\\sum\_\{v\\in S\_\{i\}\}\\widetilde\{\\alpha\}\_\{i,v\}\(\\mathbf\{g\}\)\\log\\frac\{\\widetilde\{\\alpha\}\_\{i,v\}\(\\mathbf\{g\}\)\}\{q\_\{\\theta\}\(v\\mid x,\\texttt\{<think\>\},\\widetilde\{z\}\_\{<i\}\(\\mathbf\{g\}\)\)\}\.\(30\)This is teacher\-to\-student KL: the teacher target is zero outsideSiS\_\{i\}, while the student’s normalizer includes all vocabulary entries\. Letr=\[</think\>,y,EOS\]r=\[\\texttt\{</think\>\},y,\\texttt\{EOS\}\]be the visible suffix\. Its cross\-entropy is

ℒCE\(r;𝐠\)=−1\|r\|∑j=1\|r\|logqθ\(rj∣x,<think\>,z~1:N\(𝐠\),r<j\)\.\\mathcal\{L\}\_\{\\mathrm\{CE\}\}\(r;\\mathbf\{g\}\)=\-\\frac\{1\}\{\|r\|\}\\sum\_\{j=1\}^\{\|r\|\}\\log q\_\{\\theta\}\(r\_\{j\}\\mid x,\\texttt\{<think\>\},\\widetilde\{z\}\_\{1:N\}\(\\mathbf\{g\}\),r\_\{<j\}\)\.\(31\)Prompt and latent positions are excluded from this CE term\. The stochastic objective combines the latent and explicit losses:

ℒauto=𝔼𝐠​\[ℒKL​\(𝐠\)\+ℒCE​\(r,𝐠\)\]\.\\mathcal\{L\}\_\{\\mathrm\{auto\}\}=\\mathbb\{E\}\_\{\\mathbf\{g\}\}\\\!\\left\[\\mathcal\{L\}\_\{\\mathrm\{KL\}\}\(\\mathbf\{g\}\)\+\\mathcal\{L\}\_\{\\mathrm\{CE\}\}\(r;\\mathbf\{g\}\)\\right\]\.\(32\)
Both terms depend on the sampled trajectory through teacher\-forced inputs\. Each training update estimates the expectation with one draw per example\.

At inference, the judge generates its own latent trajectory\. Given the preceding latent states, stepiipredicts

qi=qθ\(⋅∣x,<think\>,z<i\)\.q\_\{i\}=q\_\{\\theta\}\(\\cdot\\mid x,\\texttt\{<think\>\},z\_\{<i\}\)\.\(33\)Ifarg⁡maxv​qi​\(v\)=</think\>\\arg\\max\_\{v\}q\_\{i\}\(v\)=\\texttt\{</think\>\}, latent generation terminates\. Otherwise, we define

Siθ=TopK⁡\(qi\),q¯i​\(v\)=\{qi​\(v\)∑u∈Siθqi​\(u\),v∈Siθ,0,v∉Siθ\.S\_\{i\}^\{\\theta\}=\\operatorname\{TopK\}\(q\_\{i\}\),\\qquad\\bar\{q\}\_\{i\}\(v\)=\\begin\{cases\}\\displaystyle\\frac\{q\_\{i\}\(v\)\}\{\\sum\_\{u\\in S\_\{i\}^\{\\theta\}\}q\_\{i\}\(u\)\},&v\\in S\_\{i\}^\{\\theta\},\\\\\[6\.0pt\] 0,&v\\notin S\_\{i\}^\{\\theta\}\.\\end\{cases\}\(34\)Replacing\(Si,α¯i\)\(S\_\{i\},\\bar\{\\alpha\}\_\{i\}\)in Equation[29](https://arxiv.org/html/2610.09788#A3.E29)with\(Siθ,q¯i\)\(S\_\{i\}^\{\\theta\},\\bar\{q\}\_\{i\}\)yields the perturbed weightsq~i​\(𝐠\)\\widetilde\{q\}\_\{i\}\(\\mathbf\{g\}\)\. The next latent input is

zi=∑v∈Siθq~i​\(v,𝐠\)​Ein​\[v\]\.z\_\{i\}=\\sum\_\{v\\in S\_\{i\}^\{\\theta\}\}\\widetilde\{q\}\_\{i\}\(v;\\mathbf\{g\}\)E\_\{\\mathrm\{in\}\}\[v\]\.\(35\)The recurrence also ends when the latent\-length cap is reached; the judge then feeds the end\-marker embedding and greedily decodes the preference label\. Distinct Gumbel draws provide the rollout diversity used for voting\. Appendix[E](https://arxiv.org/html/2610.09788#A5)specifies generation limits, and Appendix[F\.1](https://arxiv.org/html/2610.09788#A6.SS1)describes the vLLM implementation\.

## Appendix DLatent Trace Interpreter Details

The Latent Trace Interpreter \(LTI\) reconstructs an explicit evaluation from a latent trajectory\. It is initialized from the LatentGRM\-8B Stage 1 decoder, whose pretrained weights remain frozen, and learns its own LoRA adapter\.

### D\.1Vocabulary\-Probability Readout

For each positionii, the readout consumes top\-KKvocabulary indicesSisaveS\_\{i\}^\{\\mathrm\{save\}\}and normalized weightsq¯isave\\bar\{q\}\_\{i\}^\{\\mathrm\{save\}\}, following the notation in Section[3\.4](https://arxiv.org/html/2610.09788#S3.SS4)\. For a teacher\-derived trajectory,\(Sisave,q¯isave\)=\(Si,α¯i\)\(S\_\{i\}^\{\\mathrm\{save\}\},\\bar\{q\}\_\{i\}^\{\\mathrm\{save\}\}\)=\(S\_\{i\},\\bar\{\\alpha\}\_\{i\}\)is exported by the Stage 1 encoder\. For an autonomous trajectory,\(Sisave,q¯isave\)=\(Siθ,q~i​\(𝐠\)\)\(S\_\{i\}^\{\\mathrm\{save\}\},\\bar\{q\}\_\{i\}^\{\\mathrm\{save\}\}\)=\(S\_\{i\}^\{\\theta\},\\widetilde\{q\}\_\{i\}\(\\mathbf\{g\}\)\)is saved from the inference recurrence in Equations[34](https://arxiv.org/html/2610.09788#A3.E34)–[35](https://arxiv.org/html/2610.09788#A3.E35)\.

