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+{"client_summary": "A concrete high-degree monodromy problem with an exact local model, a symbolic enumerative check, and a sharply isolated global transversality step.", "difficulty": "frontier_research", "domain": "Enumerative algebraic geometry", "evaluation": {"expert_review_required_for_full_credit": true, "hard_failure_conditions": ["Treating numerical approximation as an exact transversality certificate.", "Inferring full symmetric monodromy without proving both the transposition and alternating-group containment hypotheses.", "Ignoring planes meeting the marked plane."], "milestones": [{"milestone_id": "m1", "target": "Verify that the top Chern number on Gr(4,9) equals 321489.", "verification_mode": "exact_symbolic", "weight_percent": 10}, {"milestone_id": "m2", "target": "For the displayed normal jets, prove that the 20-by-20 multiplication map has rank 19, identify its kernel, and verify a nonzero quadratic obstruction.", "verification_mode": "exact_linear_algebra_and_CAS", "weight_percent": 25}, {"milestone_id": "m3", "target": "Prove smoothness of the cubic along the marked plane and identify the completed local Fano algebra as C[[t]]/(t^2).", "verification_mode": "formal_proof_plus_local_CAS", "weight_percent": 20}, {"milestone_id": "m4", "target": "Produce and certify one completion whose Fano section is transverse away from the marked plane, or prove that no such completion exists.", "verification_mode": "exact_CAS_or_certified_specialization_plus_expert_review", "weight_percent": 35}, {"milestone_id": "m5", "target": "Derive rigorously that alternating-group containment plus this simple ramification gives full symmetric monodromy.", "verification_mode": "formal_group_theory_review", "weight_percent": 10}], "recommended_training_modes": ["expert_supervised_process_RL", "tool_augmented_research_agent_evaluation", "preference_and_critic_training"], "rl_suitability_score_10": 9, "verification_profile": "High for the enumerative and local-algebra milestones; specialist review for global transversality."}, "license": "MIT", "prerequisites": ["intersection theory on Grassmannians", "deformation theory of Fano schemes", "computational algebraic geometry", "permutation-group monodromy"], "problem_id": "sample_1", "prompt": {"allowed_tools": ["SageMath or SymPy", "Macaulay2, Singular, or Magma", "exact finite-field computation", "certified numerical algebraic geometry followed by exact verification"], "conjecture": "There exists a smooth complex cubic sevenfold X in P^8 whose Fano scheme F_3(X) is finite of length 321489, reduced except at exactly one 3-plane Lambda, where the completed local ring is C[[t]]/(t^2). Consequently the geometric monodromy of the 321489 three-planes on a general cubic sevenfold is the full symmetric group S_321489.", "definitions": "F_3(X) is the zero scheme on Gr(4,9) of the section of Sym^3(S dual) induced by the cubic equation. An ordinary double plane is an isolated point Lambda whose completed local Fano algebra is C[[t]]/(t^2). The universal incidence over the open locus of finite reduced Fano schemes is a degree-321489 finite etale cover, and its geometric monodromy acts on those planes.", "instruction": "Prove or refute the stated conjecture for “A single simple branch among 321,489 planes”. A valid submission may instead complete one or more scored milestones. Label every theorem, computation, and literature claim 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 reproducible code, exact inputs, and machine-checkable outputs for every computational claim.", "required_deliverables": ["claim_status: proved, disproved, partial, or inconclusive", "a precise main result with hypotheses and quantifiers", "a complete proof or reproducible computation for every claimed milestone", "an adversarial check or counterexample search", "an explicit list of unresolved gaps and dependencies", "a literature boundary separating known input from new work"]}, "quality_signals": ["exact enumerative target", "explicit local algebra", "clear monodromy consequence"], "release_problem_id": "aim_ag_001", "release_schema_version": "1.0.0", "research_area": "Fano schemes, discriminants, and geometric monodromy", "research_context": {"closest_known_results": ["Debarre--Manivel prove that a general cubic sevenfold has a reduced zero-dimensional Fano scheme of length 321489.", "Hashimoto--Kadets prove A_321489 is contained in the monodromy, so only the sign character remains; they expect symmetric monodromy in all nonexceptional cases but do not decide this one.", "Harris proves that a unique double solution gives a simple transposition in enumerative monodromy.", "Yahl certifies simple transpositions and symmetric groups for nonexceptional Fano problems of degree below 75000, but 321489 is outside the theorem.", "No published source located in the targeted review produced a smooth cubic sevenfold with exactly one double 3-plane or determined the corresponding degree-321489 monodromy group."], "established_progress": [{"statement": "For the explicit five quadratic normal jets, the Fano-section differential has rank 19 with a one-dimensional kernel; the term y_3^2 x_2 gives a nonzero quadratic obstruction, the cubic is smooth along the marked plane, and the marked Fano point has completed local algebra C[[t]]/(t^2).", "type": "verified_local_result"}], "falsifiable_prediction": "In the affine family of cubics whose first normal jet along Lambda is y_0 x_0^2+y_1 x_1^2+y_2 x_2^2+y_3 x_3^2+y_4 x_3(x_0+x_1+x_2) and whose y_3^2 x_2 coefficient is nonzero, there is a smooth member transverse at every Fano point other than Lambda. It then has one length-two plane and 321487 reduced planes. A proof that every such member has a second nontransverse plane, or that the generic degree-321489 Fano discriminant is a square, refutes the prediction.", "hardness_rationale": "The local singularity can be engineered by a 20-by-20 multiplication map of corank one, but the same cubic has 321487 other geometric planes that must all remain transverse. Proving this globally requires a stratified incidence calculation for planes meeting the prescribed plane, or a certified discriminant/Frobenius sign computation in a degree-321489 finite algebra. Existing general theory stops at containment of the alternating group, and published computation covers only degrees below 75000.", "main_obstruction": "Prove the global separation lemma: among cubic completions preserving the explicit first normal jet and nonzero quadratic obstruction at Lambda, find one whose Fano section is transverse on Gr(4,9) minus Lambda. Equivalently, exclude an unavoidable second ramification point by stratifying remote planes according to their intersection with Lambda, or certify a nonsquare Fano discriminant in one arithmetic specialization.", "motivation": "Debarre--Manivel show that a general cubic sevenfold has 321489 reduced 3-planes. Hashimoto--Kadets prove that the associated monodromy contains A_321489, leaving only the alternating-versus-symmetric sign. Harris's local method shows that one fiber with exactly one ordinary double plane supplies a transposition and therefore forces the symmetric group. An explicit first and second normal jet gives the desired local length-two point on a smooth cubic, reducing the conjecture to remote transversality.", "references": [{"citation": "Olivier Debarre and Laurent Manivel, Sur la variete des espaces lineaires contenus dans une intersection complete, Mathematische Annalen 312 (1998), 549-574.", "relevance": "Primary source for generic smoothness, the top-Chern degree formula, and the table giving 321489 for 3-planes on a cubic in P^8.", "url": "https://doi.org/10.1007/s002080050235"}, {"citation": "Sachi Hashimoto and Borys Kadets, 38406501359372282063949 and All That: Monodromy of Fano Problems, International Mathematics Research Notices 2022 (2022), no. 5, 3349-3370.", "relevance": "Primary source proving that this nonexceptional Fano monodromy group contains the alternating group and stating the broad expectation of symmetric monodromy.", "url": "https://doi.org/10.1093/imrn/rnaa275"}, {"citation": "Joe Harris, Galois groups of enumerative problems, Duke Mathematical Journal 46 (1979), no. 4, 685-724.", "relevance": "Primary source for geometric monodromy of enumerative problems and the simple-transposition method from a unique double solution.", "url": "https://doi.org/10.1215/S0012-7094-79-04635-0"}, {"citation": "Thomas Yahl, Computing Galois groups of Fano problems, Journal of Symbolic Computation 119 (2023), 81-89.", "relevance": "Primary source implementing certified double-point constructions and proving full symmetric monodromy for nonexceptional Fano problems of degree below 75000, a bound not covering 321489.", "url": "https://doi.org/10.1016/j.jsc.2023.02.004"}, {"citation": "Frank Gounelas and Alexis Kouvidakis, The Fermat cubic and monodromy of lines, New York Journal of Mathematics 31 (2025), 650-667.", "relevance": "Primary modern example of full symmetric geometric monodromy arising from detailed geometry of lines on cubic threefolds and fourfolds.", "url": "https://nyjm.albany.edu/j/2025/31-23.html"}], "tractability_rationale": "One group-theoretic half is already known: A_321489 is contained in the monodromy. An explicit local calculation proves a concrete local proposition using the normal jet q_0=x_0^2, q_1=x_1^2, q_2=x_2^2, q_3=x_3^2, q_4=x_3(x_0+x_1+x_2) and the term y_3^2 x_2. Its Fano-section differential has rank 19 and nonzero quadratic obstruction, while the cubic remains smooth along the plane. Only a finite-dimensional global separation lemma remains, with algebraic incidence and arithmetic sign routes available."}, "research_status": {"classification": "candidate_open_problem", "expert_signoff_required": true, "frontier_contribution": "Construct a smooth cubic sevenfold with exactly one ordinary double 3-plane and all remaining planes reduced, thereby forcing full symmetric monodromy.", "literature_boundary": "Related results exist; the stated frontier contribution is narrower than the known inputs.", "novelty_policy": "Final novelty and attribution signoff should be performed by a domain expert before publication."}, "source_stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "A single simple branch among 321,489 planes"}
