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GraphDecide: Benchmarking System One Models on Graph Tasks
Xianliang Yang, Yapu Zhang, Li Zhao
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Abstract:Large language models (LLMs) are increasingly explored for graph understanding and decision-making, while System One models such as Jev select directly from supplied options. However, the capabilities of System One models on graph-related tasks remain unclear. We introduce GraphDecide, a model-independent benchmark that combines structural task profiles, matched graph-text input contrasts and heuristic-proposal controls to diagnose graph decision performance. We evaluate Jev and related choice-based models alongside language-model baselines, covering fourteen model-interface configurations. Jev's results illustrate the benchmark's central distinctions: accurate adjacency recognition does not guarantee broader structural correctness, joint graph-text input does not consistently improve prediction, and feasible construction does not establish high solution quality. Its task contracts, candidate interfaces and scoring rules support comparison across native selectors and language-model adapters. Code and aggregate results are available at this https URL.
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- Source: https://arxiv.org/abs/2610.06354v1
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1 GraphDecide: Benchmarking System One Models on Graph Tasks Xianliang Yang 1 , Yapu Zhang 2 , Li Zhao 1 1 Microsoft 2 Beijing University of Technology xianya@microsoft.com, zhangyapu@bjut.edu.cn, lizo@microsoft.com Abstract Large language models (LLMs) are increasingly explored for graph understanding and decision-making, while System One models such as Jev select directly from supplied options. However, the capabilities of System One models on graph-related tasks remain unclear. We introduce GraphDecide, a model-independent benchmark that combines structural task profiles, matched graph-text input contrasts and heuristic-proposal controls to diagnose graph decision performance. We evaluate Jev and related choice-based models alongside language-model baselines, covering fourteen model-interface configurations. Jev's results illustrate the benchmark's central distinctions: accurate adjacency recognition does not guarantee broader structural correctness, joint graph-text input does not consistently improve prediction, and feasible construction does not establish high solution quality. Its task contracts, candidate interfaces and scoring rules support comparison across native selectors and language-model adapters. Code and aggregate results are available at https://github.com/VictorYXL/JevGraphBench. 1 Introduction Graphs encode relationships among entities. Large language models (LLMs) are increasingly explored for graph-structure questions, node-label prediction, and graph optimization [3, 6, 26, 29, 33]. Sequential applications such as route construction require repeated choices, motivating evaluation of the complete sequence of decisions. Emerging System One models, such as Jev, select from supplied options and return probabilities to support repeated decisions [1]. We study the correctness and quality of their graph decisions. Structural judgments, graph-text information use and sequential solution construction remain distinct evaluation targets. Evaluating a model on existing graph tasks is necessary but does not by itself answer three different questions: whether success transfers across structural operations, whether an input component adds value beyond the accompanying evidence, and whether an assisted solution reflects effective model selection rather than strong external proposals. A pooled accuracy or final objective can obscure these distinctions. GraphDecide therefore organizes its evaluation around task-level profiles, matched input conditions and candidate-pool controls, rather than treating the three task families as a single capability score. We introduce GraphDecide, a benchmark addressing three complementary research questions. As shown in Figure 1, GraphDecide includes three main task families: RQ1: Graph Cognition. How accurately do models answer questions about local connections and broader graph properties involving paths, cycles, and connectivity? RQ2: Graph-Text Decision-Making. For the same classification targets or retrieval queries, how does performance differ between textual input, relational input, and their combination? RQ3: Sequential Decision-Making on Graphs. Can successive model choices produce complete, feasible solutions, and how does their quality compare with classical methods and fixed rules? Together, these questions examine how System One models understand graph structure, use graph and text information, and make sequential decisions for graph optimization. Our contributions are threefold: 1. A model-independent diagnostic benchmark. GraphDecide connects structural task profiles, graph-text input contrasts and sequential construction controls through a common choice interface. The task definitions and evaluator are independent of Jev, allowing the same evaluation contracts to be used for other decision models. 2. Controlled information comparisons. Matched graph-text conditions assess performance differences from adding explicit edges while holding text and shared label anchors or candidate evidence fixed. The comparison covers both node classification and candidate-based retrieval. 3. Candidate and solution-quality diagnostics. Direct-construction, within-pool random and fixed-rule comparisons assess the value of model selection relative to externally supplied heuristic guidance. Across these evaluations, System One and other decision models are compared with LLMs and classical controls. GraphDecide 2 Figure 1. Overview of GraphDecide's three task families. (A) Graph Cognition evaluates structural answer accuracy against algorithmically computed ground truth. (B) Graph-Text Decision-Making evaluates classification and candidate selection under controlled textual and relational inputs; the illustration shows the classification interface. (C) Sequential Decision-Making evaluates TSP, signed MaxCut, deterministic influence maximization, and binary modularity, comparing direct action selection with selection among fixed-heuristic proposals. 2 Related Work 2.1 Benchmarks for Graph Learning and Reasoning Graph benchmarks cover prediction, algorithmic reasoning, and computational problem-solving. Open Graph Benchmark provides standardized datasets and evaluation protocols for graph learning [8], while CLRS evaluates the learned execution of algorithms [28]. NLGraph tests graph reasoning through natural-language questions [29], and GraphArena extends evaluation to diverse graph computational problems with explicit assessment of feasibility and solution quality [26]. Beyond structural computation, STaRK evaluates retrieval using textual and relational information in knowledge bases [30]. GraphDecide builds on established graph problems and public data, rather than introducing new optimization objectives. Its emphasis is the combination of diagnostic comparisons: task profiles reveal uneven structural performance, matched input contrasts assess additional evidence, and proposal-pool controls assess selection relative to supplied guidance. These comparisons complement task-level benchmark scores without assuming that the underlying tasks are themselves novel. 2.2 Language Models for Graph Understanding and Optimization GraphWiz [3] and GraphGPT [25] explore graph-specific instruction tuning, while GraphLLM integrates graph representations with language models [2]. For text-attributed graphs, GLEM couples language and graph models [33], while TAPE enriches node representations with LLM-generated explanations [6]. Learned constructive policies [10] and attention-based routing models [11] build solutions through successive decisions. FunSearch [23], EoH [14], ReEvo [32], and HeurAgenix [31] explore LLM-assisted program search, heuristic evolution, and heuristic selection for computational and optimization problems. GraphDecide 3 2.3 System One Decision-Making and Evaluation Jev targets option selection with associated probabilities [1], while SemIf [13], Decider [15], Kev [18], and Laya [4] provide related decision interfaces. Existing evaluations examine complementary aspects: JevBench studies accuracy, calibration, and cost [24]; WorldTour evaluates Jev-guided route construction [12]; and JevAdvBench examines decision changes under input perturbations relative to repeated-call variability [7]. GraphDecide complements these efforts by evaluating structural correctness, graph-text information utility, and sequential solution quality within a common choice-based framework. 3 The GraphDecide Benchmark GraphDecide separates task specification from model execution. Its graph instances, answer spaces, reference evaluators and scoring rules define the benchmark, while a model adapter supplies decisions. Jev is the main empirical case study, not a component required by the task definitions or evaluator. We use System One operationally for decision configurations that return a choice without an explicitly requested deliberative reasoning stage. Native choice-based models form the main study group, with no-thinking language-model configurations providing comparison baselines. This evaluation designation does not assert a shared architecture or reveal unobserved internal computation. A choice interface, a no-reasoning setting and low latency are distinct properties; the benchmark assesses decision quality, not a general efficiency advantage. 3.1 Tasks and Data Sources Graph Cognition (RQ1). Six query tasks on simple undirected graphs cover adjacency, degree, cycle existence, connectivity, distance thresholds, and articulation points. Induced subgraphs with 8–12 vertices are drawn from Facebook [34], ca-GrQc [35], Power Grid, and Human PPI. The RQ1 evaluation set contains 800 queries: 400 degree queries and eighty queries for each of the other five tasks, with 200 queries from each data source. Models select answers from the supplied graph and question; reference answers are computed by independent graph algorithms. Explicit vertex sets retain isolated vertices. Appendix A specifies the task contracts, and Appendix C reports both aggregate and source-specific results. Graph-Text Decision-Making (RQ2). A 40-class node-classification task on ogbn-arxiv [8] and a biomedical entity-selection task on STaRK-Prime [30, 36] compare predictions under controlled textual and relational inputs. The primary panels contain 100 test targets and 100 test queries, respectively. ogbn-arxiv conditions share the same targets, candidate categories and training-label anchors. STaRK-Prime conditions share the query and forty distinct candidate entities, accepting any native correct answer. The independent 200-target ogbn-arxiv evaluation retains its own targets and historical configurations and is not pooled with the primary evaluation (Appendix D.8 and Appendix F). Sequential Decision-Making on Graphs (RQ3). We evaluate four graph-optimization tasks, each using ten public-data instances. •TSP. The model selects unvisited cities sequentially to construct a minimum-length closed tour. We use complete TSPLIB EUC_2D instances with 76–225 cities and their published proven optima as reference values [22]. •Signed MaxCut. The model assigns vertices to two sides to maximize the total signed weight of crossing edges. We use OPTSICOM Set2 graphs with 125 vertices, 375 edges and weights ±1, with historical best-known values from 2016 as references [5, 17]. •Deterministic LT influence maximization. The model selects two distinct seeds, activated simultaneously, to maximize the final number of active nodes under fixed-threshold linear-threshold diffusion [9]. We use directed induced subgraphs of SNAP email-Eu-core [37] with 50–150 vertices and exact reference optima obtained by enumerating all seed pairs. •Binary modularity maximization. The model assigns vertices to at most two communities to maximize resolution-1 modularity [16, 38]. We use induced subgraphs of SNAP ca-GrQc [35] with 50–150 vertices and numerical MILP-solver-certified optima as reference values. Appendix A lists the optimization instances and their data specifications, while Appendix H details the LT and modularity reference calculations and their per-instance values. GraphDecide 4 3.2 Decision Interface and Input Conditions Each call supplies an input x and candidate set A and returns a choice. A model-interface configuration specifies the model and its readout: native selection, constrained generation or candidate scoring. Compared configurations share task data and prompt-construction rules. For RQ1 and RQ2, the query and condition determine the supplied input and options; RQ3 inputs follow the evolving state described below. To evaluate another model under the same protocol, an adapter maps the supplied input and candidate set to the model's interface and maps its response back to a candidate identifier. The adapter must preserve candidate identities and the condition-specific evidence, and leave ground-truth answers, objective evaluation and state updates to the evaluator. Native selection, constrained generation and candidate scoring are alternative readouts, not different task definitions. An unsupported request is recorded as unsupported rather than silently reducing the answer space. The adapter and inference settings identify the configuration being compared. The evaluation unit is a query for RQ1, a target–condition or query–condition pair for RQ2, and a trajectory attempt for RQ3. Each RQ3 trajectory consists of successive decisions made through the task interface. Graph-text input conditions. RQ2 uses five matched input conditions on each dataset. For ogbn-arxiv, all conditions share graph-selected context identities and training-label anchors. Relative to this shared setting, T supplies target text; G supplies directed edges; TG supplies target text, context text and edges; BAG retains TG's nodes and text but removes edges; and A supplies anonymous anchor identifiers and class labels without entity text or edges. Because context selection uses graph relations, T is not a graph-independent text baseline. For STaRK-Prime, T, G, TG, BAG and A* retain the same query and candidate entities while varying textual and relational evidence. BAG removes explicit edges from the joint-input condition, and A* provides the no-additional-entity-text comparator for G. Each forty-entity candidate set contains at least one native correct answer, without marking it in the model input. The task therefore evaluates selection within a gold-containing candidate set rather than end-to-end retrieval. Candidate construction and preprocessing details are provided in Appendix B. Sequential construction modes. Direct construction supplies all currently legal actions, while Heuristic-proposal selection supplies distinct actions nominated by four named fixed rules. The four tasks, ten instances per task and two modes define eighty trajectory-evaluation units per configuration. Task-specific answer sets and candidate-processing details appear in Appendices A and B. In RQ3, the state s t contains the graph, task specification and