candidate_required_fields: rank dimension count

policy_visibility: public

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

prompt_kind: string

split: dev


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


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


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


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


form_type: alternating

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


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


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


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


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


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


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

allowed_tools: equivariant K-theory software where available

allowed_tools: symbolic finite-group character calculations

conjecture: For a labelled rigid genus-zero tropical type tau in a projective log-smooth simple-normal-crossings degeneration over characteristic zero, let B_tau be the derived fiber product of the vertex stable-map moduli along evaluation diagonals, let M_tau be the moduli of basic logarithmic maps of that type, and let mu_tau forget the logarithmic root choices. If A_tau is the torsion cokernel of the tropical integral matching map and G_tau is its Cartier dual, then mu_tau is canonically a G_tau-torsor after rigidification, the obstruction theory of M_tau is the finite-etale pullback of the virtual diagonal-gluing theory, and R mu_(tau,*) O^vir_(M_tau) = F_tau tensor mu_(tau,*) O_(M_tau) in G_tau-equivariant G-theory. Etale-locally the second factor is the regular representation, so every character occurs once; forgetting characters recovers only the usual scalar tropical multiplicity |A_tau|.

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

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

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

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

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


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

allowed_tools: computer algebra for complete intersections

conjecture: Let p be odd, let p not divide 2d, and let M be the moduli stack of primitively polarized K3 surfaces of degree 2d over F_p. The degree -2 cyclotomic homotopy object of THH globalizes to a base-change-compatible V-complete Cartier sheaf for the universal K3 family. For h=1,...,10, its successive V-adic Frobenius obstructions are sections a_h of the Hodge line lambda^(p^h-1), and the recursive derived zero locus obtained by imposing a_1,...,a_h has classical truncation equal to the natural scheme-theoretic height-at-least-(h+1) stratum, with height at least 11 interpreted as the supersingular locus. On the finite-height locus the successive inclusions are regular Cartier divisors, while the tenth obstruction recovers the natural multiplicity-two supersingular cycle.

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

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

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

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 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.


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

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

conjecture: Let L/K be a finite Galois extension of p-adic local fields with group G, and give their valuation rings the divisorial log structures. The p-completed log-TP descent map TP^log(O_K;Z_p) -> TP^log(O_L;Z_p)^{hG} is an equivalence if and only if L/K is tamely ramified. For a totally ramified Kummer extension pi_K=pi_L^e with p dividing e, the module O_L/e O_L generated by dlog(pi_L) is the first wild class in the Hodge-Tate linearization and survives in the filtered descent defect.

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

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

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

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 “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.