Equation[7](https://arxiv.org/html/2610.09788#S3.E7)maps the saved vocabulary probabilities through the interpreter’s embedding table\. The input is\[x,<think\>,z1:Nint\]\[x,\\texttt\{<think\>\},z^\{\\mathrm\{int\}\}\_\{1:N\}\], wherex=\(q,ℛ,a,b\)x=\(q,\\mathcal\{R\},a,b\)\. Each mixture occupies one position in the original trajectory order, with consecutive position indices and causal attention\. Text generation begins afterzNintz\_\{N\}^\{\\mathrm\{int\}\}\.

### D\.2Full\-Trajectory Reconstruction Objective

We pair each cached trajectory with the complete teacher evaluation that produced it\. Letω\\omegadenote the trainable LTI parameters, lett=\[c1:M,</think\>\]t=\[c\_\{1:M\},\\texttt\{</think\>\}\], and letLLbe its token length\. The training sequence is

\[x,<think\>,z1:Nint,c1:M,</think\>\],\[x,\\texttt\{<think\>\},z^\{\\mathrm\{int\}\}\_\{1:N\},c\_\{1:M\},\\texttt\{</think\>\}\],\(36\)with the token\-averaged teacher\-forcing objective

ℒint\(ω\)=−1L∑j=1Llogpω\(tj∣x,<think\>,z1:Nint,t<j\)\.\\mathcal\{L\}\_\{\\mathrm\{int\}\}\(\\omega\)=\-\\frac\{1\}\{L\}\\sum\_\{j=1\}^\{L\}\\log p\_\{\\omega\}\(t\_\{j\}\\mid x,\\texttt\{<think\>\},z^\{\\mathrm\{int\}\}\_\{1:N\},t\_\{<j\}\)\.\(37\)Loss covers the complete evaluation and closing marker; prompt and latent positions are masked, and the binary preference label is excluded\. Each example supplies one full trajectory\. LTI thus reconstructs the evaluation from its beginning, whereas the Stage 1 decoder predicts the suffix after a sampled frontier\.

### D\.3LTI Training Settings

For the reconstruction study, the LTI adapter is optimized with Equation[37](https://arxiv.org/html/2610.09788#A4.E37); the frozen weights include the input embeddings\. Table[8](https://arxiv.org/html/2610.09788#A4.T8)summarizes its training configuration\.

Table 8:LTI training configuration\. Effective batch size includes data\-parallel workers and gradient accumulation\.

## Appendix ESupplementary RQ1: Effectiveness and Rubric\-Based Process Evaluation

### E\.1Data, Backbones, and Evaluation Protocol

Each training record contains the pairwise judge promptxx, an explicit rubric\-based evaluationcc, and the binary labelyy\. The Semantic Chunking cache contains 35,612 OpenRubric SFT records\. It contains 27,541,067 explicit evaluation tokens and 3,458,245 latent frontiers, corresponding to 773\.36 explicit tokens and 97\.11 latent states per example on average\. The realized ratio is 7\.96 explicit tokens per latent state\. These counts are computed after applying the Qwen tokenizer and before Stage 1 frontier subsampling\.

We use Qwen3\-4B and Qwen3\-8B backbones\. Stage 1 starts from the base model and uses the dataset’s explicit evaluations as supervision\.

In the main benchmark runs, a prediction is valid only if its whitespace\-trimmed text equalsResponse AorResponse B\. Invalid generations cast no vote; ties are resolved by the earliest valid rollout\. In the dedicated efficiency runs, an equal vote count or no valid label yields an invalid prediction\.

### E\.2Evaluation\-Time Parameters and Seeded Diversity

The main benchmark runs use maximum sequence length 6,144, top\-K=10K=10, and Gumbel temperature and noise scale of one\. Prompts reserve space for the latent start/end markers, 256 latent positions, and eight answer tokens\.

Gumbel perturbations supply rollout diversity\. For example indexnnand vote indexrr, the seed is

sn,r=s0\+n\+r​\|𝒟\|,s0=42\.s\_\{n,r\}=s\_\{0\}\+n\+r\|\\mathcal\{D\}\|,\\qquad s\_\{0\}=42\.\(38\)This schedule assigns a distinct noise stream to each example–vote pair and shares seed assignments across voting budgets\.

### E\.3Protocol for Rubric\-Based Process Evaluation

This appendix specifies the LatentGRM\-8B interventions in Section[4\.2\.3](https://arxiv.org/html/2610.09788#S4.SS2.SSS3)and supplementary trajectory analyses\.

#### E\.3\.1Data and Annotation

Table 9:Example 1: brevity versus contextual detail\.Table 10:Example 2: figurative versus literal language\.##### Sampling\.

Using seed 42, we sampled 200 distinct question and response pairs from each of RewardBench Chat and Chat Hard, balancing the reference winner’s displayed position within each domain\. Semantic validation produced 398 accepted pairs, comprising 198 Chat and 200 Chat Hard pairs\. Each pair is evaluated in both response orders, and intervention scores are mapped to response identity before aggregation\.

##### Criterion generation and verification\.

GLM\-5\.2 was called separately for two distinct roles: generation and verification\. In the generation call, it proposed an equivalent paraphrase and a counterfactual for one criterion, while all other text remained fixed\. In the verification call, it independently checked the proposed edits against the rubric without benchmark labels or judge outputs\. We retain for which the beneficiary inferred in the generation stage agrees with the beneficiary inferred in the verification stage\.

##### Examples of GLM\-5\.2 criterion edits\.

Tables[9](https://arxiv.org/html/2610.09788#A5.T9)and[10](https://arxiv.org/html/2610.09788#A5.T10)show two accepted edits from the 398\-pair analysis\. Requests, responses, and criterion texts are reproduced verbatim; only the indicated criterion changes\. Directions below are the proposer and reviewer’s agreed criterion\-level predictions, not measured judge outcomes or overall winner labels\.

#### E\.3\.2Judge Analysis

For each pair, we build three rubric conditions: the original rubricℛ0\\mathcal\{R\}\_\{0\}, a paraphrase controlℛp\\mathcal\{R\}\_\{p\}that rewords one criterion without changing its meaning, and a criterion interventionℛc\\mathcal\{R\}\_\{c\}that changes that criterion to favor a designated response\. The paraphrase condition serves as a control: if a semantics\-preserving rewording also shifted the preference, the effect could not be attributed to the criterion’s meaning\. We index the trajectory source bys∈\{0,p,c\}s\\in\\\{0,p,c\\\}, meaning the trajectory was generated underℛs\\mathcal\{R\}\_\{s\}\.