current partial solution, from which the input x t and candidate set A t are constructed. The model selects an action, and the evaluator updates the state: Each selected action advances construction by one step, following the model's own decisions until completion or failure. Different models or construction modes can induce different histories, so later states and candidate sets need not coincide. Automatic completion, candidate rules and exception handling are specified in Appendices A and B. 3.3 Evaluation Protocol Reference answers and optimization reference values are maintained by the evaluator and are not supplied to models as answers. Held-out classification labels are excluded from training-label anchors. Evaluation follows the task-specific protocols below. Graph cognition. The primary score for each task is query accuracy averaged equally across the five graph sizes. We additionally report task-size macro, task macro and query-weighted accuracy, with aggregation definitions in Appendix C. Invalid outputs and failed predictions count as incorrect, while output validity and interface support are reported separately. a t = D θ (x t , t ) ∈ t , s t+ 1 = F(s t , a t ). (1) GraphDecide 5 Graph-text decision-making. We report accuracy and macro-F1 over the fixed forty classes for ogbn-arxiv, and Hit@1 for STaRK-Prime, accepting any native correct answer. We report paired performance differences between the matched input conditions in Section 3.2. Let S(C) denote ogbn-arxiv accuracy or STaRK-Prime Hit@1, expressed as a percentage, under condition C. Define C 0 = A for ogbn-arxiv and C 0 = A* for STaRK-Prime. The principal contrasts are Both differences are measured in percentage points. TG-BAG compares explicit edges while holding accompanying text and anchors or candidates fixed. G-C 0 compares relational input without additional entity text. We also report TG-T and TG-G; in particular, ogbn-arxiv TG-T adds both context text and edges and is not an edge-only contrast. Invalid outputs and failed predictions count as incorrect; unsupported interfaces are reported separately. Paired 95% bootstrap intervals preserve the pairing of conditions for each target or query. Statistical details and interpretation are provided in Appendix B.3. Sequential decision-making on graphs. Solution quality is measured by the gap to the reference value for each instance. For a completed instance i, let L i be the TSP tour length, f i the MaxCut or LT objective, and Q i the binary-modularity objective. Using the corresponding reference R i , or Q* i for modularity, we define TSP, MaxCut and LT gaps are percentages; modularity gaps are absolute differences in Q. Lower values are better in every task. The reference types are specified in Section 3.1. For each model configuration, task and construction mode, let N be the number of scheduled trajectory-evaluation units. Define c i = 1 if unit i produces a complete legal solution and c i = 0 otherwise, and let I = i : c i = 1. We report completion coverage and mean gap over completed trajectories: The mean gap is undefined when no trajectory completes. Failed and unsupported units remain in the coverage denominator and are reported separately, without assigning them artificial objective values. Paired mode comparisons use instances completed under both modes, and comparisons with random or classical controls use the same completed-instance subset. Coverage and terminal outcomes are reported in Appendix B. 4 Experiments 4.1 Experimental Setup We evaluate Jev [1, 27], Decider-2B v11 [15], Kev-4B [18] and Laya [4] as native choice-based models. No-thinking Qwen3.5-0.8B, 2B, 4B and 9B [20] provide generation baselines, while Qwen3.5-27B, Qwen3.8-27B [21] and Qwen2.5-72B-Instruct [19] extend the comparison to larger models. We also evaluate Qwen3.5-4B through candidate-token scoring to compare decision interfaces. GPT-5.4 without reasoning (reasoning_effort="none") serves as a large-model System One baseline, and GPT-6-Astra, with medium reasoning observed in response metadata, serves as a strong reference. These comparisons comprise fourteen model-interface configurations. Detailed model settings and the GPU candidate-scoring implementation appear in Appendix B.1 and Appendix F.1, respectively. The configuration set covers native choice-based interfaces, generation baselines across model sizes, an alternative scoring readout and a reasoning-enabled reference. This selection targets complementary decision configurations rather than an exhaustive model census; interface limitations remain visible instead of changing the benchmark to accommodate a particular model. Δ edge = S(TG) −S(BAG),Δ relation = S(G) −S(C 0 ). (2) Δ i = 100 L i −R i R i , TSP, Δ i = 100 R i −f i R i , MaxCut or LT, Δ i = Q * i −Q i , binary modularity. (3) Coverage = 1 N N ∑ i = 1 c i , μ gap = 1 || ∑ i ∈ Δ i . (4) GraphDecide 6 4.2 Graph Cognition (RQ1) We examine whether accurate local connection recognition extends to other structural judgments. Figure 2 presents task-level accuracy, highlighting the capability profiles of Jev and related choice-based models against language-model baselines. Figure 2. Graph-cognition accuracy across six tasks and fourteen configurations. Values are percentages, equally averaged over graph sizes 8–12 within each task; higher is better. Degree is multiclass and the other tasks are binary. Appendix C provides aggregate and source-specific results. Jev exhibits a strong but uneven structural capability profile. It answers every adjacency query correctly and performs relatively well on cycle, connectivity and distance judgments, but reaches only 67.80% on exact degree and 61.25% on articulation. Recognizing an explicitly listed connection therefore does not establish accurate neighbor aggregation or identification of a vertex whose removal changes connectivity. The contrast locates Jev's limitations at particular structural operations rather than at graph input handling in general; it does not identify the model's internal reasoning mechanism. Related choice-based models share this task dependence but not Jev's overall level of performance. Decider and Kev answer some binary structural questions relatively well, yet their exact-degree accuracies are only about 10%. The common selection interface thus does not imply comparable structural competence. Jev exceeds both on five of the six task scores, but Kev's articulation accuracy is slightly higher (62.50% versus 61.25%), ruling out a claim of uniform dominance. Laya supports these inputs but records approximately 50% on the binary tasks; these accuracies alone do not establish that it selects randomly. The language-model baselines place these results in context. Jev outperforms the evaluated smaller Qwen configurations across the six task scores and is competitive with the larger Qwen models, while remaining below the high, broad accuracy of the GPT baselines. Its comparison with larger Qwen models is task-specific: Jev matches Qwen3.5-27B on adjacency and cycle detection, whereas Qwen2.5-72B is stronger on distance and articulation. A small aggregate ordering reversal between Jev and Qwen2.5-72B under query-weighted versus task-size-macro scoring further favors interpreting the task profile rather than a single overall rank (Appendix C). 4.3 Graph-Text Decision-Making (RQ2) We examine whether relations help Jev and related choice-based models, and whether combining relations with text yields additional benefit. Table 1 compares the matched input conditions for node classification on ogbn-arxiv and candidate-based retrieval on STaRK-Prime. AdjacencyDegreeCycleConnectivityDistanceArticulation Jev-1.13.0100.067.887.588.890.061.2 Decider-2B72.59.978.850.061.250.0 Kev-4B88.810.285.060.083.862.5 Laya50.010.450.051.250.050.0 Q3.5-0.8B50.012.152.550.050.050.0 Q3.5-2B50.014.750.050.050.050.0 Q3.5-4B gen.78.840.480.050.050.051.2 Q3.5-9B57.534.781.273.875.050.0 Q3.5-27B100.063.287.587.580.070.0 Q3.8-27B96.255.785.086.272.551.2 Q2.5-72B98.867.178.888.891.272.5 Q3.5-4B score90.017.578.850.061.255.0 GPT-5.4100.091.492.596.297.575.0 GPT-6-Astra100.0100.0100.0100.0100.0100.0 0 25 50 75 100 Accuracy (%) GraphDecide 7 Table 1. Graph-text decision performance under matched input conditions for all fourteen configurations. Model / Configuration ogbn-arxiv — Accuracy (%)STaRK-Prime — Hit@1 (%) TGTGBAGATGTGBAGA* Jev-1.13.055615857514847494642 Decider-2B v1135454238242017182120 Kev-4B29612727413919263534 LayaUUUUUUUUUU Qwen3.5-0.8B323426243399978 Qwen3.5-2B25121923151610161213 Qwen3.5-4B generation21593019423617242826 Qwen3.5-9B32554136394325313833 Qwen3.5-27B46615856495044464341 Qwen3.8-27B62607170515350515148 Qwen2.5-72B-Instruct51585952453935333536 Qwen3.5-4B scoringUUUUUUUUUU GPT-5.473617372486051535656 GPT-6-Astra73607673556969677371 Values are ogbn-arxiv accuracy and STaRK-Prime Hit@1 (%); higher is better. Each dataset uses 100 fixed targets or queries per condition. U denotes unsupported, not zero. A is the ogbn-arxiv shared-label anchor condition; A* is the STaRK-Prime relation-ablation comparator. Invalid outputs count as incorrect. Appendix D provides macro-F1, paired intervals and coverage. Each input-condition column is ranked independently within its dataset; higher is better. Within each stated comparison group, best values are bold and second-distinct values underlined, including ties; U and undefined values are excluded. 4.3.1 Node Classification Jev benefits from relational input over label anchors alone, but this does not translate into a clear gain from adding explicit edges when text is already available. On ogbn-arxiv, its G-A difference is +10 percentage points with a paired 95% interval of [+3,+18], whereas TG-BAG is only +1 point [-4,+7]. These contrasts answer different questions: relations improve the anchor-only setting, but their additional contribution is less clear in the text-rich setting. The result is consistent with either overlapping evidence or limited use of the added edges; the present comparisons do not distinguish those explanations. The Jev-like models reinforce the need to distinguish relational prediction from effective graph-text combination. Kev matches Jev's 61% accuracy under G but falls to 27% under TG, while Decider falls from 45% to 42%. Jev's own TG accuracy of 58% also remains below its G result. Joint input is therefore not a generally reliable enhancement for these choice-based models, and its effect differs substantially by configuration. Kev's large decrease establishes a performance loss under added target and context text, not a specific mechanism such as distraction or input-length sensitivity. Jev's main-panel TG accuracy is 58%, while its fixed-forty-class macro-F1 is 24.8080%. Overall accuracy is consequently insufficient to characterize class-level performance. However, the difference also reflects class coverage in this sample and the convention that classes absent from both truth and predictions receive zero F1; it cannot be attributed solely to uneven performance among observed classes. The independent 200-target study likewise finds a small TG-BAG difference of +3.0 points [-1.0,+7.0], without pooling that estimate with the main panel (Appendices D.8 and F). Language-model baselines achieve higher joint-input accuracy in some configurations, but do not remove the distinction between joint performance and explicit-edge benefit. Qwen3.8-27B reaches 71% under TG while gaining only one point over BAG; Astra reaches 76% with a three-point TG-BAG difference whose interval includes zero. Conversely, Decider, Qwen3.5-4B generation and Qwen2.5-72B have positive TG-BAG intervals in the reported GraphDecide 8 pointwise analysis. The evidence therefore supports configuration-specific gains, rather than either universal usefulness or universal irrelevance of explicit edges (Appendix D.3). 4.3.2 Candidate-Based Retrieval Jev shows no clear additional relational benefit in the current STaRK-Prime candidate-selection setting. Its T, G and TG Hit@1 values are close at 48%, 47% and 49%; TG-BAG is +3 points [-6,+12] and G-A* is +5 points [-5,+14]. Both intervals include zero. This limits the evidence for an increment from relations within the answer-containing candidate sets, rather than establishing an absence of relational ability or a result about end-to-end retrieval. The related models also lack a consistent joint-input advantage, but their responses to text differ across tasks. Kev's G-to-TG change is +7 points on STaRK-Prime, reversing its large negative change on ogbn-arxiv; nevertheless, its TG-BAG difference is negative at -9 points [-17,-2]. Adding text to a relational condition and adding edges to a text-rich condition should therefore not be conflated. Decider's TG result similarly does not exceed its T or BAG result. Laya's unsupported forty-choice interface is a separate limitation, not a zero retrieval score. The baselines provide higher absolute retrieval performance without eliminating input dependence: GPT-5.4 performs best under T, whereas Astra performs best under BAG. No supported configuration has a strictly positive TG-BAG interval in this panel (Appendix D.6). Together with the ogbn-arxiv results, this shows that the usefulness of an input combination must be established for the particular model and task, rather than inferred from the availability of more information. 