data/showcase.jsonl

{"difficulty": "frontier_research", "domain": "Additive combinatorics", "expected_output": "A rigorous proof, disproof, or mathematically substantive partial result, with explicit hypotheses, gap control, and a current literature boundary.", "inspiration": "Erdős Problem #1: if an n-element set A contained in {1,...,N} has all subset sums distinct, must N be bounded below by a positive constant times 2^n?", "license": "MIT", "milestones": [], "problem_id": "erdos_001", "prompt": "Bounded-defect rigidity for distinct subset sums. Let A={a_1<...<a_n} be positive integers such that all 2^n subset sums are distinct, and define\n\nD_2(A) = sum_{i=1}^n a_i^2 - (4^n-1)/3.\n\nFor every fixed integer D>=0, does there exist r=r(D) such that, whenever D_2(A)<=D, one has\n\na_i=2^{i-1} for every i>r?\n\nMore precisely, for each D are there only finitely many possible defective initial segments (a_1,...,a_r), after which the sequence is forced to continue 2^r,2^{r+1},...,2^{n-1}?", "quality_signals": ["sharp equality case", "elementary variance lower bound", "finite-defect inverse-theorem target"], "rationale": "It studies the same dissociated or distinct-subset-sum condition as #1, but replaces the extremal maximum element by a quadratic energy and asks for a finite-defect inverse theorem. There is an exact elementary lower bound behind the normalization. If Z is the sum of a uniformly random subset of A, then Var(Z)=(1/4)sum a_i^2. Its support consists of 2^n distinct integers. Among any M distinct integers, consecutive integers minimize variance, with minimum (M^2-1)/12. Taking M=2^n gives sum a_i^2 >= (4^n-1)/3. Equality forces the subset sums to be {0,1,...,2^n-1}, and then induction forces A={1,2,4,...,2^{n-1}}. The new conjecture asks whether a bounded excess above this minimum can only alter boundedly many low binary digits. For example, {2,3,4,8,16,...} has a fixed energy excess while agreeing with the binary sequence from a bounded index onward.", "research_status": "candidate_open_problem_requires_expert_signoff", "rl_ready": false, "schema_version": "1.0.0", "source_record_index": 1, "stream": "erdos_variant", "task_type": "prove_or_refute", "title": "Bounded-defect rigidity for distinct subset sums", "verification": "Specialist mathematical review; exact computations must be reproducible."}
{"difficulty": "frontier_research", "domain": "Extremal set theory", "expected_output": "A rigorous proof, disproof, or mathematically substantive partial result, with explicit hypotheses, gap control, and a current literature boundary.", "inspiration": "Erdős Problem #20, the sunflower conjecture: for each fixed k, is the threshold f(n,k) for a k-sunflower in an n-uniform family at most c_k^n?", "license": "MIT", "milestones": [], "problem_id": "erdos_003", "prompt": "Profile-balanced sunflower threshold. Fix integers k>=3 and r>=2. Color the ground set of an n-uniform family with r colors. Call a k-sunflower S_1,...,S_k with core C profile-balanced if the r-vectors\n\n( |(S_i minus C) intersect color j| )_{j=1}^r\n\nare the same for all i. Let f_bal(n,k,r) be the least M such that every r-colored ground set and every n-uniform family of M sets contains a profile-balanced k-sunflower. If f(n,k) is the ordinary sunflower threshold, is\n\nf_bal(n,k,r) = (1+o(1)) f(n,k)\n\nfor every fixed k and r?", "quality_signals": ["classical sunflower connection", "rigorous polynomial-factor baseline", "clean asymptotic target"], "rationale": "It retains the same sunflower configuration but imposes an adversarially colored, equitable-petal condition. The question is not about the exponential growth rate alone, but whether the extra requirement costs asymptotically nothing even at the sharp threshold. There are immediate bounds. Monochromatically coloring the ground set shows f_bal(n,k,r)>=f(n,k). On the other hand, an n-set has one of C(n+r-1,r-1) possible color-count profiles. If a family has more than C(n+r-1,r-1)(f(n,k)-1) members, one profile class has at least f(n,k) sets and hence contains an ordinary sunflower. Since those sets have equal total profiles and share the same core, their petals automatically have equal profiles. Thus\n\nf_bal(n,k,r) <= C(n+r-1,r-1)(f(n,k)-1)+1.\n\nThe known elementary comparison loses a polynomial factor; the new problem asks whether that factor can be removed completely.", "research_status": "candidate_open_problem_requires_expert_signoff", "rl_ready": false, "schema_version": "1.0.0", "source_record_index": 3, "stream": "erdos_variant", "task_type": "prove_or_refute", "title": "Profile-balanced sunflower threshold", "verification": "Specialist mathematical review; exact computations must be reproducible."}
{"difficulty": "frontier_research", "domain": "Analytic and combinatorial number theory", "expected_output": "A rigorous proof, disproof, or mathematically substantive partial result, with explicit hypotheses, gap control, and a current literature boundary.", "inspiration": "Erdős Problem #208, which asks for sharp upper bounds on gaps between consecutive squarefree numbers, including the conjectural scale (π²/6)·log x/log log x.", "license": "MIT", "milestones": [], "problem_id": "erdos_025", "prompt": "Second-order certificate complexity of a squarefree-free interval. Let κ(h) be the smallest cardinality of a set Q of primes for which there exists an integer M such that every one of\n\nM+1,M+2,…,M+h\n\nis divisible by q² for at least one q∈Q. Thus Q is a square-divisor certificate that the whole interval contains no squarefree integer.\n\nDetermine the second-order asymptotic of κ(h). Is\n\nκ(h)\n=\n(6/π²)h\n+\n(4√6/π+o(1))·√h/log h?