For each sources∈\{0,p,c\}s\\in\\\{0,p,c\\\}, we probe its latent trajectory at0%,10%,…,100%0\\%,10\\%,\\ldots,100\\%of its recorded length\. This matches relative progress across sources without forcing equal lengths\. At each probe, we append the latent end marker and a sharedResponseprefix, and score the remaining label tokens\. We measure the log\-odds differenceDi​\(C\)D\_\{i\}\(C\)between the two response labels \(Equation[9](https://arxiv.org/html/2610.09788#S4.E9)\)\.

##### Criterion intervention\.

Each rubric generates its own autonomous trajectory\. Fors∈\{p,c\}s\\in\\\{p,c\\\}, we compare

Δ​Di,s​\(j\)=Di​\(ℛs,Zs≤ki,s,j\)−Di​\(ℛ0,Z0≤ki,0,j\)\.\\Delta D\_\{i,s\}\(j\)=D\_\{i\}\(\\mathcal\{R\}\_\{s\},Z\_\{s\}^\{\\leq k\_\{i,s,j\}\}\)\-D\_\{i\}\(\\mathcal\{R\}\_\{0\},Z\_\{0\}^\{\\leq k\_\{i,0,j\}\}\)\.\(39\)This contrast combines prompt and trajectory effects; its zero\-latent value captures a difference already present in the rubric prompt\.

##### Source replacement\.

We fix the recipient rubric toℛ0\\mathcal\{R\}\_\{0\}and replay the source trajectory generated underℛs\\mathcal\{R\}\_\{s\}in a fresh baseline context\. Donor prompt tokens and donor KV caches are not transferred\. Fors∈\{p,c\}s\\in\\\{p,c\\\}, we compare

Δ​Di,s​\(j\)=Di​\(ℛ0,Zs≤ki,s,j\)−Di​\(ℛ0,Z0≤ki,0,j\)\.\\Delta D\_\{i,s\}\(j\)=D\_\{i\}\(\\mathcal\{R\}\_\{0\},Z\_\{s\}^\{\\leq k\_\{i,s,j\}\}\)\-D\_\{i\}\(\\mathcal\{R\}\_\{0\},Z\_\{0\}^\{\\leq k\_\{i,0,j\}\}\)\.\(40\)
We compute signed effects per pair/order, average the two orders within each pair, then average pairs within each domain and weight the domains equally\. The interventions establish criterion\-directed preference shifts and their transfer through latent prefixes\.

### E\.4Direct Judgment and Voting

We compare DirectJudge, explicit Rubric\-RM, and LatentGRM using Qwen3\-8B on the RewardBench Chat and Chat Hard domains and the RewardBench2 Precise IF and Focus domains\. DirectJudge is initialized from the same backbone and trained for two epochs on the same OpenRubrics Dataset, supervising only the finalResponse A/Bverdict\. It receives the request, rubric, and both responses, but generates no intermediate evaluation trace\.

Table 11:Direct judgment versus explicit and latent evaluation \(%\)\.Direct judgment is a strong low\-decoding\-cost baseline\. Table[11](https://arxiv.org/html/2610.09788#A5.T11)shows competitive Chat performance and strong Focus scores, including an advantage over Rubric\-RM\. On Chat Hard, however, DirectJudge reaches only 67\.43% at vote@5, compared with 70\.50% for Rubric\-RM and 73\.00% for LatentGRM, and increasing votes from one to five improves it by only 0\.22 points\. This gap shows that direct judgment is limited on challenging preference comparisons\.

DirectJudge produces a terminal assessment without an autoregressive evaluation trajectory, and thus loses the test\-time scaling enabled by sampling multiple evaluation paths\. It also lacks the diversity needed for reinforcement learning to optimize\. LatentGRM instead retains a compact continuous trajectory that can be probed, inspected, and potentially optimized, at lower generation cost than explicit evaluation\.

### E\.5Data Revision Protocol and Evaluation Details

An evaluation model uses the full judging context, including the task\-specific rubric, to produce feedbackFFfor a candidate responsexx\. A separate reviser then generates

G⁡\(q,x,F\)→x′\.G\(q,x,F\)\\rightarrow x^\{\\prime\}\.\(41\)The reviser receives only the question, the response to revise, and the feedback interface under study; the rubric is never included in its input\.

We evaluate 240 RewardBench Rubric examples, with 120 from Chat and 120 from Chat Hard\. The rubric is used to generate evaluation feedback and to score the resulting revision, but is not exposed to the reviser\. All four conditions use the same Qwen3\-8B Base reviser in a fresh context:

1. 1\.Prompt\-only: the reviser receives only the question and the response to revise\.
2. 2\.Explicit trajectory: the textual reasoning generated by the Rubric\-RM\-8B\.
3. 3\.Latent only: the complete LatentGRM trajectory, represented at every recurrent step by the probability\-weighted mixture of its top\-K vocabulary embeddings, is inserted directly in the reviser’s context\. This is a training\-free interface: the reviser has never been trained to decode or consume LatentGRM trajectories\. It measures whether a pretrained model can directly turn latent reasoning into a better response\.
4. 4\.Interpreter\-restored trajectory: LTI converts the same soft latent trajectory into textual feedback, which is then supplied to a fresh frozen reviser context\.

We use DeepSeek\-V4\.1\-Flash\([Xu et al\., 2026](https://arxiv.org/html/2610.09788#bib.bib31)\)as the external evaluation model\. For each original or revised answer, it sees the question, the complete rubric\. It independently marks every rubric criterion as satisfied or unsatisfied and provides brief supporting evidence\. The original 240 answers contain 1,973 criterion decisions: 1,076 satisfied and 897 unsatisfied\. On a preselected 20\-example subset, reversing criterion order and paraphrasing the judging instruction yields 93\.76% criterion\-level agreement and Cohen’sκ=0\.803\\kappa=0\.803\.

For original labelssi​js\_\{ij\}and revised labelssi​j′s^\{\\prime\}\_\{ij\}, we measure

Repair\\displaystyle\\operatorname\{Repair\}=∑i​j𝟏​\[¬si​j∧si​j′\]∑i​j𝟏​\[¬si​j\],\\displaystyle=\\frac\{\\sum\_\{ij\}\\mathbf\{1\}\[\\neg s\_\{ij\}\\wedge s^\{\\prime\}\_\{ij\}\]\}\{\\sum\_\{ij\}\\mathbf\{1\}\[\\neg s\_\{ij\}\]\},\(42\)Regression\\displaystyle\\operatorname\{Regression\}=∑i​j𝟏​\[si​j∧¬si​j′\]∑i​j𝟏​\[si​j\],\\displaystyle=\\frac\{\\sum\_\{ij\}\\mathbf\{1\}\[s\_\{ij\}\\wedge\\neg s^\{\\prime\}\_\{ij\}\]\}\{\\sum\_\{ij\}\\mathbf\{1\}\[s\_\{ij\}\]\},\(43\)the percentage of answers satisfying every rubric criterion \(Full\), and overall criterion satisfaction\.