+{"client_summary": "A sharp extremal-classification problem combining incidence geometry with equality-sensitive Hodge-theoretic estimates.", "difficulty": "frontier_research", "domain": "Hodge theory and Fano geometry", "evaluation": {"expert_review_required_for_full_credit": true, "hard_failure_conditions": ["Using a parameter count alone as proof of the reverse Noether-Lefschetz inequality.", "Assuming every Hodge class is represented by a curve of bounded degree.", "Claiming equality classification from generic Noether-Lefschetz properness alone."], "milestones": [{"milestone_id": "m1", "target": "Prove irreducibility and codimension d+1-h of the line-incidence component for a fixed line-Hilbert component H.", "verification_mode": "formal_algebraic_geometry_review", "weight_percent": 20}, {"milestone_id": "m2", "target": "Verify line-regularity and the expected dimension h=iota in at least one nontrivial Fano family.", "verification_mode": "symbolic_or_literature_verified_computation", "weight_percent": 15}, {"milestone_id": "m3", "target": "Establish the sharp infinitesimal Hodge-locus lower bound for one specified family beyond projective space.", "verification_mode": "formal_proof_plus_computer_algebra_when_applicable", "weight_percent": 35}, {"milestone_id": "m4", "target": "Classify equality in that family or construct a genuine lower- or equal-codimension counterexample not arising from a line.", "verification_mode": "expert_proof_review", "weight_percent": 30}], "recommended_training_modes": ["expert_supervised_process_RL", "tool_augmented_research_agent_evaluation", "preference_and_critic_training"], "rl_suitability_score_10": 7, "verification_profile": "Moderate: incidence and family-specific algebra are checkable; the universal equality classification requires specialist proof review."}, "license": "MIT", "prerequisites": ["variation of Hodge structure", "Noether-Lefschetz theory", "Fano threefolds", "Hilbert schemes and incidence correspondences"], "problem_id": "sample_2", "prompt": {"allowed_tools": ["Macaulay2 or Singular", "Borel-Weil-Bott and Jacobian-ring calculations", "Hilbert-scheme computation", "symbolic linear algebra"], "conjecture": "Let Y be a general line-regular Picard-rank-one smooth complex Fano threefold with Pic(Y)=Z[A], A a very ample primitive generator, and -K_Y=iota A. For all sufficiently large d, every component of the Noether-Lefschetz locus of smooth surfaces in |dA| has codimension at least d-iota+1, and equality occurs exactly for the loci of surfaces containing a line from an irreducible component of the Hilbert scheme of A-lines.", "definitions": "The Noether-Lefschetz locus consists of smooth S in |dA| for which Pic(Y)->Pic(S) is not surjective. An A-line is a smooth rational curve ell with A.ell=1. Line-regular means that the line Hilbert scheme is nonempty, generically reduced, pure of the expected dimension iota, and has an unobstructed general member in each component.", "instruction": "Prove or refute the stated conjecture for “Lines as the largest Noether-Lefschetz loci on Fano threefolds”. A valid submission may instead complete one or more scored milestones. Label every theorem, computation, and literature claim 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 reproducible code, exact inputs, and machine-checkable outputs for every computational claim.", "required_deliverables": ["claim_status: proved, disproved, partial, or inconclusive", "a precise main result with hypotheses and quantifiers", "a complete proof or reproducible computation for every claimed milestone", "an adversarial check or counterexample search", "an explicit list of unresolved gaps and dependencies", "a literature boundary separating known input from new work"]}, "quality_signals": ["sharp codimension target", "proved incidence component", "family-by-family computational route"], "release_problem_id": "aim_ag_002", "release_schema_version": "1.0.0", "research_area": "Noether-Lefschetz loci, Hilbert schemes of lines, and infinitesimal Hodge theory", "research_context": {"closest_known_results": ["Green-Voisin prove for degree-d surfaces in projective three-space that every component has codimension at least d-3 and that equality occurs only for surfaces containing a line.", "Otwinowska asymptotically identifies successive low-codimension Hodge loci of high-degree projective hypersurfaces with loci containing low-degree subvarieties.", "Lopez-Maclean prove explicit regularity-dependent lower bounds for arbitrary smooth threefolds, but that theorem does not supply the sharp Fano-index equality classification proposed here.", "Bruzzo-Grassi and subsequent toric work study line components and sharp codimension bounds in simplicial projective toric threefolds; after writing dA=-K_Y+(d-iota)A, the natural toric bound matches d-iota+1.", "Joshi proves generic Noether-Lefschetz for sufficiently high powers on arbitrary smooth projective threefolds, establishing properness but not the minimum-component classification."], "established_progress": [{"proof_summary": "Uniform Serre vanishing makes the line-surface incidence a projective bundle over H, and a line imposes h0(O_P1(d))=d+1 conditions. The incidence of surfaces containing two distinct H-lines has dimension at most dim|dA|-(2d+1)+2h, smaller than the single-line incidence for d>h, so projection is generically one-to-one. Bertini gives smooth members. The line class cannot restrict from mA because intersecting with A would give 1=m d A^3.", "statement": "If H is an h-dimensional irreducible component of the Hilbert scheme of A-lines on Y, then for d sufficiently large the locus of smooth surfaces in |dA| containing an H-line is irreducible of codimension d+1-h and lies in the Noether-Lefschetz locus. Under line-regularity this equals d-iota+1.", "type": "proved_reduction"}], "falsifiable_prediction": "For a line-regular general cubic threefold, every minimum-codimension component of the Noether-Lefschetz locus in |dA| has codimension d-1 and is induced by the Fano surface of lines for all sufficiently large d. Any lower-codimension component, or any equal-codimension component whose general surface contains no line, refutes the conjecture.", "hardness_rationale": "The elementary incidence calculation supplies components of the predicted codimension, but the reverse inequality must exclude all abstract Hodge classes and curve classes whose degree or genus grows with d. Existing arbitrary-threefold estimates lose constants, while equality requires a sharp Macaulay-growth theorem in a non-polynomial subcanonical coordinate ring.", "main_obstruction": "The missing equality-sensitive theorem must show that every Hodge locus of codimension at most d-iota+1 comes from an actual A-line. Current general bounds lose constants, and a curve-extraction proof must control classes whose degree and genus can grow with d.", "motivation": "In projective three-space, the largest Noether-Lefschetz component consists exactly of surfaces containing a line. For a Fano threefold the anticanonical shift suggests replacing d-3 by d-iota+1. The conjecture says that the smallest possible curve degree, not an abstract Hodge class, universally controls the largest Picard-jumping loci.", "references": [{"citation": "C. Voisin, Composantes de petite codimension du lieu de Noether-Lefschetz, Commentarii Mathematici Helvetici 64 (1989), 515-526.", "url": "https://doi.org/10.1007/BF02564692"}, {"citation": "A. Otwinowska, Composantes de petite codimension du lieu de Noether-Lefschetz: un argument asymptotique en faveur de la conjecture de Hodge pour les hypersurfaces, Journal of Algebraic Geometry 12 (2003), 307-320.", "url": "https://doi.org/10.1090/S1056-3911-02-00349-1"}, {"citation": "K. Joshi, A general Noether-Lefschetz theorem and applications, Journal of Algebraic Geometry 4 (1995), 105-135.", "url": "https://arxiv.org/abs/alg-geom/9305001"}, {"citation": "A. F. Lopez and C. Maclean, Explicit Noether-Lefschetz for arbitrary threefolds, Mathematical Proceedings of the Cambridge Philosophical Society 143 (2007), no. 2.", "url": "https://doi.org/10.1017/S0305004107000126"}, {"citation": "U. Bruzzo and A. Grassi, The Noether-Lefschetz locus of surfaces in toric threefolds, Communications in Contemporary Mathematics 20 (2018), no. 5, 1750070.", "url": "https://doi.org/10.1142/S0219199717500705"}, {"citation": "V. Lanza and I. Martino, On the codimension of Noether-Lefschetz loci for toric threefolds, Communications in Contemporary Mathematics 22 (2020), no. 5, 1950037.", "url": "https://arxiv.org/abs/1807.11426"}, {"citation": "U. Bruzzo and W. D. Montoya, Codimension bounds for the Noether-Lefschetz components for toric varieties, European Journal of Mathematics 8 (2022), 806-814.", "url": "https://doi.org/10.1007/s40879-021-00461-0"}], "tractability_rationale": "The statement is exactly known for projective three-space, agrees with sharp toric Noether-Lefschetz bounds, and matches incidence counts on quadrics, cubic threefolds, and prime index-one Fanos. The missing statement can be attacked family by family through explicit hypersurface or complete-intersection coordinate rings."}, "research_status": {"classification": "candidate_open_problem", "expert_signoff_required": true, "frontier_contribution": "Prove the optimal codimension lower bound and classify every equality component as a line-incidence locus on a general line-regular Picard-rank-one Fano threefold.", "literature_boundary": "Related results exist; the stated frontier contribution is narrower than the known inputs.", "novelty_policy": "Final novelty and attribution signoff should be performed by a domain expert before publication."}, "source_stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "Lines as the largest Noether-Lefschetz loci on Fano threefolds"}
+{"client_summary": "A highly verifiable syzygy problem with an exact reduction, explicit Betti-table benchmarks, and determinantal rank tests.", "difficulty": "frontier_research", "domain": "Commutative algebra and projective geometry", "evaluation": {"expert_review_required_for_full_credit": true, "hard_failure_conditions": ["Replacing geometry-induced maps by arbitrary generic chain maps.", "Checking rank at one point without proving membership in the intended irreducible family.", "Ignoring triple intersections or incidence cycles excluded by the clean-tree hypotheses."], "milestones": [{"milestone_id": "m1", "target": "Prove the Mayer-Vietoris short exact sequence, the regularity bound, and the cokernel description of the nonlinear Tor strand.", "verification_mode": "formal_commutative_algebra_review", "weight_percent": 20}, {"milestone_id": "m2", "target": "Reproduce the benchmark of two quadric surfaces meeting in a conic and its unique beta_(2,4)=1.", "verification_mode": "Macaulay2_or_exact_Betti_verifier", "weight_percent": 20}, {"milestone_id": "m3", "target": "For a nontrivial irreducible incidence family, construct one member where every relevant edge map has maximal rank.", "verification_mode": "exact_CAS_plus_semicontinuity_argument", "weight_percent": 35}, {"milestone_id": "m4", "target": "Resolve the first multi-edge compatibility case or exhibit a generic rank defect that refutes the conjecture.", "verification_mode": "exact_CAS_plus_expert_review", "weight_percent": 25}], "recommended_training_modes": ["expert_supervised_process_RL", "tool_augmented_research_agent_evaluation", "preference_and_critic_training"], "rl_suitability_score_10": 9, "verification_profile": "High for reductions and bounded examples; specialist review for the generic dominance theorem."}, "license": "MIT", "prerequisites": ["graded free resolutions", "Castelnuovo-Mumford regularity", "varieties of minimal degree", "determinantal semicontinuity"], "problem_id": "sample_3", "prompt": {"allowed_tools": ["Macaulay2", "Singular", "SageMath", "exact random specialization over finite fields", "determinantal rank computation"], "conjecture": "Fix a nonempty irreducible characteristic-zero parameter family of clean tree arrangements X=union X_v in projective space, with fixed tree, Hilbert polynomials, span dimensions, and incidence data, such that every component X_v and every edge overlap D_e is a variety of minimal degree in its span. For a general member, every signed restriction map Phi_q from the direct sum of the degree-(q+1) pieces of Tor_q of the vertex coordinate rings to the corresponding direct sum for the edge coordinate rings has maximal rank.", "definitions": "A clean tree arrangement has scheme-theoretic pairwise intersections exactly along the edges of a tree, no triple intersections, and a leaf ordering in which each new component meets the previous union only in its parent overlap and the two relevant linear spans intersect in the span of that overlap. The map Phi_q is induced by the two quotient maps R_v -> R_e at every edge, with opposite signs. Maximal rank means rank equal to the minimum of the total source and target dimensions.", "instruction": "Prove or refute the stated conjecture for “Generic maximal-rank edge maps for tree-glued varieties of minimal degree”. A valid submission may instead complete one or more scored milestones. Label every theorem, computation, and literature claim 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 reproducible code, exact inputs, and machine-checkable outputs for every computational claim.", "required_deliverables": ["claim_status: