PaperlayerView exact revision on arXiv

Source-linked AI summary

ComBench: A Benchmark for Rigorous Proof Reasoning and Constructive Realization in Olympiad-Level Combinatorics

Shunkai Zhang, Haoran Zhang, Yun Luo, Qianjia Cheng, Haodi Lei, Yizhuo Li, Runzhe Zhan, Zhilin Wang, Bangjie Xu, Yucheng Su, Xinmiao Han, Xiaoye Qu, Dongrui Liu, Zhouchen Lin, Yu Qiao, Ning Ding, Yafu Li, Yu Cheng

arXiv:2606.10479v1cs.AI

TL;DR

Existing benchmarks do not fully assess the constructive nature of Olympiad-level combinatorics or whether models produce valid arguments and explicit witnesses. ComBench addresses this with rubric-guided proof grading and deterministic construction verification, finding distinct proof and construction capabilities while GPT-5.5 reaches 65.4% overall Avg.

  • Problem

    Existing benchmarks do not isolate combinatorics’ constructive nature or whether models produce valid mathematical arguments and explicit witnesses.

  • Method

    ComBench evaluates 100 Olympiad-level combinatorics problems using rubric-guided proof grading and deterministic verification of explicitly represented construction witnesses.

  • Results

    Frontier models show distinct proof and construction capabilities while Olympiad-level combinatorics remains unsaturated; GPT-5.5 reaches 65.4% overall Avg.

  • Takeaways & Limitations

    Rigorous Proof Reasoning and Constructive Realization should be evaluated as distinct capabilities in Olympiad-level combinatorics.

  • Takeaways & Limitations

    The deterministic verifier covers only construction targets formalized in advance and may exclude valid constructions expressed in alternative representations.

Abstract

from arXiv · show

Combinatorics is central to Olympiad-level mathematical problem solving, requiring deep discrete reasoning, creative constructions, and rigorous structural insight. Recent evidence suggests that even today's strongest frontier models remain uneven on Olympiad combinatorics, revealing a gap in creative mathematical reasoning. We introduce ComBench, an Olympiad-level combinatorics benchmark for evaluating and diagnosing the combinatorial reasoning capabilities of large language models. ComBench contains 100 human-annotated competition-level problems organized around two complementary settings: analysis-centric problems, which primarily require rigorous mathematical arguments, and construction-centric problems, which require explicit constructions in addition to correctness justifications. The evaluation protocol combines rubric-guided proof grading with deterministic construction verification, exposing cases where proof quality and construction validity diverge. Experiments on frontier open- and closed-source models show that ComBench is far from saturated: the strongest model reaches 65.4% overall Avg. and 75.3% overall Best@4. We further find that Rigorous Proof Reasoning and Constructive Realization are distinct capabilities: Kimi-K2.6 trails GPT-5.5 on analysis-centric proof grading but surpasses it on construction-centric Best@4, while Existence and Construction problems remain consistently hardest across representative frontier models.

1 Introduction

ComBench is a 100-problem Olympiad-level combinatorics benchmark designed to evaluate both rigorous proof reasoning and constructive realization through rubric-guided grading and deterministic construction verification. Experiments show that frontier models remain far from saturating the benchmark and exhibit distinct proof-level and construction abilities.

  • Motivation: Existing answer-based benchmarks do not establish whether a solution contains a valid argument, while natural-language construction descriptions are difficult to evaluate under global discrete constraints.The benchmark addresses these limitations by separating proof-quality assessment from executable witness validation.
  • Benchmark design: ComBench evaluates 100 Olympiad-level combinatorics problems across Rigorous Proof Reasoning and Constructive Realization.The benchmark includes analysis-centric records for mathematical solutions and construction-centric records that add explicit construction tasks.
  • Empirical findings: 65.4% overall Avg. and 75.3% overall Best@4 are achieved by the strongest model, GPT-5.5, showing that Olympiad-level combinatorics remains far from saturated.These scores use rubric-guided proof grading and verifier-gated evaluation.
  • Evaluation protocol: Its protocol combines IMO-Bench-style rubric-guided proof grading with deterministic verification of explicit, machine-checkable construction witnesses.Construction witnesses can include colorings, tilings, graphs, set families, strategies, matrices, permutations, or counterexamples.
  • Empirical findings: Frontier models show a clear separation between proof-level reasoning and explicit construction ability, alongside category-specific weaknesses in Olympiad-level combinatorics.This diagnostic behavior motivates evaluating the two capabilities separately rather than relying only on answer-based performance.