## Appendix FSupplementary RQ2: Practical Efficiency of Latent Evaluation

### F\.1vLLM Implementation for Latent Reasoning

We extend vLLM to pass a continuous embedding back into the decoder at each latent step\. The unperturbed argmax determines whether reasoning has ended; otherwise, the next input is a Gumbel\-weighted mixture of the top\-KKtoken embeddings\. After</think\>, decoding returns to discrete tokens for the preference label\. Each request keeps its own feedback and random state, allowing independent latent rollouts under continuous batching and native parallel sampling\.

For tensor\-parallel inference, workers select local candidates before forming the global top\-KKand mix their embeddings on device\. Cached input embeddings and persistent feedback buffers support the existing KV cache and CUDA\-graph decode path\. The explicit CoT baseline uses the same vLLM scheduler and request\-submission policy\.

### F\.2Equal\-Length Runtime Control

The 50\-position diagnostic covers all 716 Chat and 912 Chat Hard samples at vote@1, using the same prefix\-cache reset after warmup\. Both samplers set minimum and maximum output length to 50 and ignore EOS; the latent request uses an unreachable latent\-end ID to retain soft feedback throughout the measured trajectory\. Each model therefore produces exactly 81,400 positions\.

Table 12:Equal\-length serving throughput\.The full RewardBench Chat and Chat Hard subsets are evaluated at vote@1 with exactly 50 generated positions per sample\.LatentGRM adds one prompt position per sample for its start marker \(0\.11% of mean prompt length\)\. Total inference time is 46\.469 seconds for Rubric\-RM\-8B and 45\.714 seconds for LatentGRM\-8B, a difference of only1\.017×1\.017\\times\. Throughput uses the same total\-time boundary as natural generation\. Forced latent generation never enters visible\-answer decoding, so this control measures serving cost without producing accuracy predictions\. The near\-equal runtime at matched length shows that latent reasoning does not introduce additional per\-step computational cost\.

## Appendix GSupplementary RQ3: Recovering Rubric\-Level Evaluations

### G\.1Readout Data and Splits

The readouts use LatentGRM\-8B Stage 1 artifacts and teacher\-derived trajectories\. The seed\-42 split contains 28,430 training, 512 validation, and 3,298 test records\.

Of the test pool’s 57,097 parsed criteria, 9,768 have explicit states: 7,476Met, 1,644Not Met, and 648Partial\. We evaluate all 607 test records containing at least one such state\. These records contain 9,992 criteria; 548 are fully labeled and 59 partially labeled\.

### G\.2Readout Controls and Evaluation Protocol

To isolate the information contributed by the saved vocabulary\-probability trajectory, we compare LTI with two independently trained controls\. All three readouts are initialized from the LatentGRM\-8B Stage 1 decoder, keep its pretrained weights frozen, and learn separate LoRA adapters\.

##### Prompt\-only\.

This control uses the same reconstruction target as LTI but receives only\[x,<think\>\]\[x,\\texttt\{<think\>\}\]\. Removing the latent positions measures how much of the teacher evaluation can be reconstructed from the prompt alone\.

##### Replay\-hidden\.

We replay the latent trajectory through the frozen Stage 2 judge and extract the final\-layer statehireph\_\{i\}^\{\\mathrm\{rep\}\}at each latent position\. By causal attention,hireph\_\{i\}^\{\\mathrm\{rep\}\}summarizesxxandz1:iz\_\{1:i\}without access to the explicit evaluation\. A learned bridge maps these states into the readout’s input space:

h^irep=RMSNorm⁡\(hirep\),zihid=Wdir​h^irep\+Wup​SiLU⁡\(Wdown​h^irep\)\.\\begin\{split\}\\widehat\{h\}\_\{i\}^\{\\mathrm\{rep\}\}&=\\operatorname\{RMSNorm\}\(h\_\{i\}^\{\\mathrm\{rep\}\}\),\\\\ z\_\{i\}^\{\\mathrm\{hid\}\}&=W\_\{\\mathrm\{dir\}\}\\widehat\{h\}\_\{i\}^\{\\mathrm\{rep\}\}\+W\_\{\\mathrm\{up\}\}\\operatorname\{SiLU\}\(W\_\{\\mathrm\{down\}\}\\widehat\{h\}\_\{i\}^\{\\mathrm\{rep\}\}\)\.\\end\{split\}\(44\)HereWdir∈ℝd×dW\_\{\\mathrm\{dir\}\}\\in\\mathbb\{R\}^\{d\\times d\}is a direct linear projection, whileWdown∈ℝdb×dW\_\{\\mathrm\{down\}\}\\in\\mathbb\{R\}^\{d\_\{b\}\\times d\}andWup∈ℝd×dbW\_\{\\mathrm\{up\}\}\\in\\mathbb\{R\}^\{d\\times d\_\{b\}\}form a nonlinear bottleneck correction\. The outputszhid1:Nz^\{\\mathrm\{hid\}\}\_\{1:N\}replacezint1:Nz^\{\\mathrm\{int\}\}\_\{1:N\}in Equation[36](https://arxiv.org/html/2610.09788#A4.E36)\. The bridge and readout LoRA are trained jointly with Equation[37](https://arxiv.org/html/2610.09788#A4.E37), without propagating gradients into the Stage 2 judge\.

Prompt\-only follows the LTI training configuration in Table[8](https://arxiv.org/html/2610.09788#A4.T8)\. Replay\-hidden uses bottleneck widthdb=512d\_\{b\}=512, learning rate5×10−65\\times 10^\{\-6\}, and effective batch size 28; its remaining settings follow LTI\. All readouts use eight H20 GPUs\. Reconstruction is decoded greedily until</think\>ormin⁡\(4096,Lref\+100\)\\min\(4096,L\_\{\\mathrm\{ref\}\}\+100\)generated tokens, whereLrefL\_\{\\mathrm\{ref\}\}is the reference length\. The three outputs are evaluated with the same parser and metrics below\.

### G\.3Metrics and Parsing

State metrics use the explicit labels, Rubric Semantic uses all criterion justifications, and ROUGE\-L uses all 607 records\.

