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0b28a2c | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 | {"candidate_required_fields": ["rank", "dimension", "count"], "episode_id": "dev_sample_1_top_chern_quintic_threefold_lines", "episode_type": "exact_benchmark", "input_json": "{\"degree\": 5, \"k\": 2, \"n\": 5}", "milestone_id": "m1", "milestone_target": "", "policy_visibility": "public", "problem_id": "sample_1", "prompt_json": "\"Compute exactly ∫_Gr(2,5) c_6(Sym^5 S*) by torus localization. Return the rank, dimension, and integer count.\"", "prompt_kind": "string", "prompt_text": "Compute exactly ∫_Gr(2,5) c_6(Sym^5 S*) by torus localization. Return the rank, dimension, and integer count.", "required_artifact_policy": "", "reward_mode": "exact_machine", "schema_version": "3.0", "source_fields": ["candidate_required_fields", "episode_id", "episode_type", "input", "milestone_id", "policy_visibility", "problem_id", "prompt", "reward_mode", "schema_version", "split", "submission_schema_ref", "verifier_id"], "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "target_milestone_ids": [], "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_json": "{\"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", "milestone_target": "", "policy_visibility": "public", "problem_id": "sample_1", "prompt_json": "\"Repeat the local Fano-section calculation for a scaled fifth jet and certify whether the obstruction is nonzero.\"", "prompt_kind": "string", "prompt_text": "Repeat the local Fano-section calculation for a scaled fifth jet and certify whether the obstruction is nonzero.", "required_artifact_policy": "", "reward_mode": "exact_machine", "schema_version": "3.0", "source_fields": ["candidate_required_fields", "episode_id", "episode_type", "input", "milestone_id", "policy_visibility", "problem_id", "prompt", "reward_mode", "schema_version", "split", "submission_schema_ref", "verifier_id"], "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "target_milestone_ids": [], "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_json": "{\"curve_degree\": 1, \"curve_family_dimension\": 3, \"curve_genus\": 0, \"fano_index\": 3, \"linear_system_multiple\": 8}", "milestone_id": "m1", "milestone_target": "", "policy_visibility": "public", "problem_id": "sample_2", "prompt_json": "\"For a quadric threefold of index 3 with a 3-dimensional line family and d=8, compute the incidence codimension.\"", "prompt_kind": "string", "prompt_text": "For a quadric threefold of index 3 with a 3-dimensional line family and d=8, compute the incidence codimension.", "required_artifact_policy": "", "reward_mode": "exact_machine", "schema_version": "3.0", "source_fields": ["candidate_required_fields", "episode_id", "episode_type", "input", "milestone_id", "policy_visibility", "problem_id", "prompt", "reward_mode", "schema_version", "split", "submission_schema_ref", "verifier_id"], "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "target_milestone_ids": [], "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_json": "{\"edge_ci_degrees\": [[1, 1]], \"vertex_ci_degrees\": [[1], [1]]}", "milestone_id": "m2", "milestone_target": "", "policy_visibility": "public", "problem_id": "sample_3", "prompt_json": "\"Compute the same Tor benchmark when both vertices and the edge are cut out only by linear equations.\"", "prompt_kind": "string", "prompt_text": "Compute the same Tor benchmark when both vertices and the edge are cut out only by linear equations.", "required_artifact_policy": "", "reward_mode": "exact_machine", "schema_version": "3.0", "source_fields": ["candidate_required_fields", "episode_id", "episode_type", "input", "milestone_id", "policy_visibility", "problem_id", "prompt", "reward_mode", "schema_version", "split", "submission_schema_ref", "verifier_id"], "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "target_milestone_ids": [], "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_json": "{\"proposed_bound\": 6, \"raw_generators\": [3, 4]}", "milestone_id": "m2", "milestone_target": "", "policy_visibility": "public", "problem_id": "sample_4", "prompt_json": "\"Compute the normalized numerical semigroup generated by 3 and 4, including its Apéry set, Frobenius number, conductor, and bound test.