2 Related Work

Mathematical reasoning benchmarks have progressed from GSM8K and MATH to Olympiad- and frontier-level evaluations using answer matching and rubric-guided proof grading. Despite advances in Olympiad-level reasoning, the hardest combinatorics problems remain unsolved, motivating ComBench’s focus on disentangling this gap.

  • Evaluation of Mathematical Reasoning: Benchmarks have evolved from GSM8K and MATH to Olympiad- and frontier-level evaluations, with answer-only matching and rubric-guided proof grading among the dominant protocols.
  • LLMs for Olympiad-level Reasoning: Large-scale post-training has improved natural-language performance under contest conditions after early systems achieved medal-level IMO performance.
  • LLMs for Olympiad-level Reasoning: No model received credit on IMO 2025 P6, and ComBench is designed to measure and disentangle this persistent combinatorics gap.

3 ComBench Dataset

ComBench is a 100-problem Olympiad-level combinatorics benchmark balanced between analysis-centric proof tasks and construction-centric tasks with mechanically checkable witnesses. Its dataset design separates Rigorous Proof Reasoning from Constructive Realization while supporting rubric-based grading and executable verification.

  • Dataset composition: ComBench contains 100 Olympiad-level problems: 50 analysis-centric and 50 construction-centric, drawn from 15 competition sources spanning 2000–2025.Sources include IMO, USAMO, team selection tests, regional Olympiads, and IMO Shortlist problems.
  • Target capabilities: Analysis-centric records evaluate Rigorous Proof Reasoning, whereas construction-centric records additionally evaluate Constructive Realization through explicit, mechanically checkable witnesses.Construction witnesses are concrete discrete objects rather than proof sketches or natural-language descriptions.
  • Record annotation: Each record includes problem and solution metadata plus problem-specific 0/1/6/7-point grading guidelines, while construction records add instructions, reference witnesses, verifier plans, code, and semantic audits.The shared rubric supports proof evaluation for both record types; construction records add executable validation components.
  • Categories and quality control: ComBench uses five primary categories—extremal, existence and construction, operations and strategies, graph theory, and counting—and reports a 14% overlap rate with IMO-Bench.The overlap review identified 14 overlapping or substantially modified problems, leaving 86 without a high-confidence match.
  • Verification pipeline: Construction verification proceeds from specification and rubric construction through deterministic verifier generation, semantic audit, executable reference checking, and final human inspection.A record is accepted only when its reference witness succeeds through the same raw-standard-input interface used for model submissions.
  • Scoring design: Proof grading and construction verification remain separate signals, while construction-centric ranking combines proof and binary construction scores through verifier-gated scoring.The rule calibrates high proof scores using construction validity without erasing meaningful proof progress when construction fails.

4 Experiments

ComBench remains far from saturated: GPT-5.5 reaches only 65.4% overall Avg. and 75.3% overall Best@4, while results separate proof reasoning from constructive realization. Construction validity, category difficulty, and proof-error analyses reveal persistent weaknesses in global combinatorial planning.

  • Main results: Kimi-K2.6 attains the highest construction-centric Best@4 despite weaker analysis-centric performance, demonstrating that Constructive Realization is distinct from Rigorous Proof Reasoning.The ranking pattern shows construction ability is not merely a by-product of stronger proof reasoning.
  • Main results: 65.4% overall Avg. and 75.3% overall Best@4 are achieved by GPT-5.5, showing that Olympiad-level combinatorics remains difficult even under best-of-four sampling.Scores use rubric-guided proof grading and verifier-gated construction evaluation.
  • Construction-centric behavior: Verifier-gated scores are lower than proof scores across models, and the strongest systems reach only a 75.0% per-sample construction pass rate.Construction-centric evaluation jointly tests proof quality and whether the extracted witness passes an item-specific executable verifier.
  • Category-level behavior: Existence and Construction is the hardest category across four representative frontier models, while Counting and Graph Theory achieve the highest scores.Analysis-centric records use proof scores, whereas construction-centric records use verifier-gated scores.
  • Proof-error analysis: Dominant below-full-credit errors reflect deeper failures in global proof planning, including missing the central mechanism, proving the wrong target, or leaving global obligations unclosed.Primary error reasons are mutually exclusive, while secondary tags may overlap.
  • Category-level behavior: Harder categories often require selecting a global target, maintaining an invariant, or matching a sharp bound with a concrete construction.Operations and Strategies and Extremal Problems share these demands, unlike more enumerative or structurally reformulable tasks.