proved, disproved, partial, or inconclusive", "a precise main result with hypotheses and quantifiers", "a complete proof or reproducible computation for every claimed milestone", "an adversarial check or counterexample search", "an explicit list of unresolved gaps and dependencies", "a literature boundary separating known input from new work"]}, "quality_signals": ["proved Tor reduction", "explicit Betti prediction", "natural computer-algebra verifier"], "release_problem_id": "aim_ag_003", "release_schema_version": "1.0.0", "research_area": "Syzygies of reducible varieties and Mayer-Vietoris Tor maps", "research_context": {"closest_known_results": ["Eisenbud-Green-Hulek-Popescu characterize reduced 2-regular schemes by linearly joined sequences of varieties of minimal degree and give explicit ideal and resolution constructions for gluing along linear spaces. This is the closest result and makes novelty uncertain.", "Morales analyzes generators, depth, cohomological dimension, and resolutions of linearly joined arrangements, but the exact generic Tor maximal-rank formula was not located in that source.", "Ran proves property N_p for certain reducible curves from componentwise numerical hypotheses using Koszul-cohomology vanishing rather than the edge-map formula.", "Burnham-Rosen-Sidman-Vermeire study graph curves, their equations and property N_p, and explicitly ask for combinatorial control of graded Betti numbers; their intersections are linear rather than nonlinear minimal-degree overlaps."], "established_progress": [{"proof_summary": "The clean tree gives 0 -> R_X -> direct_sum_v R_v -> direct_sum_e R_e -> 0. Vertex and edge coordinate rings have regularity at most 1 because they are varieties of minimal degree with possible additional linear span equations. The regularity inequality gives reg R_X <= 2. In the long exact Tor sequence at internal degree q+1, the following vertex term Tor_{q-1}(R_v,k)_{q+1} vanishes by regularity, leaving the stated cokernel isomorphism.", "statement": "For every clean tree arrangement, reg_S(R_X) is at most 2 and reg_S(I_X) is at most 3. Moreover Tor^S_{q-1}(R_X,k)_{q+1} is naturally the cokernel of Phi_q for every q at least 1.", "type": "proved_reduction"}, {"statement": "For every q>=1, beta_{q-1,q+1}^S(R_X)=max(0, sum_e dim Tor_q^S(R_e,k)_{q+1} - sum_v dim Tor_q^S(R_v,k)_{q+1}).", "type": "derived_prediction"}], "falsifiable_prediction": "For every nonempty irreducible characteristic-zero family of clean tree arrangements with fixed incidence data, a general member satisfies the displayed Betti formula in every homological degree. One explicit family whose generic signed edge map Phi_q has rank below the minimum of its source and target dimensions for any q refutes the conjecture.", "hardness_rationale": "The same geometric inclusion controls the Tor maps in every homological degree, so maximal rank cannot be imposed independently degree by degree. Resolutions of scrolls, cones, quadrics, and the Veronese surface have structured morphisms, and several nonlinear edges incident to one vertex can force compatibility relations. Generic chain maps between abstract resolutions need not arise from actual projective incidences, making the required dominance statement genuinely geometric.", "main_obstruction": "Morphisms of linear resolutions induced by actual inclusions D_e into X_v occupy a constrained geometric subset of all chain maps. It is unknown whether this subset is large enough to meet the maximal-rank locus simultaneously in every homological degree, especially at a component incident to several nonlinear edges.", "motivation": "Varieties of minimal degree have completely linear ideals, but a nonlinear minimal-degree overlap can create a new 3-linear strand when such varieties are glued. A maximal-rank theorem would say that a general tree union has no hidden global dependencies: every nonlinear Betti number is determined by a simple excess of edge linear-strand dimensions over vertex dimensions. This gives a finite, computable attachment law for resolutions of a natural reducible class.", "references": [{"citation": "D. Eisenbud, M. Green, K. Hulek, and S. Popescu, Small schemes and varieties of minimal degree, American Journal of Mathematics 128 (2006), 1363-1389; arXiv:math/0404517.", "url": "https://arxiv.org/abs/math/0404517"}, {"citation": "M. Morales, Simplicial ideals, 2-linear ideals and arithmetical rank, arXiv:math/0702668.", "url": "https://arxiv.org/abs/math/0702668"}, {"citation": "Z. Ran, On the syzygies of reducible curves, arXiv:1303.6002.", "url": "https://arxiv.org/abs/1303.6002"}, {"citation": "G. Burnham, Z. Rosen, J. Sidman, and P. Vermeire, Line arrangements modeling curves of high degree: equations, syzygies and secants, arXiv:1201.5010.", "url": "https://arxiv.org/abs/1201.5010"}, {"citation": "M. Green, Koszul cohomology and the geometry of projective varieties, Journal of Differential Geometry 19 (1984), 125-171.", "url": "https://doi.org/10.4310/JDG/1214438426"}], "tractability_rationale": "A rigorous Mayer-Vietoris reduction shows that these edge maps are the only possible source of nonlinear syzygies and that every clean tree union is at most 3-regular. Rank failure is determinantal and therefore closed in a parameter family. The conjecture is vacuous for one component, recovers the known 2-regular theorem when all overlaps are linear, and matches exactly the first nonlinear model of two quadric surfaces meeting along a conic."}, "research_status": {"classification": "candidate_open_problem", "expert_signoff_required": true, "frontier_contribution": "Show that the geometry-induced edge maps have simultaneous maximal rank, making every nonlinear Betti number a computable excess of edge over vertex linear strands.", "literature_boundary": "Related results exist; the stated frontier contribution is narrower than the known inputs.", "novelty_policy": "Final novelty and attribution signoff should be performed by a domain expert before publication."}, "source_stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "Generic maximal-rank edge maps for tree-glued varieties of minimal degree"}
+{"client_summary": "A finite fan-combinatorics program that turns a frontier existence bound into exact Hilbert-basis and numerical-semigroup computations.", "difficulty": "frontier_research", "domain": "Toric geometry and numerical semigroups", "evaluation": {"expert_review_required_for_full_credit": true, "hard_failure_conditions": ["Computing only cone generators instead of the integral monoid Hilbert basis.", "Forgetting gcd normalization.", "Equating finite generation with a conductor bound without proving cofinality in the normalized gcd-one lattice."], "milestones": [{"milestone_id": "m1", "target": "Prove that free-map classes are exactly the integral curve classes nonnegative on all effective torus-invariant divisors, under the stated multiple-cover convention.", "verification_mode": "formal_toric_geometry_review", "weight_percent": 25}, {"milestone_id": "m2", "target": "Implement a reproducible pipeline from fan data to the Hilbert basis, normalized degree semigroup, Apéry set, and conductor.", "verification_mode": "deterministic_code_verifier", "weight_percent": 20}, {"milestone_id": "m3", "target": "Run the pipeline on all 18 smooth toric Fano threefolds and verify or refute c_X<=4.", "verification_mode": "deterministic_enumeration_with_artifacts", "weight_percent": 30}, {"milestone_id": "m4", "target": "Prove a dimension-only residue-filling bound for non-extremal Hilbert-basis elements, or find a higher-dimensional counterexample.", "verification_mode": "expert_proof_or_exact_counterexample_review", "weight_percent": 25}], "recommended_training_modes": ["expert_supervised_process_RL", "tool_augmented_research_agent_evaluation", "preference_and_critic_training"], "rl_suitability_score_10": 9, "verification_profile": "High for fixed dimensions and classified fan lists; specialist review for the uniform dimension bound."}, "license": "MIT", "prerequisites": ["smooth toric varieties and Cox coordinates", "Mori cones and divisor duality", "Hilbert bases of affine semigroups", "numerical semigroup conductors"], "problem_id": "sample_4", "prompt": {"allowed_tools": ["SageMath", "Normaliz", "polymake", "Macaulay2", "exact numerical-semigroup code"], "conjecture": "Let X be a smooth projective toric Fano variety of dimension n over an algebraically closed characteristic-zero field. Let M_X be the monoid of integral numerical curve classes nonnegative on every effective divisor, let g_X be the gcd of the positive anticanonical degrees -K_X.beta for beta in M_X, and normalize those degrees by g_X. The conductor c_X of the resulting numerical semigroup satisfies c_X <= floor((n+1)^2/4). Equivalently, every normalized integer at least floor((n+1)^2/4) is the anticanonical degree of a free morphism P1 -> X.", "definitions": "N_1(X)_Z is the numerical curve lattice. M_X=N_1(X)_Z intersect Eff^1(X)^dual consists of integral classes beta with D.beta>=0 for every effective divisor D. Gamma_X is {0} union {(-K_X.beta)/g_X: nonzero beta in M_X}, where g_X is the gcd of all positive degrees. Its conductor is the least c such that every integer m>=c lies in Gamma_X. A map f:P1->X is free when f^*T_X is globally generated; multiple covers are allowed.", "instruction": "Prove or refute the stated conjecture for “A conductor bound for free anticanonical degrees on toric Fano varieties”. A valid submission may instead complete one or more scored milestones. Label every theorem, computation, and literature claim 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 reproducible code, exact inputs, and machine-checkable outputs for every computational claim.", "required_deliverables": ["claim_status: proved, disproved, partial, or inconclusive", "a precise main result with hypotheses and quantifiers", "a complete proof or reproducible computation for every claimed milestone", "an adversarial check or counterexample search", "an explicit list of unresolved gaps and dependencies", "a literature boundary separating known input from new work"]}, "quality_signals": ["dimension-only quantitative target", "finite threefold benchmark", "deterministic fan-to-semigroup pipeline"], "release_problem_id": "aim_ag_004", "release_schema_version": "1.0.0", "research_area": "Free rational curves, fan monoids, Hilbert bases, and Apéry sets", "research_context": {"closest_known_results": ["Cox gives a functorial homogeneous-coordinate description of all maps from projective space to a smooth toric variety; this yields a reduction of free-map degrees to the lattice monoid dual to effective divisors.", "Cox-von Renesse and Batyrev's primitive-collection theory describe nef and related fan cones, but that source does not bound the conductor of an anticanonical degree image.", "Fu-Hwang classify minimal dominating families through centered primitive collections, controlling selected low-degree free curves rather than eventual degree saturation.", "Ilten-Levinson classify smooth embedded rational-curve components by Cayley structures for a chosen projective degree, not free maps or normalized anticanonical conductors.", "Lehmann-Riedl-Tanimoto prove asymptotic structure theorems for non-free curves on general Fano varieties, but no explicit dimension-only toric conductor appears in the cited paper."], "established_progress": [], "falsifiable_prediction": "For each of the 18 smooth toric Fano threefold fans, compute the monoid of integral beta satisfying D_rho.beta>=0, its Hilbert basis, the normalized anticanonical degree semigroup, and its conductor. The conjecture predicts c_X<=4 in every case. A single fan whose normalized semigroup omits any integer m>=4 refutes it.", "hardness_rationale": "The dual effective cone of a smooth toric variety need not be unimodular, so its lattice monoid