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APTER: Adaptive Post-Training with Expert-Grounded Rubrics
Xukai Wang, Liangqi Li, Zhiyue Xu, Jingang Zhou, Xiaoyu Shi, Jiansheng Cai, Bo Zhang, Zhe Li, Xu-Yao Zhang
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Status: succeeded | Model: Gemma-4-26B-A4B | Prompt: intel-v1 | Confidence: 93%
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Summary
The paper introduces APTER (Adaptive Post-Training with Expert-Grounded Rubrics), a framework for fine-grained evaluation and optimization of large language models in specialized domains. APTER constructs query-level rubrics from a stable, expert-defined criteria framework, enabling reusable supervision without reference answers. It employs adaptive post-training strategies, including Rubric RL, Rubric-based SFT, and Ada-IFT (Adaptive Interleaved Fine-Tuning), which uses criterion-level failure statistics to trigger targeted supervised repair. Experiments on mathematical reasoning and medical question answering demonstrate significant performance gains over base models.
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APTER → appliedto → Qwen2.5-7B-Instruct
confidence 95% · We evaluate APTER on Qwen2.5-7B-Instruct [16]... Relative to the corresponding base models, APTER improves the mathematics macro-average...
APTER → appliedto → Qwen3.5-9B
confidence 95% · We evaluate APTER on Qwen2.5-7B-Instruct [16], Qwen3-8B [17], and Qwen3.5-9B [18]... Relative to the corresponding base models, APTER improves the mathematics macro-average...
APTER → appliedto → Qwen3-8b
confidence 95% · We evaluate APTER on Qwen2.5-7B-Instruct [16], Qwen3-8B [17]... Relative to the corresponding base models, APTER improves the mathematics macro-average...
APTER → evaluatedon → HealthBench
confidence 95% · Experiments on mathematical reasoning and medical question answering show consistent gains... gains of up to 25.82 points on AIME 24 and 17.96 points on HealthBench.
APTER → evaluatedon → AIME-24
confidence 95% · Experiments on mathematical reasoning and medical question answering show consistent gains... gains of up to 25.82 points on AIME 24 and 17.96 points on HealthBench.
APTER → includes → Ada-IFT
confidence 95% · APTER supports Rubric RL, Rubric-based SFT, and Ada-IFT (Adaptive Interleaved Fine-Tuning).
APTER → uses → Expert Criteria Framework
confidence 95% · APTER consists of expert-grounded rubric construction and adaptive post-training. The first component uses an expert-constructed framework of reusable professional criteria...
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Abstract
Abstract:As large language models enter professional domains, they must satisfy domain constraints, include critical evidence, and provide complete reasoning rather than merely produce fluent responses. Existing post-training methods often rely on holistic preferences or outcome-level verification, while recent rubric-based methods usually generate rubrics independently for each query. In specialized domains, such unconstrained rubrics may omit critical requirements and vary across samples, hindering the diagnosis and targeted repair of persistent capability deficiencies. We propose APTER (Adaptive Post-Training with Expert-Grounded Rubrics), a framework that integrates structured domain knowledge into fine-grained evaluation, optimization, and diagnosis for specialized complex reasoning. First, expert-grounded rubric construction starts from an expert criteria framework built by domain experts, where each criterion represents a stable professional capability. For each query, APTER selects relevant criteria and instantiates them into query-level rubrics linked to their source criteria, turning reusable expert criteria into executable query-level supervision without reference answers. Second, adaptive post-training uses rubric verdicts as both optimization and criterion-level diagnostic signals. Aggregating low-scoring verdicts by criterion ID reveals persistent deficiencies and triggers targeted supervised fine-tuning updates during reinforcement learning. Experiments on mathematical reasoning and medical question answering show consistent gains across both domains. Across three model generations, APTER improves the mathematics and medical averages over the corresponding base models by up to 15.86 and 8.04 points, respectively. Code and rubric datasets are available at this https URL.
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- Source: https://arxiv.org/abs/2608.14212v1
- Canonical: https://arxiv.org/abs/2608.14212v1
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T E C H N O L O G I E S D I G I T A L APTER: Adaptive Post-Training with Expert-Grounded Rubrics Xukai Wang ∗ , Liangqi Li ∗ , Zhiyue Xu, Jingang Zhou, Xiaoyu Shi, Jiansheng Cai, Bo Zhang † , Zhe Li, Xu-Yao Zhang † Ant Digital Technologies, Ant Group Abstract As large language models enter professional domains, they must satisfy domain constraints, include critical evidence, and provide complete reasoning rather than merely pro- duce fluent responses. Existing post-training methods often rely on holistic preferences or outcome-level verifi- cation, while recent rubric-based methods usually gener- ate rubrics independently for each query. In specialized domains, such unconstrained rubrics may omit critical requirements and vary across samples, hindering the di- agnosis and targeted repair of persistent capability de- ficiencies. We propose APTER (Adaptive Post-Training with Expert-Grounded Rubrics), a framework that inte- grates structured domain knowledge into fine-grained evaluation, optimization, and diagnosis for specialized complex reasoning. First, expert-grounded rubric con- struction starts from an expert criteria framework built by domain experts, where each criterion represents a stable professional capability. For each query, APTER selects rele- vant criteria and instantiates them into query-level rubrics linked to their source criteria, turning reusable expert cri- teria into executable query-level supervision without refer- ence answers. Second, adaptive post-training uses rubric verdicts as both optimization and criterion-level diagnos- tic signals. Aggregating low-scoring verdicts by criterion ID reveals persistent deficiencies and triggers targeted su- pervised fine-tuning updates during reinforcement learn- ing. Experiments on mathematical reasoning and medi- cal question answering show consistent gains across both domains. Across three model generations, APTER im- proves the mathematics and medical averages over the corresponding base models by up to 15.86 and 8.04 points, respectively. Code and rubric datasets are available at https://github.com/AntDT-APTER/APTER.git. 1. Introduction Large language models (LLMs) increasingly address specialized tasks requiring expert judgment [1,2]. In such settings, a fluent response with a plausible conclusion may still fail if it violates domain con- straints, omits critical evidence, or lacks complete reasoning. Recent analyses of medical LLM bench- marks further show that evaluating such failures re- quires lifecycle-oriented, safety-aware, and clinically faithful criteria, which are often specified by domain * Equal contribution. † Corresponding to boyuan.zb@antgroup.com and xyz@nlpr.ia.ac.cn. QueryResponses Pair Reward Model 푦 ! 푦 " Reward Diff 휎(푟 ! −푟 " ) Coarse No attribution QueryResponseVerifier 0 / 1 Score Reward Outcome-only Verifiable only Query Ref Response R1 R2 R3 Rubrics 푠=[.5,.4,.7] Scores Weak expert knowledge Fragmented Hard to improve Fragmented Rewards (d) APTER: Expert-grounded Rubric Supervision Query Expert Criteria R1 R2 R3 Rubrics 푠=[.5,.4,.7] Scores Reward Diagnosis Expert criteria Capability Aligned Adaptive repairing (b) RLVR: Outcome Verification (a) RLHF: Holistic Preference (c) Existing Rubric-based Methods Figure 1: Comparison of post-training supervision signals. RLHF provides coarse response-level preferences and RLVR provides outcome verification, while existing rubric-based methods often generate query-level rubrics without explicit grounding in expert-defined criteria. APTER integrates struc- tured domain knowledge into rubric construction and uses rubric verdicts for fine-grained optimization, capability diag- nosis, and targeted repair. experts rather than captured by leaderboard-style final scores alone [3]. As illustrated in Figure 1(a) and (b), existing post-training methods often use re- inforcement learning from human feedback (RLHF) with pairwise preferences [4] or reinforcement learn- ing with verifiable rewards (RLVR) for tasks with verifiable answers [5,6]. Both have proven effective, but expert knowledge is still insufficiently encoded in these signals: they do not explicitly represent which domain criteria should be checked, nor how failures on those criteria should guide model improvement. Fine-grained supervision can better capture such professional requirements than holistic or outcome- level signals. Process-level feedback offers one op- tion [7,8,9], but it often requires expensive step-level 1 arXiv:2608.14212v1 [cs.AI] 14 Aug 2026 annotations. Rubric-based evaluation and LLM-as- a-Judge methods provide a more scalable alterna- tive [10,11,12,13,14,15]. As shown in Figure 1(c), existing rubric-based methods often generate rubrics independently for individual queries or responses. In specialized domains, however, rubrics should re- flect domain-critical requirements that may be diffi- cult for general-purpose LLMs to identify and pri- oritize from a query alone. Unconstrained gener- ation may omit critical constraints or encode inap- propriate standards. Variations in rubric granularity and domain depth also hinder aggregating failures across samples. For example, missed contraindica- tions and omitted safety constraints may appear un- related unless mapped to a shared expert criterion such as safety-constraint handling. We propose APTER (Adaptive Post-Training with Expert-Grounded Rubrics), a framework that in- tegrates structured domain knowledge into fine- grained evaluation, optimization, and capability di- agnosis for specialized complex reasoning. APTER consists of expert-grounded rubric construction and adaptive post-training. As illustrated in Figure 1(d), the first component uses an expert-constructed framework of reusable professional criteria, such as evidence coverage, constraint handling, and reason- ing completeness. It routes relevant criteria for each query and instantiates them as assessable rubrics with source criterion IDs, providing query-specific supervision without per-query expert authoring or reference answers. The second uses rubric verdicts as both optimization and criterion-level diagnostic sig- nals. APTER supports Rubric RL, Rubric-based SFT, and Ada-IFT (Adaptive Interleaved Fine-Tuning). Ada-IFT aggregates failures by criterion to iden- tify persistent capability deficiencies and trigger tar- geted supervised repair during reinforcement learn- ing, serving as the default configuration in the main experiments. We evaluate APTER in two comple- mentary regimes: mathematics tests rubric rewards against strong RLVR baselines in a verifiable setting, whereas open-ended medical question answering naturally requires multi-criterion expert evaluation and optimization. Relative to the corresponding base models, APTER improves the mathematics macro- average by 6.13–15.86 points and the medical macro- average by 5.19–8.04 points across three Qwen model generations [16,17,18], with gains of up to 25.82 points on AIME 24 and 17.96 points on HealthBench. Our main contributions are as follows: 1. We introduce an expert-grounded rubric con- struction framework that integrates structured domain knowledge into query-level rubric gen- eration by routing expert-defined criteria to each query and instantiating them as executable rubrics without requiring reference answers. 