4.4 Sequential Decision-Making on Graphs (RQ3) We examine whether Jev and related models can turn successive choices into high-quality graph solutions, and whether heuristic proposals improve their decisions beyond the benefit of the proposal mechanism itself. Figure 3 compares direct construction and heuristic-proposal selection; matched random and fixed-rule controls in Appendix E distinguish these sources of performance. Figure 3. Sequential solution quality under direct construction and heuristic-proposal selection across four tasks. Bars show mean reference gaps over completed instances; lower is better. TSP, MaxCut and LT use percentages, while modularity uses absolute Q*-Q units. Partial results are labeled with completed graphs out of ten; combinations without completions are omitted and dots retain zero or visually tiny gaps. Appendix E provides objectives, coverage and matched controls. Direct constructionHeuristic-proposal selection TSP 0200400600 Jev-1.13.0 Decider-2B Kev-4B 8/10 Q3.5-0.8B Q3.5-2B Q3.5-4B gen. Q3.5-9B Q3.5-27B Q3.8-27B Q2.5-72B GPT-5.4 GPT-6-Astra Reference gap (%); lower is better Signed MaxCut 0255075100 Jev-1.13.0 Decider-2B Kev-4B Q3.5-0.8B Q3.5-2B Q3.5-4B gen. Q3.5-9B Q3.5-27B Q3.8-27B Q2.5-72B GPT-5.4 GPT-6-Astra Reference gap (%); lower is better Deterministic LT 0255075100 Jev-1.13.0 8/10 8/10 Decider-2B Kev-4B 3/10 4/10 Laya 1/10 Q3.5-0.8B Q3.5-2B Q3.5-4B gen. Q3.5-9B Q3.5-27B Q3.8-27B Q2.5-72B Q3.5-4B score 1/10 GPT-5.4 GPT-6-Astra Reference gap (%); lower is better Binary modularity 00.20.40.6 Jev-1.13.0 Decider-2B Kev-4B Laya 3/10 Q3.5-0.8B Q3.5-2B Q3.5-4B gen. Q3.5-9B Q3.5-27B Q3.8-27B Q2.5-72B Q3.5-4B score 4/10 1/10 GPT-5.4 GPT-6-Astra 0.0002611 Q*−Q; lower is better GraphDecide 9 4.4.1 Direct Construction Jev can complete construction without producing a strong solution. It finishes all TSP, MaxCut and modularity instances, yet its mean TSP gap is 59.66%, its mean MaxCut objective is zero, and its mean modularity is also zero. Its completed LT instances have a 52.84% gap. These weaknesses span many-option route construction, binary partition decisions and a two-seed selection problem, so neither a large action space nor a long decision sequence alone explains the pattern. The common challenge is making local choices that serve the final objective, not simply identifying legal next actions. Task-specific controls show the practical size of the shortfall. Jev's TSP gap remains above the 43.89% within-proposal-pool random control and the 5.75% classical control, although the former already benefits from heuristic candidate construction and is not a full-action random policy. On signed MaxCut, positive and negative attained objectives average to zero; this does not mean every graph receives the same partition or objective. Its 99.96% mean gap contrasts with 19.73% for assigned-prefix gain. For modularity, mean Q=0 matches the single-community baseline in objective value, but does not by itself identify the produced partitions. These comparisons expose weak optimization despite successful completion. The Jev-like results separate execution coverage from decision quality. Decider completes all eighty evaluation units across both modes, but its direct TSP gap is 583.40% and its direct LT gap is 75.40%. Kev also constructs weak direct solutions and has task-dependent support limitations. Jev's LT means cover only eight graphs because both modes fail on the same two instances; on its completed subset, the matched classical gap is 6.84%, well below its direct gap. Across all configurations, the 1,120 scheduled evaluation units comprise 951 feasible completions, 165 unsupported units and four failures, all in Jev LT. Appendix B records their terminal causes and coverage. The language-model baselines confirm that the direct-construction interface permits much stronger policies, but model scale alone is not sufficient. Larger Qwen models remain weak on several tasks, especially signed MaxCut, whereas Astra obtains much smaller gaps across all four tasks and reaches the exact LT references. These baselines locate the limitations of the evaluated System One configurations; differences between distinct models and reasoning settings do not isolate the causal contribution of reasoning. 4.4.2 Heuristic-Proposal Selection Heuristic proposals consistently improve Jev on the instances completed under both modes: all ten TSP, ten MaxCut, eight LT and ten modularity instances have lower gaps under proposal selection. This is useful assistance under the evaluated configuration, but the intervention supplies candidate actions, numerical features, rule identities and sometimes forced singleton transitions together. Its benefit cannot be attributed solely to reducing the number of options, particularly in the already binary partition tasks. For Jev, beating its direct policy does not necessarily mean beating a fixed rule. Its TSP proposal gap of 25.87% improves on random selection within the pool, but is almost identical to the nearest rule's 25.86% and remains above return reserve's 15.58%. Its MaxCut gap of 9.05% is far better than the 82.72% random-proposal gap, yet slightly worse than greedy completion's 8.87%. Similarly, its modularity gap falls to 0.0555 but remains above greedy completion's 0.0361. These results show useful selection relative to random proposals without a consistent advantage over the strongest fixed rule. LT is a stronger case for Jev's assisted performance. On the same eight completed graphs, its proposal gap is 0.50%, below both the 21.91% random-candidate gap and the 6.84% classical gap. This supports effective use of the supplied proposals on that subset, not a comparison against another model's full ten-graph mean. One of these completions requires no model call because both seed choices are forced; the zero-gap LT completions of Laya and candidate scoring are also zero-call episodes and do not demonstrate successful model selection. The Jev-like models also benefit from proposals, although their assisted quality and coverage differ. Decider's large improvements from weak direct policies still leave it behind Jev on the same ten-graph TSP, MaxCut and modularity panels. For Kev's TSP comparison, quality improves by 384.2287 percentage points on the eight graphs completed under both modes, while proposal-mode coverage rises to ten graphs (Appendix E.1). This paired result, rather than subtraction of the eight-graph and ten-graph conditional means, establishes the improvement. The stronger baselines bound the generality of assistance. GPT-5.4 improves in mean gap on MaxCut, LT and modularity, but not TSP; Astra's mean gap increases on each of the four tasks. The latter is an aggregate statement, not deterioration on every instance: its LT gap increases from 0 to 0.4% because one instance worsens while nine remain GraphDecide 10 tied. Proposal restrictions may exclude useful actions, but features and decision histories also change, so the observed losses do not identify a unique mechanism. Overall, Jev benefits substantially from heuristic guidance, while surpassing a single fixed rule remains task-dependent. Related choice-based models show the same need to distinguish an improvement over a weak direct policy from strong final optimization performance. Comparisons against direct construction, random selection within the proposal pool and fixed rules reveal different levels of value; none can replace the others. Appendix G retains the independent supporting public evaluations and the anonymous-versus-named TSP diagnostic, with their own samples and historical configurations rather than pooled primary estimates. 5 Discussion and Conclusion 5.1 Main Findings and Implications Structural competence is task-specific, even within a common choice interface. Jev combines strong local connection recognition with weaker exact counting and articulation judgments. The different profiles of Decider, Kev and Laya show that related choice-based interfaces do not imply interchangeable graph capabilities. For applications, successful edge recognition is therefore insufficient evidence that a model can reliably carry out other structural operations. The relevant capability must be assessed at the operation required by the task, rather than inferred from the interface or a pooled score. Useful relational input does not guarantee effective graph-text combination. Jev benefits from relations over label anchors on ogbn-arxiv, but the additional benefit of explicit edges is less clear when text is already supplied. The Jev-like models respond differently to joint input, and the STaRK-Prime results do not reproduce the same pattern of relational gains. Input design should thus consider what additional evidence a model can use, not simply how much information can be included. Matched contrasts assess this incremental value without treating high joint-input accuracy as evidence of effective use of every input component. Heuristic guidance is useful, but its value must be distinguished from adaptive selection. Proposal assistance substantially improves Jev's construction results, while surpassing a fixed rule remains task-dependent. Related models also improve from weak direct policies without uniformly attaining strong solutions. Random selection within the proposal pool and fixed-rule controls help assess the model's additional value over the supplied guidance. Worse assisted results for stronger baselines further show that assistance is not universally preferable: candidate restrictions are one possible explanation, but changes in features and decision histories prevent a unique attribution. 5.2 Scope and Limitations The benchmark covers limited public instances, small graph-cognition inputs and specific construction tasks; the findings do not establish performance on larger or dynamic graphs. In ogbn-arxiv, shared context and label anchors induce dependence between targets. STaRK-Prime evaluates selection within answer-containing candidate sets rather than end-to-end retrieval. Independent supporting studies broaden the observations but retain their own samples and configurations; repeated conditions on a graph are not independent graphs. Comparisons concern model-interface configurations and do not isolate effects of architecture, scale or reasoning. Proposal assistance changes candidate actions, features, forced transitions and subsequent histories together, so its mechanisms are not separately identified. Classical controls may also revise solutions, unlike irreversible direct construction. Quality comparisons with partial completion require matched instance subsets. Single executions do not estimate within-instance stochastic variation, and paired target-bootstrap intervals neither replace repeated-run uncertainty estimates nor fully account for shared graph context. The reported intervals are pointwise and unadjusted for multiple comparisons. GraphDecide 11 5.3 Conclusion GraphDecide provides a model-independent benchmark for studying System One models through graph-structural judgments, graph-text information use and sequential solution construction. Jev exhibits competitive but uneven structural capabilities, condition-dependent benefits from relational input, and substantial improvements from heuristic proposals without consistently surpassing fixed rules. The differing profiles of related choice-based models show that these capabilities cannot be inferred from a shared selection interface alone. The results characterize both the promise and the limitations of the evaluated System One configurations and support assessing their usefulness under the task, information and candidate conditions in which they will be applied. Separating these diagnostic comparisons from model-specific adapters makes the benchmark applicable to future decision models as well as the configurations studied here. References [1] Almeida, D. Introducing system one models & Jev. TypeSafe AI, 2026. Accessed 2026-10-01. [link] [2] Chai, Z., Zhang, T., Wu, L., Han, K., Hu, X., Huang, X., and Yang, Y. 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In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2017. Email-Eu-core network data: SNAP. Accessed 2026-10-04. [link] [38] Newman, M. E. J. Modularity and community structure in networks. Proceedings of the National Academy of Sciences, 103(23):8577–8582, 2006. DOI: 10.1073/pnas.0601602103. [link] GraphDecide 13 Appendix A. Data and Task Specifications A.1 Structural Queries RQ1 uses simple, undirected, unweighted induced graphs with an explicit vertex set, including isolated vertices. Adjacency asks whether a specified edge is present; degree asks for the exact neighbor count; cycle asks whether a cycle exists; connectivity asks whether all vertices are mutually reachable; distance tests a specified shortest-path threshold; articulation asks whether deleting a vertex increases the number of connected components. An unreachable pair has infinite distance. Each task is scored against an algorithmically computed answer. There are five sizes, four public sources, and six tasks; only degree uses more than two answer options. Full results are in Appendix C. Binary queries supply yes/no options, while degree queries supply integers from 0 to n-1. A.2 Sequential Objectives and Feasibility TSP starts at the fixed initial city, appends unvisited cities to a partial route, and visits every city once before closing the tour. The last unvisited city and the return edge are completed automatically. EUC_2D distances round the Euclidean distance to the nearest integer as specified by TSPLIB. Signed MaxCut assigns each vertex to one of two sides and maximizes the sum of signed weights on crossing edges; negative weights remain negative. Partition construction fixes the next vertex rather than allowing a vertex-order search. The current state and partial solution are carried forward after each action. LT chooses two distinct seeds before diffusion starts and activates them simultaneously. Updates are synchronous and irreversible. Each incoming edge has weight 1/indegree and every nonseed has threshold 1/2; equality activates a vertex, and a zero-indegree nonseed never activates. The objective is the final number of active vertices. Exhaustive evaluation of all unordered seed pairs supplies the exact reference. This fixed-threshold process is not the random-threshold model underlying the usual influence-maximization approximation guarantee [9]. The lack of a diminishing-returns guarantee follows even in a small deterministic example: let a, b and d each point to c with weight 1/3. Vertex c requires two active parents. Spread is zero for the empty set, one for either seed a or b alone, and three for seeds a,b. Adding b therefore gains one at the empty set but two at a. This task property motivates evaluating marginal-cascade heuristics empirically rather than transferring the random-threshold guarantee. Binary modularity uses full-graph degrees, resolution one, and at most two communities [38]. A single-community assignment is legal and gives Q=0. Results include attained Q and gaps to the numerical solver-certified optima in Appendix H.1. A.3 Public Optimization Instances The following inventory gives every RQ3 graph, size and edge count. TSP references are proven optima from TSPLIB [22]; MaxCut references are historical 2016 best-known values from the OPTSICOM collection [5, 17]; LT references are exact seed-pair optima. The modularity graph inventory is supplemented by the certified reference values in Table H.1. Each graph contributes one condition per arm. LT and binary modularity each include two induced subgraphs at each size 50, 75, 100, 125 and 150, indexed s0 and s1. Q(c) = 1 2m ∑ u, v ( A uv − d u d v 2m ) 1[c u = c v ], c v ∈0, 1. (A.1) GraphDecide 14 Table A.1. TSP: proven optimum. InstancenEdgesReference kroA100100495021282 kroA1501501117526524 kroB100100495022141 kroB2002001990029437 kroC100100495020749 bier1271278001118282 tsp225225252003916 eil76762850538 rat999948511211 kroD100100495021294 Table A.2. Signed MaxCut: 2016 best-known. InstancenEdgesReference sg3dl051000125375110 sg3dl052000125375112 sg3dl053000125375106 sg3dl054000125375114 sg3dl055000125375112 sg3dl056000125375110 sg3dl057000125375112 sg3dl058000125375108 sg3dl059000125375110 sg3dl0510000125375112 Table A.3. Deterministic LT: exact seed-pair optimum. InstancenEdgesReference email:n50-s05015014 email:n50-s15012817 email:n75-s07531729 email:n75-s17523263 email:n100-s010034825 email:n100-s110047025 email:n125-s012550725 email:n125-s112564316 email:n150-s015084317 email:n150-s1150100215 GraphDecide 15 Table A.4. Binary-modularity graph inventory. InstancenEdges GrQc:n50-s05085 GrQc:n50-s15069 GrQc:n75-s07592 GrQc:n75-s175139 GrQc:n100-s0100226 GrQc:n100-s1100180 GrQc:n125-s0125231 GrQc:n125-s1125199 GrQc:n150-s0150204 GrQc:n150-s1150224 A.4 Reference Calculations Appendix H.1 gives the binary-modularity MILP and numerical certification; Appendix H.2 gives deterministic LT seed-pair enumeration and independent closure checks. Both sections list all ten fixed-instance references. Appendix B. Interfaces and Candidate Rules B.1 Model Configurations Table B.1. Served models, interfaces, and configured output values. ConfigurationProvider / interfaceServed modelRQ1 / RQ3RQ2 Jev-1.13.0typesafejev-1.13.0---- Decider-2B