can have non-extremal Hilbert-basis elements. Mori theory bounds lengths of extremal rational curves but does not directly control those interior elements or the residue classes of their anticanonical degrees. The conjecture requires a new bridge from Fano fan geometry to the Apery set of a numerical semigroup.", "main_obstruction": "There is no proved bound linking the anticanonical degrees and residue coverage of non-extremal Hilbert-basis elements of Eff^1(X)^dual to the dimension. Smoothness controls individual fan cones and Mori theory controls extremal curves, but neither directly controls the Apery set of the degree semigroup.", "motivation": "Free rational curves are central to deformation theory and the geometry of Fano varieties, while toric Cox coordinates make their classes explicitly combinatorial. The anticanonical degrees of free maps form an eventual sequence after gcd normalization, but no explicit dimension-only onset was found. A quadratic conductor bound would turn eventual existence into a finite fan invariant and would give a uniform low-degree threshold for constructing free maps.", "references": [{"citation": "David A. Cox, The functor of a smooth toric variety, Tohoku Mathematical Journal 47 (1995), 251-262; arXiv:alg-geom/9312001.", "relevance": "Describes maps to smooth toric varieties by compatible line bundles and nondegenerate sections, providing the Cox-coordinate reduction.", "url": "https://arxiv.org/abs/alg-geom/9312001"}, {"citation": "David A. Cox and Christine von Renesse, Primitive collections and toric varieties, arXiv:0808.1836.", "relevance": "Gives primitive-collection descriptions of nef cones and the fan inequalities adjacent to the proposed monoid calculation.", "url": "https://arxiv.org/abs/0808.1836"}, {"citation": "Baohua Fu and Jun-Muk Hwang, Minimal rational curves on complete toric manifolds, arXiv:0912.1638.", "relevance": "Identifies minimal dominating rational-curve components with special primitive collections but does not address all eventual free degrees.", "url": "https://arxiv.org/abs/0912.1638"}, {"citation": "Nathan Ilten and Jake Levinson, Rational curves in projective toric varieties, arXiv:2312.16590.", "relevance": "Classifies components of Hilbert schemes of smooth embedded rational curves using Cayley structures, a different degree and moduli problem.", "url": "https://arxiv.org/abs/2312.16590"}, {"citation": "Brian Lehmann, Eric Riedl, and Sho Tanimoto, Non-free curves on Fano varieties, Osaka Journal of Mathematics 62 (2025), 19-50; arXiv:2304.05438.", "relevance": "Studies high-degree non-free components of morphism spaces on Fano varieties but gives no toric free-degree conductor.", "url": "https://arxiv.org/abs/2304.05438"}], "tractability_rationale": "Cox's functor converts every integral class nonnegative on invariant divisors into a morphism from P1 using general homogeneous coordinates, and the toric Euler sequence makes that morphism free. Thus the conjecture reduces to exact finite polyhedral arithmetic. It holds for projective spaces, products tested symbolically, and every smooth toric Fano surface. Smooth toric Fano varieties are classified in each fixed dimension, making low-dimensional falsification practical."}, "research_status": {"classification": "candidate_open_problem", "expert_signoff_required": true, "frontier_contribution": "Prove the quadratic dimension-only conductor bound for normalized anticanonical degrees of free maps to smooth toric Fano varieties.", "literature_boundary": "No exact match was found in the targeted primary-source review.", "novelty_policy": "Final novelty and attribution signoff should be performed by a domain expert before publication."}, "source_stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "A conductor bound for free anticanonical degrees on toric Fano varieties"}
+{"client_summary": "A long-horizon proof task where integral lattice completeness, geometric vanishing cycles, and arithmetic-group criteria can be graded separately.", "difficulty": "frontier_research", "domain": "Arithmetic monodromy and Picard-Lefschetz theory", "evaluation": {"expert_review_required_for_full_credit": true, "hard_failure_conditions": ["Inferring finite index solely from Zariski density.", "Failing to saturate the integral vanishing lattice.", "Treating pencil monodromy as automatically equal to monodromy of the full smooth-divisor locus."], "milestones": [{"milestone_id": "m1", "target": "Reconstruct the finite-index implication from a complete integral vanishing lattice in the symplectic case.", "verification_mode": "formal_lattice_theory_review", "weight_percent": 15}, {"milestone_id": "m2", "target": "Prove integral generation and saturation of the vanishing lattice for one new ambient family.", "verification_mode": "expert_proof_review", "weight_percent": 25}, {"milestone_id": "m3", "target": "Realize the pairing-one or complete configurations required by the relevant vanishing-lattice theorem using high jet-ampleness.", "verification_mode": "formal_geometry_review", "weight_percent": 30}, {"milestone_id": "m4", "target": "Control the finite quadratic, spinor, or characteristic refinements in one orthogonal example.", "verification_mode": "exact_lattice_computation_plus_review", "weight_percent": 15}, {"milestone_id": "m5", "target": "Run an explicit thinness stress test and explain why Zariski density or large mod-prime images alone are insufficient.", "verification_mode": "adversarial_analysis_review", "weight_percent": 15}], "recommended_training_modes": ["expert_supervised_process_RL", "tool_augmented_research_agent_evaluation", "preference_and_critic_training"], "rl_suitability_score_10": 7, "verification_profile": "Moderate in explicit families; specialist review is required for the full arbitrary-ambient theorem."}, "license": "MIT", "prerequisites": ["Picard-Lefschetz theory", "integral lattices and congruence subgroups", "Lefschetz pencils", "vanishing-cycle monodromy"], "problem_id": "sample_5", "prompt": {"allowed_tools": ["SageMath lattice computations", "Magma", "symbolic intersection calculations", "finite congruence-image computation"], "conjecture": "Let Z be a smooth simply connected complex projective variety of dimension n+1 at least 2 and A an ample line bundle. For d sufficiently large, let U_d be the smooth-divisor locus in |A^d| and let Lambda_d be the saturated orthogonal complement of the ambient middle cohomology inside the torsion-free H^n of a smooth divisor, with its intersection form Q_d. Then the image of pi_1(U_d) in Aut(Lambda_d,Q_d) has finite index.", "definitions": "The integral vanishing lattice Lambda_d is (i^*H^n(Z,Z)_free)^{perp,sat} inside H^n(Y,Z)_free for a smooth Y in |A^d|. Its pairing is alternating for odd n and symmetric for even n. Finite index permits the monodromy to preserve a spin, quadratic, characteristic, or other finite refinement, so the conjecture does not predict surjectivity.", "instruction": "Prove or refute the stated conjecture for “Arithmeticity of vanishing cohomology in high-power linear systems”. A valid submission may instead complete one or more scored milestones. Label every theorem, computation, and literature claim 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 reproducible code, exact inputs, and machine-checkable outputs for every computational claim.", "required_deliverables": ["claim_status: proved, disproved, partial, or inconclusive", "a precise main result with hypotheses and quantifiers", "a complete proof or reproducible computation for every claimed milestone", "an adversarial check or counterexample search", "an explicit list of unresolved gaps and dependencies", "a literature boundary separating known input from new work"]}, "quality_signals": ["rigorous finite-index reduction", "clear thinness adversary", "explicit odd/even lattice split"], "release_problem_id": "aim_ag_005", "release_schema_version": "1.0.0", "research_area": "Integral vanishing lattices, transvection groups, and sufficiently ample linear systems", "research_context": {"closest_known_results": ["Beauville uses Janssen and Ebeling to compute integral monodromy for universal hypersurfaces and complete intersections in projective space, establishing the central special case and making novelty uncertain.", "Janssen's complete skew-symmetric vanishing-lattice theorem gives a congruence subgroup, while Ebeling supplies the symmetric complete-intersection-singularity theory used in the orthogonal cases.", "Banerjee-Salter identify the full mapping-class monodromy of sufficiently ample systems on arbitrary simply connected projective surfaces as an r-spin mapping class group; this implies the conjecture for curve divisors and is stronger there.", "Di Gennaro-Franco and Otwinowska-Saito establish rational irreducibility for broad high-degree divisor families in general ambients, but rational irreducibility does not exclude an infinite-index thin integral image."], "established_progress": [{"proof_summary": "Janssen's congruence-subgroup theorem says that the transvection group of a complete skew-symmetric vanishing lattice contains an integral congruence subgroup of the ambient symplectic isometry group. Congruence subgroups have finite index, and every such transvection is realized by a nodal loop in U_d. Therefore the containing full monodromy image is finite index. The unresolved work is verifying the integral completeness hypotheses geometrically.", "statement": "For odd middle dimension, if the geometric vanishing cycles of a Lefschetz pencil generate Lambda_d integrally and form a complete skew-symmetric vanishing lattice in Janssen's sense, then their Picard-Lefschetz group, and hence the full monodromy image, has finite index in Aut(Lambda_d,Q_d).", "type": "proved_reduction"}], "falsifiable_prediction": "One smooth simply connected complex projective Z, one ample A, and infinitely many arbitrarily large d for which the full U_d-monodromy is infinite index in Aut(Lambda_d,Q_d) refutes the conjecture. A particularly sharp counterexample would be Zariski dense with large mod-prime images but integrally thin.", "hardness_rationale": "Picard-Lefschetz theory supplies many transvections or reflections, but rational generation and large mod-prime images do not imply finite index. A proof must establish integral primitivity, generation, and prescribed intersection configurations among geometric vanishing cycles; the orthogonal case must also control characteristic and spinor refinements. These are global constraints on the discriminant complement, not formal consequences of high jet-ampleness.", "main_obstruction": "No general integral completeness theorem was located for the geometric vanishing cycles of high powers on arbitrary simply connected ambients. Rational irreducibility, Zariski density, or large reductions modulo primes do not establish finite index, and the even-dimensional orthogonal case also requires control of finite quadratic refinements.", "motivation": "Rational irreducibility and Zariski density do not rule out thin integral monodromy. A finite-index theorem would show that the abundance of nodal degenerations in a complete high-power linear system forces all but finitely much of the allowable integral symmetry of vanishing cohomology. It would extend a classical projective-space calculation and a recent surface theorem to arbitrary simply connected projective ambients.", "references": [{"citation": "A. Beauville, Le groupe de monodromie des familles universelles d'hypersurfaces et d'intersections completes, Complex Analysis and Algebraic Geometry (Gottingen, 1985), Lecture Notes in Mathematics 1194, Springer, 1986, 8-18.", "url": "https://doi.org/10.1007/BFb0076991"}, {"citation": "W. A. M. Janssen, Skew-Symmetric Vanishing Lattices and Their Monodromy Groups, Mathematische Annalen 266 (1984), 115-134.", "url": "https://eudml.org/doc/163844"}, {"citation": "W. Ebeling, Vanishing lattices and monodromy groups of isolated complete intersection singularities, Inventiones Mathematicae 90 (1987), 653-668.", "url": "https://doi.org/10.1007/BF01389184"}, {"citation": "I. Banerjee and N. Salter, Monodromy and vanishing cycles for sufficiently ample linear systems on simply connected surfaces, arXiv:2512.04018 (2025).", "url": "https://arxiv.org/abs/2512.04018"}, {"citation": "V. Di Gennaro and D. Franco, Monodromy of a family of hypersurfaces, Annales Scientifiques de l'Ecole Normale Superieure 42 (2009), 517-529; arXiv:0803.1627.", "url": "https://arxiv.org/abs/0803.1627"}, {"citation": "A. Otwinowska and M. Saito, Monodromy of a family of hypersurfaces containing a given subvariety, Annales Scientifiques de l'Ecole Normale Superieure 38 (2005), 365-386; arXiv:math/0404469.", "url": "https://arxiv.org/abs/math/0404469"}], "tractability_rationale": "Beauville proves the projective-space hypersurface case using the vanishing-lattice theorems of Janssen and Ebeling. Banerjee-Salter prove a stronger mapping-class result for sufficiently ample linear systems on every simply connected projective surface. Rational irreducibility is known for broad divisor families in arbitrary smooth ambients. High powers separate jets and should permit the finite nodal configurations needed by integral vanishing-lattice criteria."