We parse traces byg=\(response identifier,criterion number,criterion type\)g=\(\\text\{response identifier\},\\text\{criterion number\},\\text\{criterion type\}\), retaining the last occurrence of duplicate keys\. Let𝒢\\mathcal\{G\}contain the 9,768 gold keys with explicit statesyg∈\{Met,Not Met,Partial\}y\_\{g\}\\in\\\{\\textsc\{Met\},\\textsc\{Not Met\},\\textsc\{Partial\}\\\}, and lety^g\\hat\{y\}\_\{g\}be the parsed prediction\. Missing or unparseable predictions count as errors; unlabeled reference criteria are excluded from state scoring\.

##### Criterion State Macro\-F1\.

For classcc, precision and recall arePc=T​Pc/\(T​Pc\+F​Pc\)P\_\{c\}=TP\_\{c\}/\(TP\_\{c\}\+FP\_\{c\}\)andRc=T​Pc/\(T​Pc\+F​Nc\)R\_\{c\}=TP\_\{c\}/\(TP\_\{c\}\+FN\_\{c\}\), withF​1c=2​Pc​Rc/\(Pc\+Rc\)F1\_\{c\}=2P\_\{c\}R\_\{c\}/\(P\_\{c\}\+R\_\{c\}\)\. We report

MacroF1=13​∑c∈\{Met,Not Met,Partial\}F​1c\.\\operatorname\{MacroF1\}=\\frac\{1\}\{3\}\\sum\_\{c\\in\\\{\\textsc\{Met\},\\textsc\{Not Met\},\\textsc\{Partial\}\\\}\}F1\_\{c\}\.\(45\)The three states receive equal weight; missing predictions contribute false negatives\.

##### Criterion Vector Exact\.

Letℱ\\mathcal\{F\}be the 548 records for which every parsed gold criterion has an explicit state, and let𝒢i\\mathcal\{G\}\_\{i\}be the criterion keys in recordii\. The strict record\-level score is

VectorExact=1\|ℱ\|∑i∈ℱ\[∀g∈𝒢i,y^g=yg\]\.\\operatorname\{VectorExact\}=\\frac\{1\}\{\|\\mathcal\{F\}\|\}\\sum\_\{i\\in\\mathcal\{F\}\}\\mathbf\{1\}\\\!\\left\[\\forall g\\in\\mathcal\{G\}\_\{i\},\\ \\hat\{y\}\_\{g\}=y\_\{g\}\\right\]\.\(46\)Partially labeled records contribute to criterion\-level metrics only\.

##### Rubric\-Aligned Semantic Score\.

For each of the 9,992 parsed gold criteria, we align the predicted and reference justifications bygg, compute rescaled RoBERTa\-large BERTScore\-F1 usingbert\-score0\.3\.12, assign zero when the predicted key is missing, and macro\-average over gold criteria:

RubricSemantic=1\|𝒢all\|∑g∈𝒢all𝟏\[g∈𝒢^\]BERTScoreF1\(e^g,eg\)\.\\operatorname\{RubricSemantic\}=\\frac\{1\}\{\|\\mathcal\{G\}\_\{\\rm all\}\|\}\\sum\_\{g\\in\\mathcal\{G\}\_\{\\rm all\}\}\\mathbf\{1\}\[g\\in\\widehat\{\\mathcal\{G\}\}\]\\,\\operatorname\{BERTScoreF1\}\(\\hat\{e\}\_\{g\},e\_\{g\}\)\.\(47\)All readouts use the same alignment policy and RoBERTa\-large checkpoint\. Table[13](https://arxiv.org/html/2610.09788#A7.T13)breaks down criterion\-state F1 by class\.

##### ROUGE\-L F1\.

We macro\-average ROUGE\-L F1 between complete generated and reference traces over the 607 records\. This measures whole\-trace lexical overlap, including shared templates\.

Table 13:Class\-wise criterion\-state F1 on the 607\-record test set \(%\)\.LTI exceeds 98% F1 for every state \(Table[13](https://arxiv.org/html/2610.09788#A7.T13)\)\. The largest separation from Replay\-hidden is onPartial\(98\.46 versus 34\.36\), showing that LTI better preserves distinctions among criterion judgments\.

### G\.4Qualitative Reconstruction Examples

In test record 14270, LTI recovers all ten criterion states and the comparative rationale\. The teacher trace \(Figure[8](https://arxiv.org/html/2610.09788#A7.F8)\) and LTI reconstruction \(Figure[9](https://arxiv.org/html/2610.09788#A7.F9)\) preserve the decisive facts\. Green marks wording in the teacher reference, and purple marks the corresponding semantic paraphrase reconstructed by LTI\.

[⬇](data:text/plain;base64,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)\-\-\-ComplianceCheck\-\-\-IdentifiedGatekeeperCriterion:Criterion1:Theresponsemustdirectlyaddressthespecificquestionposedintherequest\.\-\-\-Analysis\-\-\-\*\*ResponseA:\*\*\-Criterion1\[HardRule\]:Met\.Theresponsedirectly addresses the calculation requested\.\-Criterion2\[Principle\]:Met\.Theresponsecorrectlyappliestheorderofoperations\(PEMDAS\)andperformsthecalculationaccurately\.\-Criterion3\[Principle\]:Met\.Theresponseshows the steps: \(2\-4\) = \-2, then 2 \* \-2 = \-4\.\-Criterion4\[Principle\]:Met\.Thefinalanswer,\-4,ispresentedexplicitly\.\-Criterion5\[Principle\]:Met\.TheresponseusesstandardmathematicalnotationandthedollarsignforLaTeXformatting\.\*\*ResponseB:\*\*\-Criterion1\[HardRule\]:Met\.Theresponseprovidesananswertothequestion\.\-Criterion2\[Principle\]:NotMet\.Theresponsestatestheansweris0,whichisincorrect\.Thecorrectansweris\-4\.\-Criterion3\[Principle\]:NotMet\.Noreasoningorstepsareshown\.\-Criterion4\[Principle\]:Met\.Thefinalanswer,0,ispresentedexplicitly\.\-Criterion5\[Principle\]:Met\.Theresponseusesstandardnumericalformat\.\-\-\-FinalJudgment\-\-\-Justification:ResponseAcorrectlyanswersthemathematicalquestion,showsthestepsinvolved,andadherestoallothercriteriaintherubric\.ResponseBfails to provide the correct answer and does not show any reasoning\.</think\>Figure 8:Case study, record 14270: teacher reference evaluation\. Green highlights decisive wording in the original trace\.[⬇](data:text/plain;base64,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)\-\-\-ComplianceCheck\-\-\-IdentifiedGatekeeperCriterion:Criterion1:Theresponsemustdirectlyaddressthespecificquestionposedintherequest\.\-\-\-Analysis\-\-\-\*\*ResponseA:\*\*\-Criterion1\[HardRule\]:Met\.Theresponsedirectly answers the mathematical question\.\-Criterion2\[Principle\]:Met\.Theresponsecorrectlyappliestheorderofoperations\(PEMDAS\)\.\-Criterion3\[Principle\]:Met\.Theresponseshows the steps \(2\-4 = \-2, then 2\*\-2 = \-4\)\.\-Criterion4\[Principle\]:Met\.Thefinalanswer\(\-4\)ispresentedexplicitly\.\-Criterion5\[Principle\]:Met\.Theresponseusesappropriatemathematicalnotationandformat\.\*\*ResponseB:\*\*\-Criterion1\[HardRule\]:Met\.Theresponsedirectlyanswersthemathematicalquestion\.\-Criterion2\[Principle\]:NotMet\.Theresponsestatestheansweris0,whichisincorrect\.Thecorrectansweris\-4\.\-Criterion3\[Principle\]:NotMet\.Nostepsorreasoningareshown\.\-Criterion4\[Principle\]:Met\.Thefinalanswer\(0\)ispresentedexplicitly\.\-Criterion5\[Principle\]:Met\.Theresponseusesappropriatenotation\(anumber\)\.\-\-\-FinalJudgment\-\-\-Justification:ResponseAcorrectlyanswersthemathematicalquestion,showsthecorrectsteps,andadherestoallprinciplesoftherubric\.ResponseBprovides an incorrect answer and fails to show any reasoning or steps\.</think\>Figure 9:Case study, record 14270: LTI reconstruction\. Purple highlights semantic paraphrases of the teacher trace\.#### G\.4\.1A Reconstruction Failure