v11pluginMapika/decider-2b---- Kev-4Bpluginjaredpalmer/kev-4b---- Layapluginconvaiinnovations/laya-typed-decisions---- Qwen3.5-0.8BvllmQwen3.5-0.8B6416 Qwen3.5-2BvllmQwen3.5-2B6416 Qwen3.5-4B generationvllmQwen3.5-4B6416 Qwen3.5-9BvllmQwen3.5-9B6416 Qwen3.5-27BvllmQwen3.5-27B6416 Qwen3.8-27BvllmQwen3.8-27B6416 Qwen2.5-72B-InstructvllmQwen2.5-72B6416 Qwen3.5-4B scoringpluginQwen3.5-4B---- GPT-5.4github_copilotgpt-5.440964096 GPT-6-Astragithub_copilotgpt-6-astra40964096 Q3.5/Q3.8/Q2.5 abbreviate Qwen3.5/Qwen3.8/Qwen2.5 in result tables. gen. and score distinguish generation and scoring interfaces. A dash means no value is specified for that interface. Qwen generation uses no-thinking mode with 64 output tokens for RQ1/RQ3 and 16 for RQ2. Native/scoring interfaces retain their support constraints. GPT-5.4 uses provider github_copilot, served model gpt-5.4, think=False, reasoning_effort="none", output_format=function_call and max_tokens=4096, with answer_only task instructions for RQ2. Temperature, model seed, top_p, top_k, presence_penalty and constrain_choices are unset. Astra leaves think and reasoning_effort unset in its historical request configuration; medium is observed in response metadata rather than GraphDecide 16 explicitly set in the request. Other unspecified settings retain provider defaults. The configured output value of 4,096 is not a verified limit on total reasoning tokens. Thirty LT trajectories from Qwen3.5-27B, Qwen3.8-27B and Qwen2.5-72B use 40,960-token contexts. Ten of Qwen2.5-72B's LT trajectories additionally use static YaRN factor 4 with original_max_position_embeddings=32768; four of its TSP direct-construction trajectories use BF16, a 40,960-token context and the same YaRN setting. Other trajectories retain their respective context settings. B.2 Fixed Candidate Rules and Controls RQ2 classification supplies all forty classes, and STaRK-Prime supplies forty distinct candidate entities. For RQ3 direct construction, the legal actions are unvisited cities for TSP, unselected vertices for seed selection, and either side for the current vertex in MaxCut and modularity. At each state, Heuristic-proposal selection merges duplicate actions nominated by four fixed rules and retains their rule identities. Only one action is executed. A singleton is forced and incurs no model call. The random-candidate control selects uniformly among distinct candidates, averaging five seeds within each graph before averaging across graphs. Each fixed-rule control follows that rule throughout construction; the reported classical control is listed separately. TSP. For current city c, start s, and unvisited set U, let e(v) be the distance from c to v, r(v) the distance from v to s, and h(v) the distance from v to its nearest other unvisited city. The rules nearest, one_step_lookahead, return_reserve, and isolation_priority minimize e, e+h, 2e-r, and e-h, respectively. Rules propose a next city, not a repaired or locally optimized tour. Ties use the lowest vertex identifier. Partition tasks. The four rules are assigned_gain, greedy_completion, opposite_completion, and degree_balance. They evaluate assigned-prefix gain, full greedy completion, completion with the remaining suffix on the opposite side, and degree-volume balance, respectively. Cut gains preserve signed weights, while cut volume uses absolute incident weights. For modularity, with current degree d, m full-graph edges, e_s edges to assigned side s, and pre-action volume V_s, assigned gain is proportional to 4me_s-2dV_s-d 2 . Completion estimates score a full partition but execute only the next vertex assignment. Because Direct construction already has two sides, Heuristic-proposal selection is either the same two actions or a forced singleton. LT. The rules marginal_cascade, outdegree, two_hop_reach, and threshold_deficit maximize marginal deterministic spread, outdegree, inactive two-hop reach, and threshold relief. With active closure X, threshold relief is zero for v in X. For an inactive v, it sums reciprocal residual activation deficits over its inactive out-neighbors; the deficit at t is ceil(indegree(t)/2) minus its active in-neighbor count. The two selected seeds are activated together for final scoring. These task-specific rules are fixed rather than learned. STaRK-Prime candidate construction. Candidate construction starts with BM25 top-40. If no native correct answer is present, the lowest-ranked candidate is replaced by a deterministically selected correct entity; the final set is deterministically shuffled and its text and relational evidence rebuilt without answer markers. Candidate Hit@40 is therefore 100% by design, while candidate Recall@40 is 77.7241% because a multi-answer query need not retain every correct entity. B.3 Scoring, Statistics, and Coverage RQ2 paired intervals use 1,000 target-bootstrap resamples, seed 20261001 and sorted zero-based endpoints 24 and 974. Conditions of a resampled target stay paired. For GPT-5.4, numeric query IDs are sorted and Python random.Random draws indices uniformly with replacement, using common samples across contrasts. These are pointwise 95% intervals without multiplicity adjustment. Fixed-forty-class macro-F1 gives zero F1 to classes absent from both truth and predictions. Shared ogbn-arxiv contexts and anchors induce dependence beyond target resampling. RQ3 means condition on feasible completion, while coverage retains every scheduled trajectory without objective imputation. GraphDecide 17 Table B.2. Primary RQ3 coverage: eighty scheduled evaluation units per configuration. ConfigurationCompleteUnsupportedFailedCoverage Jev-1.13.0760495.00% Decider-2B8000100.00% Kev-4B6515081.25% Laya47605.00% Q3.5-0.8B8000100.00% Q3.5-2B8000100.00% Q3.5-4B gen.8000100.00% Q3.5-9B8000100.00% Q3.5-27B8000100.00% Q3.8-27B8000100.00% Q2.5-72B8000100.00% Q3.5-4B score67407.50% GPT-5.48000100.00% GPT-6-Astra8000100.00% Coverage alone is ranked as a completion rate over the same 80 scheduled evaluation units, not as solution quality. RQ1 has 11,200 legal outputs. Supported RQ2 units have 11,998 legal and two illegal outputs, the latter both GPT-5.4 STaRK-Prime TG responses; illegal outputs count as incorrect. The two unsupported configurations remain explicit. The four RQ3 failures are Jev LT http_error terminals caused by max_tokens_exceeded, listed below. Unsupported trajectories are counted separately. Forced-singleton completion may require no model call, so the few Laya/scoring completions do not imply general interface support. Table B.3. Retained primary RQ3 failures. ConfigurationGraphModeTerminalCause Jev-1.13.0email:n150-s1Heuristic-proposal selectionhttp_errormax_tokens_exceeded Jev-1.13.0email:n150-s1Direct constructionhttp_errormax_tokens_exceeded Jev-1.13.0email:n150-s0Heuristic-proposal selectionhttp_errormax_tokens_exceeded Jev-1.13.0email:n150-s0Direct constructionhttp_errormax_tokens_exceeded B.4 Complete GPT-5.4 RQ3 Results Tables B.4a–B.4h report all eighty GPT-5.4 no-reasoning trajectories, ten graphs for each task and construction mode. Every row is feasible with legal output; objectives and gaps are rounded only for display. TSP, MaxCut and LT gaps are percentages; modularity uses absolute Q*-Q. A zero-call LT proposal trajectory follows forced singleton actions. Task summaries and matched controls appear in Appendix E. GraphDecide 18 Table B.4a. GPT-5.4: TSP, Direct construction. GraphObjectiveGap (%)Calls bier12715900134.425356125 eil7676442.00743574 kroA1002751529.28766198 kroA1503416128.792791148 kroB1002823127.50553398 kroB2003928433.451099198 kroC1002633226.90732198 kroD1002699826.78688898 rat99140115.68951397 tsp225509130.005107223 Table B.4b. GPT-5.4: TSP, Heuristic-proposal selection. GraphObjectiveGap (%)Calls bier12715766633.29669869 eil7668226.76579939 kroA1002889635.77671353 kroA1503610736.12954386 kroB1002847828.62111052 kroB2003702825.787275101 kroC1002681929.25442244 kroD1002733728.37888641 rat99163635.09496357 tsp225561343.335036126 Table B.4c. GPT-5.4: Signed MaxCut, Direct construction. GraphObjectiveGap (%)Calls sg3dl051000-6105.454545124 sg3dl0510000892.857143124 sg3dl052000-6105.357143124 sg3dl053000-10109.433962124 sg3dl054000-2101.754386124 sg3dl055000-6105.357143124 sg3dl0560002676.363636124 sg3dl0570001091.071429124 sg3dl058000496.296296124 sg3dl059000-14112.727273124 GraphDecide 19 Table B.4d. GPT-5.4: Signed MaxCut, Heuristic-proposal selection. GraphObjectiveGap (%)Calls sg3dl0510009612.72727390 sg3dl05100001047.14285786 sg3dl0520009614.285714105 sg3dl053000969.433962118 sg3dl0540001085.263158116 sg3dl0550001065.35714384 sg3dl0560009414.54545594 sg3dl05700010010.714286104 sg3dl0580001025.55555696 sg3dl0590001009.09090994 Table B.4e. GPT-5.4: Deterministic LT, Direct construction. GraphObjectiveGap (%)Calls email:n100-s0772.0000002 email:n100-s11540.0000002 email:n125-s01252.0000002 email:n125-s1756.2500002 email:n150-s01041.1764712 email:n150-s11126.6666672 email:n50-s01028.5714292 email:n50-s11417.6470592 email:n75-s0290.0000002 email:n75-s1630.0000002 Table B.4f. GPT-5.4: Deterministic LT, Heuristic-proposal selection. GraphObjectiveGap (%)Calls email:n100-s01732.0000002 email:n100-s1244.0000001 email:n125-s0250.0000001 email:n125-s11318.7500001 email:n150-s0170.0000001 email:n150-s1150.0000001 email:n50-s0140.0000002 email:n50-s1170.0000002 email:n75-s0290.0000000 email:n75-s1630.0000001 GraphDecide 20 Table B.4g. GPT-5.4: Binary modularity, Direct construction. GraphObjectiveGap (Q)Calls GrQc:n100-s00.11010063430.377134074799 GrQc:n100-s10.06479938270.390740740799 GrQc:n125-s00.00153670280.4531680441124 GrQc:n125-s10.16292517870.3270119441124 GrQc:n150-s00.05646866590.4313725490149 GrQc:n150-s10.15791414220.3286830357149 GrQc:n50-s00.04816609000.440069204249 GrQc:n50-s10.17979416090.301932367149 GrQc:n75-s00.01459120980.467391304374 GrQc:n75-s1-0.05486258480.524895191874 Table B.4h. GPT-5.4: Binary modularity, Heuristic-proposal selection. GraphObjectiveGap (Q)Calls GrQc:n100-s00.48319171430.004042994884 GrQc:n100-s10.42097222220.034567901250 GrQc:n125-s00.44604673830.008658008780 GrQc:n125-s10.40452261310.0854145097100 GrQc:n150-s00.41171664740.0761245675109 GrQc:n150-s10.43294603000.053651148098 GrQc:n50-s00.42103806230.067197231822 GrQc:n50-s10.46838899390.013337534139 GrQc:n75-s00.40211483930.079867674955 GrQc:n75-s10.47003260700.000000000067 B.5 Selected Episode Outcomes Table B.5 lists selected primary-evaluation episodes, including context-expanded LT inputs and Jev's input rejection. These rows also contribute to the aggregate results. Table B.5. Selected episode outcomes: terminal status and confirmed calls. ConfigurationGraphModeFeasibleTerminalCalls Qwen3.5-27Bemail-Eu-core-main-n75-s1Direct constructionTruecomplete2 Qwen3.5-27Bsg3dl055000Direct constructionTruecomplete124 Qwen3.5-27Bsg3dl0510000Heuristic-proposal selectionTruecomplete92 Jev-1.13.0email-Eu-core-main-n150-s1Heuristic-proposal selectionFalsehttp_error1 Qwen2.5-72Bemail-Eu-core-main-n100-s1Heuristic-proposal selectionTruecomplete2 Qwen2.5-72Bemail-Eu-core-main-n150-s1Heuristic-proposal selectionTruecomplete1 Qwen2.5-72Bemail-Eu-core-main-n125-s1Heuristic-proposal selectionTruecomplete2 Qwen2.5-72Bemail-Eu-core-main-n100-s1Direct constructionTruecomplete2 Qwen2.5-72Bemail-Eu-core-main-n150-s1Direct constructionTruecomplete2 Qwen2.5-72Bemail-Eu-core-main-n125-s1Direct constructionTruecomplete2 Qwen2.5-72Bemail-Eu-core-main-n150-s0Heuristic-proposal selectionTruecomplete1 GraphDecide 21 ConfigurationGraphModeFeasibleTerminalCalls Qwen2.5-72Bemail-Eu-core-main-n125-s0Heuristic-proposal selectionTruecomplete1 Qwen2.5-72Bemail-Eu-core-main-n125-s0Direct constructionTruecomplete2 Qwen2.5-72Bemail-Eu-core-main-n150-s0Direct constructionTruecomplete2 Qwen3.5-27Bemail-Eu-core-main-n100-s1Heuristic-proposal selectionTruecomplete2 Qwen3.5-27Bemail-Eu-core-main-n150-s1Heuristic-proposal selectionTruecomplete1 Qwen3.5-27Bemail-Eu-core-main-n125-s1Heuristic-proposal selectionTruecomplete2 Qwen3.5-27Bemail-Eu-core-main-n100-s1Direct constructionTruecomplete2 Qwen3.5-27Bemail-Eu-core-main-n150-s1Direct constructionTruecomplete2 Qwen3.5-27Bemail-Eu-core-main-n125-s1Direct constructionTruecomplete2 Qwen3.5-27Bemail-Eu-core-main-n150-s0Heuristic-proposal selectionTruecomplete1 Qwen3.5-27Bemail-Eu-core-main-n125-s0Heuristic-proposal selectionTruecomplete1 Qwen3.5-27Bemail-Eu-core-main-n125-s0Direct constructionTruecomplete2 Qwen3.5-27Bemail-Eu-core-main-n150-s0Direct constructionTruecomplete2 Qwen3.8-27Bemail-Eu-core-main-n100-s1Heuristic-proposal selectionTruecomplete2 Qwen3.8-27Bemail-Eu-core-main-n150-s1Heuristic-proposal selectionTruecomplete1 Qwen3.8-27Bemail-Eu-core-main-n125-s1Heuristic-proposal selectionTruecomplete1 Qwen3.8-27Bemail-Eu-core-main-n100-s1Direct constructionTruecomplete2 Qwen3.8-27Bemail-Eu-core-main-n150-s1Direct constructionTruecomplete2 Qwen3.8-27Bemail-Eu-core-main-n125-s1Direct constructionTruecomplete2 Qwen3.8-27Bemail-Eu-core-main-n150-s0Heuristic-proposal selectionTruecomplete1 Qwen3.8-27Bemail-Eu-core-main-n125-s0Heuristic-proposal selectionTruecomplete1 Qwen3.8-27Bemail-Eu-core-main-n125-s0Direct constructionTruecomplete2 Qwen3.8-27Bemail-Eu-core-main-n150-s0Direct constructionTruecomplete2 Appendix C. Complete RQ1 Results All fourteen configurations are retained. Task-size macro gives each task and each of its five sizes equal weight. Task macro pools queries within a task before averaging the six tasks. Query-weighted accuracy gives degree half of the total weight. Invalid choices count as incorrect. GPT-5.4 uses the no-reasoning configuration throughout the primary results: 733 correct and 800 legal answers. Its degree task has 364/400 correct, but equal averaging across sizes gives 91.4154%, rather than the pooled 91.0000%. GraphDecide 22 Table C.1. Aggregate accuracy (%) and response coverage. ConfigurationCorrect / 800Query weighted Task macroTask-size macroValidFailedU Jev-1.13.060976.125082.375082.550080000 Decider-2B28936.125053.708353.736780000 Kev-4B34443.000065.000065.033080000 Laya24230.250043.583343.605680000 Q3.5-0.8B25031.250044.083344.098480000 Q3.5-2B25832.250044.083344.124980000 Q3.5-4B gen.40650.750058.250058.399880000 Q3.5-9B40650.750061.916762.040180000 Q3.5-27B59073.750081.250081.367080000 Q3.8-27B53366.625074.375074.485380000 Q2.5-72B60876.000082.666782.847280000 Q3.5-4B score33742.125058.708358.754580000 GPT-5.473391.625092.041792.110980000 GPT-6-Astra800100.0000100.0000100.000080000 Ranked down each correctness/accuracy column; response-accounting columns are not ranked. C.2. All four sources Table C.2. All four sources: task-size averaged accuracy (%). ConfigurationAdjacencyDegreeCycleConnectivityDistanceArticulation Jev-1.13.0100.000067.799987.500088.750090.000061.2500 Decider-2B72.50009.920578.750050.000061.250050.0000 Kev-4B88.750010.198285.000060.000083.750062.5000 Laya50.000010.383850.000051.250050.000050.0000 Q3.5-0.8B50.000012.090352.500050.000050.000050.0000 Q3.5-2B50.000014.749450.000050.000050.000050.0000 Q3.5-4B gen.78.750040.399080.000050.000050.000051.2500 Q3.5-9B57.500034.740581.250073.750075.000050.0000 Q3.5-27B100.000063.202087.500087.500080.000070.0000 Q3.8-27B96.250055.661685.000086.250072.500051.2500 Q2.5-72B98.750067.083378.750088.750091.250072.5000 Q3.5-4B score90.000017.527178.750050.000061.250055.0000 GPT-5.4100.000091.415492.500096.250097.500075.0000 GPT-6-Astra100.0000100.0000100.0000100.0000100.0000100.0000 