\n\nAt minimum, is\n\nκ(h)−(6/π²)h = Θ(√h/log h)?", "quality_signals": ["proved first-order term", "explicit conjectural second-order constant", "certificate-complexity interpretation"], "rationale": "Similarity: a gap between consecutive squarefree numbers is exactly an interval in which every integer has a square prime divisor. Erdős #208 studies the length of such an interval at a given location; the new problem studies the minimum number of distinct prime-square obstructions needed to manufacture an interval of a given length.\n\nNew feature: κ(h) measures certificate complexity rather than the location or maximum length of a gap. It is an optimization over both the interval and the collection of prime squares.\n\nBasic first-order argument: fix a cutoff y. Impose M≡0 mod p² for every prime p≤y. The positions i≤h divisible by one of these p² are then covered. For each remaining position i, choose a fresh prime q_i>h and impose M≡−i mod q_i². The Chinese remainder theorem gives one M satisfying all conditions. This costs\n\nπ(y)+R_y(h),\n\nwhere R_y(h) counts integers i≤h divisible by no p² with p≤y. Taking y→∞ slowly gives\n\nκ(h)≤(6/π²+o(1))h.\n\nConversely, for any fixed y, the residue classes supplied by primes p≤y cover at most\n\n(1−∏_{p≤y}(1−1/p²))h+O_y(1)\n\npositions. Primes larger than y contribute at most h/p²+1 positions each. Letting y→∞ yields\n\nκ(h)≥(6/π²−o(1))h.\n\nHence the first-order term is already forced: κ(h)=(6/π²+o(1))h. The proposed second term comes from the heuristic optimization\n\nπ(y)+h∏_{p≤y}(1−1/p²),\n\nusing ∑_{p>y}p⁻²∼1/(y log y). The optimum occurs near y≈√((6/π²)h) and predicts the constant 4√6/π≈3.11879. Controlling finite-interval sieve errors sharply enough to confirm or refute that constant is the new problem.", "research_status": "candidate_open_problem_requires_expert_signoff", "rl_ready": false, "schema_version": "1.0.0", "source_record_index": 25, "stream": "erdos_variant", "task_type": "prove_or_refute", "title": "Second-order certificate complexity of a squarefree-free interval", "verification": "Specialist mathematical review; exact computations must be reproducible."}
{"difficulty": "frontier_research", "domain": "Discrete geometry", "expected_output": "A rigorous proof, disproof, or mathematically substantive partial result, with explicit hypotheses, gap control, and a current literature boundary.", "inspiration": "Erdős Problem #604: must every n-point set in the plane contain at least one point from which there are n^{1-o(1)} distinct distances, perhaps as many as a constant multiple of n/sqrt(log n)?", "license": "MIT", "milestones": [], "problem_id": "erdos_075", "prompt": "Median pinned-distance conjecture. Let A be a set of n distinct points in the Euclidean plane, and for x in A write\n\nd_A(x) = |{||x-y|| : y in A, y != x}|.\n\nDefine the upper median pinned-distance count by\n\nq(A) = max{D : at least ceil(n/2) points x in A satisfy d_A(x) >= D},\n\nand define q(n) = min_{|A|=n} q(A). Is\n\nq(n) = Theta(n/sqrt(log n))?\n\nA stronger version asks whether q(n) has an asymptotic constant, and whether the extremal configurations are, after deleting o(n) points and applying a Euclidean similarity (translation, rotation, reflection, and uniform scaling), essentially two-dimensional lattice patches.", "quality_signals": ["robust pinned-distance formulation", "lattice upper-bound model", "incidence-geometric proof route"], "rationale": "It uses exactly the pinned-distance statistic from #604, but replaces one exceptional good pin by a positive proportion of good pins. Thus it is a distributional or robust version rather than a change of exponent. The m by m integer grid, with n=m^2, gives the expected upper scale for every pin: every squared distance is a sum of two squares of size O(m^2), and the classical count of integers representable as two squares is O(m^2/sqrt(log m))=O(n/sqrt(log n)). For a possible lower-bound route, suppose more than half of the pins have at most D distance classes. At each such pin, Cauchy-Schwarz forces on the order of n^2/D equal-distance pairs, hence many isosceles triangles. Summing over the bad pins converts the problem into a global perpendicular-bisector incidence estimate. Problem #604 only needs that estimate to produce one good pin; the new conjecture requires enough control to rule out a large population of bad pins and also suggests a stability theorem for near-extremizers.", "research_status": "candidate_open_problem_requires_expert_signoff", "rl_ready": false, "schema_version": "1.0.0", "source_record_index": 75, "stream": "erdos_variant", "task_type": "prove_or_refute", "title": "Median pinned-distance conjecture", "verification": "Specialist mathematical review; exact computations must be reproducible."}
{"difficulty": "frontier_research", "domain": "Harmonic analysis and polynomial inequalities", "expected_output": "A rigorous proof, disproof, or mathematically substantive partial result, with explicit hypotheses, gap control, and a current literature boundary.", "inspiration": "Erdős Problem #1150, which asks whether there is an absolute c>0 such that every sufficiently high-degree polynomial with coefficients in {−1,1} has max_{|z|=1}|P(z)|>(1+c)√n. Parseval gives only the baseline √(n+1).", "license": "MIT", "milestones": [], "problem_id": "erdos_149", "prompt": "RMS-superlevel concentration for Littlewood polynomials. Fix 0<η<1. For a Littlewood polynomial\n\nP(z)=∑_{j=0}^n ε_j z^j,  ε_j∈{−1,1},\n\ndefine\n\nμ_η(P)= (1/2π) · meas{θ∈[0,2π] : |P(e^{iθ})| ≥ (1−η)√(n+1)}\n\nand\n\nμ_η(n)=min_P μ_η(P),\n\nwhere the minimum is over all degree-n Littlewood polynomials.