2.We propose adaptive post-training, where rubric verdicts serve as both optimization and criterion-level diagnostic signals, enabling Ada- IFT to identify persistent capability deficiencies and trigger targeted repair. 3. We implement APTER for mathematical reason- ing and medical question answering, construct expert criteria frameworks and rubric datasets for both domains, and demonstrate consistent improvements over strong post-training base- lines. 2. Related Work Rubric construction and evaluation criteria. With the growth of LLM-as-a-Judge [10,11] and rubric-based reward modeling, recent work studies scalable rubric generation, refinement, retrieval, and adaptive design [12,13,19,20,21]. Domain evalua- tion frameworks such as HealthBench further high- light the value of aligning evaluation criteria with expert judgment in professional settings [2]. These methods commonly generate rubrics for individual queries or responses, which is flexible but can vary in granularity and domain depth across samples. APTER instead instantiates query-level rubrics from reusable expert criteria and retains their criterion linkage for cross-query diagnosis. Rubric-based rewards for reasoning RL.Reason- ing rewards range from outcome verification and RLVR [5,6,22] to process-level supervision [8,9] and LLM- or rubric-based evaluation for open-ended responses [10,14,15]. Recent work incorporates rubric scores through reward design, rubric an- chors, or advantage decomposition [14,15,23,24,25]. APTER likewise uses rubric-level rewards, while sta- ble expert-criterion links make the same verdicts reusable for capability diagnosis. Interleaving RL with supervised fine-tuning.In- terleaving RL with SFT can repair reasoning fail- ures that exploration alone does not resolve. ReST and rejection-sampling methods alternate sampling with filtered SFT [26,27,28], while ReLIFT and it- erative self-improvement introduce SFT during or between RL stages [29,30]. Rubric-guided and self- verification methods provide related signals for near- miss responses [31,32]; APTER instead triggers re- pair from recurring failures aggregated under stable expert criterion IDs. 2 Expert-Grounded Rubrics Construction Expert Criteria Framework Domain Sub- domain Capability Criterion Query Routing C1C3C2C4C5C6C7 C2C5C7 Selected Criteria ! ! w/ weights Selection Query " Multi-role Rubric Generation Analysts (Decompose) Consolidator (Merge & Draft) Auditor (Quality Check) Refiner (Rubrics Polish) Decompose a query by multi analysts Draft and merge rubrics for each criterion Check clarity, coverage, consistency, etc. Improve wording, scales, and weights Expert-in-the-loop Calibration Expert ReviewJudge CalibrationPipeline Update Assess quality and provide feedback Align judge with expert preferences Update criteria framework, prompts, and templates C1C3C2C4C5C7 R1 R2 R3 Query-Level Rubrics Criterion Back-mapping Rubric Judge Verdicts C6 Rubric-Driven Post-Training Rubric RL Use rubric scores as rewards to optimize the policy model. GeometryAlgebraCalculus Math Condition Use Key StepVerification Policy ModelRollout Response Judge Model Weighted Reward RL Updating Rubric-based SFT Filter and select high-quality data for supervised fine-tuning. Frontier Candidates Rubric Filtering High-quality SFT Data Rubrics in CoT Answer Planning & Response RiC-SFT (CoT w/ Rubrics) Ada-IFT (Adaptive Interleaved Fine-Tuning) R1C20 R2C51 R3C70 C10.85 C30.73 C20.59 Failure Rate C1C2C3C4C5 Threshold >" !"#$ ? Low-Scoring Rubrics Aggregate by Criterion ID Criterion-Level Failure stats Trigger Check Targeted SFT Buffer Query: q Rubrics: Response: ______ Frontier Repair Sample Targeted SFT Update Improved Policy Resume RL (back to rollout) 865 R10 R21 R30 Diagnostic Signal (Low scores) Reward Signal (Optimization) Let AB=8, BC=6, AC=10 in △ABC. Find ∠B. Using the Law of Cosines ... Substitute the given ... Weighted Reward QueryResponse #= ∑ "#$ % & " ⋅( " ∑ "#$ % & " Geometry Algebra Calculus Math Condition Use Key StepVerification Select Relevant Theorem Set Up Equation Derive Target Angle R1 7 R2 6 R3 5 Judge Model Figure 2: The APTER pipeline. Expert-grounded rubric construction first builds an expert criteria framework and instantiates relevant criteria into query-level rubrics through routing, multi-role generation and refinement, and expert-in-the-loop calibration. The resulting rubric verdicts preserve criterion back-mapping and are used for Rubric RL, Rubric-based SFT, and Ada-IFT. 3. Methodology 3.1. Overview We propose APTER (Adaptive Post-Training with Expert-Grounded Rubrics), a framework combining expert-grounded rubric construction with adaptive post-training. As shown in Figure 2, the former instantiates expert-defined criteria into query-level rubrics with retained provenance; the latter uses their verdicts for Rubric RL, Rubric-based SFT, and Ada- IFT, with Ada-IFT serving as the default configura- tion. 3.2. Expert-Grounded Rubric Construction We call the resulting rubrics expert-grounded because they are instantiated from an expert-constructed cri- teria framework, retain their source criterion IDs, and are calibrated through expert feedback. 3.2.1. Expert Criteria Framework. Before constructing query-level rubrics, domain ex- perts organize recurring professional capabilities into a hierarchy of domain→sub-domain→capabil- ity category→criterion. Each reusable criterion has a persistent identifier, a concise definition, and representative examples. Experts review the frame- work for coverage, overlap, and operational clarity, merging redundant criteria and clarifying ambigu- ous boundaries. Expert-approved updates are incorporated be- tween training runs, while the framework and crite- rion IDs remain fixed within a run. The framework therefore both constrains query-level rubric genera- tion and provides a stable capability space for aggre- gating verdicts across queries. The detailed construc- tion, review, and maintenance protocol is provided in the supplementary materials. 3.2.2. Query-Level Routing and Rubric Instantia- tion. For each query, APTER selects relevant expert crite- ria and instantiates them into concrete, executable evaluation standards. This process does not require experts to write every query-level rubric or provide reference answers, making it applicable to unlabeled and open-ended tasks without tying the evaluation standard to a single response. Given a query푞and an expert criteria framework 퐶, a routing function푃(·)selects the subset of criteria relevant to the query: 퐶 푞 = 푃(푞,퐶), 퐶 푞 ⊆ 퐶.(1) For each selected criterion, APTER instantiates a query-specific rubric and outputs a weighted rubric set: 푟 푖 = 퐺(푞, 푐 푖 ), 푅 푞 =(푐 푖 ,푤 푖 ,푟 푖 ) 퐿 푞 푖=1 ,(2) 3 where퐺(·)conditions generation on the query and selected criterion,퐿 푞 is the number of rubrics, and 푤 푖 denotes relevance and importance. The router ex- cludes irrelevant criteria and preserves each criterion ID for later aggregation. 3.2.3.Multi-Role Rubric Generation and Refine- ment. After routing, Analyst LMs identify query-specific requirements under the selected expert criteria. As shown in Figure 2, the Consolidator merges their proposals, the Auditor checks clarity, assessability, relevance, and redundancy, and the Refiner updates wording, scoring scales, and weights. This separa- tion reduces dependence on a single generation pass, while all stages preserve source criterion IDs and the expert-defined capability space. 3.2.4. Expert-in-the-Loop Calibration. Experts inspect query, response, and rubric samples and analyze expert–judge disagreements. Rather than treating every disagreement as a judge error, experts distinguish among genuine policy-response failures, incorrect judge decisions, flawed query-level rubrics, and defects in the source criteria. The cor- responding actions include correcting judge super- vision, revising a rubric or its weight, and propos- ing additions, removals, or revisions in the criteria framework. This structured feedback is reused to improve rubric generation and judge calibration in subsequent iterations. The framework and its iden- tifiers remain fixed within each policy-training run; Ada-IFT uses them for targeted improvement but does not modify them automatically. The detailed an- notation decision flow and representative calibration cases are provided in the supplementary material. 3.3. Adaptive Post-Training Let휋 휃 denote the policy model being post-trained and퐽 휑 the judge model. Given a query푞, its rubric set푅 푞 , and a response푦generated by휋 휃 , the judge produces rubric-level scores conditioned on(푞,푦,푅 푞 ); human annotations can replace or calibrate these scores when available. APTER supports three ways to use this supervision. Rubric RL aggregates the scores into scalar rewards for updating휋 휃 ; Rubric- based SFT uses them to select or structure super- vised targets; and Ada-IFT combines reward-based optimization with criterion-level capability diagnosis and targeted repair. Ada-IFT is the most distinctive strategy and serves as the default configuration in the main experiments. 3.3.1. Rubric RL. Rubric RL aggregates expert-grounded rubric scores into fine-grained rewards, extending recent rubric- based reward methods for open-ended domains [14, 15] and RL methods such as GRPO for verifiable rea- soning [6]. The retained criterion provenance makes the resulting verdicts reusable for diagnosis in Ada- IFT. In the RL stage, for each query푞,휋 휃 samples푁roll- outs, denoted as푌 푞 =푦 (푛) 푁 푛=1 . For each rollout푦 (푛) and rubric(푐 푖 ,푤 푖 ,푟 푖 ) ∈ 푅 푞 , the judge model outputs: 푠 (푛) 푖 = 퐽 휑 (푞, 푦 (푛) ,푟 푖 ).(3) The푁rollouts and퐿 푞 rubrics together form a ver- dict matrix s∈R 푁×퐿 푞 . APTER then aggregates the rubric-level scores of the same response across differ- ent rubrics into a scalar reward for RL: Reward(푞, 푦 (푛) )= Í 퐿 푞 푖=1 푤 푖 푠 (푛) 푖 Í 퐿 푞 푖=1 푤 푖 .(4) Compared with RLHF or RLVR methods that rely on holistic rewards or outcome-level verification, rubric rewards provide finer-grained and more inter- pretable rubric-level supervision by scoring specific reasoning and domain-compliance criteria. 3.3.2. Rubric-based SFT. Rubric-based SFT uses expert-grounded rubrics to select reliable supervised data, following rejection- sampling-style post-training [28]. For each query, the current policy휋 휃 generates multiple candidate re- sponses푌 푞 , and퐽 휑 evaluates each candidate using푅 푞 . APTER ranks the candidates with the same weighted score aggregation as Rubric RL, retains high-scoring responses as<Query, Response>pairs, and uses them to update휋 휃 with standard supervised learn- ing. Optionally, Rubric-in-CoT SFT (RiC-SFT) also places query-level rubrics before the final answer as lightweight reasoning guidance [33], making the evaluation requirements visible in the supervised trajectory. The sample format is provided in the sup- plementary materials. 