AI and Human Approaches to Mathematical Problem Solving

• The six practices connect problem closure with method accumulation, boundary specification, agenda formation, and knowledge integration.The framework links mathematical problem solving to cumulative knowledge production.Source: §1 · p. 3
• The study compares AI and human research texts organized around the same mathematical problems across six observable research practices.Texts are treated as public records; the analysis does not recover private cognition, certify correctness, or estimate a causal effect of AI.Source: §1 · p. 3
• A closure-only evaluation can miss whether texts explain reusable methods, identify limits, or open subsequent lines of work.The paper argues that a multidimensional comparison can examine these components without assuming any one defines mathematical quality.Source: §1 · p. 3
• Within-problem comparison gives each problem equal inferential weight, limiting confounding from differences in subject matter and literature size.The study also examines which contrasts persist across text representations, matching rules, and problem composition.Source: §1 · p. 3

2.1 Sources and unit of observation

• The mathematical problem is the unit of analysis; the confirmatory sample covers 11 problems, 58 human comparator papers, and 31 AI–human matched pairs.A separate set of 497 parseable full texts supports abstract–full-text representation checks, not the final effect-estimation sample.Source: §2.1 · pp. 3–4

2.2 Mathematical problems represented

• The 11 targets include specified resolutions, disproofs, and narrower advances rather than resolutions of entire fields’ broader open problems.The unit-distance target concerns a particular asymptotic conjecture, and the Navier–Stokes entry concerns the smoothly forced Clay C/D formulation.Source: §2.2 · p. 4

2.3 Identification, cleaning and comparator audit

• The comparator audit retains human records when the observed text links a result-oriented predicate to the target statement under prespecified rules.Cleaning removes generic-name matches, disciplinary false positives, non-research items, and records not directed at the target; textual comparability is not independent certification of correctness.Source: §2.3 · p. 5

3.1 Six-dimensional research-practice framework

• Six indicators capture target closure, reusable methods, assumptions or failure boundaries, successor questions, extension beyond the focal case, and cross-field integration.The first five are semantic orientations; integration is measured from relations connecting claims, objects, methods, and fields.Source: §3.1 · p. 5

3.2 Measurement models, matching and problem-level inference

• Five constructs use prespecified semantic models, while cross-disciplinary integration uses a separate supervised model because field-word co-occurrence alone does not establish integration.The integration model uses normalized lexical and relational features, including claim–object, method–operation, condition, scope, generalization, and field-bridge features.Source: §3.2 · pp. 6–7
• The semantic models’ out-of-fold AUCs range from 0.926 to 0.990, and the separate integration model’s external known-group AUC is 0.849.The models are trained and evaluated using independent external known groups.Source: §3.2 · p. 7
• Within each problem, AI texts are matched to as many as five human texts with the closest log token count; problems then receive equal weight in aggregate inference.The design addresses the association between document length and opportunities to express a construct, and intervals resample problems.Source: §3.2 · pp. 7–8

• AI texts emphasize problem resolution and cross-disciplinary integration more, while human texts emphasize methods, boundaries, and successor questions.Figure 2 presents the aggregate contrasts alongside problem-omission stability, problem-level variation, and the human comparator distribution.Source: §4 · p. 8, Figure 2 · p. 9
• Omitting each of the 11 problems in turn preserves all six aggregate directions; generality remains close to zero and directionally, but not statistically, stable.No single problem determines the aggregate pattern.Source: §4 · p. 9
• In five problems with complete original AI texts, alternative construct-aligned measures reproduce all six main directions, with each measure’s external known-group AUC above 0.80.Source: §4 · pp. 9–10
• Across 497 human papers, abstract scores correlate positively with equal-length full-text-window scores for all six constructs, with Spearman correlations from 0.637 to 0.831.The authors interpret this as support for broad rank correspondence between the two representations.Source: §4 · p. 10
• Alternative comparator and matching specifications preserve five directional findings; generality changes sign in some specifications, with estimates remaining close to zero.The authors view this as reinforcing the five directional findings and supporting a null interpretation for generality.Source: §4 · p. 10

5. Discussion

• AI accounts emphasize closure and cross-field relations, while human papers emphasize methods, boundary conditions, and successor questions.Source: §5 · p. 10
• These differences describe public research presentation, not direct observations of private cognition.Source: §5 · p. 10
• A stronger closure orientation does not imply greater emphasis across every practice supporting cumulative inquiry.Source: §5 · p. 10
• Resolution, method, boundary specification, and successor-question generation remain separate dimensions that a single success criterion can conceal.Such a criterion can conceal differences in how results are prepared for interpretation and further use.Source: §5 · pp. 10–11
• The sampled AI texts represent an early, institutionally concentrated stage while documentation and dissemination conventions are still taking shape.Source: §5 · p. 11
• Future research can test whether methods are reused, successor questions attract work, and human-AI collaborations produce different research-practice profiles.That evidence would connect presentation differences to later knowledge accumulation.Source: §5 · p. 11

