candidate_required_fields: rank dimension count

policy_visibility: public

prompt_json: "Compute exactly ∫_Gr(2,4) c_4(Sym^3 S*) by torus localization. Return the rank, dimension, and integer count."

prompt_kind: string

split: train


prompt_json: "For the supplied five quadratic jets, compute the exact 20×20 multiplication map, its kernel, and the quadratic obstruction from the supplied second-order term."


prompt_json: "A line moves in a 4-dimensional family in P^3. For surfaces of degree 7, compute the line-incidence codimension."


prompt_json: "Use Koszul resolutions to compute the linear-strand source/target dimensions and the maximal-rank nonlinear Betti prediction."


prompt_json: "Normalize the raw free anticanonical degrees 2 and 3, compute the Apéry set and conductor, and test the stated bound 2."


form_type: alternating

prompt_json: "Verify the standard rank-two skew lattice, the span index of the two cycles, and the Picard-Lefschetz transvection in the first cycle."


prompt_json: "Verify the chain condition and integral homology of the one-cell CW chain complex for S^1."


prompt_json: "Verify that the supplied operator strictly raises a two-step filtration and compute the ideal nilpotence exponent."


prompt_json: "Compute the torsion cokernel and regular-character count for independent contact orders 2 and 3."


prompt_json: "Compute the Hodge-line exponents p^h-1 and their product for p=3 and h=1,2,3."


prompt_json: "For p=5 and ramification index e=3, determine tameness and the p-completed relative logarithmic differential module."


milestone_target: For the displayed five quadratic normal jets, prove that the 20 by 20 multiplication map has rank 19, identify its kernel, and verify a nonzero quadratic obstruction.

allowed_tools: SageMath or SymPy for Schubert-calculus coefficient extraction

allowed_tools: 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: Attempt to 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. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.

required_deliverables: claim_status: proved, disproved, partial, or inconclusive

required_deliverables: a precise main result with hypotheses and quantifiers

required_deliverables: a complete proof or reproducible computation for each claimed milestone

required_deliverables: an adversarial check or counterexample search

required_deliverables: a list of unresolved gaps and dependencies

required_deliverables: a literature boundary that distinguishes known input from new work

prompt_kind: object

instruction: Attempt to 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. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.

required_artifact_policy: hash every machine-generated artifact and provide an exact rerun command


milestone_target: Prove the irreducibility and codimension d+1-h of the line-incidence component for a fixed line-Hilbert component H.

allowed_tools: Borel-Weil-Bott and Jacobian-ring calculations

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: Attempt to 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. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.

required_deliverables: claim_status: proved, disproved, partial, or inconclusive

required_deliverables: a precise main result with hypotheses and quantifiers

required_deliverables: a complete proof or reproducible computation for each claimed milestone

required_deliverables: an adversarial check or counterexample search

required_deliverables: a list of unresolved gaps and dependencies

required_deliverables: a literature boundary that distinguishes known input from new work

instruction: Attempt to 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. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.


milestone_target: Reproduce the two-quadric-surfaces-meeting-in-a-conic benchmark and its unique beta_(2,4)=1.

allowed_tools: exact random specialization over finite fields

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: Attempt to 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. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.

required_deliverables: claim_status: proved, disproved, partial, or inconclusive

required_deliverables: a precise main result with hypotheses and quantifiers

required_deliverables: a complete proof or reproducible computation for each claimed milestone

required_deliverables: an adversarial check or counterexample search

required_deliverables: a list of unresolved gaps and dependencies

required_deliverables: a literature boundary that distinguishes known input from new work

instruction: Attempt to 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. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.


milestone_target: Implement a reproducible pipeline from fan data to the Hilbert basis, normalized degree semigroup, Apéry set, and conductor.

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: Attempt to 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. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.

required_deliverables: claim_status: proved, disproved, partial, or inconclusive

required_deliverables: a precise main result with hypotheses and quantifiers

required_deliverables: a complete proof or reproducible computation for each claimed milestone

required_deliverables: an adversarial check or counterexample search

required_deliverables: a list of unresolved gaps and dependencies

required_deliverables: a literature boundary that distinguishes known input from new work

instruction: Attempt to 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. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.


milestone_target: Reconstruct the finite-index implication from a complete integral vanishing lattice in the symplectic case.

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: Attempt to 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. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.

required_deliverables: claim_status: proved, disproved, partial, or inconclusive

required_deliverables: a precise main result with hypotheses and quantifiers

required_deliverables: a complete proof or reproducible computation for each claimed milestone

required_deliverables: an adversarial check or counterexample search

required_deliverables: a list of unresolved gaps and dependencies

required_deliverables: a literature boundary that distinguishes known input from new work

instruction: Attempt to 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. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.


milestone_target: Control orientations, sign local systems, and the first nontrivial integral differentials associated with overlapping primitive collections.

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: Attempt to 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. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.

required_deliverables: claim_status: proved, disproved, partial, or inconclusive

required_deliverables: a precise main result with hypotheses and quantifiers

required_deliverables: a complete proof or reproducible computation for each claimed milestone

required_deliverables: an adversarial check or counterexample search

required_deliverables: a list of unresolved gaps and dependencies

required_deliverables: a literature boundary that distinguishes known input from new work

instruction: Attempt to 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. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.


milestone_target: Prove the strict-filtration nilpotence lemma and state precisely the faithfulness and filtration hypotheses.

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: Attempt to prove or refute the stated conjecture for “A factorial bound for Rost nilpotence”. A valid submission may instead complete one or more scored milestones. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.

required_deliverables: claim_status: proved, disproved, partial, or inconclusive

required_deliverables: a precise main result with hypotheses and quantifiers

required_deliverables: a complete proof or reproducible computation for each claimed milestone

required_deliverables: an adversarial check or counterexample search

required_deliverables: a list of unresolved gaps and dependencies

required_deliverables: a literature boundary that distinguishes known input from new work

instruction: Attempt to prove or refute the stated conjecture for “A factorial bound for Rost nilpotence”. A valid submission may instead complete one or more scored milestones. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.