| {"candidate_required_fields": ["rank", "dimension", "count"], "episode_id": "train_sample_1_top_chern_cubic_surface_lines", "episode_type": "exact_benchmark", "input": {"degree": 3, "k": 2, "n": 4}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_1", "prompt": "Compute exactly ∫_Gr(2,4) c_4(Sym^3 S*) by torus localization. Return the rank, dimension, and integer count.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "train", "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": "train_sample_1_local_fano_canonical_fold", "episode_type": "exact_benchmark", "input": {"quadrics": ["x0**2", "x1**2", "x2**2", "x3**2", "x3*(x0+x1+x2)"], "second_order_terms": ["y3**2*x2"], "x_variables": ["x0", "x1", "x2", "x3"]}, "milestone_id": "m2", "policy_visibility": "public", "problem_id": "sample_1", "prompt": "For the supplied five quadratic jets, compute the exact 20×20 multiplication map, its kernel, and the quadratic obstruction from the supplied second-order term.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "train", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "local_fano_deformation"} | |
| {"candidate_required_fields": ["h0_restriction", "incidence_codimension", "line_formula_matches"], "episode_id": "train_sample_2_incidence_p3_lines_d7", "episode_type": "exact_benchmark", "input": {"curve_degree": 1, "curve_family_dimension": 4, "curve_genus": 0, "fano_index": 4, "linear_system_multiple": 7}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_2", "prompt": "A line moves in a 4-dimensional family in P^3. For surfaces of degree 7, compute the line-incidence codimension.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "train", "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": "train_sample_3_ci_tor_quadric_surfaces_conic", "episode_type": "exact_benchmark", "input": {"edge_ci_degrees": [[1, 1, 2]], "vertex_ci_degrees": [[1, 2], [1, 2]]}, "milestone_id": "m2", "policy_visibility": "public", "problem_id": "sample_3", "prompt": "Use Koszul resolutions to compute the linear-strand source/target dimensions and the maximal-rank nonlinear Betti prediction.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "train", "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": "train_sample_4_semigroup_f1", "episode_type": "exact_benchmark", "input": {"proposed_bound": 2, "raw_generators": [2, 3]}, "milestone_id": "m2", "policy_visibility": "public", "problem_id": "sample_4", "prompt": "Normalize the raw free anticanonical degrees 2 and 3, compute the Apéry set and conductor, and test the stated bound 2.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "train", "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": "train_sample_5_lattice_rank2_basis", "episode_type": "exact_benchmark", "input": {"cycles": [[1, 0], [0, 1]], "form_type": "alternating", "pairing_matrix": [[0, 1], [-1, 0]], "transvection_cycle": [1, 0]}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_5", "prompt": "Verify the standard rank-two skew lattice, the span index of the two cycles, and the Picard-Lefschetz transvection in the first cycle.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "train", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "integral_lattice"} | |
| {"candidate_required_fields": ["chain_condition", "homology"], "episode_id": "train_sample_6_chain_circle", "episode_type": "exact_benchmark", "input": {"boundaries": {"1": [[0]]}, "chain_dimensions": {"0": 1, "1": 1}}, "milestone_id": "m2", "policy_visibility": "public", "problem_id": "sample_6", "prompt": "Verify the chain condition and integral homology of the one-cell CW chain complex for S^1.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "train", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "integer_chain_complex"} | |
| {"candidate_required_fields": ["strictly_raises_filtration", "nilpotence_exponent"], "episode_id": "train_sample_7_filtration_two_step", "episode_type": "exact_benchmark", "input": {"basis_levels": [0, 1], "generators": [[[0, 0], [1, 0]]]}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_7", "prompt": "Verify that the supplied operator strictly raises a two-step filtration and compute the ideal nilpotence exponent.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "train", "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": "train_sample_8_smith_contacts_two_three", "episode_type": "exact_benchmark", "input": {"matching_matrix": [[2, 0], [0, 3]]}, "milestone_id": "m2", "policy_visibility": "public", "problem_id": "sample_8", "prompt": "Compute the torsion cokernel and regular-character count for independent contact orders 2 and 3.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "train", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "smith_root_group"} | |
| {"candidate_required_fields": ["weights", "cycle_coefficient_product", "codimension_increment_count"], "episode_id": "train_sample_9_hasse_p3_first_three", "episode_type": "exact_benchmark", "input": {"levels": [1, 2, 3], "prime": 3}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_9", "prompt": "Compute the Hodge-line exponents p^h-1 and their product for p=3 and h=1,2,3.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "train", "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": "train_sample_10_kummer_tame_e3_p5", "episode_type": "exact_benchmark", "input": {"prime": 5, "ramification_index": 3}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_10", "prompt": "For p=5 and ramification index e=3, determine tameness and the p-completed relative logarithmic differential module.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "train", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "kummer_log_differential"} | |