5 Conclusion

ComBench is a 100-problem Olympiad combinatorics benchmark that separates Rigorous Proof Reasoning from Constructive Realization using proof grading and deterministic witness checking. Frontier models show these capabilities can diverge.

  • Benchmark and evaluation: ComBench contains 100 Olympiad-level combinatorics problems and evaluates both Rigorous Proof Reasoning and Constructive Realization.Its verifier-gated protocol combines rubric-based proof grading with deterministic witness checking.
  • Benchmark and evaluation: The verifier-gated protocol decouples proof quality from construction validity by combining rubric proof grading with deterministic witness checking.
  • Main findings: Frontier models exhibit a clear dissociation between proof reasoning and constructive realization: strong proofs may lack valid witnesses, while weaker proof grading can coexist with top construction performance.

Limitations

ComBench’s deterministic verifier reliably evaluates prescribed witness interfaces but cannot capture every valid natural-language construction. Its Olympiad-level combinatorics focus also means results do not measure all mathematical abilities, including algebra, geometry, and number theory.

  • Verifier coverage: The deterministic verifier checks only construction targets formalized in advance, so mathematically meaningful ideas can fail when they are not expressed through the prescribed witness interface.This limits coverage of valid natural-language constructions despite making evaluation reliable for the supported interface.
  • Domain scope: ComBench targets Olympiad-level combinatorics and should not be treated as a complete measure of Olympiad mathematics ability.Algebra, geometry, and number theory require forms of abstraction, symbolic manipulation, and proof technique outside the benchmark’s scope.

Ethical Considerations

ComBench is intended as a diagnostic benchmark for combinatorial reasoning and construction, not a complete measure of mathematical intelligence or the sole basis for capability claims. Its interpretation requires care because public-problem overlap and evaluation limitations may affect conclusions, while annotators were compensated for their work.

  • Scope and limitations: ComBench contains public Olympiad-style problems without personal, demographic, or sensitive user data, so its main risks concern interpretation and use rather than privacy.Some public problems may overlap with existing benchmarks or model training data.
  • Annotation process: All annotators received compensation corresponding to their workloads for annotation, grading-guideline construction, construction specification, and verifier checking.
  • Evaluation limitations: The LLM proof judge may disagree with expert graders, while executable verifiers check only pre-specified construction targets rather than arbitrary natural-language reasoning.Accordingly, ComBench is treated as diagnostic rather than as the sole basis for claims about model capability.

A Dataset Details … C.1 Record Schema

ComBench’s appendices document the dataset composition, released record schema, and a reviewed pipeline that converts constructive problems into proof rubrics and executable witness verifiers. The design combines broad Olympiad-problem coverage with quality controls ensuring that construction instructions, reference witnesses, verifier code, and mathematical targets remain aligned.

  • A.1 Source and Year Distribution: ComBench combines canonical international, national, and selection-test problems, emphasizes recent Olympiad-style coverage while retaining classical themes, and avoids domination by one contest family.The appendix reports competition sources and year coverage in Tables 4 and 5.
  • A.2 Problem Type and Category Breakdown: The dataset uses five primary categories, with construction-centric records concentrated where explicit witnesses naturally arise, especially extremal and existence-style tasks.Tables 6 and 7 define the categories and report the problem type-by-category breakdown.
  • A Dataset Details; A.3 Released Metadata Fields: The public release will provide problem-level metadata for reproducing aggregate analyses, including identifiers, sources, statements, types, categories, reference answers, solutions, and grading guidelines.Construction-centric records additionally include instructions, reference witnesses, verifier plans, and verifier code.
  • B Construction Annotation Pipeline; C Record Schema and Quality Control: The construction annotation pipeline transforms each constructive problem into a proof-grading rubric and an executable witness-verification interface.This extends the common proof-evaluation record with construction-specific artifacts.
  • B.1 Stage 1: Specification and Rubric Construction: Stage 1 combines the original problem, reference materials, construction intent, and witness to generate an item-specific instruction, standardized reference construction, verifier plan, and 0/1/6/7-style grading guidelines.Human review checks semantic preservation, witness validity, and meaningful Olympiad-style proof progress before approving or revising the record.
  • B.2 Stage 2: Verifier Generation and Semantic Audit: Stage 2 generates deterministic Python verifiers that validate both witness representation and mathematical correctness, including global combinatorial constraints rather than syntax or local conditions alone.A semantic audit compares the full annotation chain and routes task-definition issues back to specification or checker issues back to verifier generation.
  • B.3 Stage 3: Assembly and Executable Reference Check: Stage 3 assembles audited records into JSONL and executable-checks each standardized reference construction through the same raw-input interface used for submissions.Final review emphasizes verifier strictness, coverage, consistency, diagnostic clarity, rejection of incomplete witnesses, and useful debugging feedback.
  • C.1 Record Schema: During evaluation, models receive only the construction instruction and submit one payload, while the verifier checks that payload and the withheld reference construction serves dataset validation.Thus, construction-centric fields define an executable interface rather than an additional natural-language hint.

