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


Prompt: Compute exactly ∫_Gr(2,5) c_6(Sym^5 S*) by torus localization. Return the rank, dimension, and integer count.


Prompt: Compute exactly ∫_Gr(4,9) c_20(Sym^3 S*) by torus localization. Return the rank, dimension, and integer count.


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


Prompt: Repeat the local Fano-section calculation for a scaled fifth jet and certify whether the obstruction is nonzero.


Prompt: Compute the exact corank and second-order obstruction for the weighted fifth jet.


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


Prompt: For a quadric threefold of index 3 with a 3-dimensional line family and d=8, compute the incidence codimension.


Prompt: For a cubic threefold of index 2 with a 2-dimensional Fano surface of lines and d=9, compute the incidence codimension.


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


Prompt: Compute the same Tor benchmark when both vertices and the edge are cut out only by linear equations.


Prompt: Compute the maximal-rank cokernel prediction for two quadratic hypersurface coordinate rings glued along a linear-plus-quadratic complete intersection.


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


Prompt: Compute the normalized numerical semigroup generated by 3 and 4, including its Apéry set, Frobenius number, conductor, and bound test.


Prompt: For the product benchmark with raw degrees 6 and 7, compute the exact conductor and test the quadratic bound 36.


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


Prompt: Compute the integral span index and verify the transvection for the supplied rank-four vanishing-cycle configuration.


Prompt: Verify primitivity, pairing-one connectivity data, and an exact transvection for the rank-four configuration.


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


Prompt: Compute the integral homology of the two-term complex Z --2--> Z.


Prompt: Compute integral homology, including torsion, for the supplied three-term chain complex.


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


Prompt: Verify strict filtration increase and compute the minimal exponent killing every product.


Prompt: Compute the nilpotence exponent of the ideal generated by the two exact filtration-raising matrices.


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


Prompt: Compute the Smith invariants and character count for a cyclic order-four root-choice group.


Prompt: Distinguish the order-four torsion group with invariant factors 2,2 from the cyclic order-four case.


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


Prompt: Compute the first two higher-Hasse weights and cycle coefficient for p=5.


Prompt: Compute all ten odd-characteristic Hasse weights at p=3, their exact product, and state the terminal supersingular multiplicity used by the task.


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


Prompt: For p=5 and e=5, compute the p-primary annihilator and whether the dlog class is nonzero.


Prompt: For p=3 and e=9, compute the exact p-adic valuation, tameness, and p-primary annihilator of the Kummer dlog class.


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.

SageMath or SymPy for Schubert-calculus coefficient extraction

certified numerical algebraic geometry followed by exact verification

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.

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.

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work

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


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

Borel-Weil-Bott and Jacobian-ring calculations

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.

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.

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


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

exact random specialization over finite fields

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.

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.

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


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

SageMath
Normaliz
polymake
Macaulay2
exact numerical-semigroup code

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.

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.

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


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

SageMath lattice computations
Magma
symbolic intersection calculations
finite congruence-image computation

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.

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.

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


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

SageMath
configuration-space chain complexes
spectral-sequence bookkeeping code
computer algebra for Cox presentations

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.

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.

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


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

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.

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.

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


Prove the one-edge contact-m regular-representation formula over an etale trivialization.

equivariant K-theory software where available

symbolic finite-group character calculations

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|.

The vertex moduli M_v parametrize relative or expanded stable maps associated with the vertices of the labelled genus-zero tree tau. Their derived fiber product B_tau is formed by matching evaluations for every bounded edge, and F_tau is the K-theoretic virtual pullback of the external product of their virtual structure sheaves along those diagonals. The integral matching map Phi_tau records the edge and vertex matching equations; A_tau=tors(coker Phi_tau), and G_tau=D(A_tau) is its finite diagonalizable Cartier dual. For the independent-edge benchmark with contact orders m_e, G_tau is the product of the groups mu_(m_e). Rigidification means quotienting the labelled type stack by the explicitly specified residual automorphism inertia before asserting that mu_tau is a torsor.

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

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


Reconstruct the fiberwise identification of cyclotomic degree -2 with H^2(W O_X) and the finite-height Frobenius criterion.

computer algebra for complete intersections

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.