\n\nDetermine the order of μ_η(n). Is μ_η(n)=n^{-1/2+o(1)} for every fixed η, or can Littlewood polynomials concentrate their L² mass so efficiently that μ_η(n)=n^{-1+o(1)}? Does the exponent depend on η?", "quality_signals": ["Parseval-scale formulation", "rigorous exponent gap", "explicit Dirichlet-kernel construction"], "rationale": "Similarity: both problems compare the size of a Littlewood polynomial on the unit circle with its Parseval or root-mean-square scale √(n+1). Erdős #1150 asks for one point with a fixed excess above that scale. The new problem asks how much of the circle must remain near the RMS scale, even when the polynomial is allowed to concentrate its energy.\n\nNew feature: this is a distributional concentration problem rather than a supremum problem. A single very high spike may settle a maximum question while occupying negligible measure; μ_η(n) distinguishes narrow spikes from genuinely spread-out magnitude.\n\nBasic universal lower bound: Parseval gives\n\n(1/2π)∫|P(e^{iθ})|²dθ=n+1,\n\nwhile |P(e^{iθ})|≤n+1. Put a=1−η and μ=μ_η(P). Bounding |P|² by a²(n+1) off the superlevel set and by (n+1)² on it yields\n\n1 ≤ a²(1−μ)+(n+1)μ,\n\nso\n\nμ_η(n) ≥ (1−a²)/(n+1−a²) = Θ_η(1/n).\n\nBasic upper construction: for the all-plus polynomial P(z)=1+z+⋯+z^n, the Dirichlet-kernel formula gives |P(e^{iθ})|≤1/|sin(θ/2)| away from θ=0. Hence its RMS superlevel set has measure O_η(n^{-1/2}), and μ_η(n)≤O_η(n^{-1/2}).\n\nThe new problem is to close the exponent gap between n^{-1} and n^{-1/2}. An n^{-1/2} answer would say the Dirichlet-kernel concentration pattern is essentially extremal under the ±1 coefficient constraint; an n^{-1} answer would require much sharper spike constructions. Targeted searches found extensive work on L^q norms, flatness, and subarc behavior of Littlewood polynomials, but no exact minimization of this RMS-superlevel measure.", "research_status": "candidate_open_problem_requires_expert_signoff", "rl_ready": false, "schema_version": "1.0.0", "source_record_index": 149, "stream": "erdos_variant", "task_type": "prove_or_refute", "title": "RMS-superlevel concentration for Littlewood polynomials", "verification": "Specialist mathematical review; exact computations must be reproducible."}
{"difficulty": "frontier_research", "domain": "Enumerative algebraic geometry", "expected_output": "claim_status: proved, disproved, partial, or inconclusive\na precise main result with hypotheses and quantifiers\na complete proof or reproducible computation for every claimed milestone\nan adversarial check or counterexample search\nan explicit list of unresolved gaps and dependencies\na literature boundary separating known input from new work", "inspiration": "AIM-AG research-conjecture stream", "license": "MIT", "milestones": ["m1 (10%): Verify that the top Chern number on Gr(4,9) equals 321489. [exact_symbolic]", "m2 (25%): 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. [exact_linear_algebra_and_CAS]", "m3 (20%): Prove smoothness of the cubic along the marked plane and identify the completed local Fano algebra as C[[t]]/(t^2). [formal_proof_plus_local_CAS]", "m4 (35%): Produce and certify one completion whose Fano section is transverse away from the marked plane, or prove that no such completion exists. [exact_CAS_or_certified_specialization_plus_expert_review]", "m5 (10%): Derive rigorously that alternating-group containment plus this simple ramification gives full symmetric monodromy. [formal_group_theory_review]"], "problem_id": "aim_ag_001", "prompt": "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.\n\nDefinitions:\nF_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.\n\nInstructions:\nProve 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.\n\nRequired deliverables:\n- claim_status: proved, disproved, partial, or inconclusive\n- a precise main result with hypotheses and quantifiers\n- a complete proof or reproducible computation for every claimed milestone\n- an adversarial check or counterexample search\n- an explicit list of unresolved gaps and dependencies\n- a literature boundary separating known input from new work\n\nAllowed tools:\n- SageMath or SymPy\n- Macaulay2, Singular, or Magma\n- exact finite-field computation\n- certified numerical algebraic geometry followed by exact verification", "quality_signals": ["exact enumerative target", "explicit local algebra", "clear monodromy consequence"], "rationale": "A concrete high-degree monodromy problem with an exact local model, a symbolic enumerative check, and a sharply isolated global transversality step.", "research_status": "candidate_open_problem", "rl_ready": true, "schema_version": "1.0.0", "source_record_index": 1, "stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "A single simple branch among 321,489 planes", "verification": "High for the enumerative and local-algebra milestones; specialist review for global transversality."}
{"difficulty": "frontier_research", "domain": "Hodge theory and Fano geometry", "expected_output": "claim_status: proved, disproved, partial, or inconclusive\na precise main result with hypotheses and quantifiers\na complete proof or reproducible computation for every claimed milestone\nan adversarial check or counterexample search\nan explicit list of unresolved gaps and dependencies\na literature boundary separating known input from new work", "inspiration": "AIM-AG research-conjecture stream", "license": "MIT", "milestones": ["m1 (20%): Prove irreducibility and codimension d+1-h of the line-incidence component for a fixed line-Hilbert component H. [formal_algebraic_geometry_review]", "m2 (15%): Verify line-regularity and the expected dimension h=iota in at least one nontrivial Fano family. [symbolic_or_literature_verified_computation]", "m3 (35%): Establish the sharp infinitesimal Hodge-locus lower bound for one specified family beyond projective space. [formal_proof_plus_computer_algebra_when_applicable]", "m4 (30%): Classify equality in that family or construct a genuine lower- or equal-codimension counterexample not arising from a line. [expert_proof_review]"], "problem_id": "aim_ag_002", "prompt": "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.