\"", "prompt_kind": "string", "prompt_text": "Compute the normalized numerical semigroup generated by 3 and 4, including its Apéry set, Frobenius number, conductor, and bound test.", "required_artifact_policy": "", "reward_mode": "exact_machine", "schema_version": "3.0", "source_fields": ["candidate_required_fields", "episode_id", "episode_type", "input", "milestone_id", "policy_visibility", "problem_id", "prompt", "reward_mode", "schema_version", "split", "submission_schema_ref", "verifier_id"], "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "target_milestone_ids": [], "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_json": "{\"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", "milestone_target": "", "policy_visibility": "public", "problem_id": "sample_5", "prompt_json": "\"Compute the integral span index and verify the transvection for the supplied rank-four vanishing-cycle configuration.\"", "prompt_kind": "string", "prompt_text": "Compute the integral span index and verify the transvection for the supplied rank-four vanishing-cycle configuration.", "required_artifact_policy": "", "reward_mode": "exact_machine", "schema_version": "3.0", "source_fields": ["candidate_required_fields", "episode_id", "episode_type", "input", "milestone_id", "policy_visibility", "problem_id", "prompt", "reward_mode", "schema_version", "split", "submission_schema_ref", "verifier_id"], "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "target_milestone_ids": [], "verifier_id": "integral_lattice"}
{"candidate_required_fields": ["chain_condition", "homology"], "episode_id": "dev_sample_6_chain_moore2", "episode_type": "exact_benchmark", "input_json": "{\"boundaries\": {\"1\": [[2]]}, \"chain_dimensions\": {\"0\": 1, \"1\": 1}}", "milestone_id": "m2", "milestone_target": "", "policy_visibility": "public", "problem_id": "sample_6", "prompt_json": "\"Compute the integral homology of the two-term complex Z --2--> Z.\"", "prompt_kind": "string", "prompt_text": "Compute the integral homology of the two-term complex Z --2--> Z.", "required_artifact_policy": "", "reward_mode": "exact_machine", "schema_version": "3.0", "source_fields": ["candidate_required_fields", "episode_id", "episode_type", "input", "milestone_id", "policy_visibility", "problem_id", "prompt", "reward_mode", "schema_version", "split", "submission_schema_ref", "verifier_id"], "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "target_milestone_ids": [], "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_json": "{\"basis_levels\": [0, 1, 2], \"generators\": [[[0, 0, 0], [1, 0, 0], [0, 1, 0]]]}", "milestone_id": "m1", "milestone_target": "", "policy_visibility": "public", "problem_id": "sample_7", "prompt_json": "\"Verify strict filtration increase and compute the minimal exponent killing every product.\"", "prompt_kind": "string", "prompt_text": "Verify strict filtration increase and compute the minimal exponent killing every product.", "required_artifact_policy": "", "reward_mode": "exact_machine", "schema_version": "3.0", "source_fields": ["candidate_required_fields", "episode_id", "episode_type", "input", "milestone_id", "policy_visibility", "problem_id", "prompt", "reward_mode", "schema_version", "split", "submission_schema_ref", "verifier_id"], "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "target_milestone_ids": [], "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_json": "{\"matching_matrix\": [[4]]}", "milestone_id": "m2", "milestone_target": "", "policy_visibility": "public", "problem_id": "sample_8", "prompt_json": "\"Compute the Smith invariants and character count for a cyclic order-four root-choice group.\"", "prompt_kind": "string", "prompt_text": "Compute the Smith invariants and character count for a cyclic order-four root-choice group.", "required_artifact_policy": "", "reward_mode": "exact_machine", "schema_version": "3.0", "source_fields": ["candidate_required_fields", "episode_id", "episode_type", "input", "milestone_id", "policy_visibility", "problem_id", "prompt", "reward_mode", "schema_version", "split", "submission_schema_ref", "verifier_id"], "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "target_milestone_ids": [], "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_json": "{\"levels\": [1, 2], \"prime\": 5}", "milestone_id": "m1", "milestone_target": "", "policy_visibility": "public", "problem_id": "sample_9", "prompt_json": "\"Compute the first two higher-Hasse weights and cycle coefficient for p=5.