Ranked down each task column within the same source panel; higher accuracy is better. GraphDecide 23 C.3. Facebook Table C.3. Facebook: task-size averaged accuracy (%). ConfigurationAdjacencyDegreeCycleConnectivityDistanceArticulation Jev-1.13.0100.000062.9773100.000090.000090.000060.0000 Decider-2B70.000013.686995.000050.000060.000050.0000 Kev-4B85.000014.7980100.000050.000080.000065.0000 Laya50.000014.596050.000055.000050.000050.0000 Q3.5-0.8B50.000016.901565.000050.000050.000050.0000 Q3.5-2B50.000015.219750.000050.000050.000050.0000 Q3.5-4B gen.75.000034.719795.000050.000050.000050.0000 Q3.5-9B55.000031.9066100.000070.000085.000050.0000 Q3.5-27B100.000067.298095.000095.000080.000065.0000 Q3.8-27B95.000056.7854100.000090.000075.000050.0000 Q2.5-72B100.000066.540495.000095.000085.000060.0000 Q3.5-4B score90.000019.409195.000050.000055.000055.0000 GPT-5.4100.000090.2727100.0000100.000095.000070.0000 GPT-6-Astra100.0000100.0000100.0000100.0000100.0000100.0000 Ranked down each task column within the same source panel; higher accuracy is better. C.4. ca-GrQc Table C.4. ca-GrQc: task-size averaged accuracy (%). ConfigurationAdjacencyDegreeCycleConnectivityDistanceArticulation Jev-1.13.0100.000073.204595.000095.0000100.000060.0000 Decider-2B80.000012.088485.000050.000065.000050.0000 Kev-4B100.000012.088490.000065.000075.000065.0000 Laya50.000012.164150.000050.000050.000050.0000 Q3.5-0.8B50.000013.227345.000050.000050.000050.0000 Q3.5-2B50.000014.449550.000050.000050.000050.0000 Q3.5-4B gen.75.000044.714685.000050.000050.000055.0000 Q3.5-9B55.000044.376380.000080.000070.000050.0000 Q3.5-27B100.000064.414185.000085.000090.000080.0000 Q3.8-27B100.000052.590990.000085.000085.000050.0000 Q2.5-72B100.000070.628885.0000100.000095.000080.0000 Q3.5-4B score100.000015.386485.000050.000065.000050.0000 GPT-5.4100.000087.1616100.0000100.000095.000085.0000 GPT-6-Astra100.0000100.0000100.0000100.0000100.0000100.0000 Ranked down each task column within the same source panel; higher accuracy is better. GraphDecide 24 C.5. Power Grid Table C.5. Power Grid: task-size averaged accuracy (%). ConfigurationAdjacencyDegreeCycleConnectivityDistanceArticulation Jev-1.13.0100.000063.853565.000080.000085.000065.0000 Decider-2B80.00005.353555.000050.000055.000050.0000 Kev-4B95.00005.353560.000060.000085.000065.0000 Laya50.00007.298050.000050.000050.000050.0000 Q3.5-0.8B50.00009.096050.000050.000050.000050.0000 Q3.5-2B50.000015.358650.000050.000050.000050.0000 Q3.5-4B gen.75.000042.027860.000050.000050.000050.0000 Q3.5-9B55.000029.636465.000070.000060.000050.0000 Q3.5-27B100.000056.833380.000090.000085.000065.0000 Q3.8-27B95.000052.563165.000090.000070.000055.0000 Q2.5-72B100.000065.532855.000075.000090.000075.0000 Q3.5-4B score90.000018.073255.000050.000050.000060.0000 GPT-5.4100.000096.348585.000090.0000100.000075.0000 GPT-6-Astra100.0000100.0000100.0000100.0000100.0000100.0000 Ranked down each task column within the same source panel; higher accuracy is better. C.6. Human PPI Table C.6. Human PPI: task-size averaged accuracy (%). ConfigurationAdjacencyDegreeCycleConnectivityDistanceArticulation Jev-1.13.0100.000071.164190.000090.000085.000060.0000 Decider-2B60.00008.553080.000050.000065.000050.0000 Kev-4B75.00008.553090.000065.000095.000055.0000 Laya50.00007.477350.000050.000050.000050.0000 Q3.5-0.8B50.00009.136450.000050.000050.000050.0000 Q3.5-2B50.000013.969750.000050.000050.000050.0000 Q3.5-4B gen.90.000040.133880.000050.000050.000050.0000 Q3.5-9B65.000033.042980.000075.000085.000050.0000 Q3.5-27B100.000064.262690.000080.000065.000070.0000 Q3.8-27B95.000060.707185.000080.000060.000050.0000 Q2.5-72B95.000065.631380.000085.000095.000075.0000 Q3.5-4B score80.000017.239980.000050.000075.000055.0000 GPT-5.4100.000091.878885.000095.0000100.000070.0000 GPT-6-Astra100.0000100.0000100.0000100.0000100.0000100.0000 Ranked down each task column within the same source panel; higher accuracy is better. GraphDecide 25 C.7. GPT-5.4 Source and Size Counts Table C.7. GPT-5.4: correct/scheduled queries at each source and size. SourcenAdjacencyDegreeCycleConnectivityDistanceArticulation Facebook84/414/164/44/44/44/4 Facebook94/418/184/44/44/43/4 Facebook104/418/204/44/44/42/4 Facebook114/419/224/44/43/42/4 Facebook124/421/244/44/44/43/4 ca-GrQc84/414/164/44/44/42/4 ca-GrQc94/417/184/44/44/44/4 ca-GrQc104/416/204/44/44/43/4 ca-GrQc114/419/224/44/43/44/4 ca-GrQc124/421/244/44/44/44/4 Power Grid84/416/164/44/44/44/4 Power Grid94/418/184/44/44/44/4 Power Grid104/419/203/43/44/43/4 Power Grid114/420/223/44/44/42/4 Power Grid124/423/243/43/44/42/4 Human PPI84/416/164/44/44/43/4 Human PPI94/418/183/44/44/43/4 Human PPI104/419/204/44/44/43/4 Human PPI114/416/223/44/44/42/4 Human PPI124/422/243/43/44/43/4 Appendix D. Complete RQ2 Results Each primary panel contains 100 targets or queries and five matched conditions. U denotes unsupported, not zero accuracy. ogbn-arxiv macro-F1 averages over the fixed forty classes. Paired intervals are pointwise 95% target-bootstrap intervals. Coverage cells give valid/failed/unsupported out of 100. GraphDecide 26 D.1. ogbn-arxiv accuracy (%) Table D.1. ogbn-arxiv accuracy (%). ConfigurationTGTGBAGA Jev-1.13.055.000061.000058.000057.000051.0000 Decider-2B35.000045.000042.000038.000024.0000 Kev-4B29.000061.000027.000027.000041.0000 LayaUUUUU Q3.5-0.8B32.000034.000026.000024.000033.0000 Q3.5-2B25.000012.000019.000023.000015.0000 Q3.5-4B gen.21.000059.000030.000019.000042.0000 Q3.5-9B32.000055.000041.000036.000039.0000 Q3.5-27B46.000061.000058.000056.000049.0000 Q3.8-27B62.000060.000071.000070.000051.0000 Q2.5-72B51.000058.000059.000052.000045.0000 Q3.5-4B scoreUUUUU GPT-5.473.000061.000073.000072.000048.0000 GPT-6-Astra73.000060.000076.000073.000055.0000 Ranked down each condition column on this dataset's fixed targets; higher is better. D.2. ogbn-arxiv macro-F1 (%) Table D.2. ogbn-arxiv macro-F1 (%). ConfigurationTGTGBAGA Jev-1.13.022.163225.823024.808024.544520.1305 Decider-2B12.933817.627717.197813.37725.8756 Kev-4B5.903125.40736.69616.579810.0644 LayaUUUUU Q3.5-0.8B7.77868.02596.69677.23349.0932 Q3.5-2B5.25783.71645.23964.49653.0843 Q3.5-4B gen.6.517021.480311.05207.738614.1785 Q3.5-9B13.057023.681817.340712.019513.5623 Q3.5-27B18.641025.732623.704120.619318.5902 Q3.8-27B22.251225.938129.789329.910819.4896 Q2.5-72B20.994024.389525.573621.758419.5813 Q3.5-4B scoreUUUUU GPT-5.435.834625.885930.303828.969913.8617 GPT-6-Astra33.157224.968436.420935.493522.6972 Ranked down each condition column on this dataset's fixed targets; higher is better. GraphDecide 27 D.3. ogbn-arxiv paired differences (p), 95% intervals Table D.3. ogbn-arxiv paired differences (p), 95% intervals. ConfigurationTG-BAGG-ATG-TTG-G Jev-1.13.0+1 [-4,+7] +10 [+3,+18] +3 [-5,+10] -3 [-10,+4] Decider-2B+4 [+1,+8] +21 [+12,+30] +7 [+1,+13] -3 [-14,+8] Kev-4B+0 [-3,+3] +20 [+12,+29] -2 [-10,+5] -34 [-44,-24] LayaUUUU Q3.5-0.8B+2 [-3,+8] +1 [-5,+7] -6 [-13,+0] -8 [-14,-2] Q3.5-2B-4 [-10,+1] -3 [-8,+1] -6 [-15,+2] +7 [+0,+14] Q3.5-4B gen.+11 [+6,+18] +17 [+9,+25] +9 [+1,+17] -29 [-39,-20] Q3.5-9B+5 [-1,+11] +16 [+9,+23] +9 [+2,+17] -14 [-23,-5] Q3.5-27B+2 [-3,+8] +12 [+5,+20] +12 [+4,+21] -3 [-10,+5] Q3.8-27B+1 [-3,+6] +9 [+3,+16] +9 [+2,+16] +11 [+5,+18] Q2.5-72B+7 [+2,+12] +13 [+7,+20] +8 [+3,+13] +1 [-5,+7] Q3.5-4B scoreUUUU GPT-5.4+1 [-2, +4] +13 [+3, +22] +0 [-6, +5] +12 [+4, +20] GPT-6-Astra+3 [+0,+7] +5 [-2,+12] +3 [-1,+7] +16 [+9,+23] GraphDecide 28 D.4. ogbn-arxiv coverage Table D.4. ogbn-arxiv coverage. ConfigurationTGTGBAGA Jev-1.13.0100/0/0100/0/0100/0/0100/0/0100/0/0 Decider-2B100/0/0100/0/0100/0/0100/0/0100/0/0 Kev-4B100/0/0100/0/0100/0/0100/0/0100/0/0 Laya0/0/1000/0/1000/0/1000/0/1000/0/100 Q3.5-0.8B100/0/0100/0/0100/0/0100/0/0100/0/0 Q3.5-2B100/0/0100/0/0100/0/0100/0/0100/0/0 Q3.5-4B gen.100/0/0100/0/0100/0/0100/0/0100/0/0 Q3.5-9B100/0/0100/0/0100/0/0100/0/0100/0/0 Q3.5-27B100/0/0100/0/0100/0/0100/0/0100/0/0 Q3.8-27B100/0/0100/0/0100/0/0100/0/0100/0/0 Q2.5-72B100/0/0100/0/0100/0/0100/0/0100/0/0 Q3.5-4B score0/0/1000/0/1000/0/1000/0/1000/0/100 GPT-5.4100/0/0100/0/0100/0/0100/0/0100/0/0 GPT-6-Astra100/0/0100/0/0100/0/0100/0/0100/0/0 D.5. STaRK-Prime Hit@1 (%) Table D.5. STaRK-Prime Hit@1 (%). ConfigurationTGTGBAGA* Jev-1.13.048.000047.000049.000046.000042.0000 Decider-2B20.000017.000018.000021.000020.0000 Kev-4B39.000019.000026.000035.000034.0000 LayaUUUUU Q3.5-0.8B9.00009.00009.00007.00008.0000 Q3.5-2B16.000010.000016.000012.000013.0000 Q3.5-4B gen.36.000017.000024.000028.000026.0000 Q3.5-9B43.000025.000031.000038.000033.0000 Q3.5-27B50.000044.000046.000043.000041.0000 Q3.8-27B53.000050.000051.000051.000048.0000 Q2.5-72B39.000035.000033.000035.000036.0000 Q3.5-4B scoreUUUUU GPT-5.460.000051.000053.000056.000056.0000 GPT-6-Astra69.000069.000067.000073.000071.0000 Ranked down each condition column on this dataset's fixed targets; higher is better. GraphDecide 29 D.6. STaRK-Prime paired differences (p), 95% intervals Table D.6. STaRK-Prime paired differences (p), 95% intervals. ConfigurationTG-BAGG-A*TG-TTG-G Jev-1.13.0+3 [-6,+12] +5 [-5,+14] +1 [-8,+10] +2 [-3,+8] Decider-2B-3 [-10,+4] -3 [-13,+7] -2 [-9,+4] +1 [-5,+7] Kev-4B-9 [-17,-2] -15 [-24,-6] -13 [-22,-4] +7 [+0,+15] LayaUUUU Q3.5-0.8B+2 [-2,+6] +1 [-4,+6] +0 [-5,+5] +0 [-4,+4] Q3.5-2B+4 [+0,+9] -3 [-11,+4] +0 [-5,+6] +6 [+0,+13] Q3.5-4B gen.-4 [-11,+3] -9 [-16,-1] -12 [-21,-4] +7 [+1,+13] Q3.5-9B-7 [-16,+0] -8 [-18,+1] -12 [-21,-4] +6 [-2,+15] Q3.5-27B+3 [-6,+12] +3 [-4,+10] -4 [-12,+4] +2 [-4,+8] Q3.8-27B+0 [-9,+8] +2 [-7,+12] -2 [-11,+7] +1 [-4,+7] Q2.5-72B-2 [-8,+4] -1 [-11,+9] -6 [-14,+2] -2 [-8,+5] Q3.5-4B scoreUUUU GPT-5.4-3 [-11, +4] -5 [-13, +3] -7 [-15, +1] +2 [-4, +8] GPT-6-Astra-6 [-13,+1] -2 [-9,+5] -2 [-10,+5] -2 [-6,+1] GraphDecide 30 D.7. STaRK-Prime coverage Table D.7. STaRK-Prime coverage. ConfigurationTGTGBAGA* Jev-1.13.0100/0/0100/0/0100/0/0100/0/0100/0/0 Decider-2B100/0/0100/0/0100/0/0100/0/0100/0/0 Kev-4B100/0/0100/0/0100/0/0100/0/0100/0/0 Laya0/0/1000/0/1000/0/1000/0/1000/0/100 Q3.5-0.8B100/0/0100/0/0100/0/0100/0/0100/0/0 Q3.5-2B100/0/0100/0/0100/0/0100/0/0100/0/0 Q3.5-4B gen.100/0/0100/0/0100/0/0100/0/0100/0/0 Q3.5-9B100/0/0100/0/0100/0/0100/0/0100/0/0 Q3.5-27B100/0/0100/0/0100/0/0100/0/0100/0/0 Q3.8-27B100/0/0100/0/0100/0/0100/0/0100/0/0 Q2.5-72B100/0/0100/0/0100/0/0100/0/0100/0/0 Q3.5-4B score0/0/1000/0/1000/0/1000/0/1000/0/100 GPT-5.4100/0/0100/0/098/2/0100/0/0100/0/0 GPT-6-Astra100/0/0100/0/0100/0/0100/0/0100/0/0 D.8. Independent 200-Target ogbn-arxiv Evaluation This separate public-data evaluation is not pooled with either 100-target primary panel. It provides supporting accuracy results on its own 200-target set; it does not add targets to the primary paired comparisons. Table D.8. Independent 200-target ogbn-arxiv accuracy (%). ConfigurationTGTGBAGA Decider-2B31.500044.000039.000035.000026.5000 GPT-5.4 (reasoning reference)74.000058.500075.500074.000049.0000 GPT-6-Astra74.500059.500077.500077.000051.0000 Jev-1.13.054.500059.000057.500054.500047.5000 Kev-4B28.500058.000028.000025.500037.0000 LayaUUUUU Q3.5-27B48.000058.000056.000053.000045.5000 Q3.5-4B gen.22.000056.000029.500021.000041.5000 Q2.5-72B53.500056.500059.000056.500043.0000 Q3.5-9B31.000054.000042.000038.500040.0000 Ranked down each condition column on this dataset's fixed targets; higher is better. GraphDecide 31 Appendix E. Complete RQ3 Results and Controls Every configuration has ten scheduled graphs per task and construction mode. Direct construction supplies all legal next actions; heuristic-proposal selection supplies deduplicated named fixed-rule proposals. Means condition on feasible completion. Paired change is heuristic-proposal selection minus direct construction on jointly completed graphs; it is not a subtraction of marginal means when completed subsets differ. Matched random and classical controls use each mode's completed subset. Calls are those of the single evaluated trajectory. U and F mean unsupported and failed; -- indicates no defined mean. E.1. TSP Table E.1. TSP: reference gaps (%), completion and paired change. ConfigurationDirect construction Gap (done/10) Heuristic-proposal selection Gap (done/10) Paired changePairs Jev-1.13.059.6641 (10/10)25.8731 (10/10)-33.791110 Decider-2B583.3950 (10/10)40.4541 (10/10)-542.940910 Kev-4B420.8638 (8/10)37.4815 (10/10)-384.22878 Laya-- (0/10)-- (0/10)--0 Q3.5-0.8B576.7172 (10/10)33.8624 (10/10)-542.854810 Q3.5-2B539.5417 (10/10)34.7646 (10/10)-504.777210 Q3.5-4B gen.242.7966 (10/10)38.4328 (10/10)-204.363810 Q3.5-9B252.0041 (10/10)24.0653 (10/10)-227.938810 Q3.5-27B50.4023 (10/10)27.2948 (10/10)-23.107510 Q3.8-27B51.4061 (10/10)28.8632 (10/10)-22.542910 Q2.5-72B116.9987 (10/10)29.2820 (10/10)-87.716710 Q3.5-4B score-- (0/10)-- (0/10)--0 GPT-5.429.4859 (10/10)32.2440 (10/10)2.758210 GPT-6-Astra2.2117 (10/10)6.2354 (10/10)4.023710 Paired change: heuristic-proposal selection - direct construction; lower is better. Percent-task changes are percentage points; modularity changes use Q units. Ranked down each mode/task column only among full-ten-graph results; partial subsets remain unranked. Paired changes are diagnostics, not absolute-quality rankings. GraphDecide 32 Table E.2. TSP, Direct construction: objectives and matched controls. ConfigurationObjectiveGapRandom gapClassical gapDone/U/FCalls Jev-1.13.043034.900059.664143.88505.753210/0/01257 Decider-2B171889.2000583.395043.88505.753210/0/01257 Kev-4B140365.2500420.863843.98125.06888/2/0838 Laya--------0/10/00 Q3.5-0.8B172152.4000576.717243.88505.753210/0/01257 Q3.5-2B167230.3000539.541743.88505.753210/0/01257 Q3.5-4B gen.86291.9000242.796643.88505.753210/0/01257 Q3.5-9B91894.1000252.004143.88505.753210/0/01257 Q3.5-27B38203.700050.402343.88505.753210/0/01257 Q3.8-27B38473.700051.406143.88505.753210/0/01257 Q2.5-72B51646.5000116.998743.88505.753210/0/01257 Q3.5-4B score--------0/10/00 GPT-5.434877.800029.485943.88505.753210/0/01257 GPT-6-Astra27176.60002.211743.88505.753210/0/01257 Calls include confirmed requests on incomplete or unsupported trajectories; forced singleton actions require none. Objectives: ranked down full-coverage model rows (TSP lower; other tasks higher). Gaps: ranked jointly across model/random/classical entries on the same ten graphs, lower is better. Each partial row instead ranks only its three matched gaps, never against other rows; partial objectives and calls are unranked. Table E.3. TSP, Heuristic-proposal selection: objectives and matched controls. ConfigurationObjectiveGapRandom gapClassical gapDone/U/FCalls Jev-1.13.032388.600025.873143.88505.753210/0/0655 Decider-2B38014.600040.454143.88505.753210/0/0681 Kev-4B37155.000037.481543.88505.753210/0/0695 Laya--------0/10/010 Q3.5-0.8B34821.600033.862443.88505.753210/0/0673 Q3.5-2B36700.300034.764643.88505.753210/0/0673 Q3.5-4B gen.36498.600038.432843.88505.753210/0/0686 Q3.5-9B31982.700024.065343.88505.753210/0/0658 Q3.5-27B32620.600027.294843.88505.753210/0/0655 Q3.8-27B34598.100028.863243.88505.753210/0/0680 Q2.5-72B33904.700029.282043.88505.753210/0/0647 Q3.5-4B score--------0/10/010 GPT-5.435026.200032.244043.88505.753210/0/0668 GPT-6-Astra28078.80006.235443.88505.753210/0/0547 Calls include confirmed requests on incomplete or unsupported trajectories; forced singleton actions require none. Objectives: ranked down full-coverage model rows (TSP lower; other tasks higher). Gaps: ranked jointly across model/random/classical entries on the same ten graphs, lower is better. Each