\n\nDefinitions:\nThe 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.\n\nInstructions:\nProve 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.\n\nRequired deliverables:\n- claim_status: proved, disproved, partial, or inconclusive\n- a precise main result with hypotheses and quantifiers\n- a complete proof or reproducible computation for every claimed milestone\n- an adversarial check or counterexample search\n- an explicit list of unresolved gaps and dependencies\n- a literature boundary separating known input from new work\n\nAllowed tools:\n- Macaulay2 or Singular\n- Borel-Weil-Bott and Jacobian-ring calculations\n- Hilbert-scheme computation\n- symbolic linear algebra", "quality_signals": ["sharp codimension target", "proved incidence component", "family-by-family computational route"], "rationale": "A sharp extremal-classification problem combining incidence geometry with equality-sensitive Hodge-theoretic estimates.", "research_status": "candidate_open_problem", "rl_ready": true, "schema_version": "1.0.0", "source_record_index": 2, "stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "Lines as the largest Noether-Lefschetz loci on Fano threefolds", "verification": "Moderate: incidence and family-specific algebra are checkable; the universal equality classification requires specialist proof review."}
{"difficulty": "frontier_research", "domain": "Commutative algebra and projective geometry", "expected_output": "claim_status: proved, disproved, partial, or inconclusive\na precise main result with hypotheses and quantifiers\na complete proof or reproducible computation for every claimed milestone\nan adversarial check or counterexample search\nan explicit list of unresolved gaps and dependencies\na literature boundary separating known input from new work", "inspiration": "AIM-AG research-conjecture stream", "license": "MIT", "milestones": ["m1 (20%): Prove the Mayer-Vietoris short exact sequence, the regularity bound, and the cokernel description of the nonlinear Tor strand. [formal_commutative_algebra_review]", "m2 (20%): Reproduce the benchmark of two quadric surfaces meeting in a conic and its unique beta_(2,4)=1. [Macaulay2_or_exact_Betti_verifier]", "m3 (35%): For a nontrivial irreducible incidence family, construct one member where every relevant edge map has maximal rank. [exact_CAS_plus_semicontinuity_argument]", "m4 (25%): Resolve the first multi-edge compatibility case or exhibit a generic rank defect that refutes the conjecture. [exact_CAS_plus_expert_review]"], "problem_id": "aim_ag_003", "prompt": "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.\n\nDefinitions:\nA 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.\n\nInstructions:\nProve 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.\n\nRequired deliverables:\n- claim_status: proved, disproved, partial, or inconclusive\n- a precise main result with hypotheses and quantifiers\n- a complete proof or reproducible computation for every claimed milestone\n- an adversarial check or counterexample search\n- an explicit list of unresolved gaps and dependencies\n- a literature boundary separating known input from new work\n\nAllowed tools:\n- Macaulay2\n- Singular\n- SageMath\n- exact random specialization over finite fields\n- determinantal rank computation", "quality_signals": ["proved Tor reduction", "explicit Betti prediction", "natural computer-algebra verifier"], "rationale": "A highly verifiable syzygy problem with an exact reduction, explicit Betti-table benchmarks, and determinantal rank tests.", "research_status": "candidate_open_problem", "rl_ready": true, "schema_version": "1.0.0", "source_record_index": 3, "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", "verification": "High for reductions and bounded examples; specialist review for the generic dominance theorem."}
{"difficulty": "frontier_research", "domain": "Toric geometry and numerical semigroups", "expected_output": "claim_status: proved, disproved, partial, or inconclusive\na precise main result with hypotheses and quantifiers\na complete proof or reproducible computation for every claimed milestone\nan adversarial check or counterexample search\nan explicit list of unresolved gaps and dependencies\na literature boundary separating known input from new work", "inspiration": "AIM-AG research-conjecture stream", "license": "MIT", "milestones": ["m1 (25%): Prove that free-map classes are exactly the integral curve classes nonnegative on all effective torus-invariant divisors, under the stated multiple-cover convention. [formal_toric_geometry_review]", "m2 (20%): Implement a reproducible pipeline from fan data to the Hilbert basis, normalized degree semigroup, Apéry set, and conductor. [deterministic_code_verifier]", "m3 (30%): Run the pipeline on all 18 smooth toric Fano threefolds and verify or refute c_X<=4. [deterministic_enumeration_with_artifacts]", "m4 (25%): Prove a dimension-only residue-filling bound for non-extremal Hilbert-basis elements, or find a higher-dimensional counterexample. [expert_proof_or_exact_counterexample_review]"], "problem_id": "aim_ag_004", "prompt": "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\nDefinitions:\nN_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.\n\nInstructions:\nProve 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.