C.2 Example Construction-Centric Record … D.3 Construction Object Types

The construction-centric records pair explicit witness tasks with deterministic executable verification, while quality-control and human-audit procedures preserve the original mathematical target. The verifier separately evaluates construction payloads, leaving proof quality and witness validity as distinct outcomes.

  • C.3 Representative Witness Forms: Construction-centric witnesses appear in multiple representations, including binary matrices, ordered-pair lists, comprehension-defined tables, dictionaries, and rule-based objects.These representative forms are associated with different verifier checks summarized for recurring construction object types.
  • C.4 Quality-Control Checks: Quality control checks construction instructions and verifiers against the original problem and reference answer, requiring interface compatibility and enforcement of global mathematical constraints.Dataset-integrity checks also cross-check identifiers, labels, categories, types, and construction-field availability against released JSONL records.
  • C.5 Human Audit Checklist: A final human audit follows automatic generation and executable reference checking to verify the mathematical target, intended verifier constraints, and consistency of released metadata.The audit questions are summarized in Table 16.
  • D Executable Verifier Details: The executable verifier is deterministic Python code for each construction-centric record and checks only the submitted construction payload, unlike the rubric-guided judge for natural-language proofs.This separation defines distinct roles for executable witness validation and proof evaluation.
  • D.1 Input and Output Contract: Models must provide exactly one payload inside a <construct>...</construct> block, which is passed as raw standard input; malformed or missing blocks receive construction score 0.The verifier returns a binary result, passing only when it prints True, and reference constructions use the same interface.
  • D.2 Verifier Scope: The verifier is not a theorem prover: it formalizes only the item-specific construction target and separates witness validity from natural-language proof quality.A model may earn partial proof credit while failing verification, while a valid witness does not guarantee a complete proof.
  • D.3 Construction Object Types: Construction-centric records use recurring witness object types, with Table 19 summarizing each representation and its corresponding verifier checks.The object-type taxonomy supports verifier checks tailored to the submitted construction form.

E Evaluation Protocol … E.4 Scoring and Metrics

The evaluation protocol specifies reproducible sampling, prompting, parsing, failure handling, judging, verification, and metric computation. It separates rubric-guided proof assessment from deterministic construction validation while reporting aggregate performance over four sampled solutions.

  • E Evaluation Protocol: The appendix defines the sampling setup, prompt templates, parsing rules, failure handling, and metric definitions used to reproduce the reported results.
  • E.1 Response Models and Sampling: Each record receives four independent solutions at temperature 0.6, with model-specific token limits and a shared downstream evaluation protocol.
  • E.1 Response Models and Sampling: Proofs are graded by one unified rubric-guided Gemini-3.1-Pro-Preview-Thinking judge, while construction payloads are checked by item-specific deterministic Python verifiers.
  • E.2 Prompt Templates: Analysis-centric prompts request one complete rigorous solution, whereas construction-centric prompts require separate proof and construction sections with exactly one <construct>...</construct> block.
  • E.3 Parsing and Failure Handling: Construction responses lacking, duplicating, or malformed construction blocks receive construction score 0 without verifier execution, while parse failures remain in aggregate denominators.
  • E.4 Scoring and Metrics: Proof scores use a four-level 0/1/6/7 rubric, construction verification is binary, and construction-centric aggregates apply the verifier-gated rule.
  • E.4 Scoring and Metrics: Avg. averages normalized scores over all generated solutions, whereas Best@4 averages each record’s highest score across four samples; supplementary metrics include Pass@4, Passˆ4, and construction pass rate.