For a K3 surface X over a perfect field, C(X)=pi_{-2}^{cyc} THH(X) is a derived V-complete p-typical Cartier module. Antieau-Nikolaus identify it with H^2(X,W O_X). Finite V-quotients recover finite Witt cohomology, and van der Geer-Katsura characterize the formal Brauer height as the least Witt level at which Frobenius is nonzero. A relative cyclotomic Cartier sheaf is a sheafified family of these objects with strong base change. After lower Frobenius components vanish, the next semilinear coefficient defines a higher Hasse section in lambda^(p^h-1).

Instruction: Attempt to prove or refute the stated conjecture for “A cyclotomic Hasse tower for K3 moduli”. A valid submission may instead complete one or more scored milestones. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


Derive the relative logarithmic differential module B/eB generated by dlog(t) for the Kummer chart and verify tame vanishing.

spectral-sequence computation
derived log-cotangent calculations
group cohomology software
exact Kummer-extension arithmetic

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.

TP^log(O_K;Z_p) is the circle Tate construction on p-completed cyclotomic logarithmic THH of the divisorial pre-log ring (O_K,M_K). The descent defect D_TP^log(L/K) is the fiber of the map from the base log-TP spectrum to G-homotopy fixed points of the extension spectrum. Tame means that the ramification index is prime to p; residue extensions of p-adic local fields are automatically separable.

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

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


SageMath or SymPy for Schubert-calculus coefficient extraction

certified numerical algebraic geometry followed by exact verification

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.

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.

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


Borel-Weil-Bott and Jacobian-ring calculations

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.

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.

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


exact random specialization over finite fields

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.

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.

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


SageMath
Normaliz
polymake
Macaulay2
exact numerical-semigroup code

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.

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.

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


SageMath lattice computations
Magma
symbolic intersection calculations
finite congruence-image computation

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.

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.

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


SageMath
configuration-space chain complexes
spectral-sequence bookkeeping code
computer algebra for Cox presentations

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.

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.

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


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.

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.

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


equivariant K-theory software where available

symbolic finite-group character calculations

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|.

The vertex moduli M_v parametrize relative or expanded stable maps associated with the vertices of the labelled genus-zero tree tau. Their derived fiber product B_tau is formed by matching evaluations for every bounded edge, and F_tau is the K-theoretic virtual pullback of the external product of their virtual structure sheaves along those diagonals. The integral matching map Phi_tau records the edge and vertex matching equations; A_tau=tors(coker Phi_tau), and G_tau=D(A_tau) is its finite diagonalizable Cartier dual. For the independent-edge benchmark with contact orders m_e, G_tau is the product of the groups mu_(m_e). Rigidification means quotienting the labelled type stack by the explicitly specified residual automorphism inertia before asserting that mu_tau is a torsor.

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

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


computer algebra for complete intersections

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.

For a K3 surface X over a perfect field, C(X)=pi_{-2}^{cyc} THH(X) is a derived V-complete p-typical Cartier module. Antieau-Nikolaus identify it with H^2(X,W O_X). Finite V-quotients recover finite Witt cohomology, and van der Geer-Katsura characterize the formal Brauer height as the least Witt level at which Frobenius is nonzero. A relative cyclotomic Cartier sheaf is a sheafified family of these objects with strong base change. After lower Frobenius components vanish, the next semilinear coefficient defines a higher Hasse section in lambda^(p^h-1).

Instruction: Attempt to prove or refute the stated conjecture for “A cyclotomic Hasse tower for K3 moduli”. A valid submission may instead complete one or more scored milestones. Every theorem, computation, and literature claim must be labeled as proved, reproduced, inferred, or conjectural. Do not convert a heuristic, dimension count, numerical approximation, or known special case into a claim about the full conjecture. Return all reproducible code, exact inputs, and machine-checkable outputs used.

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work


spectral-sequence computation
derived log-cotangent calculations
group cohomology software
exact Kummer-extension arithmetic

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.

TP^log(O_K;Z_p) is the circle Tate construction on p-completed cyclotomic logarithmic THH of the divisorial pre-log ring (O_K,M_K). The descent defect D_TP^log(L/K) is the fiber of the map from the base log-TP spectrum to G-homotopy fixed points of the extension spectrum. Tame means that the ramification index is prime to p; residue extensions of p-adic local fields are automatically separable.

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

claim_status: proved, disproved, partial, or inconclusive

a precise main result with hypotheses and quantifiers

a complete proof or reproducible computation for each claimed milestone

an adversarial check or counterexample search

a list of unresolved gaps and dependencies

a literature boundary that distinguishes known input from new work