\n\nRequired deliverables:\n- claim_status: proved, disproved, partial, or inconclusive\n- a precise main result with hypotheses and quantifiers\n- a complete proof or reproducible computation for every claimed milestone\n- an adversarial check or counterexample search\n- an explicit list of unresolved gaps and dependencies\n- a literature boundary separating known input from new work\n\nAllowed tools:\n- SageMath\n- Normaliz\n- polymake\n- Macaulay2\n- exact numerical-semigroup code", "quality_signals": ["dimension-only quantitative target", "finite threefold benchmark", "deterministic fan-to-semigroup pipeline"], "rationale": "A finite fan-combinatorics program that turns a frontier existence bound into exact Hilbert-basis and numerical-semigroup computations.", "research_status": "candidate_open_problem", "rl_ready": true, "schema_version": "1.0.0", "source_record_index": 4, "stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "A conductor bound for free anticanonical degrees on toric Fano varieties", "verification": "High for fixed dimensions and classified fan lists; specialist review for the uniform dimension bound."}
{"difficulty": "frontier_research", "domain": "Arithmetic monodromy and Picard-Lefschetz theory", "expected_output": "claim_status: proved, disproved, partial, or inconclusive\na precise main result with hypotheses and quantifiers\na complete proof or reproducible computation for every claimed milestone\nan adversarial check or counterexample search\nan explicit list of unresolved gaps and dependencies\na literature boundary separating known input from new work", "inspiration": "AIM-AG research-conjecture stream", "license": "MIT", "milestones": ["m1 (15%): Reconstruct the finite-index implication from a complete integral vanishing lattice in the symplectic case. [formal_lattice_theory_review]", "m2 (25%): Prove integral generation and saturation of the vanishing lattice for one new ambient family. [expert_proof_review]", "m3 (30%): Realize the pairing-one or complete configurations required by the relevant vanishing-lattice theorem using high jet-ampleness. [formal_geometry_review]", "m4 (15%): Control the finite quadratic, spinor, or characteristic refinements in one orthogonal example. [exact_lattice_computation_plus_review]", "m5 (15%): Run an explicit thinness stress test and explain why Zariski density or large mod-prime images alone are insufficient. [adversarial_analysis_review]"], "problem_id": "aim_ag_005", "prompt": "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.\n\nDefinitions:\nThe 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.\n\nInstructions:\nProve 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.\n\nRequired deliverables:\n- claim_status: proved, disproved, partial, or inconclusive\n- a precise main result with hypotheses and quantifiers\n- a complete proof or reproducible computation for every claimed milestone\n- an adversarial check or counterexample search\n- an explicit list of unresolved gaps and dependencies\n- a literature boundary separating known input from new work\n\nAllowed tools:\n- SageMath lattice computations\n- Magma\n- symbolic intersection calculations\n- finite congruence-image computation", "quality_signals": ["rigorous finite-index reduction", "clear thinness adversary", "explicit odd/even lattice split"], "rationale": "A long-horizon proof task where integral lattice completeness, geometric vanishing cycles, and arithmetic-group criteria can be graded separately.", "research_status": "candidate_open_problem", "rl_ready": true, "schema_version": "1.0.0", "source_record_index": 5, "stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "Arithmeticity of vanishing cohomology in high-power linear systems", "verification": "Moderate in explicit families; specialist review is required for the full arbitrary-ambient theorem."}
{"difficulty": "frontier_research", "domain": "Algebraic topology and toric geometry", "expected_output": "claim_status: proved, disproved, partial, or inconclusive\na precise main result with hypotheses and quantifiers\na complete proof or reproducible computation for every claimed milestone\nan adversarial check or counterexample search\nan explicit list of unresolved gaps and dependencies\na literature boundary separating known input from new work", "inspiration": "AIM-AG research-conjecture stream", "license": "MIT", "milestones": ["m1 (20%): Construct the based Cox-coordinate discriminant resolution over the relevant Picard or Jacobian base and verify its augmentation. [formal_topology_and_geometry_review]", "m2 (25%): Control orientations, sign local systems, and the first nontrivial integral differentials associated with overlapping primitive collections. [chain_complex_or_spectral_sequence_verifier]", "m3 (30%): Compute the H_1 and H_2 comparison for an elliptic source and the Hirzebruch surface F_1 in an explicit high-degree chamber. [reproducible_chain_computation_plus_review]", "m4 (25%): Prove the required integral scanning or group-completion statement, or exhibit persistent torsion obstructing it. [expert_proof_or_counterexample_review]"], "problem_id": "aim_ag_006", "prompt": "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.\n\nDefinitions:\nMor^*_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.\n\nInstructions:\nProve 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.\n\nRequired deliverables:\n- claim_status: proved, disproved, partial, or inconclusive\n- a precise main result with hypotheses and quantifiers\n- a complete proof or reproducible computation for every claimed milestone\n- an adversarial check or counterexample search\n- an explicit list of unresolved gaps and dependencies\n- a literature boundary separating known input from new work\n\nAllowed tools:\n- SageMath\n- configuration-space chain complexes\n- spectral-sequence bookkeeping code\n- computer algebra for Cox presentations", "quality_signals": ["classical boundary cases", "explicit low-degree torsion test", "spectral-sequence milestones"], "rationale": "A synthesis task connecting classical mapping-space stability to integral configuration-space and Cox-coordinate calculations.", "research_status": "candidate_open_problem", "rl_ready": true, "schema_version": "1.0.0", "source_record_index": 6, "stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "An integral Segal theorem for toric targets", "verification": "Moderate: low-degree spectral-sequence pages and benchmark targets are computable; full integral stability needs specialist review."}