\"", "prompt_kind": "string", "prompt_text": "Compute the first two higher-Hasse weights and cycle coefficient for p=5.", "required_artifact_policy": "", "reward_mode": "exact_machine", "schema_version": "3.0", "source_fields": ["candidate_required_fields", "episode_id", "episode_type", "input", "milestone_id", "policy_visibility", "problem_id", "prompt", "reward_mode", "schema_version", "split", "submission_schema_ref", "verifier_id"], "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "target_milestone_ids": [], "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_json": "{\"prime\": 5, \"ramification_index\": 5}", "milestone_id": "m1", "milestone_target": "", "policy_visibility": "public", "problem_id": "sample_10", "prompt_json": "\"For p=5 and e=5, compute the p-primary annihilator and whether the dlog class is nonzero.\"", "prompt_kind": "string", "prompt_text": "For p=5 and e=5, compute the p-primary annihilator and whether the dlog class is nonzero.", "required_artifact_policy": "", "reward_mode": "exact_machine", "schema_version": "3.0", "source_fields": ["candidate_required_fields", "episode_id", "episode_type", "input", "milestone_id", "policy_visibility", "problem_id", "prompt", "reward_mode", "schema_version", "split", "submission_schema_ref", "verifier_id"], "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "target_milestone_ids": [], "verifier_id": "kummer_log_differential"}
{"candidate_required_fields": [], "episode_id": "milestone_sample_8_m2", "episode_type": "targeted_frontier_milestone", "input_json": "", "milestone_id": "", "milestone_target": "Prove the one-edge contact-m regular-representation formula over an etale trivialization.", "policy_visibility": "public", "problem_id": "sample_8", "prompt_json": "{\"allowed_tools\": [\"Smith normal form\", \"derived fiber-product calculations\", \"equivariant K-theory software where available\", \"symbolic finite-group character calculations\"], \"conjecture\": \"For a labelled rigid genus-zero tropical type tau in a projective log-smooth simple-normal-crossings degeneration over characteristic zero, let B_tau be the derived fiber product of the vertex stable-map moduli along evaluation diagonals, let M_tau be the moduli of basic logarithmic maps of that type, and let mu_tau forget the logarithmic root choices. If A_tau is the torsion cokernel of the tropical integral matching map and G_tau is its Cartier dual, then mu_tau is canonically a G_tau-torsor after rigidification, the obstruction theory of M_tau is the finite-etale pullback of the virtual diagonal-gluing theory, and R mu_(tau,*) O^vir_(M_tau) = F_tau tensor mu_(tau,*) O_(M_tau) in G_tau-equivariant G-theory. Etale-locally the second factor is the regular representation, so every character occurs once; forgetting characters recovers only the usual scalar tropical multiplicity |A_tau|.\", \"definitions\": \"The vertex moduli M_v parametrize relative or expanded stable maps associated with the vertices of the labelled genus-zero tree tau. Their derived fiber product B_tau is formed by matching evaluations for every bounded edge, and F_tau is the K-theoretic virtual pullback of the external product of their virtual structure sheaves along those diagonals. The integral matching map Phi_tau records the edge and vertex matching equations; A_tau=tors(coker Phi_tau), and G_tau=D(A_tau) is its finite diagonalizable Cartier dual. For the independent-edge benchmark with contact orders m_e, G_tau is the product of the groups mu_(m_e). Rigidification means quotienting the labelled type stack by the explicitly specified residual automorphism inertia before asserting that mu_tau is a torsor.\", \"instruction\": \"Attempt to prove or refute the stated conjecture for “A character-valued logarithmic gluing formula”. A valid submission may instead complete one or more scored milestones. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.