F Additional Experimental Results … G.3 Global Distribution

The supplementary results identify category- and problem-level difficulty, sampling instability in construction tasks, and the dominant causes of proof-side failures. Across these analyses, existence and construction problems and missing core mechanisms emerge as recurring challenges.

  • F Additional Experimental Results: Supplementary analyses extend the main results with category-level, problem-level, and repeated-sampling summaries.The appendix provides additional evidence for the main-result and subset-analysis claims.
  • F.1 Category-Level Summary: Existence and Construction is the lowest-scoring category, whereas Counting and Graph Theory achieve substantially higher average scores.The scoring rule uses proof scores for analysis-centric records and verifier-gated scores for construction-centric records.
  • F.2 Problem-Level Difficulty: The easiest analysis problems include IMO2008 5 and TSTST2023 4 with 32 full-score samples each, while eight listed problems receive zero.Analysis success requires a proof score of 7.
  • F.2 Problem-Level Difficulty: RMO-2019-P15 is the easiest strict construction problem with 31 successful samples, while eight listed construction problems receive zero strict successes.Strict success requires both proof score 7 and construction score 1.
  • F.3 Construction-Centric Stability: GPT-5.5 reaches 68.0% verifier-gated Pass@4 but only 42.0% Passˆ4 on construction-centric records.The gap shows that some solutions succeed at least once without succeeding consistently across all four samples.
  • G Proof Error Taxonomy: The proof-error taxonomy diagnoses Rigorous Proof Reasoning, while Constructive Realization failures are assessed separately through construction pass rates, verifier-gated scores, and case studies.The taxonomy applies only to proof-side critiques.
  • G.1 Scope and Annotation Rule: Every below-full-credit proof sample receives one mutually exclusive primary reason, with low-frequency mathematical categories merged into Other Mathematical Error.Secondary tags may overlap, and Format or Unscored remains separate.
  • G.3 Global Distribution: Missing Core Mechanism is the largest proof-error category, followed by Wrong Mathematical Target, indicating predominantly substantive rather than formatting failures.Table 22 reports the primary-reason distribution after merging low-frequency mathematical categories into Other Mathematical Error.

G.4 Model-Level Error Profiles … I.2 Audit Decision Labels

The appendices profile model-level proof errors, document ComBench’s 14/100 overlap with IMO-Bench, and audit automatic proof judging through sampled manual labels and acceptance rates.

  • G.4 Model-Level Error Profiles: Stronger models still primarily fail by missing the core mechanism or targeting the wrong mathematical claim, whereas format or parsing failures indicate a different error profile.Tables 23 and 24 report below-full-credit samples and primary error categories, including MCM, WMT, false lemmas, faulty induction, unjustified leaps, incomplete case analysis, and format or unscored outputs.
  • H.1 Matching Protocol: The overlap analysis compares all 100 ComBench problems with IMO-Bench using source metadata, statement-similarity search, and manual inspection of high-similarity candidates.A problem counts as overlapping only when it is an exact source match or substantially modified version; shared themes with different mathematical targets are excluded.
  • H.2 Overlap Summary: ComBench overlaps with IMO-Bench at a rate of 14/100, leaving 86 problems without a high-confidence match and providing largely complementary benchmark coverage.The overlap judgments are summarized in Table 25 after source matching, similarity search, and manual review.
  • H.3 Released Metadata and Filtering: Released metadata records the best-matching IMO-Bench entry, match type, similarity score, and manual decision note for each inspected problem when applicable.The overlap analysis characterizes coverage rather than excluding records, because overlapping problems retain ComBench’s construction instructions, reference witnesses, and deterministic verifier-gated scoring.
  • I Manual Audit of Automatic Proof Evaluation: A sampled manual audit checks whether the rubric-guided automatic proof judge produces reasonable scores, without replacing expert mathematical review.The audit evaluates proof-judge decisions against the problem statement, reference solution when available, and problem-specific grading guidelines.
  • I.1 Audit Sampling Protocol: The audit samples five scored proof evaluations from each of 10 response models, yielding 50 samples across analysis-centric and construction-centric records.Null or unscored outputs caused by parsing or evaluation failure are excluded, and construction-verifier decisions are analyzed separately.
  • I.2 Audit Decision Labels: Each audited proof evaluation receives one of three labels: Accepted, Minor disagreement, or Rejected, reflecting increasing inconsistency with the rubric or mathematical requirements.Accepted denotes a mathematically reasonable rubric-consistent decision; Minor disagreement covers debatable boundary cases; Rejected marks clear rubric inconsistency or a missed major issue.
  • I.2 Audit Decision Labels: The audit reports both a strict acceptance rate and an accepted-or-minor rate, treating boundary disagreements as acceptable for aggregate trend analysis.The strict rate counts only samples labeled Accepted, while the broader rate also includes Minor disagreement.