{"difficulty": "frontier_research", "domain": "Algebraic cycles and motives", "expected_output": "claim_status: proved, disproved, partial, or inconclusive\na precise main result with hypotheses and quantifiers\na complete proof or reproducible computation for every claimed milestone\nan adversarial check or counterexample search\nan explicit list of unresolved gaps and dependencies\na literature boundary separating known input from new work", "inspiration": "AIM-AG research-conjecture stream", "license": "MIT", "milestones": ["m1 (15%): Prove the strict-filtration nilpotence lemma and state precisely the faithfulness and filtration hypotheses. [formal_algebra_review]", "m2 (25%): Compute or sharply bound the geometric kernel ideal for conics, Severi-Brauer surfaces, or another explicit low-dimensional homogeneous variety. [exact_correspondence_or_motive_review]", "m3 (35%): Control mixed products through same-dimensional off-diagonal motivic summands in one nontrivial splitting tower. [expert_motivic_review]", "m4 (25%): Derive the factorial recurrence without summand-count dependence, or construct a counterexample to that recurrence. [expert_proof_or_exact_counterexample_review]"], "problem_id": "aim_ag_007", "prompt": "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.\n\nDefinitions:\nI_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.\n\nInstructions:\nProve 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.\n\nRequired deliverables:\n- claim_status: proved, disproved, partial, or inconclusive\n- a precise main result with hypotheses and quantifiers\n- a complete proof or reproducible computation for every claimed milestone\n- an adversarial check or counterexample search\n- an explicit list of unresolved gaps and dependencies\n- a literature boundary separating known input from new work\n\nAllowed tools:\n- SageMath or Magma for finite algebras\n- symbolic correspondence matrices\n- formal motive calculations\n- computer-assisted ring-theoretic exploration", "quality_signals": ["simple universal bound", "proved filtration lemma", "explicit Severi-Brauer benchmark"], "rationale": "A quantitative motivic nilpotence problem with a clean universal exponent, low-dimensional test cases, and an abstract filtration lemma.", "research_status": "candidate_open_problem", "rl_ready": true, "schema_version": "1.0.0", "source_record_index": 7, "stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "A factorial bound for Rost nilpotence", "verification": "Moderate for low-dimensional motives and abstract filtration lemmas; specialist review for the uniform factorial estimate."}
{"difficulty": "frontier_research", "domain": "Logarithmic enumerative geometry", "expected_output": "claim_status: proved, disproved, partial, or inconclusive\na precise main result with hypotheses and quantifiers\na complete proof or reproducible computation for every claimed milestone\nan adversarial check or counterexample search\nan explicit list of unresolved gaps and dependencies\na literature boundary separating known input from new work", "inspiration": "AIM-AG research-conjecture stream", "license": "MIT", "milestones": ["m1 (25%): Define the labelled tree-type stacks, rigidification, matching lattice, deck action, and source and target of every virtual K-class without ambiguity. [expert_definitional_review]", "m2 (20%): Prove the one-edge contact-m regular-representation formula over an etale trivialization. [formal_local_log_geometry_review]", "m3 (20%): Verify the two independent contacts 2 and 3 benchmark, including all six characters and no character-dependent virtual sign. [finite_group_and_virtual_pullback_review]", "m4 (35%): Establish equivariant finite-etale pullback of obstruction theories and descent across boundary expansions for the stated tree class, or exhibit a stack-inertia counterexample. [expert_stack_and_virtual_K_review]"], "problem_id": "aim_ag_008", "prompt": "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|.\n\nDefinitions:\nThe 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.\n\nInstructions:\nProve 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.\n\nRequired deliverables:\n- claim_status: proved, disproved, partial, or inconclusive\n- a precise main result with hypotheses and quantifiers\n- a complete proof or reproducible computation for every claimed milestone\n- an adversarial check or counterexample search\n- an explicit list of unresolved gaps and dependencies\n- a literature boundary separating known input from new work\n\nAllowed tools:\n- Smith normal form\n- derived fiber-product calculations\n- equivariant K-theory software where available\n- symbolic finite-group character calculations", "quality_signals": ["character-level refinement", "six-character benchmark", "explicit rigidification checks"], "rationale": "A representation-valued refinement of logarithmic degeneration formulas with exact finite-group benchmarks and stack-theoretic failure modes.", "research_status": "candidate_open_problem", "rl_ready": true, "schema_version": "1.0.0", "source_record_index": 8, "stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "A character-valued logarithmic gluing formula", "verification": "Moderate for local labelled tree types; specialist review for global stack descent and obstruction-theory compatibility."}