3.3.3. Adaptive Interleaved Fine-Tuning. Ada-IFT extends Rubric RL with criterion-level ca- pability diagnosis and dynamically triggered super- vised repair. Although Rubric RL provides fine- grained reward signals, RL exploration may be in- sufficient for persistent capability deficiencies. Inter- leaved RL-SFT training has recently been explored 4 RL Optimization Rollout Generation A₁ ⋮ A₂ A₃ A & PolicyModel Rubrics Scoring RL Policy Update Policy Update RewardSignal Query Response $= ∑ "#$ % ' " ⋅) " ∑ "#$ % ' " Capability Diagnose Low High Score JudgeModel RubricsJudgeVerdicts Target skill passes? Threshold CriteriaSignal (Cross multi-query) Yes Continue RL (No SFT triggered) No C1C2C3C4 Interleaved SFT Targeted Improvement SFT Data Interleaved SFT Update Policy Model Using Rubrics Reward RL Optimization Rollout Generation A₁ ⋮ A₂ A₃ A & PolicyModel Rubric Scoring RL Policy Update RewardSignal Query Response $= ∑ "#$ % ' " ⋅) " ∑ "#$ % ' " Capability Diagnosis Low High Score JudgeModel RubricJudgeVerdicts Target Criteria passes? Threshold Criteria Failure Statistics (Cross multi-query) Yes Continue RL (No SFT triggered) No C1C2C3C4 Interleaved SFT Targeted Improvement SFT Data AdaptiveInterleavedFine-Tuning Update Policy Model with Rubric Reward Ada-IFT Update Single-step per RL iteration Figure 3: Ada-IFT training flow. Rubric verdicts provide both a scalar reward for the RL update and criterion-level diagnostic signals. Persistent criterion failures trigger targeted SFT before training returns to RL. for difficult reasoning questions and iterative reason- ing improvement [29,30]. APTER differs from these approaches by using expert-grounded criterion-level failure statistics as the trigger signal: targeted SFT is not activated merely by question difficulty or a fixed training curriculum, but by recurring failures on stable expert criteria. Ada-IFT therefore uses the same judge verdicts both as scalar rewards for RL and as criterion-level diagnostic signals for targeted SFT. Figure 3 summarizes the two update channels acting on the same policy휋 휃 : RL improves the policy continuously, while SFT is activated only when the diagnostic channel identifies a persistent criterion- level deficiency. Capability Diagnosis.The criterion ID attached to each rubric allows low-scoring verdicts to be aggre- gated across queries. Ada-IFT uses two thresholds at different granularities. Because the judge verdicts are binary, a rubric fails locally when푠 (푛) 푖 =0. The step threshold휏 step determines whether the failure rate of a criterion within the current training step is high enough to increment its persistent counter. The trigger threshold휏 trig specifies how many such step-level events must accumulate before targeted repair. For training step푡, APTER first collects failed rubric instances: E (푡) =(푞, 푦 (푛) , 푐 푖 ,푟 푖 ) | 푠 (푛) 푖 = 0.(5) For criterion푐, letO (푡) 푐 denote all of its judged occur- rences in the step. Its step-level failure rate is 푓 (푡) 푐 = Í (푞,푦 (푛) ,푐 푖 ,푟 푖 )∈E (푡) 1[푐 푖 = 푐] |O (푡) 푐 | .(6) Among criteria with푓 (푡) 푐 ⩾ 휏 step , APTER increments the counters of the top-퐾failure rates. If a counter reaches휏 trig , APTER treats that criterion as a persis- tent capability deficiency. Targeted SFT.Before policy training, a frontier model generates candidate repair responses condi- tioned on the query, low-scoring rubric, source cri- terion, representative policy failure, and diagnostic information; only verified responses are stored in a criterion-indexed SFT database. When the counter of criterion푐reaches휏 trig , APTER pauses RL, retrieves the corresponding targeted samples(푞,푦 ∗ ,푐), and performs an SFT update. For medicine, the retriever first attempts to use a verified response for the same query; for mathematics, it samples criterion-matched records without replacement. This update modifies 휋 휃 ; training then returns to RL to test persistence and discover other deficiencies. Hyperparameters are reported in the supplementary material. 4. Experiment 4.1. Experimental Setup Models.We evaluate APTER on Qwen2.5-7B- Instruct [16], Qwen3-8B [17], and Qwen3.5-9B [18]; all ablations use Qwen3-8B. We use full-parameter fine-tuning in the non-thinking setting and keep prompts and output formats fixed across methods. Training data and domains.For mathematics, we use 17K problems from DAPO-Math [34]. The main medical training set contains 14,713 queries from LiveMedBench [35], SpeechMedDataset [36], and I-Medical queries [37] distributed through RubricHub [38].We use only the queries from RubricHub and construct their rubrics with APTER. Main medical runs train for up to two epochs and select checkpoints on HealthBench-500, a fixed ran- dom subset of 500 HealthBench examples [2]; Ta- ble 1 reports the full 5,000-item HealthBench set. For the mathematics ablations, we randomly sample 5,000 queries from DAPO-Math; the medical abla- tions use the complete 4,865-query LiveMedBench set. Each mathematics ablation variant is trained for two epochs, whereas each medical ablation variant is trained for one epoch. 5 (a) Mathematical Reasoning Model FamilyVariantAIME24AIME25MATH500OlympiadAvg. Qwen2.5-7B-InstructBase13.334.6168.8438.7131.37 APTER34.27 (+20.94)12.74 (+8.13)86.10 (+17.26)52.34 (+13.63)46.36 (+14.99) Qwen3-8BBase23.3318.3384.2655.8945.45 APTER49.15 (+25.82)35.00 (+16.67)92.60 (+8.34)68.50 (+12.61)61.31 (+15.86) Qwen3.5-9BBase62.0053.1397.2078.6472.74 APTER77.50 (+15.50)56.58 (+3.45)99.20 (+2.00)82.20 (+3.56)78.87 (+6.13) (b) Medical Question Answering Model FamilyVariantHealthBenchMedQAMedMCQAAvg. Qwen2.5-7B-InstructBase31.9558.9954.8248.59 APTER49.91 (+17.96)60.35 (+1.36)55.39 (+0.57)55.22 (+6.63) Qwen3-8BBase44.9063.9458.6455.83 APTER60.32 (+15.42)64.02 (+0.08)58.70 (+0.06)61.01 (+5.19) Qwen3.5-9BBase54.2676.4268.0166.23 APTER70.71 (+16.45)83.62 (+7.20)68.47 (+0.46)74.27 (+8.04) Table 1: Main results across the Qwen2.5, Qwen3, and Qwen3.5 model generations. Each APTER row reports the final score followed by its absolute improvement over the corresponding unmodified model in green parentheses. Mathematical reasoning and medical question answering averages use four and three benchmarks, respectively. All metrics are higher-is-better. Experts organize criteria as domain→sub-domain →capability category→criterion. The mathematics framework has 7 domains, 24 capability categories, and 103 leaf criteria; physicians independently con- struct the medical framework. Framework details are in the supplementary material. Benchmarks and metrics.Mathematics evalu- ation uses the competition-style AIME 24 and AIME 25 sets, the broad-coverage MATH500 [39], and the olympiad-level OlympiadBench [1]. Medical evaluation uses HealthBench for open-ended clinical responses [2] and the exam-style MedQA [40] and MedMCQA [41]. For AIME 24 and AIME 25, we re- port avg@32: for each problem, we independently sample 32 responses and average their binary correct- ness. We report each score and the macro-average within each domain. Judge and reward.Qwen3-Max evaluates all LLM- judged benchmarks. During training, Qwen3.7-Plus judges mathematics and a locally deployed Qwen3.6- 35B-A3B judges medicine [42,43,44]. Judges return per-rubric binary verdicts, weight-aggregated into the scalar reward in Equation 4; the ORM ablation uses binary outcome verification. 4.2. Experimental Results 4.2.1. Main Results. Table 1 compares APTER with the corresponding unmodified model across three Qwen model gener- ations; matched post-training baselines and compo- nent controls are reported in Table 2 on Qwen3-8B. APTER improves every reported mathematics and medical benchmark, showing that its gains are not tied to a single model generation or task format. The mathematics macro-average rises by 14.99, 15.86, and 6.13 points for Qwen2.5-7B-Instruct, Qwen3-8B, and Qwen3.5-9B, respectively. The largest improvement is 25.82 points on AIME 24 for Qwen3-8B. For the already strong Qwen3.5-9B, APTER still gains 15.50 points on AIME 24, whereas the nearly saturated MATH500 score rises by 2.00 points. This pattern in- dicates larger benefits on difficult competition-style reasoning tasks with greater headroom. Medical results are likewise consistent across the three Qwen generations: the macro-average im- proves by 6.63, 5.19, and 8.04 points, and Health- Bench gains 17.96, 15.42, and 16.45 points. The im- provements on the exam-style MedQA and MedM- CQA benchmarks are generally smaller. The concen- tration of gains on HealthBench is consistent with APTER’s design: open-ended clinical responses re- quire satisfying multiple query-specific professional criteria, whereas multiple-choice benchmarks are dominated by final-answer accuracy. Together, the 6 Variant Ablation ConfigurationDownstream Performance Rubric ConfigurationReward TypeTraining Strategy MATH500 AIME24 Olymp. Health (A) Post-training Components Rubric data, rubric reward, and Ada-IFT Qwen3-8B—Base84.2623.3355.8945.43 SFTPlain CoT data—SFT84.3026.7955.7144.62 RiC-SFTRubric-augmented CoT data —SFT88.8628.3757.0547.10 Rubric RLQuery-level rubric dataRubric Reward GRPO90.9241.6358.9750.54 APTERQuery-level rubric dataRubric Reward GRPO + Ada-IFT91.3444.6061.8753.48 (B) Reward Type Outcome-level ORM versus query-level rubric reward GRPONo rubricsORM RewardGRPO89.1637.7558.65— Rubric RLQuery-level rubricsRubric Reward GRPO90.9241.6358.9750.54 (C) Rubric Granularity No, generic, and query-level rubrics GRPONo rubricsORM RewardGRPO89.1637.7558.65— Generic CriteriaGeneric criteriaRubric Reward GRPO87.6239.3157.7547.84 Query-level Rubrics Query-level rubricsRubric Reward GRPO90.9241.6358.9750.54 (D) Rubric Provenance Framework-free versus expert-grounded query-level rubrics Framework-freeFramework-free rubricsRubric Reward GRPO89.7432.9857.6447.27 Expert-groundedExpert-grounded rubricsRubric Reward GRPO90.9241.6358.9750.54 (E) Rubric Refinement Before versus after the Auditor–Refiner stage Before RefineRaw rubricsRubric Reward GRPO86.9236.6756.1048.18 After RefineRefined rubricsRubric Reward GRPO90.9241.6358.9750.54 Table 2: Unified summary of the five performance ablations on Qwen3-8B. Light-blue section rows identify the factor isolated in each block. The best result among fully reported comparable variants within each block is highlighted in bold. results support APTER’s effectiveness across both verifiable and expert-constrained reasoning. 