| {"candidate_required_fields": ["rank", "dimension", "count"], "episode_id": "dev_sample_1_top_chern_quintic_threefold_lines", "episode_type": "exact_benchmark", "input": {"degree": 5, "k": 2, "n": 5}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_1", "prompt": "Compute exactly ∫_Gr(2,5) c_6(Sym^5 S*) by torus localization. Return the rank, dimension, and integer count.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "dev", "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": "dev_sample_1_local_fano_scaled_fold", "episode_type": "exact_benchmark", "input": {"quadrics": ["x0**2", "x1**2", "x2**2", "x3**2", "2*x3*(x0+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": "Repeat the local Fano-section calculation for a scaled fifth jet and certify whether the obstruction is nonzero.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "local_fano_deformation"} | |
| {"candidate_required_fields": ["h0_restriction", "incidence_codimension", "line_formula_matches"], "episode_id": "dev_sample_2_incidence_quadric_lines_d8", "episode_type": "exact_benchmark", "input": {"curve_degree": 1, "curve_family_dimension": 3, "curve_genus": 0, "fano_index": 3, "linear_system_multiple": 8}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_2", "prompt": "For a quadric threefold of index 3 with a 3-dimensional line family and d=8, compute the incidence codimension.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "dev", "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": "dev_sample_3_ci_tor_linear_planes", "episode_type": "exact_benchmark", "input": {"edge_ci_degrees": [[1, 1]], "vertex_ci_degrees": [[1], [1]]}, "milestone_id": "m2", "policy_visibility": "public", "problem_id": "sample_3", "prompt": "Compute the same Tor benchmark when both vertices and the edge are cut out only by linear equations.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "dev", "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": "dev_sample_4_semigroup_three_four", "episode_type": "exact_benchmark", "input": {"proposed_bound": 6, "raw_generators": [3, 4]}, "milestone_id": "m2", "policy_visibility": "public", "problem_id": "sample_4", "prompt": "Compute the normalized numerical semigroup generated by 3 and 4, including its Apéry set, Frobenius number, conductor, and bound test.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "dev", "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": "dev_sample_5_lattice_rank4_index2", "episode_type": "exact_benchmark", "input": {"cycles": [[2, 0, 0, 0], [0, 1, 0, 0], [0, 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, 0, 1, 0]}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_5", "prompt": "Compute the integral span index and verify the transvection for the supplied rank-four vanishing-cycle configuration.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "integral_lattice"} | |
| {"candidate_required_fields": ["chain_condition", "homology"], "episode_id": "dev_sample_6_chain_moore2", "episode_type": "exact_benchmark", "input": {"boundaries": {"1": [[2]]}, "chain_dimensions": {"0": 1, "1": 1}}, "milestone_id": "m2", "policy_visibility": "public", "problem_id": "sample_6", "prompt": "Compute the integral homology of the two-term complex Z --2--> Z.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "integer_chain_complex"} | |
| {"candidate_required_fields": ["strictly_raises_filtration", "nilpotence_exponent"], "episode_id": "dev_sample_7_filtration_three_step", "episode_type": "exact_benchmark", "input": {"basis_levels": [0, 1, 2], "generators": [[[0, 0, 0], [1, 0, 0], [0, 1, 0]]]}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_7", "prompt": "Verify strict filtration increase and compute the minimal exponent killing every product.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "dev", "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": "dev_sample_8_smith_cyclic_four", "episode_type": "exact_benchmark", "input": {"matching_matrix": [[4]]}, "milestone_id": "m2", "policy_visibility": "public", "problem_id": "sample_8", "prompt": "Compute the Smith invariants and character count for a cyclic order-four root-choice group.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "smith_root_group"} | |
| {"candidate_required_fields": ["weights", "cycle_coefficient_product", "codimension_increment_count"], "episode_id": "dev_sample_9_hasse_p5_first_two", "episode_type": "exact_benchmark", "input": {"levels": [1, 2], "prime": 5}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_9", "prompt": "Compute the first two higher-Hasse weights and cycle coefficient for p=5.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "dev", "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": "dev_sample_10_kummer_wild_e5_p5", "episode_type": "exact_benchmark", "input": {"prime": 5, "ramification_index": 5}, "milestone_id": "m1", "policy_visibility": "public", "problem_id": "sample_10", "prompt": "For p=5 and e=5, compute the p-primary annihilator and whether the dlog class is nonzero.", "reward_mode": "exact_machine", "schema_version": "3.0", "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "verifier_id": "kummer_log_differential"} | |