I.3 Audit Summary · J Case Studies for Verifier-Gated Scoring

The audit finds high agreement between automatic proof-judge decisions and manual review, supporting aggregate benchmarking while retaining documented limitations. Construction-centric case studies illustrate the rationale for verifier-gated scoring across varied proof scores, construction outcomes, and response models.

  • I.3 Audit Summary: The released audit metadata records sampled model outputs, proof-judge rationales, assigned proof scores, and manual-audit decisions for each sampled case.
  • I.3 Audit Summary: 45 of 50 automatic proof-judge decisions were accepted by manual audit, with four minor boundary disagreements and one rejection.
  • I.3 Audit Summary: The strict acceptance rate is 90.0%, while the accepted-or-minor rate is 98.0%.
  • I.3 Audit Summary: These audit results support using the automatic judge for aggregate benchmarking, although boundary disagreements and occasional grading errors remain possible.
  • I.3 Audit Summary: The audit samples proof evaluations and does not guarantee that the automatic judge is error-free on every record.
  • J Case Studies for Verifier-Gated Scoring: Construction-centric case studies illustrate verifier-gated scoring across different proof scores, construction outcomes, and response models.

J.1 Case 1: Proof Score 6 and Construction Score 0 … J.6 Case 6: Proof Score 0 and Construction Score 0

The case studies show that proof quality and constructive realization can diverge: complete or partial proofs may receive construction score 0, while accepted witnesses can coexist with incomplete proofs. They also include one fully successful proof-and-construction case and failures involving incorrect targets or arguments.

  • J.1 Case 1: Proof Score 6 and Construction Score 0: The Nordic-square response proves the lower bound 2n(n −1) + 1 and sketches an attaining construction, earning proof score 6/7 but construction score 0 because the witness is underspecified.The argument uses an independent set whose deletion leaves a tree, but does not fully specify or verify the claimed zig-zag construction.
  • J.2 Case 2: Proof Score 7 and Construction Score 0: The domino-tiling response gives a complete height-function proof of existence and uniqueness, earning proof score 7/7, but construction score 0 because its explicit witness payload is rejected.The proof identifies tasteful tilings with height functions admitting no upward flip and establishes that exactly one such tiling exists.
  • J.3 Case 3: Proof Score 7 and Construction Score 1: The cable-car response proves the sharp threshold k = n2 −n + 1 and supplies an accepted row-column witness, receiving both proof score 7/7 and construction score 1.The upper bound follows from path-component counting and the pigeonhole principle, while rows and columns realize sharpness at k = n2 −n.
  • J.4 Case 4: Proof Score 1 and Construction Score 0: The determinant-pair response correctly observes that only primitive vectors participate, but its false tree argument yields N = 99 instead of the required 197, with proof score 1/7 and construction score 0.Its claimed 99-edge construction is attainable under its incorrect model, but the global argument and verifier submission fail.
  • J.5 Case 5: Proof Score 1 and Construction Score 1: The tromino response gives an accepted construction for every multiple of 3, but its impossibility argument for 3 ∤n is incomplete, producing proof score 1/7 and construction score 1.The supplied solution states that the possible values are precisely the multiples of 3 and constructs an emptying sequence when n = 3m.
  • J.6 Case 6: Proof Score 0 and Construction Score 0: The 2025 × 2025 tiling response claims the minimum is 4048, but selects the wrong target and fails both the required lower-bound reasoning and construction verification, receiving score 0/7.The grading interpretation states that its diagonal construction uses too many tiles and does not establish the sharp construction.
Loading 2606.10479v1…