4.2.2. Ablation Experiments. Ablation protocols.All ablations use Qwen3-8B and keep the evaluation protocol fixed within each comparison block. The mathematics ablations use 5,000 queries randomly sampled from DAPO-Math, whereas the medical ablations use all 4,865 LiveMed- Bench queries without subsampling. Each mathe- matics ablation variant is trained for two epochs, whereas each medical ablation variant is trained for one epoch. The medical ablations in Table 2 are eval- uated on HealthBench-500, whereas Table 1 reports the full 5,000-item HealthBench set. Block A: Post-training components. SFT and RiC- SFT use identical queries, candidate responses, sam- ple counts, and training budgets. SFT trains on <Query, Response>, whereas RiC-SFT trains on <Query, Rubrics, Response>. Rubric RL uses query-level rubrics as online GRPO rewards, and APTER further adds Ada-IFT. Block B: Reward type. We hold the GRPO configu- ration fixed and change only the reward from binary outcome verification to query-level rubric evalua- tion. Block C: Rubric granularity. We hold the data, judge, GRPO hyperparameters, and training budget fixed, and replace query-level rubrics with five equally weighted generic criteria: computational accuracy, logical coherence, case-splitting completeness, vari- able constraint handling, and key-step coverage. Block D: Rubric provenance. For the framework- free variant, we use the question-level rubrics pro- vided in the original training data: SRaR for math- ematics [45] and LiveMedBench for medicine [35]. These rubrics are constructed independently for each question and are not routed from a shared, expert- defined criterion framework. The expert-grounded variant instead uses APTER rubrics routed from the corresponding domain framework, while keeping the judge, reward formulation, and GRPO configura- tion fixed. Block E: Rubric refinement.We conduct two matched GRPO runs using either the raw rubrics produced by the Consolidator or the refined rubrics produced after the Auditor–Refiner stage. The runs use identical queries, model, judge, reward formula- tion, GRPO hyperparameters, and training budget; the rubric sets average 4.82 and 4.09 rubrics per ques- tion before and after refinement, respectively. Post-training method ablation.Block A shows that SFT alone produces mixed changes, while adding rubrics to the supervised trajectories yields 7 020406080 0.0 0.2 0.4 0.6 0.8 1.0 Criterion Failure Rate (↓) Final gap -0.28 Failure-rate AUC ↓ 32% (a) Implicit Boundary and Principal-Value Constraints 020406080 Final gap -0.12 Failure-rate AUC ↓ 19% (b) Logical Consistency in Number-Theoretic Reasoning 020406080 Final gap -0.18 Failure-rate AUC ↓ 32% (c) Content Accuracy in Trigonometric Reasoning Training Step APTER w Ada-IFTAPTER w/o Ada-IFT Figure 4: Criterion-level repair dynamics with and without Ada-IFT. Curves show exponential moving averages of criterion failure rates for three randomly selected criteria; lower is better. Percentages report the relative reduction in normalized failure-rate AUC. modest but consistent gains. Using rubrics as online rewards produces the main jump: relative to RiC- SFT, Rubric RL raises AIME 24 from 28.37 to 41.63 and HealthBench from 47.10 to 50.54. Ada-IFT fur- ther reaches 44.60 and 53.48, respectively, showing that criterion-triggered repair complements rubric- guided RL. Criterion-level repair dynamics.Figure 4 exam- ines whether Ada-IFT’s aggregate gains correspond to targeted recovery. It lowers normalized failure- rate AUC by 32%, 19%, and 32% on three randomly selected criteria, indicating sustained rather than endpoint-only repair. Reward type and rubric granularity.Under matched GRPO training, Block B shows that rubric rewards outperform binary outcome verification on all three mathematics benchmarks, with the largest margin on AIME 24 (37.75→41.63). Block C shows that query-level rubrics also outperform five generic criteria on every benchmark, including HealthBench (47.84→50.54), demonstrating the value of query- specific feedback over requirements that may be too generic for the question. Rubric provenance.Expert grounding improves all reported results; the larger gap on AIME 24 (32.98→41.63) and the HealthBench gain (47.27→ 50.54) show the value of routing rubrics from a shared expert criterion framework rather than con- structing them independently for each question. Rubric Refiner effectiveness.Block E isolates the Auditor–Refiner stage. Refined rubrics yield gains of 4.00, 4.96, 2.87, and 2.36 points on MATH500, AIME 24, OlympiadBench, and HealthBench-500, respectively, indicating improved reward quality rather than merely a changed rubric count. The gains are consistent across all four benchmarks; detailed statistics are provided in the supplementary mate- rial. 5. Conclusion We presented APTER (Adaptive Post-Training with Expert-Grounded Rubrics), a framework that in- tegrates structured domain knowledge into fine- grained evaluation, optimization, and capability di- agnosis for specialized complex reasoning. Its expert- grounded rubric construction component organizes domain expertise into a reusable criteria frame- work and instantiates relevant criteria as query-level rubrics with retained criterion provenance, without requiring per-query expert authoring or reference answers. Its adaptive post-training component uses rubric verdicts as both optimization and criterion- level diagnostic signals, allowing Ada-IFT to ag- gregate recurring failures and trigger targeted su- pervised repair during reinforcement learning. Ex- periments on mathematical reasoning and medi- cal question answering show that the full APTER pipeline consistently improves performance across three Qwen model generations and applies to both verifiable and expert-constrained open-ended tasks. 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The reward channel aggregates rubric- level scores into scalar rewards, while the diagnostic channel uses the same scores to compute criterion- level failure rates and counters. The diagnostic chan- nel therefore adds only lightweight aggregation and indexing during ordinary RL steps. When a persis- tent criterion-level deficiency triggers repair, Ada-IFT additionally invokes a frontier model to generate can- didate repair responses, verifies the candidates, and performs a local SFT update on the accepted samples. These extra calls occur only at triggered repair events rather than for every rollout. A.2. Targeted SFT Sample Fields When repair is triggered for criterion푐, the frontier model receives the query, current policy response, low-scoring query-level rubric, source criterion and diagnostic information. Only a verified improved re- sponse푦 ★ is accepted. The stored supervised training tuple is(푞,푦 ★ ,푐), together with provenance metadata needed for auditing. The accepted samples are added to the criterion-indexed SFT buffer and used for the targeted update before training returns to RL. A.3. Medical Training Data Composition Table 3 reports the exact query counts after construct- ing the training files. For the I-Medical subset dis- tributed through RubricHub, APTER uses only the queries and constructs new expert-grounded rubrics. SourceQueries LiveMedBench4,865 SpeechMedDataset4,933 I-Medical (via RubricHub)4,915 Main medical training set14,713 Medical ablation set (complete LiveMedBench)4,865 Table 3: Medical training data composition. A.4. Expert Criteria Framework Construction protocol.The mathematics frame- work was constructed by doctoral students with ex- perience in mathematical reasoning, while the medi- cal framework was independently constructed by a group of practicing physicians. These domain experts first enumerate recurring capabilities that should remain meaningful across questions. They organize these capabilities into the hierarchy domain →sub-domain→capability→leaf criterion. Each leaf stores a persistent criterion ID, a scope defini- tion, and a representative positive or boundary ex- ample. A second review pass checks (i) coverage of common solution or clinical-response requirements, (i) overlap between neighboring leaves, and (i) whether a criterion can be instantiated as an observ- able, query-specific binary test. Redundant leaves are merged and ambiguous scopes are rewritten. Ap- proved versions are frozen before a policy-training run; proposed changes from expert calibration are applied only between runs, preserving the meaning of criterion-level failure counts within a run. For compactness, the case-study boxes later in this sup- plement display only the capability and leaf crite- rion from a stored path; the criterion ID and prove- nance metadata retain the omitted mathematics sub- domain. Mathematics framework.Table 4 gives the exact hierarchy stored in the released criteria knowledge base. The seven mathematics sub-domains contain 24 capabilities and 103 leaf criteria. Some capabil- ity names recur across sub-domains, but their def- initions and examples are specialized to the math- ematical context. For example, variable constraints and boundary handling requires a solution to retain do- main restrictions and verify equality cases, whereas completeness of case analysis requires an exhaustive, non-overlapping partition for a counting problem. Medical framework.The medical framework, in- dependently designed by the participating physi- cians, contains a single sub-domain, Medical Question Answering. This sub-domain comprises 10 capabili- ties and 50 criteria, listed in Table 5. These criteria are persistent framework entries rather than query- specific rubrics. For each query, APTER routes rele- vant criteria and instantiates each as an observable binary rubric. Retaining the source criterion ID al- lows related failures across questions to accumulate under the same capability. 12 Mathematics sub-domainCapabilitiesLeaf criteria Analysis416 Number theory416 Discrete mathematics416 Arithmetic313 Applied mathematics314 Trigonometry314 Geometry314 Total24103 Table 4: Composition of the mathematics expert criteria frame- work. A.5. Training Configuration For reproducibility, Table 6 reports the complete Qwen3-8B recipe used by the controlled ablations and the Ada-IFT case study.The two domain columns are separated because context length, sam- pling, KL regularization, and batching differ. The “1 epoch” entry in the Repair SFT optimizer row refers to the local pass over the accepted buffer samples at a triggered repair event; it does not denote the end-to-end ablation budget. Model-specificexceptions.ForQwen2.5-7B- Instruct, mathematics uses a 1,024-token prompt limit, a 3,072-token response limit, and PPO mini-batch size 48, whereas medicine uses the 4,096/8,192-token limits and PPO mini-batch size 32; both use repair-batch cap 16.Qwen3.5-9B mathematics uses the Qwen3-family context and batch sizes but disables rollout prefix caching for its hybrid recurrent architecture. Main mathematics runs are capped at 200 RL steps; main medical runs train for at most two epochs and select checkpoints on the fixed HealthBench-500 subset. Randomness and number of runs.Data- construction and SFT train–validation splitting use seed 42. The RL dataloader uses seed 1, and the vLLM rollout engine uses seed 0 before data-parallel worker offsets are applied. Each model–domain– method entry has one independent random-seed run; this run count is distinct from the training budget, which is two epochs for each mathematics ablation and one epoch for each medical ablation. We therefore do not report across-run standard deviations. AIME scores areavg@32: each problem is sampled 32 times and the reported accuracy averages those responses. Other benchmark entries use one evaluation pass of the selected checkpoint. HealthBench-500 is a single fixed 500-sample subset rather than a newly resampled subset for each method. A.6. Rubric Refiner Statistics The Auditor–Refiner stage screens consolidated rubrics for redundancy, surface bias, and boundary