\", \"required_deliverables\": [\"claim_status: proved, disproved, partial, or inconclusive\", \"a precise main result with hypotheses and quantifiers\", \"a complete proof or reproducible computation for each claimed milestone\", \"an adversarial check or counterexample search\", \"a list of unresolved gaps and dependencies\", \"a literature boundary that distinguishes known input from new work\"]}", "prompt_kind": "object", "prompt_text": "{\"allowed_tools\": [\"Smith normal form\", \"derived fiber-product calculations\", \"equivariant K-theory software where available\", \"symbolic finite-group character calculations\"], \"conjecture\": \"For a labelled rigid genus-zero tropical type tau in a projective log-smooth simple-normal-crossings degeneration over characteristic zero, let B_tau be the derived fiber product of the vertex stable-map moduli along evaluation diagonals, let M_tau be the moduli of basic logarithmic maps of that type, and let mu_tau forget the logarithmic root choices. If A_tau is the torsion cokernel of the tropical integral matching map and G_tau is its Cartier dual, then mu_tau is canonically a G_tau-torsor after rigidification, the obstruction theory of M_tau is the finite-etale pullback of the virtual diagonal-gluing theory, and R mu_(tau,*) O^vir_(M_tau) = F_tau tensor mu_(tau,*) O_(M_tau) in G_tau-equivariant G-theory. Etale-locally the second factor is the regular representation, so every character occurs once; forgetting characters recovers only the usual scalar tropical multiplicity |A_tau|.\", \"definitions\": \"The vertex moduli M_v parametrize relative or expanded stable maps associated with the vertices of the labelled genus-zero tree tau. Their derived fiber product B_tau is formed by matching evaluations for every bounded edge, and F_tau is the K-theoretic virtual pullback of the external product of their virtual structure sheaves along those diagonals. The integral matching map Phi_tau records the edge and vertex matching equations; A_tau=tors(coker Phi_tau), and G_tau=D(A_tau) is its finite diagonalizable Cartier dual. For the independent-edge benchmark with contact orders m_e, G_tau is the product of the groups mu_(m_e). Rigidification means quotienting the labelled type stack by the explicitly specified residual automorphism inertia before asserting that mu_tau is a torsor.\", \"instruction\": \"Attempt to prove or refute the stated conjecture for “A character-valued logarithmic gluing formula”. A valid submission may instead complete one or more scored milestones. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.\", \"required_deliverables\": [\"claim_status: proved, disproved, partial, or inconclusive\", \"a precise main result with hypotheses and quantifiers\", \"a complete proof or reproducible computation for each claimed milestone\", \"an adversarial check or counterexample search\", \"a list of unresolved gaps and dependencies\", \"a literature boundary that distinguishes known input from new work\"]}", "required_artifact_policy": "hash every machine-generated artifact and provide an exact rerun command", "reward_mode": "hybrid_expert_gated", "schema_version": "3.0", "source_fields": ["episode_id", "episode_type", "milestone_target", "policy_visibility", "problem_id", "prompt", "required_artifact_policy", "reward_mode", "schema_version", "split", "submission_schema_ref", "target_milestone_ids"], "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "target_milestone_ids": ["m2"], "verifier_id": ""}
{"candidate_required_fields": [], "episode_id": "milestone_sample_9_m1", "episode_type": "targeted_frontier_milestone", "input_json": "", "milestone_id": "", "milestone_target": "Reconstruct the fiberwise identification of cyclotomic degree -2 with H^2(W O_X) and the finite-height Frobenius criterion.", "policy_visibility": "public", "problem_id": "sample_9", "prompt_json": "{\"allowed_tools\": [\"spectral-sequence bookkeeping\", \"de Rham-Witt calculations\", \"local deformation-ring computation\", \"computer algebra for complete intersections\"], \"conjecture\": \"Let p be odd, let p not divide 2d, and let M be the moduli stack of primitively polarized K3 surfaces of degree 2d over F_p. The degree -2 cyclotomic homotopy object of THH globalizes to a base-change-compatible V-complete Cartier sheaf for the universal K3 family. For h=1,...,10, its successive V-adic Frobenius obstructions are sections a_h of the Hodge line lambda^(p^h-1), and the recursive derived zero locus obtained by imposing a_1,...,a_h has classical truncation equal to the natural scheme-theoretic height-at-least-(h+1) stratum, with height at least 11 interpreted as the supersingular locus. On the finite-height locus the successive inclusions are regular Cartier divisors, while the tenth obstruction recovers the natural multiplicity-two supersingular cycle.