}, "research_status": {"classification": "candidate_open_problem", "expert_signoff_required": true, "frontier_contribution": "Extend finite-index integral monodromy from classical and surface cases to high powers on arbitrary simply connected projective ambients.", "literature_boundary": "Related results exist; the stated frontier contribution is narrower than the known inputs.", "novelty_policy": "Final novelty and attribution signoff should be performed by a domain expert before publication."}, "source_stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "Arithmeticity of vanishing cohomology in high-power linear systems"}
+{"client_summary": "A synthesis task connecting classical mapping-space stability to integral configuration-space and Cox-coordinate calculations.", "difficulty": "frontier_research", "domain": "Algebraic topology and toric geometry", "evaluation": {"expert_review_required_for_full_credit": true, "hard_failure_conditions": ["Replacing componentwise positivity by growth of one ample degree.", "Using rational stabilization to claim an integral isomorphism.", "Dropping local coefficient systems created by configuration-space monodromy."], "milestones": [{"milestone_id": "m1", "target": "Construct the based Cox-coordinate discriminant resolution over the relevant Picard or Jacobian base and verify its augmentation.", "verification_mode": "formal_topology_and_geometry_review", "weight_percent": 20}, {"milestone_id": "m2", "target": "Control orientations, sign local systems, and the first nontrivial integral differentials associated with overlapping primitive collections.", "verification_mode": "chain_complex_or_spectral_sequence_verifier", "weight_percent": 25}, {"milestone_id": "m3", "target": "Compute the H_1 and H_2 comparison for an elliptic source and the Hirzebruch surface F_1 in an explicit high-degree chamber.", "verification_mode": "reproducible_chain_computation_plus_review", "weight_percent": 30}, {"milestone_id": "m4", "target": "Prove the required integral scanning or group-completion statement, or exhibit persistent torsion obstructing it.", "verification_mode": "expert_proof_or_counterexample_review", "weight_percent": 25}], "recommended_training_modes": ["expert_supervised_process_RL", "tool_augmented_research_agent_evaluation", "preference_and_critic_training"], "rl_suitability_score_10": 8, "verification_profile": "Moderate: low-degree spectral-sequence pages and benchmark targets are computable; full integral stability needs specialist review."}, "license": "MIT", "prerequisites": ["spaces of holomorphic maps", "smooth projective toric varieties", "Vassiliev resolutions", "configuration-space local systems"], "problem_id": "sample_6", "prompt": {"allowed_tools": ["SageMath", "configuration-space chain complexes", "spectral-sequence bookkeeping code", "computer algebra for Cox presentations"], "conjecture": "Let C be a fixed smooth projective complex curve of genus g and X a smooth projective toric variety. For every i there is B(i,g,Sigma) such that, if a curve class beta satisfies beta.D_rho >= B for every invariant prime divisor, then the inclusion of the based algebraic mapping space Mor^*_beta(C,X) into the corresponding based continuous mapping-space component induces an isomorphism on integral homology in every degree at most i.", "definitions": "Mor^*_beta(C,X) consists of algebraic maps f:C->X taking a fixed c_0 to a fixed dense-torus point x_0 and representing beta, with its complex-analytic topology. Map^*_beta is the corresponding component of the based continuous mapping space. Componentwise positivity means that every d_rho=beta.D_rho tends to infinity, not merely one chosen ample degree.", "instruction": "Prove or refute the stated conjecture for “An integral Segal theorem for toric targets”. A valid submission may instead complete one or more scored milestones. Label every theorem, computation, and literature claim 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 reproducible code, exact inputs, and machine-checkable outputs for every computational claim.", "required_deliverables": ["claim_status: proved, disproved, partial, or inconclusive", "a precise main result with hypotheses and quantifiers", "a complete proof or reproducible computation for every claimed milestone", "an adversarial check or counterexample search", "an explicit list of unresolved gaps and dependencies", "a literature boundary separating known input from new work"]}, "quality_signals": ["classical boundary cases", "explicit low-degree torsion test", "spectral-sequence milestones"], "release_problem_id": "aim_ag_006", "release_schema_version": "1.0.0", "research_area": "Holomorphic mapping spaces, Cox discriminants, and integral homological stability", "research_context": {"closest_known_results": ["Segal proves an integral stable-range comparison for maps from a Riemann surface to projective space, but not the proposed arbitrary toric target statement.", "Guest proves the toric comparison for source P1 and notes that singular toric targets can obstruct an unbounded stable range.", "Halic proves smoothness and irreducibility of high-degree higher-genus toric morphism spaces and constructs a compactification, but does not identify their stable integral homology with continuous mapping spaces.", "Banerjee proves stabilization of cohomology for maps from a fixed curve to a complete simplicial toric variety and obtains arithmetic applications; the cited statement does not supply the proposed integral comparison."], "established_progress": [{"statement": "Halic proves that if every d_rho>2g-1, Mor_beta(C,X) is smooth, irreducible, and has dimension sum_rho d_rho - dim(X)(g-1). The toric Euler sequence gives H^1(C,f^*T_X)=0 and the expected dimension; Cox's vector-bundle quotient over Jac(C)^rkPic(X) supplies the global irreducibility mechanism.", "type": "proved_supporting_result"}], "falsifiable_prediction": "For an elliptic curve C and the Hirzebruch surface F_1, any sequence beta_m with min_rho(beta_m.D_rho)->infinity eventually induces isomorphisms in integral H_1 and H_2 from Mor^*_{beta_m}(C,F_1) to Map^*_{beta_m}(C,F_1). Persistent torsion in a kernel or cokernel refutes the conjecture; a discriminant resolution truncated at two bad points should detect it.", "hardness_rationale": "A Cox-discriminant resolution contains configuration spaces of points on C, sign local systems, and incidence morphisms between overlapping primitive collections. Rational transfer arguments can suppress finite monodromy, but integral torsion differentials and extensions may remain. Fan smoothness removes lattice-index torsion without automatically orienting the whole semisimplicial resolution.", "main_obstruction": "The unresolved issue is integral control of configuration-space monodromy and torsion differentials in the primitive-collection discriminant resolution. Rational stabilization does not automatically lift to Z coefficients.", "motivation": "Segal proves the integral comparison for projective-space targets, Guest proves a toric comparison for rational source curves, Halic describes high-degree maps from higher-genus curves to smooth toric varieties, and Banerjee proves higher-genus toric cohomological stabilization. The conjecture asks for the missing synthesis: arbitrary smooth projective toric target, positive-genus source, comparison with the continuous mapping space, and integral coefficients.", "references": [{"citation": "G. Segal, The topology of spaces of rational functions, Acta Mathematica 143 (1979), 39-72.", "relevance": "Provides the integral higher-genus comparison for projective-space targets.", "url": "https://doi.org/10.1007/BF02392088"}, {"citation": "D. A. Cox, The functor of a smooth toric variety, Tohoku Mathematical Journal 47 (1995), 251-262; arXiv:alg-geom/9312001.", "relevance": "Supplies the line-bundle-and-section coordinates used to construct the toric discriminant resolution.", "url": "https://arxiv.org/abs/alg-geom/9312001"}, {"citation": "M. A. Guest, The topology of the space of rational curves on a toric variety, Acta Mathematica 174 (1995), 119-145; arXiv:alg-geom/9301005.", "relevance": "Proves the genus-zero toric comparison with continuous mapping spaces through an increasing range.", "url": "https://arxiv.org/abs/alg-geom/9301005"}, {"citation": "M. Halic, Higher genus curves on toric varieties, arXiv:math/0101100 (2001).", "relevance": "Proves smoothness, irreducibility, expected dimension, and constructs a compactification in the componentwise high-degree chamber.", "url": "https://arxiv.org/abs/math/0101100"}, {"citation": "I. Ciocan-Fontanine and B. Kim, Moduli stacks of stable toric quasimaps, Advances in Mathematics 225 (2010), 3022-3051; arXiv:0908.4446.", "relevance": "Develops Picard-stack and section-bundle constructions for maps and quasimaps to toric targets.", "url": "https://arxiv.org/abs/0908.4446"}, {"citation": "O. Banerjee, Moduli of curves on toric varieties and their stable cohomology, arXiv:2210.05826 (2022).", "relevance": "Proves the closest higher-genus stable-cohomology result for complete simplicial toric varieties.", "url": "https://arxiv.org/abs/2210.05826"}], "tractability_rationale": "The conjecture is a theorem for projective spaces by Segal and is compatible with products of projective spaces. Guest proves the genus-zero toric face. Halic proves that the relevant higher-genus morphism spaces are smooth and irreducible of expected dimension once every invariant degree exceeds 2g-1, and Banerjee proves rational cohomological stabilization for complete simplicial toric targets."}, "research_status": {"classification": "candidate_open_problem", "expert_signoff_required": true, "frontier_contribution": "Prove an integral higher-genus Segal comparison for arbitrary smooth projective toric targets under componentwise positivity.", "literature_boundary": "No exact match was found in the targeted primary-source review.", "novelty_policy": "Final novelty and attribution signoff should be performed by a domain expert before publication."}, "source_stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "An integral Segal theorem for toric targets"}