ambiguity. On a 5k-question subset of the mathe- matics training set, only 7.0% of rubric sets remain unchanged, while the average number of criteria per question decreases from 4.82 to 4.09. Table 7 gives the full action statistics underlying the main-paper analysis. StatisticValue Rubric sets unchanged7.0% Sets with⩾ 1 deletion61.4% Sets with⩾ 1 modification82.4% Avg. criteria before Refiner4.82 Avg. criteria after Refiner4.09 Overlap / redundancy issues5,424 Surface-bias issues1,439 Boundary-ambiguity issues456 Table 7: Refiner action statistics on a 5k-question subset of the mathematics training set. A.7. Expert Annotation Decision Flow This section summarizes the decision flow used by experts when annotating model responses with query-level rubrics.The goal is to avoid treat- ing every disagreement as a single generic expert– judge mismatch. Instead, the expert first determines whether the current response can be assessed under the current criterion and rubric, then decides whether the issue comes from the model response, the judge model, or the evaluation standard itself. Figure 5 shows the decision process. The key de- sign choice is that both unscorable cases and flawed- standard cases enter the same evaluation standard cal- ibration module. This is because an unscorable re- sponse usually indicates that the current criterion or rubric is irrelevant, underspecified, or otherwise unsuitable for the item. APTER therefore exposes a shared set of standard-calibration actions: revise the query-level rubric, revise the criterion definition or example, remove an irrelevant criterion or add a missing one, and adjust the criterion weight. In this flow, score confirmation and score overrid- ing are reserved for cases where the evaluation stan- dard is valid. When the standard itself is flawed, the expert records a structured calibration action rather 13 CapabilityCriteria Criterion definitions Medical-knowledge accuracy7Factual correctness; medical-terminology accuracy; guideline and consensus adherence; avoid- ance of harmful advice; examination and test interpretation; medication and dosage accuracy; differential-diagnosis plausibility. Clinical reasoning and judg- ment 5 Multi-factor integrated judgment; reasoning-chain clarity; severity assessment; diagnostic and treatment prioritization; risk–benefit analysis. Information completeness6Coverage of key information; provision of alternatives; treatment-plan completeness; follow- up and monitoring guidance; prevention and health education; risk warnings. Uncertainty handling5Identification and communication of uncertainty; questions that reduce uncertainty; transpar- ent disclosure of irreducible uncertainty; balance between a direct answer and appropriate qualification; avoidance of overconfidence. Emergency recognition and re- ferral 5Safety-netting advice; first-aid guidance; management of panic; emergency recognition; appro- priate referral timing. Context awareness and follow- up questions 6Individualized advice; proactive follow-up questions; disclosure of assumptions and premises; multi-turn coherence; use of existing context; avoidance of redundant questions. Instruction following3 Task-requirement fulfillment; scope control; output-format compliance. Communication quality and ex- pression 5 Information-density control; empathy and tone; audience adaptation; structure and organiza- tion; emphasis of key points. Health-data handling4 Accurate clinical-data extraction; identification of missing information; preservation of data fidelity; data transformation and formatting. Cultural and regional adapta- tion 4Awareness of regional medical-practice differences; cultural sensitivity; language and expres- sion adaptation; consideration of resource accessibility. Total50 Table 5: Composition of the medical expert criteria framework. than merely changing the score. The resulting an- notation can later support judge calibration, rubric- generation repair, criterion-framework refinement, or weight adjustment. 14 HyperparameterMathematicsMedicine Training data (main)DAPO-Math-17K, 17,415 queriesLiveMedBench+SpeechMed- Dataset + I-Medical, 14,713 queries Ablation data (seed question)5,000 queries sampled from DAPO- Math-17K Complete LiveMedBench training set, 4,865 queries Ablation budget2 epochs1 epoch RL algorithm / advantage esti- mator GRPO / GRPOGRPO / GRPO RewardWeighted binary rubric rewardWeighted binary rubric reward Rollouts per query 푁88 RL batch / PPO mini-batch128 / 12864 / 32 OptimizerAdamW, 훽=(0.9, 0.999)AdamW, 훽=(0.9, 0.999) RL learning rate / schedule 1×10 −6 ; constant; 10 warm-up steps1×10 −6 ; constant; no warm-up RL weight decay / gradient clip 0.1 / 1.00.01 / 1.0 GRPO clip (low / high / dual-푐) 0.2 / 0.28 / 10.00.2 / 0.2 / 3.0 KL loss / coefficientDisabled / 0Enabled (low-variance) / 0.001 Entropy coefficient (RL)00 Prompt / response length2,048 / 8,1924,096 / 8,192 Training sampling(푇 , 푝, 푘) (1.0, 1.0,−1)(0.7, 0.8, 20) Validation sampling(푇 , 푝, 푘) (0.7, 0.8,−1)(0.7, 0.8, 20) Thinking modeDisabledDisabled Training judgeQwen3.7-PlusQwen3.6-35B-A3B (local) Evaluation judgeQwen3-MaxQwen3-Max Local low-score ruleBinary verdict 푠= 0Binary verdict 푠= 0 Current-step failure-rate cutoff 휌 step 0.750.75 Top-퐾criteria / trigger count휏 trig 8 / 88 / 8 Repair warm-up / minimum samples 3 steps / 83 steps / 8 Repair target construction Online frontier-model generation and verification Online frontier-model generation and verification Repair SFT batch cap / max length 64 / 6,000 tokens32 / 6,000 tokens Repair SFT optimizerIndependent AdamW; lr 1×10 −6 ; 푤 푑 = 0.01; 1 epoch; entropy 10 −4 Independent AdamW; lr 1×10 −6 ; 푤 푑 = 0.01; 1 epoch; entropy 10 −4 Hardware8× NVIDIA A800-SXM4-80GB Table 6: Full Qwen3-8B training configuration for controlled ablations and the Ada-IFT rollout analysis. 15 Algorithm 1 Ada-IFT with criterion-indexed targeted repair Input: policy 휋 휃 ; training setD=(푞, 푅 푞 ); judge 퐽 휑 ; repair generator 퐹; verifier 푉; low-score threshold 휏 step ; current-step failure-rate cutoff 휌 step ; trigger count 휏 trig ; top-퐾 criteria; warm-up 푇 0 . Initialize: persistent counter 퐻 푐 ← 0 for each criterion 푐; criterion-indexed SFT bufferB sft ←∅. for training step 푡= 1, 2, . . . do Sample a query batch and draw 푁 rollouts푦 (푛) 푁 푛=1 ∼ 휋 휃 (· | 푞). for each(푐 푖 ,푤 푖 ,푟 푖 ) ∈ 푅 푞 and rollout 푦 (푛) do Obtain 푠 (푛) 푖 = 퐽 휑 (푞, 푦 (푛) ,푟 푖 ) ∈ 0, 1. end for Compute 푅(푞, 푦 (푛) )= Í 푖 푤 푖 푠 (푛) 푖 / Í 푖 푤 푖 and update 휋 휃 with GRPO. Mark local failures by 푠 (푛) 푖 < 휏 step ; for every observed criterion 푐, compute 푓 푐 . LetC 푡 be the top-퐾 criteria satisfying 푓 푐 ⩾ 휌 step . for each 푐 ∈ C 푡 do 퐻 푐 ← 퐻 푐 + 1. if 푡 > 푇 0 and some 퐻 푐 ⩾ 휏 trig then Select the highest-count eligible criterion 푐 ★ . For representative failures of 푐 ★ , generate candidates 푦 ★ ∼ 퐹(푞, 푦,푟, 푐 ★ , diagnostic information). Verify candidates with 푉 and add accepted(푞, 푦 ★ , 푐 ★ ) toB sft . if at least the minimum number of samples is available then Apply one targeted SFT update on the buffered samples and reset 퐻 푐 ★ ← 0. end if end if end for Inspect query, model response, criterion/rubric, and judge score Scorable under current criterion / rubric? Is the rubric or criterion flawed? Does the response satisfy the rubric? Does the judge agree with the expert? Expert score: Pass Expert score: Fail Evaluation standard calibration Revise query- level rubric Revise criterion definition/example Remove irrelevant or add missing criterion Adjust criterion weight Confirm judge score Override judge score Save structured expert annotation YesNo NoYes Yes No No Yes Case 1: model answer defectCase 2: judge scoring defect Case 3: rubric direction defectCase 4: criterion defect Record calibration Figure 5: Expert annotation decision flow. Invalid evaluation standards enter the shared calibration module; valid standards proceed to expert–judge score comparison. 16 B. Prompt Templates This section lists the prompt templates used at each stage of the APTER pipeline. We report them here to make the data-construction and post-training process reproducible. For conciseness, only the mathematics versions are provided as representative examples. Placeholders such asquery,criterion,rubric, andresponsedenote the fields that are filled in at runtime. The exact prompt bodies are provided below. In these implementation prompts, dimension is the serialized name of a selected framework criterion, not an additional level in the expert-criteria hierarchy. B.1. Query-level Routing Prompt Given a query and the expert criteria framework, the routing prompt selects the relevant candidate criteria and assigns each a weight (see the rubric-generation section of the main paper). Query-level Routing Prompt # 1. Role You are a top-tier AI evaluation architect with exceptional systems thinking and analytical ability. Your task is the initial design of an evaluation framework for solutions to complex mathematics problems. # 2. Core Mission Act as the "chief planner" of the evaluation framework and carry out a highly focused FIRST-STAGE task. Analyze the user-provided [query] and, based on it, accomplish two things: 1. Dimension selection: You MUST select ONLY from the [candidate dimension list] of the specified mathematics domain the subset of evaluation dimensions most relevant to this [query]. Do NOT create, modify, combine, or rename any dimension. 2. Weight assignment and ranking: For each selected dimension, assign an importance weight in [0, 10] according to how important it is for judging the quality of a response to this [query]. # 3. Instructions and Procedure Follow this reasoning chain strictly: 1. Analyze the query in depth: identify the mathematical concepts, solution methods, and evaluation needs it involves. 2. Select and weight: a. Quickly scan the user-provided [candidate dimension list]. b. Based on your analysis, SELECT 5-10 of the most relevant dimensions from the list. c. Assign each selected dimension a weight in [0, 10]: - High (9-10): core, decisive criteria for this response, possibly "veto" items. - Medium (6-8): important but non-core checkpoints. - Low (1-5): relevant but secondary checkpoints, or "bonus" items. 