We additionally examine a reconstruction error from the same frozen 607\-record test set\. Among complete generations that recover every response–criterion key, record 17234 differs from the teacher reference on exactly one of 16 explicit states\. The reconstruction preserves the other states and the final preference for Response A, but it is locally inconsistent: it marks Response B’s Criterion 3 asMetafter noting the correct permutation formula, even though both its later analysis and final judgment recognize that the formula is applied incorrectly\. Figures[10](https://arxiv.org/html/2610.09788#A7.F10)and[11](https://arxiv.org/html/2610.09788#A7.F11)show the teacher trace and reconstruction, respectively; green marks the decisive reference evidence, while red marks the erroneous reconstructed judgment\.

[⬇](data:text/plain;base64,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)\-\-\-ComplianceCheck\-\-\-IdentifiedGatekeeperCriterion:Criterion1:Theresponsemustprovideafinalnumericalanswertothequestionposed\.\[HardRule\]\-\-\-Analysis\-\-\-\*\*ResponseA:\*\*\-Criterion1\[HardRule\]:Met\.Theresponseprovidesthenumericalanswer"30"\.\-Criterion2\[HardRule\]:Met\.Theresponseexplicitlystates"thewordMADAMhastwoA’sandtwoM’s"andmentions"fiveletters"\.\-Criterion3\[HardRule\]:Met\.Theresponsecorrectlyappliestheformulaforpermutationswithrepetitions:5\!/\(2\!\*2\!\)\.\-Criterion4\[Principle\]:Met\.Theresponseexplainswhydivisionisneeded\(toavoidduplicates\)andhowthefactorialtermsarederived\(arrangingidenticalletters\)\.\-Criterion5\[Principle\]:Met\.Themathematicalreasoningforapplyingtheformulaandcalculatingthefactorialsisconsistent\.\-Criterion6\[Principle\]:Met\.Thecalculation5\!/\(2\!\*2\!\)=120/\(2\*2\)=120/4=30isaccurate\.\-Criterion7\[Principle\]:Met\.Theresponseclearlystates"twoA’sandtwoM’s,whichareindistinguishablefromeachother"andexplainstheneedtodividebythearrangementsoftheseidenticalletters\.\-Criterion8\[Principle\]:Met\.Theresponsefollowsalogicalflow,startingwithidentifyingrepetitions,explainingtheproblemofduplicates,introducingtheformula,andthencalculatingtheresult\.\*\*ResponseB:\*\*\-Criterion1\[HardRule\]:Met\.Theresponseprovidesthenumericalanswer"60"\.\-Criterion2\[HardRule\]:Met\.Theresponsestates"MADAMhasfiveletters,butsomeofthemarerepeated\.TherearetwoA’sandtwoM’s\."\-Criterion3\[HardRule\]:Not Met\.Whiletheresponsestatesthecorrectformula\(n\!/\(r1\!r2\!\.\.\.rk\!\)\),itincorrectly applies itbystatingtheresultis60\.Thecalculationisflawed\.\-Criterion4\[Principle\]:Met\.Theresponseexplainstheformulaanditscomponents\.\-Criterion5\[Principle\]:NotMet\.Themathematicalreasoningleadstoanincorrectconclusionduetoacalculationerror\.\-Criterion6\[Principle\]:NotMet\.Thecalculation\(5times4times3times2times1\)/\(2times1times2times1\)simplifiesto120/4=30,not60\.Thecancellationstepisalsodescribedincorrectly\("canceloutsomefactors\.\.\.suchas2and2,and1and1\.Thisleavesmewith\(5times4times3\)/1,whichis60"\)\.Thissimplificationisincorrect\.\-Criterion7\[Principle\]:Met\.Theresponseidentifiestherepeatedletters\.\-Criterion8\[Principle\]:Met\.Theresponsepresentsthereasoninginastep\-by\-stepmanner\.\-\-\-FinalJudgment\-\-\-Justification:ResponseAcorrectlyidentifiestheproblem,appliestheappropriatecombinatorialformulaforpermutationswithrepetitions,andperformsthecalculationaccurately,arrivingatthecorrectanswerof30\.ResponseBcorrectlyidentifiestheproblemandtheformulabutmakesasignificantcomputationalerrorinitssimplificationandcalculation,leadingtoanincorrectanswerof60\.Therefore,ResponseAissuperiorduetoitsmathematicalaccuracy\.</think\>Figure 10:Reconstruction failure, record 17234: teacher reference evaluation\. Green marks the evidence that Response B’s Criterion 3 is not met\.[⬇](data:text/plain;base64,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)\-\-\-ComplianceCheck\-\-\-IdentifiedGatekeeperCriterion:Criterion1:Theresponsemustprovideafinalnumericalanswertothequestionposed\.\-\-\-Analysis\-\-\-\*\*ResponseA:\*\*\-Criterion1\[HardRule\]:Met\.Theresponseprovidesthenumericalanswer"30"\.\-Criterion2\[HardRule\]:Met\.Theresponseexplicitlystates"MADAMhastwoA’sandtwoM’s"andidentifiesthemasindistinguishable\.\-Criterion3\[HardRule\]:Met\.Theresponsecorrectlyappliestheformulaforpermutationswithrepeatedelements:5\!/\(2\!\*2\!\)\.\-Criterion4\[Principle\]:Met\.Theresponseexplainswhydivisionbyfactorialsisneededtoaccountforidenticalelementsandhowtheformulaisderived\.\-Criterion5\[Principle\]:Met\.Themathematicalreasoningisaccurateandconsistent,leadingtothecorrectcalculation\.\-Criterion6\[Principle\]:Met\.Thecalculation5\!/\(2\!\*2\!\)=120/4=30iscorrect\.\-Criterion7\[Principle\]:Met\.Theresponseclearlydistinguishesbetweenidentical\(A’sandM’s\)anddistinctelementsbystating"indistinguishable"andexplainingtheneedtodividebythenumberofwaystoarrangeidenticalletters\.\-Criterion8\[Principle\]:Met\.Thereasoningispresentedinalogical,step\-by\-stepmanner,startingwithidentifyingrepetitions,explainingtheneedfordivision,andthenperformingthecalculation\.\*\*ResponseB:\*\*\-Criterion1\[HardRule\]:Met\.Theresponseprovidesthenumericalanswer"60"\.\-Criterion2\[HardRule\]:Met\.Theresponsestates"MADAMhasfiveletters,butsomeofthemarerepeated\.TherearetwoA’sandtwoM’s\."\-Criterion3\[HardRule\]:Met\.Theresponsecorrectly identifies the formulaforpermutationswithrepeatedelements:n\!/\(r1\!r2\!\.\.\.rk\!\)\.\-Criterion4\[Principle\]:Met\.Theresponseexplainstheformulaanditscomponents\.\-Criterion5\[Principle\]:NotMet\.Themathematicalreasoningisincorrect\.Thecalculation5\!/\(2\!\*2\!\)is120/4=30,not60\.\-Criterion6\[Principle\]:NotMet\.Thecalculation\(5\*4\*3\*2\*1\)/\(2\*1\*2\*1\)simplifiesto120/4=30,not60\.Theexplanationofcancelingfactorsisalsoincorrectinthiscontext\.\-Criterion7\[Principle\]:Met\.Theresponsecorrectlyidentifiestherepeatedelements\.\-Criterion8\[Principle\]:Met\.Thereasoningispresentedinastep\-by\-stepmanner,althoughthefinalcalculationisincorrect\.\-\-\-FinalJudgment\-\-\-Justification:ResponseAcorrectlyidentifiestheproblem,appliestheappropriatecombinatorialformula,andperformsthecalculationaccurately,arrivingatthecorrectanswerof30\.ResponseBcorrectlyidentifiestheformulabutmakesasignificantcomputationalerrorinthefinalcalculation,leadingtoanincorrectanswerof60\.Therefore,ResponseAissuperiorduetoitsaccuracy\.</think\>Figure 11:Reconstruction failure, record 17234: LTI reconstruction\. Red marks the erroneousMetstate for Response B’s Criterion 3\.