+{"client_summary": "A quantitative motivic nilpotence problem with a clean universal exponent, low-dimensional test cases, and an abstract filtration lemma.", "difficulty": "frontier_research", "domain": "Algebraic cycles and motives", "evaluation": {"expert_review_required_for_full_credit": true, "hard_failure_conditions": ["Presenting elementwise nilpotence as ideal nilpotence.", "Assuming direct sums preserve an exponent without controlling off-diagonal products.", "Claiming novelty for strong Rost nilpotence rather than for the explicit dimension-only bound."], "milestones": [{"milestone_id": "m1", "target": "Prove the strict-filtration nilpotence lemma and state precisely the faithfulness and filtration hypotheses.", "verification_mode": "formal_algebra_review", "weight_percent": 15}, {"milestone_id": "m2", "target": "Compute or sharply bound the geometric kernel ideal for conics, Severi-Brauer surfaces, or another explicit low-dimensional homogeneous variety.", "verification_mode": "exact_correspondence_or_motive_review", "weight_percent": 25}, {"milestone_id": "m3", "target": "Control mixed products through same-dimensional off-diagonal motivic summands in one nontrivial splitting tower.", "verification_mode": "expert_motivic_review", "weight_percent": 35}, {"milestone_id": "m4", "target": "Derive the factorial recurrence without summand-count dependence, or construct a counterexample to that recurrence.", "verification_mode": "expert_proof_or_exact_counterexample_review", "weight_percent": 25}], "recommended_training_modes": ["expert_supervised_process_RL", "tool_augmented_research_agent_evaluation", "preference_and_critic_training"], "rl_suitability_score_10": 7, "verification_profile": "Moderate for low-dimensional motives and abstract filtration lemmas; specialist review for the uniform factorial estimate."}, "license": "MIT", "prerequisites": ["Chow motives and correspondences", "projective homogeneous varieties", "Rost nilpotence", "motivic decompositions and PI rings"], "problem_id": "sample_7", "prompt": {"allowed_tools": ["SageMath or Magma for finite algebras", "symbolic correspondence matrices", "formal motive calculations", "computer-assisted ring-theoretic exploration"], "conjecture": "Let X be a d-dimensional projective homogeneous variety under a semisimple algebraic group over a field k, and let p be a prime. In Chow motives with F_p coefficients, the ideal I_X,p of endomorphisms of M(X) that vanish after base change to an algebraic closure satisfies I_X,p^((d+1)!)=0.", "definitions": "I_X,p is the kernel of End(M(X))->End(M(X_bar)) in the category of Chow motives with F_p coefficients. Its elements are degree-zero correspondences in CH^d(X times X;F_p), and ideal multiplication is composition of correspondences. Strong Rost nilpotence asks that one exponent kill every mixed product in this ideal, not only powers of each individual element.", "instruction": "Prove or refute the stated conjecture for “A factorial bound for Rost nilpotence”. A valid submission may instead complete one or more scored milestones. Label every theorem, computation, and literature claim 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 reproducible code, exact inputs, and machine-checkable outputs for every computational claim.", "required_deliverables": ["claim_status: proved, disproved, partial, or inconclusive", "a precise main result with hypotheses and quantifiers", "a complete proof or reproducible computation for every claimed milestone", "an adversarial check or counterexample search", "an explicit list of unresolved gaps and dependencies", "a literature boundary separating known input from new work"]}, "quality_signals": ["simple universal bound", "proved filtration lemma", "explicit Severi-Brauer benchmark"], "release_problem_id": "aim_ag_007", "release_schema_version": "1.0.0", "research_area": "Strong Rost nilpotence, projective homogeneous motives, and coniveau filtrations", "research_context": {"closest_known_results": ["Brosnan's version of Rost's lemma gives a d+1 exponent once residue-field actions vanish along a d-dimensional support filtration, but does not by itself organize a full projective-homogeneous splitting tower.", "Chernousov-Gille-Merkurjev prove elementwise Rost nilpotence using decompositions into twists of smaller homogeneous motives.", "Gille 2026 upgrades the projective-homogeneous result to strong ideal nilpotence and bounded exponent, but the displayed proof retains dependence on the number of isotropic summands in addition to dimension.", "Vishik's quadric estimate, quoted by Gille, trades powers of two multiplying the ideal against powers of the ideal and is compatible with, but does not state, the factorial dimension-only bound."], "established_progress": [{"statement": "An abstract strict-filtration lemma is proved: if an ideal J acts faithfully on a filtration of length d+1 and sends F^q into F^(q+1), then J^(d+1)=0. Combined with Brosnan's geometric support lemma, this reduces the conjecture to constructing a multiplicity-free coniveau filtration compatible with the Chernousov-Gille-Merkurjev motivic decomposition.", "type": "proved_supporting_result"}], "falsifiable_prediction": "For a central simple algebra A of degree 3 and its Severi-Brauer surface X=SB(A), the F_3-geometric kernel ideal in End(M(X)) satisfies I_X,3^6=0. Six geometrically trivial codimension-two correspondences on X times X with nonzero composite would refute the conjecture.", "hardness_rationale": "Known proofs alternate Rost's d+1-step support lemma with field extensions that increase the number of motivic summands before lowering dimension. The unresolved issue is to organize off-diagonal compositions through same-dimensional summands without paying once for every summand. Existing termination and bounded-exponent results do not provide the proposed factorial bound.", "main_obstruction": "The known motivic decomposition can create many same-dimensional summands. It is unknown whether mixed products can traverse these off-diagonal blocks long enough to force an exponent larger than the factorial dimension chain.", "motivation": "Chernousov-Gille-Merkurjev prove Rost nilpotence for projective homogeneous varieties, and Gille's 2026 refinement observes that their argument gives a nilpotent geometric kernel ideal with an exponent depending on dimension and isotropic decomposition data. A dimension-only explicit exponent would make motivic descent quantitatively uniform across all groups and Tits indices and would isolate the real cost of support dimension from decomposition branching.", "references": [{"citation": "P. Brosnan, A short proof of Rost nilpotence via refined correspondences, Documenta Mathematica 8 (2003), 69-78.", "relevance": "Provides the refined-correspondence support lemma with a d+1 filtration exponent.", "url": "https://www.maths.tcd.ie/EMIS/journals/DMJDMV/vol-08/04.html"}, {"citation": "V. Chernousov, S. Gille, and A. Merkurjev, Motivic decomposition of isotropic projective homogeneous varieties, Duke Mathematical Journal 126 (2005), 137-159.", "relevance": "Proves the motivic decomposition and Rost nilpotence theorem for arbitrary projective homogeneous varieties.", "url": "https://doi.org/10.1215/S0012-7094-04-12614-4"}, {"citation": "S. Gille and A. Vishik, Rost nilpotence and free theories, Documenta Mathematica 23 (2018), 1635-1657; arXiv:1804.04703.", "relevance": "Extends Rost nilpotence to generically constant coherent and free oriented theories, motivating the cobordism approach.", "url": "https://arxiv.org/abs/1804.04703"}, {"citation": "A. Rosenschon and A. Sawant, Rost nilpotence and etale motivic cohomology, Advances in Mathematics 348 (2019), 1037-1083; arXiv:1706.06386.", "relevance": "Develops an etale-motivic nilpotence mechanism and discusses improvements to low-dimensional exponent bounds.", "url": "https://arxiv.org/abs/1706.06386"}, {"citation": "S. Gille, Direct sums of Chow motives and Rost nilpotence, Canadian Mathematical Bulletin, First View (2026), 1-14.", "relevance": "Proves strong Rost nilpotence in the known geometric cases and explicitly obtains a projective-homogeneous exponent depending on dimension and the number of motivic summands.", "url": "https://doi.org/10.4153/S0008439526101660"}], "tractability_rationale": "Projective homogeneous varieties are geometrically cellular; isotropic ones decompose into Tate twists of smaller projective homogeneous motives. Rost's refined-correspondence lemma gives a sharp d+1 support-filtration cost, and a chain of dimension drops has factorial total cost. The conjecture is automatic for split varieties, agrees with low-dimensional quadric behavior, and is compatible with Vishik's independent exponent estimates for quadrics."}, "research_status": {"classification": "candidate_open_problem", "expert_signoff_required": true, "frontier_contribution": "Remove decomposition-multiplicity dependence from strong Rost nilpotence by proving the explicit exponent (dim X+1)! for projective homogeneous varieties.", "literature_boundary": "No exact match was found in the targeted primary-source review.", "novelty_policy": "Final novelty and attribution signoff should be performed by a domain expert before publication."}, "source_stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "A factorial bound for Rost nilpotence"}
+{"client_summary": "A representation-valued refinement of logarithmic degeneration formulas with exact finite-group benchmarks and stack-theoretic failure modes.", "difficulty": "frontier_research", "domain": "Logarithmic enumerative geometry", "evaluation": {"expert_review_required_for_full_credit": true, "hard_failure_conditions": ["Using cyclic tropical graphs inside the genus-zero tree statement.", "Ignoring graph automorphisms or stack inertia when claiming a torsor.", "Inferring a virtual structure-sheaf identity from equality of Chow degrees alone."], "milestones": [{"milestone_id": "m1", "target": "Define the labelled tree-type stacks, rigidification, matching lattice, deck action, and source and target of every virtual K-class without ambiguity.", "verification_mode": "expert_definitional_review", "weight_percent": 25}, {"milestone_id": "m2", "target": "Prove the one-edge contact-m regular-representation formula over an etale trivialization.", "verification_mode": "formal_local_log_geometry_review", "weight_percent": 20}, {"milestone_id": "m3", "target": "Verify the two independent contacts 2 and 3 benchmark, including all six characters and no character-dependent virtual sign.", "verification_mode": "finite_group_and_virtual_pullback_review", "weight_percent": 20}, {"milestone_id": "m4", "target": "Establish equivariant finite-etale pullback of obstruction theories and descent across boundary expansions for the stated tree class, or exhibit a stack-inertia counterexample.", "verification_mode": "expert_stack_and_virtual_K_review", "weight_percent": 35}], "recommended_training_modes": ["expert_supervised_process_RL", "tool_augmented_research_agent_evaluation", "preference_and_critic_training"], "rl_suitability_score_10": 6, "verification_profile": "Moderate for local labelled tree types; specialist review for global stack descent and obstruction-theory compatibility."