3. [Reinforced] Final check: before producing output, verify that every "dimension" value in the final JSON list comes VERBATIM from the user-provided list. 4. Rank and format: sort all selected, weighted dimensions in descending order of weight and output strictly in the required format. # 4. Required Output Format Output strictly as a Python list of JSON objects, with no other explanatory text or code- block markers: "dimensions": [ "dimension": "name of the highest-weight dimension", "weight": score, "dimension": "name of the second most important dimension", "weight": score, ... ] # 5. Example 17 Suppose the problem belongs to the "Analysis" domain; a possible result: "dimensions": [ "dimension": "Analytical rigor", "weight": 10, "dimension": "Quantitative computation accuracy", "weight": 9 ] B.2. Multi-role Rubric Generation Prompts APTER instantiates the selected criteria into query-level rubrics through a multi-role process (Analyst, Consolidator, Auditor, Refiner). The prompt for each role is given below. Analyst Prompt # Role You are a senior mathematics curriculum researcher and assessment-design expert with decades of experience setting and grading exams. Your strength: given a problem, you quickly see through its core testing points, key steps, and where most people go wrong. # Core Mission You will receive a mathematics problem and an expert evaluation framework (a set of candidate dimensions). Produce a DEEP, FREE-FORM analysis of the problem. Your analysis will be used by another expert to design scoring dimensions, so quality is critical. Write your analysis freely in natural language; do NOT output any JSON or structured format. Develop your thinking fully. # Guiding Questions The following are directions worth considering; you need not answer each one, but let them inspire a deep analysis: - How would you approach solving this problem? Try to work it through completely and give your final answer. - What are the key intermediate conclusions, i.e. steps such that if one is wrong, everything after it is necessarily wrong? - Are there "looks-valid-but-is-a-trap" solution paths? What trap might the problem setter be hinting at? - Which mathematical concepts or theorems are the real pillars for solving this problem? Someone who does not understand them cannot get it right. - Conversely: what distinguishes someone who can truly solve this problem from someone who "writes a lot but misses the point"? Which dimension best separates the two? - Among the candidate dimensions in the framework, which are truly discriminative FOR THIS SPECIFIC PROBLEM, and which are irrelevant here? - If you could pick only 3 dimensions to judge an answer, which 3 would you pick, and why ? # Key Principles - The dimensions you recommend MUST be crucial to correctly solving this problem. If a dimension does not affect answer correctness, it is not worth being a scoring criterion. You may pick 5-7 dimensions. - Every number, formula, or theorem in your analysis must be correct. If unsure about a conclusion, explicitly flag your uncertainty rather than fabricating a plausible- sounding claim. - If the problem admits multiple valid solution paths, identify them and note this in your analysis. 18 Consolidator Prompt # Role You are a rubric-synthesis expert. Your task is to distill high-quality scoring dimensions from multiple independent analysis reports. # Core Mission You will receive a mathematics problem, multiple independent in-depth analysis reports from different analysts, and the expert evaluation framework. Synthesize all information into a final list of scoring dimensions. # Workflow ## Step 1: Cross-validate facts Compare the concrete numbers, intermediate conclusions, and theorem citations across the reports: - If most reports agree, adopt the majority view. - If they disagree, prefer the report with the most complete and rigorous derivation, and verify it repeatedly yourself. - If one report makes a mathematical claim no other report mentions, treat it with caution: do not adopt it unless its derivation is convincing. ## Step 2: Identify the truly critical dimensions From the analyses, distill the dimensions most critical to CORRECTLY solving this problem : - Prefer dimensions emphasized by multiple reports; consensus among independent analysts is a strong signal. - Focus on the "get-it-wrong-and-all-is-lost" key steps and concepts. - Discard generic dimensions that are "correct" but non-discriminative for this problem. ## Step 3: Check logical consistency Before final output, check for contradictions among the selected dimensions: - Are there two dimensions whose requirements cannot logically hold simultaneously? - If the problem admits multiple valid solution paths, the criterion must not hard-code one path as the only correct way. - The dimensions must be mutually independent: do not test the same computation step or logical point in two dimensions. # Output Guidelines 1. Precise selection: usually around 5 dimensions, but analyze the specific problem, pick more if needed, fewer if fewer suffice; never pad the count. 2. Weighting: 1-10; larger means more central to correct solving. 3. Criteria must be concrete: include the specific numbers, formulas, and step anchors of this problem; no vague generalities. 4. Binary objective judgment: each criterion must support a clear "satisfied / not satisfied" verdict; no fuzzy wording. 5. Dimension names: must be chosen from the candidate dimensions of the expert framework, using their full names. Auditor Prompt # Role You are a senior rubric quality-control expert, responsible for auditing and diagnosing the reliability, atomicity, and discriminativeness of automatically generated scoring criteria. # Core Mission Your task is highly focused: given a [Query] and a set of [candidate Rubrics] generated for it (a list[dict], each element a dimension criterion), act as a strict examiner. 19 Check for logical overlap among the rubrics, diagnose the internal defects of each rubric, and finally route their "dimension names" into three groups: Keep, Modify, or Delete. # Required Input 1. [Query]: the original query text or task instruction to be evaluated. 2. [Candidate Rubrics]: the list of scoring criteria generated for the query (JSON list[ dict]; each element contains at least dimension / weight / criterion). # Instructions 1. Atomicity and overlap detection: high-quality criteria must be atomic. Carefully compare all candidate criteria and find those that test the same step, repeat a checkpoint, or logically cross over heavily. Group the overlapping dimension names into the same list. 2. Per-criterion defect diagnosis: for each dimension’s criterion, check for the following fatal defects and state the issue concisely (write "no issue" if flawless): - Non-atomic: one criterion mixes multiple points that need independent scoring. - Off-instruction: generic, not strictly derived from the specific information in this Query. - Surface-bias tendency: over-focused on surface features such as length or formatting , easily misled by a model’s verbosity, lacking substantive logical checking. - Boundary ambiguity: contains subjective or ambiguous wording, lacking an objective decision anchor. 3. Action categorization: based on the diagnosis, route every input dimension name into exactly one of three groups: - Keep: perfectly fits the Query, unambiguous, non-overlapping, and atomic; ready for use. - Modify: direction is correct but has boundary ambiguity, compound dimensions, or slight overlap (e.g. several dimensions checking the same addition); needs later polishing or merging/splitting. - Delete: generic filler with no discriminativeness, severely off-Query, or fully redundant with a Kept item. Refiner Prompt # Role You are a top-tier rubric optimization and reconstruction expert. Your specialty is targeted repair of flawed draft criteria, turning them into high-quality, atomic rubrics strictly aligned with human-expert standards, thereby defending against surface-level biases in LLM evaluation. # Core Mission You receive the original [Query], the [original Rubrics], and the full [diagnostic feedback] (the exact Keep / Modify / Delete lists and defect notes). Act as an assembly-line inspector: firmly discard items to delete, directly inherit items to keep, and deeply reconstruct items to modify and those with overlaps, finally outputting a flawless set of rubrics. # Required Input 1. [Query]: the original query text or task instruction to be evaluated. 2. [Original Rubrics]: the initially generated rubric set (JSON, with each dimension’s weight and criterion). 3. [Diagnostic feedback]: the detailed diagnosis from the quality-check module, including : - keep_rubrics: dimension names recommended to keep as-is. - delete_rubrics: dimension names recommended to delete. - modify_rubrics: dimension names recommended to modify. - rubric_issues: specific defect notes (for the modify items). 20 - overlaps: groups of logically overlapping dimensions. # Refinement & Synthesis Guidelines (must follow strictly) 1. Naming invariants: - The dimension names in keep_rubrics / modify_rubrics / delete_rubrics must be echoed and referenced verbatim; no rewriting, abbreviation, keeping only the second-level name, or synonym translation. - These names are already in "level-1 dimension - level-2 dimension" full-name format; keep that full-name format unchanged throughout. 2. Execute categorization precisely: - Absolute delete: any dimension in delete_rubrics is erased from your reasoning and final output; never keep it. - Keep as-is: any dimension in keep_rubrics has its name, weight, and criterion copied verbatim into the final output. - Targeted modify: only dimensions in modify_rubrics are re-polished and decomposed according to the defects in rubric_issues (e.g. compound dimension, boundary ambiguity, off-instruction). 3. Resolve overlaps: inspect the overlaps list. If dimensions A and B overlap, you must decide: either MERGE them into a single new dimension covering that checkpoint ( delete the original A and B, and the new dimension’s name must be chosen from A or B, do not invent a new name), or strictly revise their criteria to draw a clear boundary so they test entirely independent steps. 4. Enforce atomicity & mitigate bias: each reconstructed modify item must test one indivisible logical node. The criterion must be derived strictly from the specific information in the Query, must include concrete numbers, formulas, or step anchors as objective decision bases, and must eliminate subjective vague wording such as " basically correct" or "logically clear" that is easily misled by verbosity. B.3. Rubric Judging Prompt The implementation batches all rubrics associated with one response. The following is the medical judging template; the mathematics variant retains the same input fields and Boolean output contract but removes explanations and asks the judge to ignore superficial formatting differences. Rubric Judging Prompt Your job is to look at a conversation and a list of rubric items, and score the last assistant response on how well it follows each item. # Conversation conversation # Rubric items rubric_items # Instructions Return a JSON object with one field, "results", whose value is a list of objects in the same order as the rubric items. Each object contains "explanation" and "criteria_met". - "criteria_met" is Boolean. Return true only if the response satisfies the entire rubric item; if any required part is missing or incorrect, return false. - Text following "such as", "for example", or "including" is illustrative rather than an exhaustive checklist. - If a rubric describes an undesirable behavior with negative weight, judge whether that behavior is present; do not invert the verdict. - The result list must contain exactly one object per rubric item. 21 Return only the JSON object. Do not include other text. Mathematics Judge Output Addendum You are a strict grader for competition math. Judge each rubric item independently against the student’s solution. Ignore superficial formatting differences such as boxed versus plain answers, LaTeX versus plain text, units, trailing zeros, and mathematically equivalent forms. Return ONLY: "results":[true,false,...] The array must contain exactly one Boolean per rubric item, in order. Do not explain and do not output Markdown. B.4. Candidate Response Generation Prompt Candidate generation receives the original query and its complete weighted rubric set. Multiple sampled re- sponses are judged with the preceding prompt; the highest-scoring verified response becomes the supervised target. The runtime user-message template is: Candidate Response Generation Prompt # Question query # Evaluation Criteria weighted_query_level_rubrics Provide a complete answer to the question. Address the evaluation criteria without mentioning, quoting, or revealing that a rubric was provided. Use the same language as the question. For mathematics, show a rigorous derivation and clearly mark the final answer. B.5. Targeted Repair Prompt (Ada-IFT) The following prompt is used after Ada-IFT triggers a criterion-indexed repair event during policy training. When a diagnostic report is available, the failed variant is used; a shorter variant omits the report and negative example when those fields are unavailable. Targeted Repair Prompt ## Role You are a senior domain expert with exceptionally rigorous reasoning. Produce a high-quality solution that excels on one scoring dimension where the policy currently struggles. ## Task You receive a query, a scoring dimension with its binary criterion, a diagnostic report of common errors, and a typical failing response. Solve the query from scratch and produce a complete exemplary response. ## Critical Rules 1. The response must particularly excel on the specified dimension and 22 strictly satisfy its binary criterion. 