| {"candidate_required_fields": ["rank", "dimension", "count"], "episode_id": "eval_sample_1_top_chern_cubic_sevenfold_planes", "episode_type": "exact_benchmark", "input": {"degree": 3, "k": 4, "n": 9}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_1", "prompt": "Compute exactly ∫_Gr(4,9) c_20(Sym^3 S*) by torus localization. Return the rank, dimension, and integer count.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "eval", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "grassmannian_top_chern"} |
| {"candidate_required_fields": ["rank", "kernel_dimension", "kernel_vector", "nonzero_quadratic_obstruction", "smooth_along_plane"], "episode_id": "eval_sample_1_local_fano_weighted_fold", "episode_type": "exact_benchmark", "input": {"quadrics": ["x0**2", "x1**2", "x2**2", "x3**2", "x3*(x0+2*x1+x2)"], "second_order_terms": ["y3**2*x1"], "x_variables": ["x0", "x1", "x2", "x3"]}, "milestone_id": "m2", "policy_visibility": "public", "problem_id": "sample_1", "prompt": "Compute the exact corank and second-order obstruction for the weighted fifth jet.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "eval", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "local_fano_deformation"} |
| {"candidate_required_fields": ["h0_restriction", "incidence_codimension", "line_formula_matches"], "episode_id": "eval_sample_2_incidence_cubic_lines_d9", "episode_type": "exact_benchmark", "input": {"curve_degree": 1, "curve_family_dimension": 2, "curve_genus": 0, "fano_index": 2, "linear_system_multiple": 9}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_2", "prompt": "For a cubic threefold of index 2 with a 2-dimensional Fano surface of lines and d=9, compute the incidence codimension.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "eval", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "curve_incidence_codimension"} |
| {"candidate_required_fields": ["source_linear_strand_dimensions", "target_linear_strand_dimensions", "maximal_rank_cokernel_dimensions", "predicted_nonlinear_betti"], "episode_id": "eval_sample_3_ci_tor_quadric_hypersurfaces_conic", "episode_type": "exact_benchmark", "input": {"edge_ci_degrees": [[1, 2]], "vertex_ci_degrees": [[2], [2]]}, "milestone_id": "m2", "policy_visibility": "public", "problem_id": "sample_3", "prompt": "Compute the maximal-rank cokernel prediction for two quadratic hypersurface coordinate rings glued along a linear-plus-quadratic complete intersection.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "eval", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "complete_intersection_tor"} |
| {"candidate_required_fields": ["gcd", "normalized_generators", "apery_set", "conductor", "frobenius_number", "satisfies_bound"], "episode_id": "eval_sample_4_semigroup_p5_p6", "episode_type": "exact_benchmark", "input": {"proposed_bound": 36, "raw_generators": [6, 7]}, "milestone_id": "m2", "policy_visibility": "public", "problem_id": "sample_4", "prompt": "For the product benchmark with raw degrees 6 and 7, compute the exact conductor and test the quadratic bound 36.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "eval", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "numerical_semigroup"} |
| {"candidate_required_fields": ["form_valid", "pairing_determinant", "cycle_span_rank", "cycle_span_index", "transvection_isometry", "transvection_determinant", "contains_pairing_one"], "episode_id": "eval_sample_5_lattice_rank4_primitive", "episode_type": "exact_benchmark", "input": {"cycles": [[1, 0, 0, 0], [0, 1, 0, 0], [1, 0, 1, 0], [0, 0, 0, 1]], "form_type": "alternating", "pairing_matrix": [[0, 1, 0, 0], [-1, 0, 0, 0], [0, 0, 0, 1], [0, 0, -1, 0]], "transvection_cycle": [1, 1, 0, 0]}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_5", "prompt": "Verify primitivity, pairing-one connectivity data, and an exact transvection for the rank-four configuration.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "eval", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "integral_lattice"} |
| {"candidate_required_fields": ["chain_condition", "homology"], "episode_id": "eval_sample_6_chain_free_plus_torsion", "episode_type": "exact_benchmark", "input": {"boundaries": {"1": [[0, 0]], "2": [[0], [2]]}, "chain_dimensions": {"0": 1, "1": 2, "2": 1}}, "milestone_id": "m2", "policy_visibility": "public", "problem_id": "sample_6", "prompt": "Compute integral homology, including torsion, for the supplied three-term chain complex.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "eval", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "integer_chain_complex"} |
| {"candidate_required_fields": ["strictly_raises_filtration", "nilpotence_exponent"], "episode_id": "eval_sample_7_filtration_four_step_two_generators", "episode_type": "exact_benchmark", "input": {"basis_levels": [0, 1, 2, 3], "generators": [[[0, 0, 0, 0], [1, 0, 0, 0], [0, 1, 0, 0], [0, 0, 1, 0]], [[0, 0, 0, 0], [0, 0, 0, 0], [1, 0, 0, 0], [0, 1, 0, 0]]]}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_7", "prompt": "Compute the nilpotence exponent of the ideal generated by the two exact filtration-raising matrices.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "eval", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "filtration_nilpotence"} |
| {"candidate_required_fields": ["torsion_invariant_factors", "free_rank", "finite_torsion_cokernel", "torsion_order", "regular_character_count", "every_character_multiplicity_one"], "episode_id": "eval_sample_8_smith_klein_four", "episode_type": "exact_benchmark", "input": {"matching_matrix": [[2, 0], [0, 2]]}, "milestone_id": "m2", "policy_visibility": "public", "problem_id": "sample_8", "prompt": "Distinguish the order-four torsion group with invariant factors 2,2 from the cyclic order-four case.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "eval", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "smith_root_group"} |
| {"candidate_required_fields": ["weights", "cycle_coefficient_product", "codimension_increment_count", "predicted_terminal_multiplicity"], "episode_id": "eval_sample_9_hasse_p3_terminal_ten", "episode_type": "exact_benchmark", "input": {"levels": [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], "prime": 3, "terminal_supersingular_level": true}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_9", "prompt": "Compute all ten odd-characteristic Hasse weights at p=3, their exact product, and state the terminal supersingular multiplicity used by the task.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "eval", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "k3_hasse_arithmetic"} |
| {"candidate_required_fields": ["p_adic_valuation_of_e", "tame", "relative_log_differential_p_complete_zero", "relative_log_differential_p_complete_nonzero", "module_p_primary_annihilator"], "episode_id": "eval_sample_10_kummer_wild_e9_p3", "episode_type": "exact_benchmark", "input": {"prime": 3, "ramification_index": 9}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_10", "prompt": "For p=3 and e=9, compute the exact p-adic valuation, tameness, and p-primary annihilator of the Kummer dlog class.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "eval", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "kummer_log_differential"} |
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