}, "license": "MIT", "prerequisites": ["logarithmic Gromov-Witten theory", "perfect obstruction theories", "virtual structure sheaves", "equivariant G-theory and diagonalizable groups"], "problem_id": "sample_8", "prompt": {"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": "Prove or refute the stated conjecture for “A character-valued logarithmic gluing formula”. A valid submission may instead complete one or more scored milestones. Label every theorem, computation, and literature claim 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 reproducible code, exact inputs, and machine-checkable outputs for every computational claim.", "required_deliverables": ["claim_status: proved, disproved, partial, or inconclusive", "a precise main result with hypotheses and quantifiers", "a complete proof or reproducible computation for every claimed milestone", "an adversarial check or counterexample search", "an explicit list of unresolved gaps and dependencies", "a literature boundary separating known input from new work"]}, "quality_signals": ["character-level refinement", "six-character benchmark", "explicit rigidification checks"], "release_problem_id": "aim_ag_008", "release_schema_version": "1.0.0", "research_area": "Logarithmic stable maps, virtual K-theory, and finite root-choice groups", "research_context": {"closest_known_results": ["Abramovich-Chen-Gross-Siebert decompose the special-fiber virtual fundamental class by rigid tropical type, but retain a scalar multiplicity rather than a deck-group character.", "Ranganathan proves finite-etale root-choice counting of degree product_e m_e and compares the Chow obstruction theory with vertex gluing.", "Lee and Qu separately supply virtual structure sheaves and virtual pullback in K-theory, but that sources do not combine them with logarithmic root-choice equivariance.", "No published source located in the targeted review states that the typewise virtual structure sheaf is a regular-representation-valued refinement determined by the Smith normal form of the tropical matching map."], "established_progress": [], "falsifiable_prediction": "For a rigid labelled genus-zero type whose vertex fiber product is a reduced point and whose two independent bounded edges have contact orders 2 and 3, the G_tau = mu_2 times mu_3 equivariant virtual structure sheaf has each of the six characters with multiplicity exactly one. A missing character, a repeated character, or a virtual sign on only some characters refutes the conjecture while leaving the ordinary degree-six gluing coefficient unchanged.", "hardness_rationale": "The local root-choice cover is elementary, but a global theorem must make expansions, basic logarithmic structures, the finite deck action, and perfect obstruction theories compatible in equivariant G-theory. Chow-level equality does not imply equality of virtual structure sheaves because K-theory retains extensions, torsion line bundles, and character monodromy. Boundary expansions and graph automorphisms can also turn an apparently free action into stack inertia unless rigidification is handled exactly.", "main_obstruction": "One must construct a canonical finite deck-group action on the logarithmic type stack, including boundary expansions, and prove that its perfect obstruction theory is the equivariant finite-etale pullback of the derived vertex-gluing theory. Chow-level equality and degree counting do not control the character local systems or extensions retained by virtual G-theory.", "motivation": "Chow-valued logarithmic degeneration formulae weight a rigid tropical type by a positive integer, typically the degree of a finite-etale logarithmic-lift map. That integer forgets the finite abelian root-choice group: gluing maps with torsion cokernels Z/4 and (Z/2)^2 have the same order but different character theories. A K-theoretic virtual structure sheaf can retain the regular representation and its globally twisted character eigensheaves, potentially separating degenerations with identical ordinary logarithmic Gromov-Witten multiplicities.", "references": [{"citation": "M. Gross and B. Siebert, Logarithmic Gromov-Witten invariants, Journal of the American Mathematical Society 26 (2013), 451-510; arXiv:1102.4322.", "relevance": "Constructs logarithmic stable-map moduli and the obstruction theory underlying logarithmic Gromov-Witten invariants.", "url": "https://arxiv.org/abs/1102.4322"}, {"citation": "D. Abramovich, Q. Chen, M. Gross, and B. Siebert, Decomposition of degenerate Gromov-Witten invariants, Compositio Mathematica 156 (2020), 2020-2075; arXiv:1709.09864.", "relevance": "Proves a Chow-valued virtual decomposition indexed by rigid tropical curves with scalar multiplicities.", "url": "https://arxiv.org/abs/1709.09864"}, {"citation": "D. Ranganathan, Logarithmic Gromov-Witten theory with expansions, Algebraic Geometry 9 (2022), 714-761; arXiv:1903.09006.", "relevance": "Reconstructs type contributions from vertex theories and proves that forgetting logarithmic structures at gluing nodes is finite etale of degree given by the product of edge expansion factors in the expanded setting.", "url": "https://arxiv.org/abs/1903.09006"}, {"citation": "Y.-P. Lee, Quantum K-theory I: Foundations, Duke Mathematical Journal 121 (2004), 389-424; arXiv:math/0105014.", "relevance": "Develops quantum K-theory using virtual structure sheaves of stable-map moduli.", "url": "https://arxiv.org/abs/math/0105014"}, {"citation": "B. Fantechi and L. Goettsche, Riemann-Roch theorems and elliptic genus for virtually smooth schemes, Geometry & Topology 14 (2010), 83-115; arXiv:0706.0988.", "relevance": "Relates virtual structure sheaves to virtual fundamental classes through virtual Riemann-Roch.", "url": "https://arxiv.org/abs/0706.0988"}, {"citation": "F. Qu, Virtual pullbacks in K-theory, Annales de l'Institut Fourier 68 (2018), 1609-1641; arXiv:1608.02524.", "relevance": "Provides bivariant and functorial virtual K-pullbacks needed for the evaluation-diagonal gluing class.", "url": "https://arxiv.org/abs/1608.02524"}], "tractability_rationale": "The necessary pieces exist separately: finite-etale logarithmic lift counting and Chow-level obstruction comparisons are proved in logarithmic degeneration formulae; Lee's virtual structure sheaf gives the K-theoretic invariant; and bivariant virtual pullbacks in K-theory handle derived evaluation diagonals. Locally, the obstruction complex is independent of the chosen root of unity. The conjecture therefore asks for an equivariant finite-etale enhancement of an existing comparison rather than a new foundation for logarithmic Gromov-Witten theory."}, "research_status": {"classification": "candidate_open_problem", "expert_signoff_required": true, "frontier_contribution": "Refine scalar tropical multiplicities to a character-valued virtual structure-sheaf formula retaining the full finite root-choice group.", "literature_boundary": "Related results exist; the stated frontier contribution is narrower than the known inputs.", "novelty_policy": "Final novelty and attribution signoff should be performed by a domain expert before publication."}, "source_stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "A character-valued logarithmic gluing formula"}
+{"client_summary": "A relative comparison problem linking cyclotomic homotopy, formal Brauer height, higher Hasse sections, and scheme-theoretic multiplicity.", "difficulty": "frontier_research", "domain": "Cyclotomic homotopy theory and K3 geometry", "evaluation": {"expert_review_required_for_full_credit": true, "hard_failure_conditions": ["Conflating fiberwise equality with a base-change-compatible relative sheaf.", "Assuming agreement of associated graded objects identifies scheme-theoretic zero loci.", "Using inconsistent indexing for height-at-least-h strata and obstruction sections."], "milestones": [{"milestone_id": "m1", "target": "Reconstruct the fiberwise identification of cyclotomic degree -2 with H^2(W O_X) and the finite-height Frobenius criterion.", "verification_mode": "formal_literature_and_proof_review", "weight_percent": 15}, {"milestone_id": "m2", "target": "Construct relative finite V-quotients with base change on an explicit smooth chart of polarized K3 moduli.", "verification_mode": "derived_geometry_expert_review", "weight_percent": 25}, {"milestone_id": "m3", "target": "Identify the next Frobenius coefficient with the classical higher Hasse section and prove a simple zero at a generic finite-height transition.", "verification_mode": "local_deformation_computation_plus_review", "weight_percent": 25}, {"milestone_id": "m4", "target": "Recover the terminal supersingular scheme structure with multiplicity exactly two, or find additional embedded or derived structure.", "verification_mode": "expert_local_intersection_review", "weight_percent": 35}], "recommended_training_modes": ["expert_supervised_process_RL", "tool_augmented_research_agent_evaluation", "preference_and_critic_training"], "rl_suitability_score_10": 7, "verification_profile": "Moderate for fiberwise and finite-Witt milestones; specialist review for relative cyclotomic descent and terminal multiplicity."}, "license": "MIT", "prerequisites": ["p-typical cyclotomic spectra", "Cartier modules and de Rham-Witt cohomology", "formal Brauer groups of K3 surfaces", "derived zero loci"], "problem_id": "sample_9", "prompt": {"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": "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. Label every theorem, computation, and literature claim 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 reproducible code, exact inputs, and machine-checkable outputs for every computational claim.", "required_deliverables": ["claim_status: proved, disproved, partial, or inconclusive", "a precise main result with hypotheses and quantifiers", "a complete proof or reproducible computation for every claimed milestone", "an adversarial check or counterexample search", "an explicit list of unresolved gaps and dependencies", "a literature boundary separating known input from new work"]}, "quality_signals": ["known fiberwise detector", "finite-height local tests", "decisive supersingular multiplicity"], "release_problem_id": "aim_ag_009", "release_schema_version": "1.0.0", "research_area": "Cartier modules, de Rham-Witt cohomology, and height strata", "research_context": {"closest_known_results": ["Antieau-Nikolaus already prove the fibrewise identification of the exact cyclotomic homotopy object proposed here and analyze the supersingular K3 Cartier module, but do not state a relative construction of all scheme-theoretic height strata.", "Van der Geer-Katsura already construct the classical height strata and higher Hasse sections from Witt and de Rham cohomology, but do not formulate them as derived zero loci extracted from relative THH.", "Bhatt-Morrow-Scholze relate filtered THH, TC-minus, and TP to crystalline and Nygaard data, but the cited paper does not identify the K3 higher Hasse tower with cyclotomic truncation obstructions.", "Illusie's de Rham-Witt theory supplies the underlying F,V module and crystalline comparisons, so the conjecture may be an implicit relative repackaging rather than globally new."], "established_progress": [], "falsifiable_prediction": "For p at least 3, at a generic supersingular point of the polarized K3 moduli stack, the ten obstruction sections extracted from V-adic Frobenius on relative pi_{-2}^{cyc} THH cut out a local complete-intersection cycle of Hilbert-Samuel multiplicity exactly two. At a transverse generic height-(h+1) point for h<10, the h-th obstruction has a simple zero. Multiplicity one or greater than two at the supersingular point, or a nonsimple finite-height zero, refutes the conjecture.", "hardness_rationale": "The required cyclotomic truncation must commute with descent and non-affine base change over the K3 moduli stack. The BMS and cyclotomic filtrations retain different secondary filtrations, so agreement on TP does not automatically identify higher Hasse equations. At supersingular fibres the relevant Cartier module is not finitely presented and cyclotomic perfectness fails, exactly where the construction must retain multiplicity two.", "main_obstruction": "Construct a relative cyclotomic degree -2 Cartier object with sufficiently strong descent and base change, then prove that its V-adic Frobenius edge coefficients equal the classical higher Hasse sections as infinitesimal equations. The nonperfect supersingular fibre must yield exactly the known multiplicity two rather than extra embedded or derived structure.", "motivation": "The classical K3 height strata and their higher Hasse equations are known, and the fibrewise cyclotomic homotopy object is known to equal the Cartier module of the formal Brauer group. A relative comparison would show that THH constructs the entire height stratification intrinsically, including