2. Deliberately avoid the error patterns in the diagnostic report. 3. Do not mention or critique the failing response. 4. Maintain domain correctness and use the same language as the query. 5. Give a clear final answer or recommendation. ## Input ### Scoring Dimension - Dimension Name: criterion_id - Category: capability_category - Binary Criterion: query_level_rubric ### Query query ### Diagnostic Report - Policy Pass Rate: pass_rate - Common Error Patterns: common_errors - Failure Summary: failure_summary ### Typical Failing Response policy_response B.6. RiC-SFT Sample Format Rubric-in-CoT SFT (RiC-SFT) uses query-level rubrics as lightweight reasoning guidance before answer generation. A training sample can be formatted as follows: query: ... response: <think> I should first consider the rubrics for this query before answering. The response should satisfy the following criteria: 1. ... 2. ... Based on these rubrics, I need to ... </think> ... C. Case Studies C.1. Expert Annotation and Calibration This supplementary section presents a small expert-annotation case study. The purpose of this case study is not to evaluate mathematical difficulty itself, but to illustrate how APTER supports expert calibration at multiple levels of the evaluation pipeline. In particular, the cases demonstrate how experts handle model-answer defects, judge-model scoring defects, rubric-direction defects, and criterion-level defects. C.1.1. Case Study Design The case study contains four mathematics items. Each item is designed to represent a distinct failure mode in rubric-based evaluation. For each item, APTER presents the query, model response, criteria and query-level rubrics to be annotated, per-rubric binary verdicts, their weighted aggregate, expert scores, and expert resolution actions. This makes the annotation process auditable: readers can inspect not only the final score, but also which part of the evaluation pipeline required expert intervention. 23 Table 8 summarizes the four cases. CaseIssue TypeJudge Ag- gregate Expert Score Expert Calibration Action 1Model answer defect00Confirm the judge-model low score because the model answer is genuinely incorrect. 2Judge scoring defect0.51 Override the judge-model score while keep- ing the rubric unchanged. 3Rubric direction defect10Mark the rubric as flawed and rewrite it to- ward the correct proof-oriented evaluation direction. 4Criterion defect0.71Mark a single criterion as flawed and revise its definition and example, while keeping the overall rubric direction. Table 8: Overview of the expert-annotation case study. Each case corresponds to a different calibration level in APTER. C.1.2. Case 1: Model Answer Defect Case 1 — Judge 0 / Expert 0 — Model Answer Defect [Query] A jacket originally costs USD 80. It is discounted by 25%, and then an 8% sales tax is applied to the discounted price. What is the final price? [Model Response] The model computes the 25% discount correctly and obtains USD 60 as the discounted price. However, it then states that an 8% tax means adding USD 8, and reports the final answer as USD 68. [Criteria & Rubrics to Annotate] R1 (Foundational Mathematical Modeling -- Constraint Extraction & Transformation) weight 5 Judge: 0 Expert: 0 The response must identify that the 8% sales tax is applied to the discounted price, not to the original price. R2 (Foundational Mathematical Modeling -- Calculation Accuracy) weight 5 Judge: 0 Expert: 0 The response must compute the tax as 0.08 * 60 = 4.80 and report the final price as USD 64.80. [Expert Calibration Action] Confirm the judge-model low score. The response applies the tax incorrectly and reports the wrong final price, so the defect lies in the model answer, not in the judge model or the rubric. This is the only pure agreement case in the set. Analysis.This case demonstrates model-answer validation: the judge and the expert agree, and the low score reflects a genuine error in the response rather than any flaw in the evaluation standard. 24 C.1.3. Case 2: Judge Scoring Defect Case 2 — Judge 0.5 / Expert 1 — Judge Scoring Defect [Query] Solve for x: 2(x - 3) = 10. [Model Response] The model divides both sides by 2, obtains x - 3 = 5, adds 3 to both sides, and reports the final answer x = 8. [Criteria & Rubrics to Annotate] R1 (Foundational Cognition & Knowledge -- Content & Method Accuracy) weight 5 Judge: 1 Expert: 1 The response must use valid algebraic operations to solve the equation and obtain x = 8. R2 (Mathematical Communication -- Step Clarity & Granularity) weight 5 Judge: 0 Expert: 1 The response must include enough intermediate reasoning to verify the solution. Expanding 2(x - 3) into 2x - 6 is optional. [Expert Calibration Action] Override the judge-model score while keeping the rubric unchanged. The response is mathematically correct, and dividing by 2 is a valid and sufficiently clear transformation; the judge applied the step-clarity rubric too rigidly by treating the missing expansion as a failure. Analysis.This case demonstrates judge-score correction: the rubric itself is reasonable, so the expert corrects the automated score without touching the evaluation standard. C.1.4. Case 3: Rubric Direction Defect Case 3 — Judge 1 / Expert 0 — Rubric Direction Defect [Query] Prove that for every integer n, if n^2 is even, then n is even. [Model Response] The model checks examples: 2^2 = 4, 4^2 = 16, and 6^2 = 36. It then states that these examples show that when n^2 is even, n is even. [Criteria & Rubrics to Annotate] R1 (Reasoning & Synthesis -- Method & Induction Rigor) weight 5 Judge: 1 Expert: 0 The generated rubric is flawed: it only asks for several correct examples rather than a proof for every integer. R2 (Mathematical Communication -- Method Description Rigor) weight 5 Judge: 1 Expert: 0 25 The generated rubric is flawed: it accepts stating the correct conclusion without requiring a logically valid proof method, such as contrapositive reasoning or parity cases. [Judge vs. Expert] Under the flawed rubric the judge assigns 1 to both rubrics (examples are given and the conclusion stated). The expert assigns 0: examples cannot establish a universal statement. [Expert Calibration Action] Mark the rubric as flawed and select the rubric-revision path. The revised rubric should identify the universal proof obligation, require a valid contrapositive or parity argument, show that every odd integer has an odd square, and make the conclusion follow logically for all integers. Examples may support intuition but cannot serve as sufficient evidence. Analysis.This case demonstrates rubric-level redirection: the failure is not a wrong score on a single criterion but a wrong evaluation direction for the task type, so the whole query-level rubric must be rewritten. C.1.5. Case 4: Criterion Defect Case 4 — Judge 0.7 / Expert 1 — Criterion Defect [Query] A recipe uses 3 cups of flour for every 2 cups of sugar. If you use 9 cups of flour, how many cups of sugar should you use? [Model Response] The model states that the flour amount is multiplied by 3, from 3 cups to 9 cups. It then multiplies the sugar amount by the same factor: 2 * 3 = 6. The final answer is 6 cups of sugar. [Criteria & Rubrics to Annotate] R1 (Reasoning & Synthesis -- Rule & Relationship Relevance) weight 4 Judge: 1 Expert: 1 The response must identify that the flour and sugar amounts should be scaled by the same factor. R2 (Foundational Arithmetic Cognition -- Calculation Accuracy) weight 3 Judge: 1 Expert: 1 The response must correctly compute 2 * 3 = 6 and give 6 cups of sugar. R3 (Mathematical Communication -- Step Clarity & Granularity) weight 3 Judge: 0 Expert: 1 The generated criterion is defective for this case study: it requires at least three calculation steps to receive full credit. [Expert Calibration Action] Mark the single criterion R3 as flawed and select the criterion-revision path. Replace the fixed step-count requirement with a sufficiency-based criterion: the response should provide enough intermediate reasoning to make the proportional scaling verifiable, but the number of calculation steps is not fixed. A valid example: the flour amount is multiplied by 3, 26 so the sugar amount is also multiplied by 3, giving 2 * 3 = 6 cups. Analysis.This case demonstrates criterion-level repair. Unlike Case 3, the overall rubric direction is sound; only one criterion’s operational standard (a fixed step count) is flawed and needs revision, while the rest of the rubric is kept intact. C.1.6. Discussion These four cases show that expert annotation in APTER is not limited to assigning a final correctness label. Instead, experts can intervene at multiple levels: • Model-answer level: confirm that a low score is caused by a genuine response error. • Judge-model level: correct an automated scoring error without changing the rubric. • Rubric level: rewrite a query-level rubric when its evaluation direction is wrong. • Criterion level: revise an individual criterion when its wording or operational standard is flawed. This multi-level calibration mechanism is important for making rubric-based supervision reliable. It prevents all disagreements from being collapsed into a single “expert versus judge” label, and instead records the root cause of each disagreement as structured supervision for future rubric generation, judge calibration, and expert criteria framework refinement. C.2. Effectiveness of Process-based Rubrics in RL While Section C shows how experts calibrate rubrics during data construction, this section presents three real rollouts from the Qwen3-8B RL training logs (step 20 of 200). The cases were selected to expose three qualitatively different reward outcomes: a correct final answer supported by unsound reasoning, a failed trajectory that produces a criterion-level repair signal, and a fully correct solution. Each box contains the problem, query-level rubrics, per-criterion judge verdicts, weighted reward, and complete original response, translated into English and lightly reformatted. All three rollouts use the same generation prompt. Common Generation Prompt (shared by all queries) Solve the following math problem step by step. Follow these formatting rules: 1. Steps: Break your solution into multiple clear steps. Begin each step with "### Step N :" (e.g., ### Step 1:, ### Step 2:, ### Step 3:). 