{"difficulty": "frontier_research", "domain": "Cyclotomic homotopy theory and K3 geometry", "expected_output": "claim_status: proved, disproved, partial, or inconclusive\na precise main result with hypotheses and quantifiers\na complete proof or reproducible computation for every claimed milestone\nan adversarial check or counterexample search\nan explicit list of unresolved gaps and dependencies\na literature boundary separating known input from new work", "inspiration": "AIM-AG research-conjecture stream", "license": "MIT", "milestones": ["m1 (15%): Reconstruct the fiberwise identification of cyclotomic degree -2 with H^2(W O_X) and the finite-height Frobenius criterion. [formal_literature_and_proof_review]", "m2 (25%): Construct relative finite V-quotients with base change on an explicit smooth chart of polarized K3 moduli. [derived_geometry_expert_review]", "m3 (25%): Identify the next Frobenius coefficient with the classical higher Hasse section and prove a simple zero at a generic finite-height transition. [local_deformation_computation_plus_review]", "m4 (35%): Recover the terminal supersingular scheme structure with multiplicity exactly two, or find additional embedded or derived structure. [expert_local_intersection_review]"], "problem_id": "aim_ag_009", "prompt": "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.\n\nDefinitions:\nFor 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).\n\nInstructions:\nProve 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.\n\nRequired deliverables:\n- claim_status: proved, disproved, partial, or inconclusive\n- a precise main result with hypotheses and quantifiers\n- a complete proof or reproducible computation for every claimed milestone\n- an adversarial check or counterexample search\n- an explicit list of unresolved gaps and dependencies\n- a literature boundary separating known input from new work\n\nAllowed tools:\n- spectral-sequence bookkeeping\n- de Rham-Witt calculations\n- local deformation-ring computation\n- computer algebra for complete intersections", "quality_signals": ["known fiberwise detector", "finite-height local tests", "decisive supersingular multiplicity"], "rationale": "A relative comparison problem linking cyclotomic homotopy, formal Brauer height, higher Hasse sections, and scheme-theoretic multiplicity.", "research_status": "candidate_open_problem", "rl_ready": true, "schema_version": "1.0.0", "source_record_index": 9, "stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "A cyclotomic Hasse tower for K3 moduli", "verification": "Moderate for fiberwise and finite-Witt milestones; specialist review for relative cyclotomic descent and terminal multiplicity."}
{"difficulty": "frontier_research", "domain": "p-adic homotopy theory and logarithmic geometry", "expected_output": "claim_status: proved, disproved, partial, or inconclusive\na precise main result with hypotheses and quantifiers\na complete proof or reproducible computation for every claimed milestone\nan adversarial check or counterexample search\nan explicit list of unresolved gaps and dependencies\na literature boundary separating known input from new work", "inspiration": "AIM-AG research-conjecture stream", "license": "MIT", "milestones": ["m1 (15%): Derive the relative logarithmic differential module B/eB generated by dlog(t) for the Kummer chart and verify tame vanishing. [formal_log_algebra_review]", "m2 (25%): Prove Galois descent on every complete log-prismatic or Hodge-Tate associated graded in a tame control family. [filtered_derived_review]", "m3 (25%): Justify interchange of the circle Tate construction with the required Galois totalization, or isolate an exact obstruction. [expert_homotopy_limit_review]", "m4 (35%): 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. [spectral_sequence_artifact_plus_expert_review]"], "problem_id": "aim_ag_010", "prompt": "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.\n\nDefinitions:\nTP^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.\n\nInstructions:\nProve 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.\n\nRequired deliverables:\n- claim_status: proved, disproved, partial, or inconclusive\n- a precise main result with hypotheses and quantifiers\n- a complete proof or reproducible computation for every claimed milestone\n- an adversarial check or counterexample search\n- an explicit list of unresolved gaps and dependencies\n- a literature boundary separating known input from new work\n\nAllowed tools:\n- spectral-sequence computation\n- derived log-cotangent calculations\n- group cohomology software\n- exact Kummer-extension arithmetic", "quality_signals": ["clean tame/wild criterion", "proved log-differential calculation", "explicit first wild class"], "rationale": "An if-and-only-if ramification detector with a computable logarithmic differential, filtered descent milestones, and a concrete wild Kummer test.", "research_status": "candidate_open_problem", "rl_ready": true, "schema_version": "1.0.0", "source_record_index": 10, "stream": "aim_ag_rl", "task_type": "prove_or_refute_with_milestone_credit", "title": "Logarithmic TP as a tame-ramification detector", "verification": "High for logarithmic differentials and associated-graded tests; specialist review for Tate-totalization interchange and permanence."}