partial row instead ranks only its three matched gaps, never against other rows; partial objectives and calls are unranked. GraphDecide 33 Table E.4. TSP: full-ten-graph control gaps (%). ControlMean gapGraphs classical5.753210 isolation_priority29.308110 nearest25.858610 one_step_lookahead52.232710 random_candidate43.885010 return_reserve15.580510 Control gaps are ranked on the same full ten-graph cohort; lower is better. E.2. Signed MaxCut Table E.5. Signed MaxCut: reference gaps (%), completion and paired change. ConfigurationDirect construction Gap (done/10) Heuristic-proposal selection Gap (done/10) Paired changePairs Jev-1.13.099.9614 (10/10)9.0517 (10/10)-90.909710 Decider-2B99.8284 (10/10)19.0108 (10/10)-80.817610 Kev-4B99.8284 (10/10)21.9226 (10/10)-77.905810 Laya-- (0/10)-- (0/10)--0 Q3.5-0.8B99.8054 (10/10)70.6073 (10/10)-29.198110 Q3.5-2B99.8284 (10/10)40.1330 (10/10)-59.695410 Q3.5-4B gen.99.8284 (10/10)12.3026 (10/10)-87.525910 Q3.5-9B99.3003 (10/10)19.9961 (10/10)-79.304110 Q3.5-27B104.0532 (10/10)10.4834 (10/10)-93.569810 Q3.8-27B99.8284 (10/10)23.1526 (10/10)-76.675810 Q2.5-72B105.9990 (10/10)11.5578 (10/10)-94.441110 Q3.5-4B score-- (0/10)-- (0/10)--0 GPT-5.499.6673 (10/10)9.4116 (10/10)-90.255710 GPT-6-Astra6.3073 (10/10)7.7818 (10/10)1.474410 Paired change: heuristic-proposal selection - direct construction; lower is better. Percent-task changes are percentage points; modularity changes use Q units. Ranked down each mode/task column only among full-ten-graph results; partial subsets remain unranked. Paired changes are diagnostics, not absolute-quality rankings. GraphDecide 34 Table E.6. Signed MaxCut, Direct construction: objectives and matched controls. ConfigurationObjectiveGapRandom gapClassical gapDone/U/FCalls Jev-1.13.00.000099.961482.720615.409610/0/01240 Decider-2B0.200099.828482.720615.409610/0/01240 Kev-4B0.200099.828482.720615.409610/0/01240 Laya--------0/10/010 Q3.5-0.8B0.200099.805482.720615.409610/0/01240 Q3.5-2B0.200099.828482.720615.409610/0/01240 Q3.5-4B gen.0.200099.828482.720615.409610/0/01240 Q3.5-9B0.800099.300382.720615.409610/0/01240 Q3.5-27B-4.6000104.053282.720615.409610/0/01240 Q3.8-27B0.200099.828482.720615.409610/0/01240 Q2.5-72B-6.8000105.999082.720615.409610/0/01240 Q3.5-4B score--------0/10/010 GPT-5.40.400099.667382.720615.409610/0/01240 GPT-6-Astra103.60006.307382.720615.409610/0/01240 Calls include confirmed requests on incomplete or unsupported trajectories; forced singleton actions require none. Objectives: ranked down full-coverage model rows (TSP lower; other tasks higher). Gaps: ranked jointly across model/random/classical entries on the same ten graphs, lower is better. Each partial row instead ranks only its three matched gaps, never against other rows; partial objectives and calls are unranked. Table E.7. Signed MaxCut, Heuristic-proposal selection: objectives and matched controls. ConfigurationObjectiveGapRandom gapClassical gapDone/U/FCalls Jev-1.13.0100.60009.051782.720615.409610/0/0955 Decider-2B89.600019.010882.720615.409610/0/01016 Kev-4B86.400021.922682.720615.409610/0/0946 Laya--------0/10/010 Q3.5-0.8B32.600070.607382.720615.409610/0/01023 Q3.5-2B66.400040.133082.720615.409610/0/01025 Q3.5-4B gen.97.000012.302682.720615.409610/0/0989 Q3.5-9B88.600019.996182.720615.409610/0/01005 Q3.5-27B99.000010.483482.720615.409610/0/0982 Q3.8-27B85.000023.152682.720615.409610/0/0992 Q2.5-72B97.800011.557882.720615.409610/0/01016 Q3.5-4B score--------0/10/010 GPT-5.4100.20009.411682.720615.409610/0/0987 GPT-6-Astra102.00007.781882.720615.409610/0/0945 Calls include confirmed requests on incomplete or unsupported trajectories; forced singleton actions require none. Objectives: ranked down full-coverage model rows (TSP lower; other tasks higher). Gaps: ranked jointly across model/random/classical entries on the same ten graphs, lower is better. Each partial row instead ranks only its three matched gaps, never against other rows; partial objectives and calls are unranked. GraphDecide 35 Table E.8. Signed MaxCut: full-ten-graph control gaps (%). ControlMean gapGraphs assigned_gain19.731610 classical15.409610 degree_balance101.292510 greedy_completion8.872710 opposite_completion79.074510 random_candidate82.720610 Control gaps are ranked on the same full ten-graph cohort; lower is better. E.3. Deterministic LT Table E.9. Deterministic LT: reference gaps (%), completion and paired change. ConfigurationDirect construction Gap (done/10) Heuristic-proposal selection Gap (done/10) Paired changePairs Jev-1.13.052.8359 (8/10)0.5000 (8/10)-52.33598 Decider-2B75.4013 (10/10)9.8681 (10/10)-65.533210 Kev-4B87.2393 (3/10)4.4118 (4/10)-81.35703 Laya-- (0/10)0.0000 (1/10)--0 Q3.5-0.8B84.1737 (10/10)16.3237 (10/10)-67.849910 Q3.5-2B77.6601 (10/10)22.4095 (10/10)-55.250610 Q3.5-4B gen.79.6852 (10/10)17.4011 (10/10)-62.284110 Q3.5-9B47.1873 (10/10)15.2482 (10/10)-31.939110 Q3.5-27B46.5092 (10/10)5.6717 (10/10)-40.837510 Q3.8-27B43.1821 (10/10)15.9568 (10/10)-27.225310 Q2.5-72B39.9037 (10/10)18.0717 (10/10)-21.832010 Q3.5-4B score-- (0/10)0.0000 (1/10)--0 GPT-5.433.4312 (10/10)5.4750 (10/10)-27.956210 GPT-6-Astra0.0000 (10/10)0.4000 (10/10)0.400010 Paired change: heuristic-proposal selection - direct construction; lower is better. Percent-task changes are percentage points; modularity changes use Q units. Ranked down each mode/task column only among full-ten-graph results; partial subsets remain unranked. Paired changes are diagnostics, not absolute-quality rankings. GraphDecide 36 Table E.10. Deterministic LT, Direct construction: objectives and matched controls. ConfigurationObjectiveGapRandom gapClassical gapDone/U/FCalls Jev-1.13.011.375052.835921.91236.84388/0/218 Decider-2B4.800075.401321.24755.475010/0/020 Kev-4B3.000087.239313.94960.00003/7/013 Laya--------0/10/00 Q3.5-0.8B3.400084.173721.24755.475010/0/020 Q3.5-2B4.400077.660121.24755.475010/0/020 Q3.5-4B gen.4.400079.685221.24755.475010/0/020 Q3.5-9B14.700047.187321.24755.475010/0/020 Q3.5-27B14.600046.509221.24755.475010/0/020 Q3.8-27B15.200043.182121.24755.475010/0/020 Q2.5-72B15.600039.903721.24755.475010/0/020 Q3.5-4B score--------0/10/00 GPT-5.417.800033.431221.24755.475010/0/020 GPT-6-Astra24.60000.000021.24755.475010/0/020 Calls include confirmed requests on incomplete or unsupported trajectories; forced singleton actions require none. Objectives: ranked down full-coverage model rows (TSP lower; other tasks higher). Gaps: ranked jointly across model/random/classical entries on the same ten graphs, lower is better. Each partial row instead ranks only its three matched gaps, never against other rows; partial objectives and calls are unranked. Table E.11. Deterministic LT, Heuristic-proposal selection: objectives and matched controls. ConfigurationObjectiveGapRandom gapClassical gapDone/U/FCalls Jev-1.13.026.62500.500021.91236.84388/0/213 Decider-2B23.00009.868121.24755.475010/0/013 Kev-4B30.00004.411810.46220.00004/6/011 Laya29.00000.00000.00000.00001/9/09 Q3.5-0.8B21.300016.323721.24755.475010/0/012 Q3.5-2B19.700022.409521.24755.475010/0/013 Q3.5-4B gen.21.500017.401121.24755.475010/0/014 Q3.5-9B21.400015.248221.24755.475010/0/014 Q3.5-27B23.70005.671721.24755.475010/0/014 Q3.8-27B21.800015.956821.24755.475010/0/013 Q2.5-72B20.600018.071721.24755.475010/0/014 Q3.5-4B score29.00000.00000.00000.00001/9/09 GPT-5.423.40005.475021.24755.475010/0/012 GPT-6-Astra24.50000.400021.24755.475010/0/013 Calls include confirmed requests on incomplete or unsupported trajectories; forced singleton actions require none. Objectives: ranked down full-coverage model rows (TSP lower; other tasks higher). Gaps: ranked jointly across model/random/classical entries on the same ten graphs, lower is better. Each partial row instead ranks only its three matched gaps, never against other rows; partial objectives and calls are unranked. GraphDecide 37 Table E.12. Deterministic LT: full-ten-graph control gaps (%). ControlMean gapGraphs classical5.475010 marginal_cascade5.475010 outdegree30.693310 random_candidate21.247510 threshold_deficit20.248210 two_hop_reach31.740810 Control gaps are ranked on the same full ten-graph cohort; lower is better. E.4. Binary modularity Table E.13. Binary modularity: reference gaps (Q), completion and paired change. ConfigurationDirect construction Gap (done/10) Heuristic-proposal selection Gap (done/10) Paired changePairs Jev-1.13.00.4783832 (10/10)0.0555132 (10/10)-0.422870010 Decider-2B0.4783832 (10/10)0.1465732 (10/10)-0.331810010 Kev-4B0.4783832 (10/10)0.0964632 (10/10)-0.381920010 Laya0.5105148 (3/10)-- (0/10)--0 Q3.5-0.8B0.4783832 (10/10)0.1936732 (10/10)-0.284710010 Q3.5-2B0.4783832 (10/10)0.3656432 (10/10)-0.112740010 Q3.5-4B gen.0.4654832 (10/10)0.0885532 (10/10)-0.376930010 Q3.5-9B0.4783832 (10/10)0.1158132 (10/10)-0.362570010 Q3.5-27B0.4783832 (10/10)0.0892932 (10/10)-0.389090010 Q3.8-27B0.4752832 (10/10)0.1086932 (10/10)-0.366590010 Q2.5-72B0.3758732 (10/10)0.0849032 (10/10)-0.290970010 Q3.5-4B score0.4804942 (4/10)0.1562265 (1/10)-0.32550001 GPT-5.40.4042398 (10/10)0.0422862 (10/10)-0.361953710 GPT-6-Astra0.0002611 (10/10)0.0129828 (10/10)0.012721710 Paired change: heuristic-proposal selection - direct construction; lower is better. Percent-task changes are percentage points; modularity changes use Q units. Ranked down each mode/task column only among full-ten-graph results; partial subsets remain unranked. Paired changes are diagnostics, not absolute-quality rankings. GraphDecide 38 Table E.14. Binary modularity, Direct construction: objectives and matched controls. ConfigurationObjectiveGapRandom gapClassical gapDone/U/FCalls Jev-1.13.00.00000000.47838320.41750.478410/0/0990 Decider-2B0.00000000.47838320.41750.478410/0/0990 Kev-4B0.00000000.47838320.41750.478410/0/0990 Laya-0.02660000.51051480.41020.48393/7/0186 Q3.5-0.8B0.00000000.47838320.41750.478410/0/0990 Q3.5-2B0.00000000.47838320.41750.478410/0/0990 Q3.5-4B gen.0.01290000.46548320.41750.478410/0/0990 Q3.5-9B0.00000000.47838320.41750.478410/0/0990 Q3.5-27B0.00000000.47838320.41750.478410/0/0990 Q3.8-27B0.00310000.47528320.41750.478410/0/0990 Q2.5-72B0.10250000.37587320.41750.478410/0/0990 Q3.5-4B score0.00000000.48049420.41110.48054/6/0265 GPT-5.40.07414340.40423980.41750.478410/0/0990 GPT-6-Astra0.47810000.00026110.41750.478410/0/0990 Calls include confirmed requests on incomplete or unsupported trajectories; forced singleton actions require none. Objectives: ranked down full-coverage model rows (TSP lower; other tasks higher). Gaps: ranked jointly across model/random/classical entries on the same ten graphs, lower is better. Each partial row instead ranks only its three matched gaps, never against other rows; partial objectives and calls are unranked. Table E.15. Binary modularity, Heuristic-proposal selection: objectives and matched controls. ConfigurationObjectiveGapRandom gapClassical gapDone/U/FCalls Jev-1.13.00.42290000.05551320.41750.478410/0/0737 Decider-2B0.33180000.14657320.41750.478410/0/0729 Kev-4B0.38190000.09646320.41750.478410/0/0729 Laya--------0/10/010 Q3.5-0.8B0.28470000.19367320.41750.478410/0/0778 Q3.5-2B0.11270000.36564320.41750.478410/0/0750 Q3.5-4B gen.0.38980000.08855320.41750.478410/0/0727 Q3.5-9B0.36260000.11581320.41750.478410/0/0769 Q3.5-27B0.38910000.08929320.41750.478410/0/0706 Q3.8-27B0.36970000.10869320.41750.478410/0/0745 Q2.5-72B0.39350000.08490320.41750.478410/0/0725 Q3.5-4B score0.32550000.15622650.43160.48171/9/048 GPT-5.40.43609700.04228620.41750.478410/0/0704 GPT-6-Astra0.46540000.01298280.41750.478410/0/0743 Calls include confirmed requests on incomplete or unsupported trajectories; forced singleton actions require none. Objectives: ranked down full-coverage model rows (TSP lower; other tasks higher). Gaps: ranked jointly across model/random/classical entries on the same ten graphs, lower is better. Each partial row instead ranks only its three matched gaps, never against other rows; partial objectives and calls are unranked. GraphDecide 39 Table E.16. Binary modularity: full-ten-graph control gaps (Q). ControlMean gapGraphs assigned_gain0.120210 classical0.478410 degree_balance0.508010 greedy_completion0.036110 opposite_completion0.406410 random_candidate0.417510 Control gaps are ranked on the same full ten-graph cohort; lower is better. Modularity uses the rational Q* references in Appendix H.1. GPT-5.4's per-graph full-precision objectives and Astra's supplied high-precision aggregate gaps avoid the four-decimal input rounding of the other configurations. For the latter, objective rounding contributes at most 0.00005 to a mode mean and 0.0001 to a paired change. Tiny signed per-graph residuals within that rounding range do not establish an improvement over Q*. E.5. Cross-Task Quality and Completion Table E.17. Direct construction: reference gaps and completion. ConfigurationTSP gap (%)MaxCut gap (%)LT gap (%)Modularity gap (Q) Jev-1.13.059.6641 (10/10)99.9614 (10/10)52.8359 (8/10)0.4783832 (10/10) Decider-2B583.3950 (10/10)99.8284 (10/10)75.4013 (10/10)0.4783832 (10/10) Kev-4B420.8638 (8/10)99.8284 (10/10)87.2393 (3/10)0.4783832 (10/10) Laya-- (0/10)-- (0/10)-- (0/10)0.5105148 (3/10) Q3.5-0.8B576.7172 (10/10)99.8054 (10/10)84.1737 (10/10)0.4783832 (10/10) Q3.5-2B539.5417 (10/10)99.8284 (10/10)77.6601 (10/10)0.4783832 (10/10) Q3.5-4B gen.242.7966 (10/10)99.8284 (10/10)79.6852 (10/10)0.4654832 (10/10) Q3.5-9B252.0041 (10/10)99.3003 (10/10)47.1873 (10/10)0.4783832 (10/10) Q3.5-27B50.4023 (10/10)104.0532 (10/10)46.5092 (10/10)0.4783832 (10/10) Q3.8-27B51.4061 (10/10)99.8284 (10/10)43.1821 (10/10)0.4752832 (10/10) Q2.5-72B116.9987 (10/10)105.9990 (10/10)39.9037 (10/10)0.3758732 (10/10) Q3.5-4B score-- (0/10)-- (0/10)-- (0/10)0.4804942 (4/10) GPT-5.429.4859 (10/10)99.6673 (10/10)33.4312 (10/10)0.4042398 (10/10) GPT-6-Astra2.2117 (10/10)6.3073 (10/10)0.0000 (10/10)0.0002611 (10/10) Cells show gap (completed/scheduled); all gaps are lower-is-better. -- denotes no feasible completion, not zero. Ranked down each mode/task column only among full-ten-graph results; partial subsets remain unranked. GraphDecide 40 Table E.18. Heuristic-proposal selection: reference gaps and completion. ConfigurationTSP gap (%)MaxCut gap (%)LT gap (%)Modularity gap (Q) Jev-1.13.025.8731 (10/10)9.0517 (10/10)0.5000 (8/10)0.0555132 (10/10) Decider-2B40.4541 (10/10)19.0108 (10/10)9.8681 (10/10)0.1465732 (10/10) Kev-4B37.4815 (10/10)21.9226 (10/10)4.4118 (4/10)0.0964632 (10/10) Laya-- (0/10)-- (0/10)0.0000 (1/10)-- (0/10) Q3.5-0.8B33.8624 (10/10)70.6073 (10/10)16.3237 (10/10)0.1936732 (10/10) Q3.5-2B34.7646 (10/10)40.1330 (10/10)22.4095 (10/10)0.3656432 (10/10) Q3.5-4B gen.38.4328 (10/10)12.3026 (10/10)17.4011 (10/10)0.0885532 (10/10) Q3.5-9B24.0653 (10/10)19.9961 (10/10)15.2482 (10/10)0.1158132 (10/10) Q3.5-27B27.2948 (10/10)10.4834 (10/10)5.6717 (10/10)0.0892932 (10/10) Q3.8-27B28.8632 (10/10)23.1526 (10/10)15.9568 (10/10)0.1086932 (10/10) Q2.5-72B29.2820 (10/10)11.5578 (10/10)18.0717 (10/10)0.0849032 (10/10) Q3.5-4B score-- (0/10)-- (0/10)0.0000 (1/10)0.1562265 (1/10) GPT-5.432.2440 (10/10)9.4116 (10/10)5.4750 (10/10)0.0422862 (10/10) GPT-6-Astra6.2354 (10/10)7.7818 (10/10)0.4000 (10/10)0.0129828 (10/10) Cells show gap (completed/scheduled); all gaps are lower-is-better. -- denotes no feasible completion, not zero. Ranked down each mode/task column only among full-ten-graph results; partial subsets remain unranked. Appendix F. Implementation and Supporting Graph-Text Studies F.1 