\", \"definitions\": \"For a K3 surface X over a perfect field, C(X)=pi_{-2}^{cyc} THH(X) is a derived V-complete p-typical Cartier module. Antieau-Nikolaus identify it with H^2(X,W O_X). Finite V-quotients recover finite Witt cohomology, and van der Geer-Katsura characterize the formal Brauer height as the least Witt level at which Frobenius is nonzero. A relative cyclotomic Cartier sheaf is a sheafified family of these objects with strong base change. After lower Frobenius components vanish, the next semilinear coefficient defines a higher Hasse section in lambda^(p^h-1).\", \"instruction\": \"Attempt to prove or refute the stated conjecture for “A cyclotomic Hasse tower for K3 moduli”. A valid submission may instead complete one or more scored milestones. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.\", \"required_deliverables\": [\"claim_status: proved, disproved, partial, or inconclusive\", \"a precise main result with hypotheses and quantifiers\", \"a complete proof or reproducible computation for each claimed milestone\", \"an adversarial check or counterexample search\", \"a list of unresolved gaps and dependencies\", \"a literature boundary that distinguishes known input from new work\"]}", "prompt_kind": "object", "prompt_text": "{\"allowed_tools\": [\"spectral-sequence bookkeeping\", \"de Rham-Witt calculations\", \"local deformation-ring computation\", \"computer algebra for complete intersections\"], \"conjecture\": \"Let p be odd, let p not divide 2d, and let M be the moduli stack of primitively polarized K3 surfaces of degree 2d over F_p. The degree -2 cyclotomic homotopy object of THH globalizes to a base-change-compatible V-complete Cartier sheaf for the universal K3 family. For h=1,...,10, its successive V-adic Frobenius obstructions are sections a_h of the Hodge line lambda^(p^h-1), and the recursive derived zero locus obtained by imposing a_1,...,a_h has classical truncation equal to the natural scheme-theoretic height-at-least-(h+1) stratum, with height at least 11 interpreted as the supersingular locus. On the finite-height locus the successive inclusions are regular Cartier divisors, while the tenth obstruction recovers the natural multiplicity-two supersingular cycle.\", \"definitions\": \"For a K3 surface X over a perfect field, C(X)=pi_{-2}^{cyc} THH(X) is a derived V-complete p-typical Cartier module. Antieau-Nikolaus identify it with H^2(X,W O_X). Finite V-quotients recover finite Witt cohomology, and van der Geer-Katsura characterize the formal Brauer height as the least Witt level at which Frobenius is nonzero. A relative cyclotomic Cartier sheaf is a sheafified family of these objects with strong base change. After lower Frobenius components vanish, the next semilinear coefficient defines a higher Hasse section in lambda^(p^h-1).\", \"instruction\": \"Attempt to prove or refute the stated conjecture for “A cyclotomic Hasse tower for K3 moduli”. A valid submission may instead complete one or more scored milestones. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.\", \"required_deliverables\": [\"claim_status: proved, disproved, partial, or inconclusive\", \"a precise main result with hypotheses and quantifiers\", \"a complete proof or reproducible computation for each claimed milestone\", \"an adversarial check or counterexample search\", \"a list of unresolved gaps and dependencies\", \"a literature boundary that distinguishes known input from new work\"]}", "required_artifact_policy": "hash every machine-generated artifact and provide an exact rerun command", "reward_mode": "hybrid_expert_gated", "schema_version": "3.0", "source_fields": ["episode_id", "episode_type", "milestone_target", "policy_visibility", "problem_id", "prompt", "required_artifact_policy", "reward_mode", "schema_version", "split", "submission_schema_ref", "target_milestone_ids"], "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "target_milestone_ids": ["m1"], "verifier_id": ""}