Appendix A. Problem identification and retrieval

• Each problem receives a canonical name, aliases, statement, and intended question, while Table A1 provides an audit trail for target scope and source-claim boundaries.Source: Appendix A · p. 14, Table A1 · p. 14
• Target retrieval combines name queries, anchor co-occurrence queries, and semantic recall, retaining records that connect to the target statement in a title or abstract.Source: Appendix A · p. 14
• The listed permanent-specific targets include superquadratic lower bounds for division-free circuits and order-n4/log n formula lower bounds, while VP versus VNP remains open.Source: Appendix A · p. 14

Appendix B. Comparator audit and sample construction

• 58 of 440 broad candidates satisfy the prespecified textual rule across 11 problems.The audit requires a result-oriented predicate, target anchor, and affirmative relation in the same sentence, while flagging specified qualifications and exclusions.Source: Appendix B · pp. 14–15
• Euler and Navier-Stokes supply most comparator records, while five problems contribute one record each.Table B1 lists human comparator papers passing the target-linkage audit.Source: Appendix B · p. 15, Table B1 · p. 15

Appendix C. Measurement models and construct validity

• Five semantic constructs use separate TF-IDF logistic regressions trained on independent known groups, with documents held out from their scoring fold.The constructs are resolution, method, boundary, successor questions, and generality; focal-document scores use models refitted on the complete external corpus.Source: Appendix C · pp. 15–16
• Cross-disciplinary integration is scored with a histogram-based gradient-boosting model using 28 normalized lexical and relational features.The features capture claim-object, method-operation, scope, and mathematical-field relations.Source: Appendix C · pp. 15–16
• Five-fold out-of-fold AUCs evaluate discrimination in the external known groups.Table C1 reports the validation; focal AI-human scores use models refitted on the complete external corpus.Source: Appendix C · p. 16, Table C1 · p. 16

Appendix D. Leave-one-problem-out analysis

• Each specification omits one problem and recomputes the equal-weight mean over the remaining ten.The equation defines the leave-one-problem-out calculation.Source: Appendix D · p. 16
• The aggregate standardized mean effect size compares AI and human mean scores within each problem for a construct.The human-paper standard deviation within each problem supplies the standardization; P−1 accounts for the sample-size adjustment after omission.Source: Appendix D · pp. 16–17
• None of the six leave-one-problem-out effect ranges crosses zero; generality comes closest, with a maximum of -0.013.Table D1 reports the minimum and maximum across all omissions.Source: Appendix D · p. 17, Table D1 · p. 17

Appendix E. Construct-aligned hybrid measurement

• The hybrid analysis remeasures all six constructs on available complete original texts, which can be matched to human papers for five mathematical problems.It evaluates directional and representation sensitivity rather than replacing the 11-problem main analysis.Source: Appendix E · p. 17
• Different constructs have different observable text structures, so one shared model could create construct-estimator mismatch.The semantic models address resolution, method articulation, boundaries, and successor questions; the relation-based models address generality and integration.Source: Appendix E · pp. 17–18
• Semantic-component logistic models capture diffuse discourse functions, while claim-relation models assess generality and cross-disciplinary integration.The latter require scope-extending predicates linked to focal results or distinct-field concepts participating in the same assertion; vocabulary co-occurrence alone does not establish those relations.Source: Appendix E · pp. 17–18
• All six estimates match the main analysis in direction, every external AUC exceeds 0.80, and the result is a complete directional replication.The generality interval excludes zero, while the other five intervals span zero with only five problem clusters.Source: Appendix E · p. 18, Table E1 · p. 18
• Because the construct-model combination followed preliminary representation checks, the appendix treats this as an exploratory robustness analysis.Source: Appendix E · p. 18

Appendix F. Abstract and full-text representations

• All six constructs show statistically detectable positive abstract–full-text correlations, with the weakest agreement for cross-disciplinary integration.Table F1 reports the correlations.Source: Appendix F · p. 18, Table F1 · p. 18
• Across 497 human papers, abstract scores correlate 0.637 to 0.831 with equal-length full-text windows, supporting broad ranking similarity rather than equality.Source: Appendix F · p. 18
• The lower abstract–full-text correlation for cross-disciplinary integration is consistent with relational evidence appearing outside abstracts.Source: Appendix F · p. 18

Appendix G. Matching, influence and uncertainty

• The inverse-distance weighting assigns normalized weights to matched human papers according to their log token-count distance from the focal AI document.A small positive smoothing constant avoids division by zero, and the comparator set contains the matched human papers for that AI document.Source: Appendix G · p. 19
• Directions for problem resolution, method articulation, uncertainty and boundaries, successor questions, and cross-disciplinary integration remain stable across matching specifications.Table G1 reports matching-specification sensitivity.Source: Appendix G · p. 19, Table G1 · p. 19
• Generality changes sign in three alternatives, consistent with its imprecise main estimate.Table G1 presents matching-specification sensitivity.Source: Appendix G · p. 19, Table G1 · p. 19