## Appendix HTraining Setup and Compute

We train separate Qwen3\-4B and Qwen3\-8B LatentGRM models on the same 35,612 OpenRubrics records\. Each run uses one node with eight NVIDIA RTX 6000D GPUs \(85,651 MiB of visible memory per GPU\)\. Training uses bfloat16 precision, DeepSpeed ZeRO Stage 2, gradient checkpointing, and LoRA\. Stage 1 uses SDPA for its four\-dimensional encoder attention mask, whereas Stage 2 uses FlashAttention\-2\. Table[14](https://arxiv.org/html/2610.09788#A8.T14)summarizes the remaining settings\.

Table 14:Training configuration for LatentGRM\. Effective batch size includes eight data\-parallel workers and gradient accumulation\.The encoder LoRA targets the attention projections, MLP projections, andembed\_tokens; the last module learns the newly introduced<\|compress\_token\|\>embedding\. Decoder and Stage 2 adapters target the attention and MLP projections\. The vocabulary output matrix and input embedding basis are accessed separately, which is necessary for the untied Qwen3\-8B backbone\. Soft labels are generated in bfloat16 with batch size 16 and stored in shards of 1,000 examples\. Deterministic tokenization, Semantic Chunking boundaries, and frontier indices are cached once; only the uniformly sampled Stage 1 supervision frontier remains dynamic across epochs\.