GPU Candidate Scoring and Interface Settings The implementation specification distinguishes native selection from generation and candidate scoring. Decider uses a restricted-letter one-pass BF16 readout with temperature 1.164. Kev uses an FP32 pointer head on Qwen3.5-4B-Base with merged LoRA and temperature approximately 2.40605. Laya uses an FP32 encoder/choice head with at most twelve options and 1,024 tokens. Jev's serving precision is undisclosed. The evaluated Qwen candidate-scoring implementation runs on GPU and uses the conditional argmax over supplied candidate-token probabilities; its one generated token is counted but not used for the choice. At a matched task state, generation and scoring use the same task prompt and candidate options but differ in output readout; sequential trajectories may reach different states. F.2 Independent 200-Target ogbn-arxiv Protocol The independent 200-target ogbn-arxiv study retains its historical configurations and is reported separately from the primary 100-target evaluation. Its protocol, input lengths and uncertainty estimates are specified below. The ogbn-arxiv [8] snapshot contains 169,343 papers and 1,166,243 directed citation edges. Official temporal splits contain 90,941 training papers (through 2017), 29,799 validation papers (2018) and 48,603 test papers (2019 onward). Context is drawn from the full static graph under transductive evaluation. Seed 20261001 selects 200 targets without replacement from sorted official test IDs, independently of labels. Seed 20261002 selects 12 validation targets for interface-support checks, without performance-based prompt or model tuning. Titles and abstracts are joined to graph nodes by MAG paper ID. The text collection's 179,719 unique records cover all 1,612 selected distinct nodes. Unicode whitespace is collapsed, and each node is capped at 256 title characters and 1,600 abstract characters. All models receive the same resulting strings. For each target, select up to four training neighbors from the union of incoming and outgoing neighbors and exactly four training non-neighbors. Selection and presentation order use deterministic SHA-256 rankings with seed 20261001 and separate neighbor, control and display domains; ties use numeric node IDs. Neighbor shortages remain, yielding four to eight context nodes. All induced directed edges are retained. Context identifiers are anonymized, and original IDs and neighbor/control roles are hidden. Training-node class labels are shared across all conditions; held-out target labels are excluded. GraphDecide 41 All five conditions share training-label anchors. T adds target text, G edges, TG target/context text and edges, BAG target/context text without edges, and A anonymous anchors and labels only. The question asks for exactly one of forty class IDs; a directed pair [u,v] means u cites v. Omitted text or edges indicate unavailable information, not evidence of absence. Table F.1. Frozen class descriptions; inference follows numeric ID order. IDDescriptionIDDescription 0arxiv cs na10arxiv cs ai 1arxiv cs m11arxiv cs ma 2arxiv cs lo12arxiv cs gl 3arxiv cs cy13arxiv cs ne 4arxiv cs cr14arxiv cs sc 5arxiv cs dc15arxiv cs ar 6arxiv cs hc16arxiv cs cv 7arxiv cs ce17arxiv cs gr 8arxiv cs ni18arxiv cs et 9arxiv cs c19arxiv cs sy 20arxiv cs cg30arxiv cs cl 21arxiv cs oh31arxiv cs ir 22arxiv cs pl32arxiv cs ms 23arxiv cs se33arxiv cs fl 24arxiv cs lg34arxiv cs ds 25arxiv cs sd35arxiv cs os 26arxiv cs si36arxiv cs gt 27arxiv cs ro37arxiv cs db 28arxiv cs it38arxiv cs dl 29arxiv cs pf39arxiv cs dm Table F.2. Independent 200-target token limits and maximum observed input lengths. Configuration Maximum observed input Context limit Configured output value Reasoning mode / metadata Jev4564UndisclosedNative choiceN/A Decider-2B398732768Native choiceN/A Kev-4B38978192Native choiceN/A Qwen3.5-4B41491638416Thinking disabled Qwen3.5-9B41491638416Thinking disabled Qwen3.5-27B41491638416Thinking disabled Qwen2.5-72B-Instruct41541638416Standard instruct GPT-5.4 (reasoning reference)4104Undisclosed4096Medium GPT-6-Astra4104Undisclosed4096Medium For the GPT rows in Table F.2, Medium denotes observed response metadata rather than an explicitly submitted reasoning setting. The output value of 4,096 is a configured value, not a verified limit on total reasoning tokens. The nine supported configurations complete 60 validation decisions and 1,000 test decisions each: 540 validation and 9,000 test decisions overall. Laya is unsupported. Jev's 18 validation and 287 test probability vectors with nonunit GraphDecide 42 sums retain their otherwise legal selected answers. Validation is used for interface checks, not included in test accuracy. F.3 Supporting Macro-F1 and Paired Intervals Macro-F1 uses all 40 classes and is multiplied by 100; classes absent from both truth and prediction contribute zero. Accuracy changes are in percentage points, while macro-F1 changes are multiplied by 100. Both analyses use the same 200 target pairs and unadjusted 95% paired target-bootstrap intervals. Dashes denote unavailable results. The original paired analysis uses 1,000 target resamples, seed 20261001, and sorted zero-based endpoints 24 and 974 for the 95% interval. Repeated targets keep their paired conditions together. The 1,408 test-anchor slots contain 1,321 distinct training anchors, so these target-level intervals do not capture all network dependence. Table F.3. Independent 200-target fixed-forty-class macro-F1 (%). ConfigurationTGTGBAGA Jev27.832.932.529.520.0 Decider-2B12.727.118.415.69.8 Kev-4B7.630.211.210.811.2 Laya (unsupported)— Qwen3.5-4B9.425.716.013.715.2 Qwen3.5-9B15.527.220.215.415.7 Qwen3.5-27B24.832.530.327.319.2 Qwen2.5-72B-Instruct28.631.132.331.619.8 GPT-5.4 (reasoning reference)46.533.246.646.219.6 GPT-6-Astra44.031.947.446.722.7 Ranked down each condition column on this dataset's fixed targets; higher is better. Table F.4. Independent 200-target paired accuracy changes (p), with 95% intervals. ConfigurationTG-TTG-GTG-BAGG-A Jev+3.0 [-1.5,+7.5]-1.5 [-6.0,+3.0]+3.0 [-1.0,+7.0]+11.5 [+6.0,+17.0] Decider-2B+7.5 [+2.5,+12.5]-5.0 [-12.0,+1.5]+4.0 [+1.5,+7.0]+17.5 [+11.0,+23.5] Kev-4B-0.5 [-5.0,+4.0]-30.0 [-37.0,-23.0]+2.5 [+0.0,+5.5]+21.0 [+15.5,+27.5] Qwen3.5-4B+7.5 [+1.5,+13.5]-26.5 [-33.5,-19.5]+8.5 [+4.5,+13.0]+14.5 [+9.0,+20.5] Qwen3.5-9B+11.0 [+5.5,+16.0]-12.0 [-18.5,-6.0]+3.5 [+0.0,+7.0]+14.0 [+8.5,+19.0] Qwen3.5-27B+8.0 [+1.5,+14.0]-2.0 [-7.5,+3.5]+3.0 [-0.5,+6.5]+12.5 [+7.5,+18.0] Qwen2.5-72B-Instruct+5.5 [+2.0,+9.5]+2.5 [-1.5,+6.5]+2.5 [-0.5,+6.0]+13.5 [+9.0,+18.5] GPT-5.4 (reasoning reference)+1.5 [-3.0,+5.5]+17.0 [+12.0,+23.0]+1.5 [-1.5,+4.5]+9.5 [+3.5,+16.0] GPT-6-Astra+3.0 [-1.0,+7.0]+18.0 [+12.5,+23.5]+0.5 [-2.0,+3.0]+8.5 [+3.0,+14.0] GraphDecide 43 Table F.5. Independent 200-target paired macro-F1 changes (×100), with 95% intervals. ConfigurationTG-TTG-GTG-BAGG-A Jev+4.6 [-1.2,+6.6]-0.4 [-3.9,+2.7]+2.9 [-0.7,+4.8]+12.9 [+5.8,+15.7] Decider-2B+5.8 [+0.7,+9.4]-8.7 [-12.4,-1.5]+2.8 [+0.2,+4.8]+17.3 [+7.5,+20.2] Kev-4B+3.6 [-0.4,+5.9]-18.9 [-22.3,-10.3]+0.5 [+0.1,+1.0]+19.0 [+10.6,+21.9] Qwen3.5-4B+6.6 [+1.8,+10.4]-9.7 [-14.1,-5.1]+2.3 [+0.0,+4.8]+10.5 [+5.3,+14.0] Qwen3.5-9B+4.7 [-1.1,+10.5]-7.0 [-11.6,-1.1]+4.9 [+0.9,+7.3]+11.5 [+4.9,+14.8] Qwen3.5-27B+5.5 [-0.3,+9.0]-2.2 [-5.9,+4.1]+3.0 [+0.2,+5.0]+13.3 [+6.4,+16.2] Qwen2.5-72B-Instruct+3.7 [-0.0,+7.5]+1.2 [-0.6,+3.3]+0.7 [-2.8,+3.9]+11.3 [+5.1,+13.7] GPT-5.4 (reasoning reference)+0.1 [-2.8,+3.2]+13.4 [+7.3,+16.7]+0.3 [-2.3,+2.7]+13.6 [+5.8,+16.4] GPT-6-Astra+3.5 [-0.4,+5.7]+15.5 [+8.7,+19.9]+0.7 [-1.4,+1.8]+9.2 [+3.1,+11.7] Jev's TG-BAG accuracy difference is +3.0 percentage points [-1.0, +7.0], leaving its explicit-edge benefit unresolved by this interval. Decider and Qwen3.5-4B have positive TG-BAG accuracy intervals; the other seven configurations, including Jev, touch or cross zero. Every G-A accuracy interval is positive, with point estimates of 8.5–21.0 percentage points. The positive G-A contrasts show the value of edges when only label anchors are otherwise available. GPT-5.4 (reasoning reference) and Astra achieve TG accuracies of 75.5% and 77.5%, respectively. Table F.6. Independent 1,000-test-call token usage. ConfigurationInput tokensOutput tokensReasoning tokens Jev2,093,713327,869— Decider-2B1,669,5480— Kev-4B1,579,5480— Qwen3.5-4B1,777,5492,8170 Qwen3.5-9B1,777,5492,8780 Qwen3.5-27B1,777,5492,8600 Qwen2.5-72B-Instruct1,774,7612,907— GPT-5.4 (reasoning reference)1,753,284205,790198,790 GPT-6-Astra1,753,28430,53224,600 Appendix G. Supporting Public Optimization Studies This appendix reports independent public optimization studies. Each panel specifies its instance set, conditions and completion denominator; its observations are not pooled with the primary evaluation. G.1 Three-Condition Public Protocol The supporting full-action and model-expansion studies share seven TSPLIB EUC_2D instances [22] and ten OPTSICOM signed spin-glass graphs [5, 17]. Each cut graph has 125 vertices, 375 edges and weights in -1,+1. Cut gaps use the attainable historical best-known values from the 2016 collection. Tour distances follow the TSPLIB nearest-integer EUC_2D convention, with reference 3,916 for tsp225. All coordinates and current-to-unvisited-city distances are supplied; n-2 model decisions are followed by forced closure. Signed cuts fix vertex zero in the first side and use n-1 decisions. Each model in this supporting study receives three relabeling/start conditions per graph, or 21 tours and 30 cuts. Controls include sampled uniform legal rollouts, nearest-neighbor construction with optional 2-opt, and incremental greedy cut construction with optional single-vertex-flip refinement. GraphDecide 44 G.2 Supporting Full-Action Results and Matched Subsets Table G.1. Supporting public graph-macro gaps (%) and completion. MethodTSP gapCompleteMaxCut gapComplete Jev (native action)66.0121/21104.2630/30 Qwen3.5-0.8B750.5321/21100.6430/30 Qwen3.5-2B715.9621/21100.2330/30 Qwen3.5-4B307.1121/21100.0530/30 Qwen3.5-9B381.8221/2197.3130/30 Uniform random772.6621/21100.2830/30 Nearest neighbor / incremental greedy27.6521/2129.2030/30 2-opt / single-flip search5.9321/2122.0230/30 Ranked separately by task over the full common graph-condition set, including listed controls; lower gap is better. Table G.2. Supporting public expansion; gaps (%) condition on completion. ConfigurationTSP completeTSP gapCut completeCut gap Decider-2B v1121/21728.4430/30100.23 Kev-4B15/21348.5930/30100.23 Laya0/21—0/30— Qwen3.5-27B21/2145.1330/30102.87 Qwen2.5-72B-Instruct21/21119.5630/30104.57 Ranked separately by task among complete common-cohort rows; Kev's partial TSP subset is unranked here (see Table G.3). Across the supporting full-action and model-expansion panels, 453/510 episodes complete; 57 capacity exclusions remain in coverage. Kev's six unsupported tour conditions are the kroB200/tsp225 requests exceeding 8,192 tokens; Laya rejects all 51 conditions before inference. Jev's 6,684 public reports contain 5,497 passing probability checks, 1,186 nonunit sums and one choice–argmax disagreement; every returned action remains legal. Table G.3. Matched five-graph tour subset; three conditions per graph. MethodSame conditionsTSP gap (%) Jev15/1558.75 Decider-2B v1115/15638.63 Kev-4B15/15348.59 Qwen3.5-27B15/1543.53 Qwen2.5-72B-Instruct15/1596.75 Sampled random15/15682.52 Nearest neighbor15/1527.08 N + 2-opt15/155.53 Ranked across all listed methods on the same 15 Kev-supported TSP conditions; lower gap is better. Table G.3 compares every method on the same five TSP graphs supported by Kev: kroA100, kroB100, kroC100, bier127 and kroA150, each under three conditions. kroB200 and tsp225 are excluded from every row. GraphDecide 45 Table G.4. Separate supporting GPT panel; one condition per graph. ConfigurationTaskCompleteIncompleteGap (%) GPT-5.4 (reasoning reference)TSP0/77Undefined GPT-5.4 (reasoning reference)MaxCut0/1010Undefined GPT-6-AstraTSP7/702.156480 GPT-6-AstraMaxCut4/1065.911212 These GPT results belong to the separate supporting panel. Undefined means no feasible completion. Their graph sets, one-condition-per-graph schedules and completion subsets are reported separately from the primary estimates. Table G.5. Supporting Jev paired losses/ties/wins against classical controls. TaskSampled uniformGreedy constructionLocal improvement TSP (21 conditions)0/0/2121/0/021/0/0 MaxCut (30 conditions)17/4/930/0/030/0/0 G.3 Seven-Tour Anonymous/Named Proposal Diagnostic This diagnostic evaluates Jev, Qwen3.5-4B and Qwen3.5-9B on seven public tours under three conditions. Direct construction reuses 63 full-action trajectories; Anonymous heuristic-proposal selection/Heuristic-proposal selection add anonymous or rule-semantic proposals as specified in Table G.6. The design was fixed after observing Direct construction and before new Anonymous heuristic-proposal selection/Heuristic-proposal selection inference. The separate TSP and MaxCut extension is reported in G.5. Qwen uses greedy no-thinking generation with 16 output tokens, and Jev uses native action selection. Inclusion requires at least 95% completion in each model/arm. The descriptive improvement criterion is a graph-macro gain of at least one percentage point and improvement on at least five of seven graphs; it is not a significance test. Of 126 new Anonymous heuristic-proposal selection/Heuristic-proposal selection episodes, 125 complete. One service failure leaves Jev Anonymous heuristic-proposal selection on kroB200, condition 1, incomplete. Jev Anonymous heuristic-proposal selection completes 20/21 episodes (95.24%); all other combinations complete 21/21. Four successful Jev calls have nonunit probability sums, two per arm; their legal actions remain included without probability normalization. Table G.6. Matched-history anonymous and named proposal inputs. ComponentAnonymous heuristic-proposal selection: anonymous proposals Heuristic-proposal selection: named proposals Underlying graph and historyUnchangedSame Distinct proposed citiesShared candidate setSame Numeric candidate features and option order SharedSame Rule names, formulas and rule-to-city mapping Not suppliedSupplied Cities proposed by multiple rules are merged into one option. Rule ties select the smallest relabeled city ID; distinct cities are shuffled deterministically using the seed, instance and selected-city history. A singleton proposal is appended without a model call. Subsequent Anonymous heuristic-proposal selection/Heuristic-proposal selection histories may diverge, so their comparison does not hold all later inputs fixed. GraphDecide 46 Table G.7. Seven-tour Direct construction/Anonymous heuristic-proposal selection/Heuristic-proposal selection graph-macro gaps (%). ModelModeGap (%)Complete JevDirect construction66.0121/21 JevAnonymous heuristic-proposal selection31.1220/21 JevHeuristic-proposal selection27.8021/21 Qwen3.5-4BDirect construction307.1121/21 Qwen3.5-4BAnonymous heuristic-proposal selection43.7021/21 Qwen3.5-4BHeuristic-proposal selection36.3621/21 Qwen3.5-9BDirect construction381.8221/21 Qwen3.5-9BAnonymous heuristic-proposal selection39.0921/21 Qwen3.5-9BHeuristic-proposal selection26.7621/21 Ranked among configurations within each mode on all 21 conditions; Jev's 20-condition anonymous mean remains unranked. Uniform distinct-proposal selection averages five frozen seeds (1101–1105) within