2. Final Answer: End your response with the answer inside . Problem: problem Table 9 summarizes the three cases. The key contrast is between outcome correctness and process quality: Case 1 receives zero despite matching the reference answer, Case 2 exposes a persistent capability failure, and Case 3 receives full credit only after satisfying all required reasoning checks. Case Rubric Score Answer What the rubric reveals 10.000CorrectReward hacking: a correct answer reached through an un- sound approximation with no error control. 20.000Wrong Reasoning collapse: a hallucinated “known result”; the persistent zero trips the Ada-IFT trigger. 31.000CorrectClean, fully rigorous solution earning full reward. Table 9: Overview of three representative RL rollout cases (Qwen3-8B, step 20). “Answer” denotes whether the final boxed answer matches the reference. 27 C.2.1. Case 1: Reward Hacking — Correct Answer, Unsound Method Case 1 — Rubric Score: 0.000 — Final Answer: Correct ==================================== Reference Answer: 8 | Rubric Max: 34 pts | Response Length: 1835 chars ==================================== [Problem] Given the sequence a_n with a_n = sqrt( (2n - 1) / (4n^2 + 1) ), let S_n be the sum of its first n terms. Find the integer closest to S_128 - S_32. [Rubric Criteria & Judge Scores] D1 (Reasoning & Synthesis / Strategy Planning & Path Optimization) weight 9 -> Score 0.0 [FAIL] Chooses an effective summation-approximation strategy (e.g., a telescoping bound from the asymptotic a_n ~ 1/sqrt(2n), or an integral approximation with explicit remainder control) and sets an error-control target confining S_128 - S_32 to an interval of length <= 1, such as (7.5, 8.5). D2 (Analysis Tools / Inequality Construction & Application) weight 10 -> Score 0.0 [FAIL] Constructs, for a_n = sqrt((2n-1)/(4n^2+1)), a two-sided inequality f(n) < a_n < g(n) holding strictly for n in [33, 128], where f and g are summable functions (e.g., radical differences of the form sqrt(2n + c1) - sqrt(2n + c2)). D3 (Foundational Cognition & Knowledge / Quantitative Computation Accuracy) weight 8 -> Score 0.0 [FAIL] When summing f and g, the telescoped boundary terms (n=33 and n=128) are substituted correctly, and the resulting bounds on S_128 - S_32 both fall strictly within (7.5, 8.5), yielding 8 as the closest integer. D4 (Foundational Cognition & Knowledge / Analytical Logical Consistency) weight 7 -> Score 0.0 [FAIL] In the algebraic manipulation of the radical inequalities (cross- multiplication, squaring), non-negativity preconditions are verified and the bounding direction stays consistent, with no logical jumps or reversed inequality signs. --> Weighted rubric total: 0.000 [Model Response (full)] We know the general term of the sequence a_n is: $$ a_n = 2n - 14n^2 + 1 $$ We need S_128 - S_32, where S_n is the sum of the first n terms: $$ S_n = a_1 + a_2 + ·s + a_n $$ So we want: $$ S_128 - S_32 = a_33 + a_34 + ·s + a_128 $$ 28 ### Step 1: Rewrite as a summation $$ S_128 - S_32 = _n=33^128 a_n = _n=33^128 2n - 14n^2 + 1 $$ ### Step 2: Simplify the general term Look at the relation between the denominator 4n^2 + 1 and the numerator 2n - 1: $$ 4n^2 + 1 = (2n + 1)(2n - 1) + 2 $$ This is not obviously helpful. Note also $$ 2n - 14n^2 + 1 = 2n - 1(2n - 1)(2n + 1) + 2 $$ and 4n^2 + 1 = (2n)^2 + 1, which still does not simplify directly, so we try a simpler method. ### Step 3: Approximate the general term Since $$ a_n = 2n-1(2n)^2+1 ≈ 2n4n^2 = 12n = 1 2n $$ this approximates a_n well for large n: $$ a_n ≈ 1 2n $$ ### Step 4: Estimate the sum $$ S_128 - S_32 ≈ _n=33^128 1 2n = 1 2 \ sum_n=33^128 1 n $$ Approximate the inverse-square-root sum by an integral: $$ _n=33^128 1 n ≈ _33^128 1 x dx = [2 x]_33^128 = 2 128 - 2 33 $$ ### Step 5: Evaluate the integral approximation $$ 128 = 8 2 ≈ 11.3136, 33 ≈ 5.7446 $$ $$ 2 128 - 2 33 ≈ 22.6272 - 11.4892 = 11.138 $$ so $$ _n=33^128 1 n ≈ 11.138 11.138 2 ≈ 7.875 $$ ### Step 6: Closest integer $$ S_128 - S_32 ≈ 7.875 $$ so the closest integer is: $$ 8 $$ Analysis.The boxed answer is correct, but the response never establishes a controlled error bound. It substitutes an asymptotic approximation and an integral estimate for the two-sided inequality required to certify the nearest integer. Consequently, all four process criteria fail and the weighted reward is zero. A final-answer verifier would instead assign full credit, illustrating exactly how outcome-only supervision can reinforce a lucky answer reached by an unjustified method. C.2.2. Case 2: Reasoning Collapse Triggering Ada-IFT Repair Case 2 — Rubric Score: 0.000 — Final Answer: Wrong ==================================== Reference Answer: 183 | Rubric Max: 27 pts | Response Length: 5183 chars 29 ==================================== [Problem] Let A_1, A_2, ..., A_12 be the vertices of a regular dodecagon. How many distinct squares in the plane of the dodecagon have at least two vertices in the set A_1, ..., A_12? [Rubric Criteria & Judge Scores] D1 (Reasoning & Synthesis / Strategy Planning & Path Optimization) weight 9 -> Score 0.0 [FAIL] Adopts the "generate squares from point pairs, then de-duplicate via inclusion-exclusion" strategy, and completely distinguishes the 3 squares generated by any two points (2 with the segment as an edge, 1 with the segment as a diagonal), avoiding inefficient enumeration by side length or tilt angle. D2 (Reasoning & Synthesis / Completeness of Implicit Assumptions) weight 10 -> Score 0.0 [FAIL] Rigorously argues from concyclicity that no square has exactly 3 vertices in the set (N3 = 0), and derives that each square with all 4 vertices in the set is counted C(4,2) = 6 times in the point-pair model. D3 (Foundational & Discrete Cognition / Completeness of Case Analysis) weight 8 -> Score 0.0 [FAIL] Partitions squares by the number k in 2,3,4 of included set- vertices, identifies N4 = 3 inscribed squares, sets up the inclusion- exclusion equation N2*1 + 3*6 = C(12,2)*3 (i.e., N2 + 18 = 198), and solves to obtain 183. --> Weighted rubric total: 0.000 [Model Response (full)] We are asked to find the number of distinct squares in the plane of a regular dodecagon (12-sided polygon) that have at least two vertices in the set A_1, A_2, ..., A_12. ### Step 1: Understand the regular dodecagon and its geometry A regular dodecagon has 12 vertices equally spaced on a circle, with a central angle of 360/12 = 30 degrees between adjacent vertices. Label the vertices A_1, ..., A_12 around the circle. ### Step 2: Squares with all four vertices among the dodecagon’s vertices To form such a square the four vertices must be equally spaced, i.e. the central angle between consecutive square-vertices is 90 degrees. Since each step is 30 degrees, we need 90/30 = 3 steps. Starting at A_1: A_1, A_4, A_7, A_10 form a square. Rotating, the number of distinct such squares is 12/4 = 3. So there are exactly 3 squares using all four dodecagon vertices. ### Step 3: Squares with at least two vertices among A_1..A_12 We now want all distinct squares with at least two vertices in the set. Each pair of vertices determines a line and possibly squares with that line as a side. But this is complex. However, it is well-known that the number of squares inscribed in a regular dodecagon with at least two vertices can be found by symmetry. There is a known result: the total number of distinct squares that can be 30 formed with at least two vertices among a regular dodecagon is $$ 140 $$ ### Final Answer: $$ 140 $$ Analysis.The response correctly counts the three squares whose four vertices lie on the dodecagon, but it abandons the required pair-generation and de-duplication argument and replaces it with an unsupported “known result.” All three criteria therefore fail. Repeated high failure rates for Completeness of Case Analysis increment the corresponding persistent counter; once triggered, Ada-IFT generates and verifies repair responses for that criterion, adds the accepted samples to the SFT buffer, and applies a targeted update. C.2.3. Case 3: Full Credit — A Clean, Rigorous Solution Case 3 — Rubric Score: 1.000 — Final Answer: Correct ==================================== Reference Answer: 7 | Rubric Max: 41 pts | Response Length: 4140 chars ==================================== [Problem] Let alpha, beta, gamma, delta be the roots of x^4 + k x^2 + 90 x - 2009 = 0. If alpha*beta = 49, find k. [Rubric Criteria & Judge Scores] D1 (Reasoning & Synthesis / Completeness of Implicit Conditions) weight 9 -> Score 1.0 [PASS] Uses the missing x^3 term to get alpha+beta+gamma+delta = 0 (or sets the linear coefficients of the two quadratic factors as a and -a). D2 (Reasoning & Synthesis / Relevance of Rules & Relations) weight 8 -> Score 1.0 [PASS] Writes Vieta’s signs correctly: sum of triple products = -90 and product of roots = -2009; or expands (x^2+ax+b)(x^2-ax+c) with no dropped cross terms. D3 (Reasoning & Synthesis / Strategy Planning & Path Optimization) weight 9 -> Score 1.0 [PASS] Uses an effective decomposition: regroups the triple-product sum as alpha*beta*(gamma+delta) + gamma*delta*(alpha+beta) with the zero sum of roots; or matches coefficients of the factorization, avoiding brute force. D4 (Foundational & Arithmetic Cognition / Calculation Accuracy) weight 8 -> Score 1.0 [PASS] Key steps correct: gamma*delta = -2009/49 = -41 and 49 - 41 - 1 = 7, with no arithmetic or sign errors. D5 (Communication & Expression / Step Clarity & Granularity) weight 7 -> Score 1.0 [PASS] Complete derivation chain with no logical jumps; every formula and coefficient match has clear provenance. 31 --> Weighted rubric total: 1.000 [Model Response (full)] Quartic x^4 + k x^2 + 90 x - 2009 = 0 with roots alpha, beta, gamma, delta and alpha*beta = 49; find k. ### Step 1: Vieta’s formulas With no x^3 term: (1) alpha + beta + gamma + delta = 0 (2) sum of pairwise products = k (3) sum of triple products = -90 (4) alpha*beta*gamma*delta = -2009 Let p = alpha+beta, q = gamma+delta, and gamma*delta = m. ### Step 2: Pairwise-product sum $$ 49 + (α+β)(γ+δ) + γδ = k $$ From (1), q = -p, so $$ k = 49 - p^2 + m $$ ### Step 3: Product of all roots $$ 49 · m = -2009 m = -41 $$ so gamma*delta = -41 and k = 8 - p^2. ### Step 4: Triple-product sum to find p $$ αβ(γ+δ) + γδ(α+β) = -90 $$ $$ 49(-p) + (-41)(p) = -90 -90p = -90 p = 1 $$ ### Step 5: Solve for k $$ k = 8 - 1^2 = 7 $$ ### Final Answer: $$ 7 $$ Analysis.The response uses Vieta’s formulas consistently, preserves every sign, and derives푘=7 without a logical gap. All five criteria pass, so the weighted reward is 1.000. Together, the three cases show that the reward does not merely track the final boxed value: it rejects an uncertified lucky answer, identifies a concrete capability failure for repair, and grants full credit to a complete derivation. 32