## Appendix IPrompt Templates

`\\iow\_now:NeΞ\\iow\_now:NeΞYour task is to extract a set of rubric\-style instructions from a user’s request\. These rubrics will be used as evaluation criteria to check if a response fully meets the request\.\\iow\_now:NeΞ\\iow\_now:NeΞRubric items should be broadly reusable whenever possible\. Topic\-specific details \(e\.g\., names, places, numbers, or required content\) are allowed in a \[Hard Rule\] only when they are necessary to preserve an explicit requirement from the request\. Unnecessary topic\-specific details are invalid\.\\iow\_now:NeΞ\\iow\_now:NeΞ\- \*\*Two Distinct Categories:\*\*\\iow\_now:NeΞ \- \[Hard Rule\]: Preserve explicit requirements stated in the ¡request¿ exactly \(format, length, structure, names, numbers, forbidden/required elements, etc\.\)\.\\iow\_now:NeΞ \- \[Principle\]: Derived by abstracting any concrete cues into domain\-agnostic quality criteria \(e\.g\., clarity, correctness, sound reasoning, pedagogy\)\.\\iow\_now:NeΞ\- \*\*Comprehensiveness:\*\* The rubric must cover all critical aspects implied by the request \(including any examples contained in it\), spanning explicit requirements and implicit quality standards\.\\iow\_now:NeΞ\- \*\*Conciseness & Uniqueness:\*\* Each rubric must capture a distinct evaluation criterion\. Overlapping or redundant criteria must be merged into a single rubric\. Wording must be precise and free of repetition\.\\iow\_now:NeΞ\- \*\*Format Requirements:\*\*\\iow\_now:NeΞ \- Use a numbered list\.\\iow\_now:NeΞ \- Each item starts with ”The response” phrased in third person\.\\iow\_now:NeΞ \- Append \[Hard Rule\] or \[Principle\] at the end of each item\.\\iow\_now:NeΞ \- Do not include reasoning, explanations, or examples in the final output, only the rubrics\.\\iow\_now:NeΞ\\iow\_now:NeΞHere is the request:\\iow\_now:NeΞ–instruction˝\\iow\_now:NeΞ\\iow\_now:NeΞPlease generate the rubrics for the above request\. Rubric Generation Prompt`Figure 12:Prompt used to construct task\-specific rubric criteria\.`\\iow\_now:NeΞ\\iow\_now:NeΞYou are a fair and impartial judge\. Your task is to evaluate ’Response A’ and ’Response B’ based on a given instruction and a rubric\. You will conduct this evaluation in distinct phases as outlined below\.\\iow\_now:NeΞ\\iow\_now:NeΞ\#\#\# Phase 1: Compliance Check Instructions\\iow\_now:NeΞFirst, identify the single most important, objective ’Gatekeeper Criterion’ from the rubric\.\\iow\_now:NeΞ\- \*\*A rule is objective \(and likely a Gatekeeper\) if it can be verified without opinion\. Key examples are: word/paragraph limits, required output format \(e\.g\., JSON validity\), required/forbidden sections, or forbidden content\.\*\*\\iow\_now:NeΞ\- \*\*Conversely, a rule is subjective if it requires interpretation or qualitative judgment\. Subjective rules about quality are NOT Gatekeepers\. Examples include criteria like ”be creative,” ”write clearly,” ”be engaging,” or ”use a professional tone\.”\*\*\\iow\_now:NeΞThink step\-by\-step to determine this single most important Gatekeeper\.\\iow\_now:NeΞ\\iow\_now:NeΞ\#\#\# Phase 2: Analyze Each Response\\iow\_now:NeΞNext, for each Gatekeeper Criterion and all other criteria in the rubric, evaluate each response item by item\. For each item, think step\-by\-step and cite concrete evidence from the response before assigning your judgment\.\\iow\_now:NeΞ\\iow\_now:NeΞ\#\#\# Phase 3: Final Judgment Instructions\\iow\_now:NeΞBased on the results from the previous phases, determine the winner using these simple rules\. Provide a final justification explaining your decision first and then give your decision\. Think step\-by\-step to aggregate the findings and make the decision; keep the reasoning explicit and concise\.\\iow\_now:NeΞ\\iow\_now:NeΞ—\\iow\_now:NeΞ\#\#\# REQUIRED OUTPUT FORMAT\\iow\_now:NeΞYou must follow this exact output format below\.\\iow\_now:NeΞ— Compliance Check —\\iow\_now:NeΞIdentified Gatekeeper Criterion: ¡e\.g\., Criterion 1: Must be under 50 words\.¿\\iow\_now:NeΞ\\iow\_now:NeΞ— Analysis —\\iow\_now:NeΞ\*\*Response A:\*\*\\iow\_now:NeΞ\- Criterion 1 \[Hard Rule\]: Justification: ¡…¿\\iow\_now:NeΞ\- Criterion 2 \[Hard Rule\]: Justification: ¡…¿\\iow\_now:NeΞ\- Criterion 3 \[Principle\]: Justification: ¡…¿\\iow\_now:NeΞ\- … \(and so on for all other criteria\)\\iow\_now:NeΞ\\iow\_now:NeΞ\*\*Response B:\*\*\\iow\_now:NeΞ\- Criterion 1 \[Hard Rule\]: Justification: ¡…¿\\iow\_now:NeΞ\- Criterion 2 \[Hard Rule\]: Justification: ¡…¿\\iow\_now:NeΞ\- Criterion 3 \[Principle\]: Justification: ¡…¿\\iow\_now:NeΞ\- … \(and so on for all other criteria\)\\iow\_now:NeΞ\\iow\_now:NeΞ— Final Judgment —\\iow\_now:NeΞJustification: ¡…¿\\iow\_now:NeΞWinner: ¡Response A / Response B¿\\iow\_now:NeΞ\\iow\_now:NeΞTask to Evaluate:\\iow\_now:NeΞInstruction:\\iow\_now:NeΞ–instruction˝\\iow\_now:NeΞRubric:\\iow\_now:NeΞ–rubric˝\\iow\_now:NeΞResponse A:\\iow\_now:NeΞ–response˙a˝\\iow\_now:NeΞResponse B:\\iow\_now:NeΞ–response˙b˝ Explicit Evaluation\-Trace Prompt`Figure 13:Prompt used to elicit a structured explicit evaluation trace\.`\\iow\_now:NeΞ\\iow\_now:NeΞYou are a fair and impartial judge\. Your task is to evaluate ’Response A’ and ’Response B’ based on a given instruction and a rubric\. You will conduct this evaluation in distinct phases as outlined below\.\\iow\_now:NeΞ\\iow\_now:NeΞ\#\#\# Phase 1: Compliance Check Instructions\\iow\_now:NeΞFirst, internally identify the single most important, objective ’Gatekeeper Criterion’ from the rubric\.\\iow\_now:NeΞ\- \*\*A rule is objective \(and likely a Gatekeeper\) if it can be verified without opinion\. Key examples are: word/paragraph limits, required output format \(e\.g\., JSON validity\), required/forbidden sections, or forbidden content\.\*\*\\iow\_now:NeΞ\- \*\*Conversely, a rule is subjective if it requires interpretation or qualitative judgment\. Subjective rules about quality are NOT Gatekeepers\. Examples include criteria like ”be creative,” ”write clearly,” ”be engaging,” or ”use a professional tone\.”\*\*\\iow\_now:NeΞReason step\-by\-step internally to determine this single most important Gatekeeper\.\\iow\_now:NeΞ\\iow\_now:NeΞ\#\#\# Phase 2: Analyze Each Response\\iow\_now:NeΞInternally evaluate Response A and Response B against the Gatekeeper Criterion and every other criterion in the rubric\. For each criterion, reason step\-by\-step and use concrete evidence from the responses\.\\iow\_now:NeΞ\\iow\_now:NeΞ\#\#\# Phase 3: Final Judgment Instructions\\iow\_now:NeΞInternally aggregate the findings and determine the better response\. Keep the reasoning precise and consistent with the rubric\.\\iow\_now:NeΞ\\iow\_now:NeΞ\#\#\# REQUIRED OUTPUT FORMAT\\iow\_now:NeΞOutput exactly one label and nothing else: Response A or Response B\.\\iow\_now:NeΞ\\iow\_now:NeΞTask to Evaluate:\\iow\_now:NeΞInstruction:\\iow\_now:NeΞ–instruction˝\\iow\_now:NeΞRubric:\\iow\_now:NeΞ–rubric˝\\iow\_now:NeΞResponse A:\\iow\_now:NeΞ–response˙a˝\\iow\_now:NeΞResponse B:\\iow\_now:NeΞ–response˙b˝ LatentGRM Training and Evaluation Prompt`Figure 14:Prompt used for LatentGRM training and inference\.