condition, then conditions within graph; 105 trajectories are not 105 independent graphs. Its 43.18% gap already beats all three full-action Direct construction means, showing that external heuristics supply substantial Direct construction → Anonymous heuristic-proposal selection / Heuristic-proposal selection improvement. Every model arm remains worse than fixed return reserve (16.01%) and N+2-opt (5.93%). These controls are essential to interpreting the model's incremental contribution. Table G.8. All fixed, random and classical proposal controls. ReferenceGap (%)Complete Nearest rule27.6521/21 One-step lookahead56.8621/21 Return reserve16.0121/21 Isolation priority32.1621/21 Uniform distinct proposal (five seeds)43.18105/105 Shortest-edge proposal selector (= N)27.6521/21 Nearest neighbor (= nearest rule)27.6521/21 N + 2-opt5.9321/21 Ranked across control policies on the same seven tours and three conditions. Random selection averages five frozen seeds within each condition; 105 trajectories do not change the target cohort. GraphDecide 47 Table G.9. Paired gap changes (p), with graph and condition counts. ModelContrastChange (p)Improved graphs Pairs JevHeuristic-proposal selection-Anonymous heuristic-proposal selection -3.225/720/21 JevAnonymous heuristic-proposal selection-Direct construction-33.057/720/21 JevHeuristic-proposal selection-Direct construction-38.227/721/21 Qwen3.5-4BHeuristic-proposal selection-Anonymous heuristic-proposal selection -7.347/721/21 Qwen3.5-4BAnonymous heuristic-proposal selection-Direct construction-263.427/721/21 Qwen3.5-4BHeuristic-proposal selection-Direct construction-270.767/721/21 Qwen3.5-9BHeuristic-proposal selection-Anonymous heuristic-proposal selection -12.326/721/21 Qwen3.5-9BAnonymous heuristic-proposal selection-Direct construction-342.747/721/21 Qwen3.5-9BHeuristic-proposal selection-Direct construction-355.067/721/21 Jev's paired diagnostic Heuristic-proposal selection-Anonymous heuristic-proposal selection is -3.22 p over 20 pairs, not -3.32 p obtained by subtracting the marginal means. Named proposals improve mean paired quality for all three configurations, but semantic content, prompt length, future histories and candidate pools can diverge. The paired improvement therefore reflects the combined change in rule information and subsequent decisions. Approximately 44–48% of observed proposal steps have a single distinct candidate and therefore require no model call. G.4 Supporting Public Call and Token Accounting Token accounting is provider-specific: a native zero output-token count does not imply zero computation, and reasoning tokens may already be included in output totals. Unreported fields are shown as dashes. Kev has 5,421 successful forwards and six pre-forward rejections across the original public expansion; fifteen tour conditions are supported. Laya has zero successful forwards and 51 pre-forward rejections. Each of the other three expansion configurations has 6,684 successful forwards and no rejections. Table G.10. Supporting expansion forwards and rejections. ConfigurationForwardsRejects Decider-2B v116,6840 Kev-4B5,4216 Laya051 Qwen3.5-27B6,6840 Qwen2.5-72B-Instruct6,6840 G.5 Independent Three-Arm Public Extension In the historical extension tables, grammar denotes grammar-constrained generation and scores denotes candidate-token scoring; these labels preserve the evaluated interface configurations. This extension evaluates Direct construction/Anonymous heuristic-proposal selection/Heuristic-proposal selection on the original 51 public graph–condition pairs for each configuration. Direct construction supplies all legal actions; Anonymous heuristic-proposal selection supplies anonymous fixed-rule proposals and numerical features; Heuristic-proposal selection adds names and rule mappings at a matched history. The three arms then follow their own histories. It is distinct from the reused Direct construction seven-tour diagnostic and from the primary one-condition Direct construction/Heuristic-proposal selection panel. Qwen uses greedy no-thinking generation with a 64-token ceiling, rather than the diagnostic's historical 16. Native interface limits remain in force. GraphDecide 48 The tables retain every configuration, conditional quality and complete/unsupported/failed/never-started counts. NR means unreported, and a dash means undefined. Public means average completed conditions within each graph, then graphs equally; paired Heuristic-proposal selection-Anonymous heuristic-proposal selection uses jointly completed conditions. Proposal completion estimates execute only the next action. Fixed-rule and random controls use the full schedule, whereas comparisons to an incomplete model arm require its matched successful subset. Table G.11a. Three-arm public TSP extension: gaps (%) and paired Heuristic-proposal selection-Anonymous heuristic-proposal selection (p). ConfigurationDirect construction Anonymous heuristic-proposal selection Heuristic-proposal selection Paired Heuristic-proposal selection-Anonymous heuristic-proposal selection Jev64.44830.01527.933-2.082 Qwen3.5-4B grammar306.69444.56135.891-8.486 Qwen3.5-9B grammar363.33239.39824.282-15.116 Qwen3.5-27BNRNRNRNR Qwen3.5-4B scores— Decider-2B722.84441.60445.8274.223 Kev-4BNRNRNRNR Laya— Qwen3.8-27BNRNRNRNR Qwen2.5-72BNRNRNRNR Qwen3.5-0.8B760.96845.74938.197-7.552 Qwen3.5-2B746.21446.78747.9801.871 GPT-5.4 (reasoning reference)NRNRNRNR GPT-6-Astra (reasoning)NRNRNRNR Ranked down each mode column only for the complete common schedule; partial means remain unranked (coverage in the following table). Paired changes are diagnostics, not absolute-quality rankings. Table G.11b. Three-arm public TSP extension: complete/U/F/never-started; pairs/graphs. ConfigurationDirect constructionAnonymous heuristic-proposal selection Heuristic-proposal selection Pairs/graphs Jev21/0/0/021/0/0/021/0/0/021/7 Qwen3.5-4B grammar21/0/0/021/0/0/020/0/0/120/7 Qwen3.5-9B grammar21/0/0/021/0/0/021/0/0/021/7 Qwen3.5-27BNRNRNRNR/NR Qwen3.5-4B scores0/21/0/00/21/0/00/21/0/00/0 Decider-2B21/0/0/021/0/0/021/0/0/021/7 Kev-4BNRNRNRNR/NR Laya0/21/0/00/21/0/00/21/0/00/0 Qwen3.8-27BNRNRNRNR/NR Qwen2.5-72BNRNRNRNR/NR Qwen3.5-0.8B21/0/0/021/0/0/021/0/0/021/7 Qwen3.5-2B5/0/0/167/0/2/124/0/0/173/3 GPT-5.4 (reasoning reference)NRNRNRNR/NR GPT-6-Astra (reasoning)NRNRNRNR/NR GraphDecide 49 Table G.12a. Three-arm signed MaxCut extension: gaps (%) and paired Heuristic-proposal selection-Anonymous heuristic-proposal selection (p). ConfigurationDirect constructionAnonymous heuristic-proposal selection Heuristic-proposal selection Paired Heuristic-proposal selection-Anonymous heuristic-proposal selection Jev104.49211.57010.231-1.339 Qwen3.5-4B grammar99.80686.35617.411-68.884 Qwen3.5-9B grammar100.62931.43820.444-10.994 Qwen3.5-27BNRNRNRNR Qwen3.5-4B scores— Decider-2B100.22740.91029.020-11.889 Kev-4BNRNRNRNR Laya— Qwen3.8-27BNRNRNRNR Qwen2.5-72BNRNRNRNR Qwen3.5-0.8B100.81985.27970.474-14.805 Qwen3.5-2B101.79260.00055.512-7.094 GPT-5.4 (reasoning reference)NRNRNRNR GPT-6-Astra (reasoning)NRNRNRNR Ranked down each mode column only for the complete common schedule; partial means remain unranked (coverage in the following table). Paired changes are diagnostics, not absolute-quality rankings. Table G.12b. Three-arm signed MaxCut extension: complete/U/F/never-started; pairs/graphs. ConfigurationDirect construction Anonymous heuristic-proposal selection Heuristic-proposal selection Pairs/graphs Jev30/0/0/030/0/0/030/0/0/030/10 Qwen3.5-4B grammar27/0/2/129/0/1/030/0/0/029/10 Qwen3.5-9B grammar30/0/0/030/0/0/030/0/0/030/10 Qwen3.5-27BNRNRNRNR/NR Qwen3.5-4B scores0/30/0/00/30/0/00/30/0/00/0 Decider-2B30/0/0/030/0/0/030/0/0/030/10 Kev-4BNRNRNRNR/NR Laya0/30/0/00/30/0/00/30/0/00/0 Qwen3.8-27BNRNRNRNR/NR Qwen2.5-72BNRNRNRNR/NR Qwen3.5-0.8B30/0/0/030/0/0/030/0/0/030/10 Qwen3.5-2B7/0/0/235/0/0/2511/0/1/182/2 GPT-5.4 (reasoning reference)NRNRNRNR/NR GPT-6-Astra (reasoning)NRNRNRNR/NR An additional cohort uses the same protocol on previously unattempted slots; failed or partly executed episodes remain in the original cohort. Qwen3.5-2B results are listed below. The Qwen3.5-4B cohort contains one TSP Heuristic-proposal selection completion at 35.272% and one signed MaxCut Direct construction completion at 101.786%, each with counts 1/0/0/0. Qwen3.5-27B results in this cohort are NR. GraphDecide 50 Table G.13a. Separate public Qwen3.5-2B supplement: gaps (%). TaskOriginal extension completions (all modes) Supp. Direct construction Supp. Anonymous heuristic-proposal selection Supp. Heuristic-proposal selection Public TSP (%)16/63714.66546.64834.417 Public cut (%)23/9099.83061.18256.022 Table G.13b. Separate public Qwen3.5-2B supplement: complete/U/F/never-started. TaskOriginal extension completions (all modes) Supp. Direct construction Supp. Anonymous heuristic-proposal selection Supp. Heuristic-proposal selection Public TSP (%)16/6316/0/0/012/0/0/017/0/0/0 Public cut (%)23/9023/0/0/025/0/0/018/0/0/0 G.6 Additional Public Outcomes Table G.14 presents seven separate completion outcomes using 64 output tokens and a 40,960-token context for Qwen. This cohort is analyzed separately from the three-arm extension and its unattempted-slot cohort in Section G.5, and from the primary Direct construction/Heuristic-proposal selection panel; its outcomes do not enter those paired contrasts. Table G.14. Seven separate public outcomes. ModelGraph Conditi on ModeObjectiveGap (%) Feasibl e Jev-1.13.0kroB2001Anonymous heuristic-proposal selection4189142.3073True Qwen3.5-2Bbier1270Anonymous heuristic-proposal selection16469139.2359True Qwen3.5-2BkroA1501Anonymous heuristic-proposal selection3937948.4655True Qwen3.5-2Bsg3dl0570000Heuristic-proposal selection5848.2143True Qwen3.5-4B gen.sg3dl0540000Direct construction-4103.5088True Qwen3.5-4B gen.sg3dl0560000Anonymous heuristic-proposal selection2081.8182True Qwen3.5-4B gen.sg3dl0590002Direct construction0100.0000True GraphDecide 51 Appendix H. Reference Calculations for the Induced-Graph Tasks The references below apply to the frozen benchmark instances, not the full source networks or other partition and diffusion variants. TSP retains published proven optima and MaxCut retains historical best-known values; this appendix specifies the two induced-graph tasks. H.1 Binary Modularity The ten induced instances from SNAP ca-GrQc [35] are simple, undirected and unweighted, with resolution γ=1 and at most two communities, without size or volume balance constraints. Let m be the number of edges, d_v the degree of vertex v, x_v indicate membership in A, V_A=Σ_v d_v x_v, and c be the number of crossing edges. Newman's modularity [38] becomes Maximizing Q is equivalent to minimizing the integer energy E=(V_A-m) 2 +2mc. The MILP formulation uses binary x_v, continuous 0≤y_uv≤1 for each edge, continuous -m≤d≤m, and continuous 0≤z≤m 2 . It minimizes z+2mΣ_(u,v) y_uv subject to Although d is continuous in the formulation, binary x makes it integer. At every integer d in [-m,m], the maximum of the consecutive-integer secants equals d 2 , so minimization sets z=d 2 . Positive coefficients likewise set y_uv=|x_u-x_v|. Fixing x_0=0 removes only label symmetry. An empty community remains legal and the one-community partition has Q=0. The reference calculation uses scipy.optimize.milp with bundled HiGHS (SciPy 1.17.1, NumPy 2.4.6, NetworkX 3.6.1), one solver thread and requested relative MIP gap zero. All ten instances terminate with optimal status. For each integer incumbent E*, the numerical dual bound satisfies ceil(dual-10 -5 )=E*, and returned vertex values are integral within 10 -5 . Witness Q is recomputed by exact rational arithmetic and checked against NetworkX modularity within 10 -12 ; the solver objective matches witness energy within 10 -4 . The reported validation also checks small path, cycle, complete, disconnected and seeded-random graphs against exhaustive binary assignments, including a one-community optimum. Optimality certification rests on the numerical primal/dual bounds and integer-energy checks; exact rational arithmetic evaluates the witnesses. Q= V A (2m−V A ) 2m 2 − c m = 1 2 − (V A −m) 2 + 2mc 2m 2 (H.1) d=∑ v d v x v −m, y uv ≥x u −x v , y uv ≥x v −x u (H.2) z≥(2k+ 1)d−k(k+ 1) (k= −m, ..., m−1), x 0 = 0 (H.3) GraphDecide 52 Table H.1. Binary-modularity solver-certified references. Instance (ca-GrQc-main-)nmExact rational Q*E* n50-s0508583/170170 n50-s150691529/3174174 n75-s075928159/16928305 n75-s17513918163/386421158 n100-s010022612443/255381304 n100-s110018029519/648002881 n125-s012523148527/1067224834 n125-s112519919402/39601797 n150-s015020410151/208081012 n150-s115022448831/1003521345 All modularity references use Q*=1/2-E*/(2m 2 ). Astra's high-precision mean gaps are 0.0002611075272071162 for direct construction (8/10 optimal) and 0.012982818923280088 for heuristic-proposal selection (1/10 optimal). GPT-5.4's full-precision objectives yield mean gaps 0.4042398456 and 0.0422861571, respectively, with 0/10 and 1/10 optimal. Optimal-instance counts use a 10 -10 absolute numerical tolerance against the rational reference. H.2 Deterministic LT Influence Maximization Each frozen directed graph induced from SNAP email-Eu-core [37] uses exactly two distinct seeds, activated simultaneously. Every incoming edge of v has weight 1/indegree(v), and each nonseed has threshold 1/2. An inactive nonseed of positive indegree activates exactly when twice its active incoming-neighbor count is at least its indegree. Activation is irreversible; a zero-indegree nonseed never activates spontaneously. The objective f(S) counts all finally active vertices, including both seeds. No random threshold sampling or Monte Carlo estimation is used. Enumerate every unordered distinct seed pair. An event queue starts with both seeds; each newly active vertex is processed once, incrementing predecessor counts at its outgoing neighbors and enqueuing a neighbor once its threshold is reached. An independent synchronous implementation represents active vertices and incoming-neighbor sets as bitsets, evaluates every threshold against the previous round, and activates eligible vertices together until a fixed point. Exhaustive enumeration compares the complete final active sets of the queue and synchronous implementations for every pair, not only their sizes. Nonnegative weights and irreversible activation yield the same least closure above the seed set. Exhaustion of all n(n-1)/2 pairs covers the feasible space, so the largest final count is the global optimum; the reported lower and upper bounds both equal it. Queue evaluation costs O(n+m) per pair, giving O(n(n-1)(n+m)/2) overall, plus the independent synchronous validation. The enumeration complexity is specific to the fixed seed count two. f * = max u, v⊆V, u≠v f(u, v), gap(%) = 100 f * −f(S) f * (H.4) GraphDecide 53 Table H.2. Deterministic LT exact references. Instance (email-Eu-core-main-)nEdgesPairsf*Witness n50-s05015012251418,36 n50-s15012812251732,34 n75-s07531727752959,69 n75-s17523227756310,25 n100-s010034849502551,65 n100-s110047049502553,87 n125-s01255077750259,97 n125-s112564377501668,114 n150-s0150843111751729,61 n150-s11501002111751520,72 Witness seed IDs are zero-based local IDs of the frozen graph, not original SNAP identifiers. A listed pair is one maximizer and need not be unique. These LT optima are the references used for the primary results.