its infinitesimal equations and the exceptional supersingular multiplicity, rather than merely detecting the height of individual fibres.", "references": [{"citation": "Benjamin Antieau and Thomas Nikolaus, Cartier modules and cyclotomic spectra, Journal of the American Mathematical Society 34 (2021), 1-78, arXiv:1809.01714.", "relevance": "Primary closest result: constructs the cyclotomic t-structure, identifies its heart with V-complete Cartier modules, and explicitly identifies the degree -2 cyclotomic homotopy object of a K3 surface with H^2(W O), including the supersingular example.", "url": "https://arxiv.org/abs/1809.01714"}, {"citation": "Gerard van der Geer and Toshiyuki Katsura, On a stratification of the moduli of K3 surfaces, Journal of the European Mathematical Society 2 (2000), 259-290.", "relevance": "Primary source for the Witt-Frobenius height criterion, higher Hasse equations, codimensions, cycle classes, and terminal multiplicity of K3 height strata.", "url": "https://doi.org/10.1007/S100970000021"}, {"citation": "Bhargav Bhatt, Matthew Morrow, and Peter Scholze, Topological Hochschild homology and integral p-adic Hodge theory, Publications Mathematiques de l'IHES 129 (2019), 199-310, arXiv:1802.03261.", "relevance": "Primary source constructing filtrations on THH and its variants with crystalline and Nygaard-filtered graded pieces, providing the likely relative comparison mechanism.", "url": "https://arxiv.org/abs/1802.03261"}, {"citation": "Luc Illusie, Complexe de de Rham-Witt et cohomologie cristalline, Annales scientifiques de l'Ecole Normale Superieure 12 (1979), 501-661.", "relevance": "Primary foundation for de Rham-Witt cohomology, Cartier modules, and the crystalline description used in the K3 formal Brauer calculation.", "url": "https://www.numdam.org/item/ASENS_1979_4_12_4_501_0/"}], "tractability_rationale": "Antieau-Nikolaus prove the exact fibrewise identification pi_{-2}^{cyc} THH(X)=H^2(X,W O_X) as Cartier modules. Van der Geer-Katsura identify finite-height jumps with successive Frobenius vanishings and determine their Hodge-line weights and cycle classes. Bhatt-Morrow-Scholze provide a relative filtered cyclotomic-to-crystalline bridge. These ingredients agree on every geometric fibre and force the predicted line-bundle exponents."}, "research_status": {"classification": "candidate_open_problem", "expert_signoff_required": true, "frontier_contribution": "Construct the full scheme-theoretic K3 height tower from relative cyclotomic degree -2 data, including the terminal multiplicity-two supersingular cycle.", "literature_boundary": "Related results exist; the stated frontier contribution is narrower than the known inputs.", "novelty_policy": "Final novelty and attribution signoff should be performed by a domain expert before publication."}, "source_stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "A cyclotomic Hasse tower for K3 moduli"}
+{"client_summary": "An if-and-only-if ramification detector with a computable logarithmic differential, filtered descent milestones, and a concrete wild Kummer test.", "difficulty": "frontier_research", "domain": "p-adic homotopy theory and logarithmic geometry", "evaluation": {"expert_review_required_for_full_credit": true, "hard_failure_conditions": ["Treating nonzero log differentials as automatic proof of a nonzero TP descent defect.", "Arguing from p dividing the group order rather than from ramification.", "Using ordinary unlogged TP while claiming a logarithmic tame or wild criterion."], "milestones": [{"milestone_id": "m1", "target": "Derive the relative logarithmic differential module B/eB generated by dlog(t) for the Kummer chart and verify tame vanishing.", "verification_mode": "formal_log_algebra_review", "weight_percent": 15}, {"milestone_id": "m2", "target": "Prove Galois descent on every complete log-prismatic or Hodge-Tate associated graded in a tame control family.", "verification_mode": "filtered_derived_review", "weight_percent": 25}, {"milestone_id": "m3", "target": "Justify interchange of the circle Tate construction with the required Galois totalization, or isolate an exact obstruction.", "verification_mode": "expert_homotopy_limit_review", "weight_percent": 25}, {"milestone_id": "m4", "target": "Show that the wild dlog class survives to the descent defect in the cyclic degree-p Kummer example, or exhibit the actual first surviving obstruction if it dies.", "verification_mode": "spectral_sequence_artifact_plus_expert_review", "weight_percent": 35}], "recommended_training_modes": ["expert_supervised_process_RL", "tool_augmented_research_agent_evaluation", "preference_and_critic_training"], "rl_suitability_score_10": 8, "verification_profile": "High for logarithmic differentials and associated-graded tests; specialist review for Tate-totalization interchange and permanence."}, "license": "MIT", "prerequisites": ["logarithmic cotangent complexes", "topological Hochschild and periodic cyclic homology", "homotopy fixed points and Tate constructions", "ramification theory of local fields"], "problem_id": "sample_10", "prompt": {"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": "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. Label every theorem, computation, and literature claim 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 reproducible code, exact inputs, and machine-checkable outputs for every computational claim.", "required_deliverables": ["claim_status: proved, disproved, partial, or inconclusive", "a precise main result with hypotheses and quantifiers", "a complete proof or reproducible computation for every claimed milestone", "an adversarial check or counterexample search", "an explicit list of unresolved gaps and dependencies", "a literature boundary separating known input from new work"]}, "quality_signals": ["clean tame/wild criterion", "proved log-differential calculation", "explicit first wild class"], "release_problem_id": "aim_ag_010", "release_schema_version": "1.0.0", "research_area": "Logarithmic THH and TP, Galois descent, and tame versus wild ramification", "research_context": {"closest_known_results": ["Binda-Lundemo-Park-Oestvaer construct log TP and its motivic/log-prismatic filtration, but do not state the tame Galois homotopy-fixed-point criterion proposed here.", "Binda-Lundemo-Merici-Park prove saturated descent and an infinite-root-stack model for log TC; this may imply part of the tame direction after controlling TP and totalization.", "Hoening-Richter show that Tate spectra detect wild ramification for structured ring spectra, but their detector is not the stated log-TP Galois descent fiber for local fields.", "Bayindir-Peroux give ramified THH descent, not an if-and-only-if for p-completed logarithmic TP or permanence of the Kummer dlog class.", "The integral normal-basis criterion and vanishing of relative log differentials characterize tameness classically; the candidate joins them through a cyclotomic filtration."], "established_progress": [{"statement": "Two supporting statements are proved. First, for the Kummer extension B=A[t]/(t^e-pi) with divisorial charts, Omega^1_log(B/A) is B/eB generated by dlog(t), which vanishes for tame e and is nonzero when p divides e. Second, a map of complete separated filtered spectra that is an equivalence on every associated graded is an equivalence.", "type": "proved_supporting_result"}], "falsifiable_prediction": "For K=Q_p(zeta_p) and L=K(pi_K^(1/p)), the cyclic degree-p wild Kummer extension, the map TP^log(O_K;Z_p) -> TP^log(O_L;Z_p)^{hC_p} is not an equivalence, and dlog(pi_L) in O_L/p generated log differentials survives in the Hodge-Tate-to-log-TP descent spectral sequence. An equivalence, or death of this class without another descent obstruction, refutes the sharpened prediction. For the tame control L=K(pi_K^(1/e)) with e dividing p-1, the map is predicted to be an equivalence.", "hardness_rationale": "The circle Tate construction does not commute with arbitrary Galois Cech totalizations, so saturated descent for log TC does not formally give the claimed log-TP equivalence. Conversely, a nonzero log cotangent class can die in the circle Tate or G-homotopy-fixed-point spectral sequence. The wild direction therefore requires a new permanence theorem, while the tame direction requires filtered base change compatible with two completions.", "main_obstruction": "Control the interchange of the circle Tate construction with Galois totalization in the tame case, and prove that the lowest nonzero wild log-differential class is a permanent class in the combined Hodge-Tate, log-prismatic, and homotopy-fixed-point filtrations.", "motivation": "Tameness has two classical signatures: the ring of integers has an integral normal basis, and the divisorial log cotangent defect vanishes. Modern log-TP simultaneously carries a log-prismatic filtration sensitive to cotangent data and a homotopy-fixed-point descent problem sensitive to group cohomology. An exact tame criterion would unify these signatures in one cyclotomic invariant and provide a computational obstruction for wild extensions.", "references": [{"citation": "F. Binda, T. Lundemo, D. Park, and P. A. Oestvaer, Logarithmic Prismatic Cohomology via Logarithmic THH, arXiv:2306.01368.", "relevance": "Constructs cyclotomic log THH, log TP, and complete filtrations with log-prismatic graded pieces and log-cotangent refinements.", "url": "https://arxiv.org/abs/2306.01368"}, {"citation": "F. Binda, T. Lundemo, A. Merici, and D. Park, Logarithmic TC via the Infinite Root Stack and the Beilinson Fiber Square, arXiv:2408.15627.", "relevance": "Expresses logarithmic cyclic theories through infinite root stacks using saturated descent, supplying the proposed route for tame Galois covers.", "url": "https://arxiv.org/abs/2408.15627"}, {"citation": "J. Rognes, S. Sagave, and C. Schlichtkrull, Logarithmic topological Hochschild homology of topological K-theory spectra, Journal of the European Mathematical Society 20 (2018), 489-527; arXiv:1410.2170.", "relevance": "Proves a central formally log-THH-etale example modeling tame ramification in structured ring spectra.", "url": "https://arxiv.org/abs/1410.2170"}, {"citation": "E. Hoening and B. Richter, Detecting and describing ramification for structured ring spectra, arXiv:2101.12655.", "relevance": "Uses trace and Tate cohomology spectra to distinguish tame and wild ramification in structured ring spectra.", "url": "https://arxiv.org/abs/2101.12655"}, {"citation": "H. Bayindir and M. Peroux, Ramification and descent in homotopy theory and derived algebraic geometry, arXiv:2112.14568.", "relevance": "Develops ramified descent for ordinary THH of rings of integers and clarifies the unramified/totally ramified factorization.", "url": "https://arxiv.org/abs/2112.14568"}], "tractability_rationale": "Log-TP and its complete Nygaard/log-prismatic filtration are constructed in the literature; log TC is described by saturated descent on infinite root stacks; tame extensions are log etale and have integral normal bases; and an explicit Kummer calculation produces the proposed first wild class. Both directions reduce to bounded filtered descent and a low-filtration permanence calculation rather than to an unrestricted computation of homotopy groups."}, "research_status": {"classification": "candidate_open_problem", "expert_signoff_required": true, "frontier_contribution": "Characterize tame local-field extensions exactly by p-completed divisorial log-TP Galois descent and identify the first permanent wild obstruction.", "literature_boundary": "Related results exist; the stated frontier contribution is narrower than the known inputs.", "novelty_policy": "Final novelty and attribution signoff should be performed by a domain expert before publication."}, "source_stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "Logarithmic TP as a tame-ramification detector"}