{"candidate_required_fields": [], "episode_id": "milestone_sample_10_m1", "episode_type": "targeted_frontier_milestone", "input_json": "", "milestone_id": "", "milestone_target": "Derive the relative logarithmic differential module B/eB generated by dlog(t) for the Kummer chart and verify tame vanishing.", "policy_visibility": "public", "problem_id": "sample_10", "prompt_json": "{\"allowed_tools\": [\"spectral-sequence computation\", \"derived log-cotangent calculations\", \"group cohomology software\", \"exact Kummer-extension arithmetic\"], \"conjecture\": \"Let L/K be a finite Galois extension of p-adic local fields with group G, and give their valuation rings the divisorial log structures. The p-completed log-TP descent map TP^log(O_K;Z_p) -> TP^log(O_L;Z_p)^{hG} is an equivalence if and only if L/K is tamely ramified. For a totally ramified Kummer extension pi_K=pi_L^e with p dividing e, the module O_L/e O_L generated by dlog(pi_L) is the first wild class in the Hodge-Tate linearization and survives in the filtered descent defect.\", \"definitions\": \"TP^log(O_K;Z_p) is the circle Tate construction on p-completed cyclotomic logarithmic THH of the divisorial pre-log ring (O_K,M_K). The descent defect D_TP^log(L/K) is the fiber of the map from the base log-TP spectrum to G-homotopy fixed points of the extension spectrum. Tame means that the ramification index is prime to p; residue extensions of p-adic local fields are automatically separable.\", \"instruction\": \"Attempt to prove or refute the stated conjecture for “Logarithmic TP as a tame-ramification detector”. A valid submission may instead complete one or more scored milestones. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.\", \"required_deliverables\": [\"claim_status: proved, disproved, partial, or inconclusive\", \"a precise main result with hypotheses and quantifiers\", \"a complete proof or reproducible computation for each claimed milestone\", \"an adversarial check or counterexample search\", \"a list of unresolved gaps and dependencies\", \"a literature boundary that distinguishes known input from new work\"]}", "prompt_kind": "object", "prompt_text": "{\"allowed_tools\": [\"spectral-sequence computation\", \"derived log-cotangent calculations\", \"group cohomology software\", \"exact Kummer-extension arithmetic\"], \"conjecture\": \"Let L/K be a finite Galois extension of p-adic local fields with group G, and give their valuation rings the divisorial log structures. The p-completed log-TP descent map TP^log(O_K;Z_p) -> TP^log(O_L;Z_p)^{hG} is an equivalence if and only if L/K is tamely ramified. For a totally ramified Kummer extension pi_K=pi_L^e with p dividing e, the module O_L/e O_L generated by dlog(pi_L) is the first wild class in the Hodge-Tate linearization and survives in the filtered descent defect.\", \"definitions\": \"TP^log(O_K;Z_p) is the circle Tate construction on p-completed cyclotomic logarithmic THH of the divisorial pre-log ring (O_K,M_K). The descent defect D_TP^log(L/K) is the fiber of the map from the base log-TP spectrum to G-homotopy fixed points of the extension spectrum. Tame means that the ramification index is prime to p; residue extensions of p-adic local fields are automatically separable.\", \"instruction\": \"Attempt to prove or refute the stated conjecture for “Logarithmic TP as a tame-ramification detector”. A valid submission may instead complete one or more scored milestones. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.\", \"required_deliverables\": [\"claim_status: proved, disproved, partial, or inconclusive\", \"a precise main result with hypotheses and quantifiers\", \"a complete proof or reproducible computation for each claimed milestone\", \"an adversarial check or counterexample search\", \"a list of unresolved gaps and dependencies\", \"a literature boundary that distinguishes known input from new work\"]}", "required_artifact_policy": "hash every machine-generated artifact and provide an exact rerun command", "reward_mode": "hybrid_expert_gated", "schema_version": "3.0", "source_fields": ["episode_id", "episode_type", "milestone_target", "policy_visibility", "problem_id", "prompt", "required_artifact_policy", "reward_mode", "schema_version", "split", "submission_schema_ref", "target_milestone_ids"], "split": "dev", "submission_schema_ref": "schemas/submission.schema.json", "target_milestone_ids": ["m1"], "verifier_id": ""}
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