{"difficulty": "frontier_computational_search", "domain": "Computational number theory", "expected_output": "An exact witness with every requested invariant, plus an independently checkable certificate for feasibility and global optimality.", "inspiration": "C114 — Giuga's primality conjecture", "license": "MIT", "milestones": [], "problem_id": "counterexample_114", "prompt": "Let P_30 be the set of the first 30 odd primes. Choose a 9-element subset S of P_30 and put n=product_{p in S} p. For p in S define G_p=1 when n/p is congruent to 1 modulo p, and C_p=1 when n is congruent to 1 modulo p-1. Put G(n)=sum_p G_p, C(n)=sum_p C_p, and B(n)=sum_p G_p C_p. Find S maximizing the lexicographic score (min(G(n),C(n)), B(n), G(n)+C(n)); subject to that, minimize n. Output S, n, the complete table of residues (n/p-1 mod p) and (n-1 mod p-1), the three scores, and the sets of primes satisfying each local condition. Supply an independently checkable branch-and-bound or pseudo-Boolean certificate proving optimality over all 9-subsets. Determine the two violation counts 9-G(n) and 9-C(n), the number of primes satisfying both local conditions, and the largest prime factor of n.", "quality_signals": ["near-counterexample score", "small exact modular checks", "pseudo-Boolean optimality certificate"], "rationale": "For a squarefree composite n, the equations p | n/p-1 for every p | n are the local Giuga conditions, while p-1 | n-1 for every p | n are Korselt's local conditions for being Carmichael. A composite counterexample to Giuga's primality criterion would satisfy both systems at every prime factor. The search is restricted to odd primes because any composite Carmichael number is odd. Rather than demanding a presently unknown counterexample, this variant asks for the closest joint local fit in a fixed finite prime universe. Once a subset is supplied, all scores are verified by small modular reductions; the difficult step is proving that none of the roughly fourteen million competing 9-subsets has a better joint score. The lexicographic objective first balances the two systems, then rewards conditions holding at the same prime, so separate partial successes cannot masquerade as a near-counterexample.", "research_status": "finite_search_task_existence_or_optimum_not_preasserted", "rl_ready": false, "schema_version": "1.0.0", "source_record_index": 114, "stream": "counterexample_variant", "task_type": "find_and_certify_optimal_object", "title": "Closest bounded Giuga–Carmichael local fit", "verification": "Exact witness checks plus a machine-checkable exhaustive certificate."}
{"difficulty": "frontier_computational_search", "domain": "Structural and computational graph theory", "expected_output": "An exact witness with every requested invariant, plus an independently checkable certificate for feasibility and global optimality.", "inspiration": "C128 — Hajós's conjecture for t=5 and t=6", "license": "MIT", "milestones": [], "problem_id": "counterexample_128", "prompt": "Let G range over simple graphs on thirteen vertices that are 5-critical, meaning chi(G)=5 while chi(G−e)=4 for every edge e, and require omega(G)=4. Let tau_5(G) be the number of edge subsets whose edge-subgraph is homeomorphic to K_5, with each subdivided K_5 counted once as an edge set. Find G minimizing tau_5(G). Subject to that, maximize girth, then minimize the number of edges, then minimize |Aut(G)|, and finally choose the canonical adjacency matrix. Output the edge list, a DRAT certificate that G is not 4-colorable, an explicit 4-coloring of G−e for every edge, and the complete list of topological K_5 subgraphs with their five branch vertices and ten branch paths. Supply an isomorph-free exhaustive certificate for the minimum. Determine tau_5(G), the orbit-size distribution of the subdivisions, girth, edge-connectivity, degree sequence, and automorphism-group order.", "quality_signals": ["first open Hajós chromatic boundary", "zero optimum would be a genuine counterexample", "DRAT and isomorph-free enumeration certificates"], "rationale": "This searches for the most subdivision-poor positive instance at the first open chromatic boundary, rather than directly demanding a counterexample. If the optimum were zero, the winner would be a genuine 5-chromatic graph with no K_5 subdivision. Otherwise the exact minimum measures how deeply the required subdivision can be hidden. The admissible class is nonempty: two iterated Ore compositions of K_5 produce a thirteen-vertex 5-critical graph with clique number four. Verification is finite and transparent. Edge-criticality is certified by the deletion colorings, non-4-colorability by DRAT, and every topological K_5 can be enumerated as a subgraph with five degree-four branch vertices and degree-two internal vertices. The hard part is canonical enumeration of all admissible thirteen-vertex graphs and proof that no graph has fewer models.", "research_status": "finite_search_task_existence_or_optimum_not_preasserted", "rl_ready": false, "schema_version": "1.0.0", "source_record_index": 128, "stream": "counterexample_variant", "task_type": "find_and_certify_optimal_object", "title": "Subdivision-poor five-critical graph", "verification": "Exact witness checks plus a machine-checkable exhaustive certificate."}
{"difficulty": "frontier_computational_search", "domain": "Spectral graph theory and discrete geometry", "expected_output": "An exact witness with every requested invariant, plus an independently checkable certificate for feasibility and global optimality.", "inspiration": "C131 — Hot Spots conjecture for convex planar domains", "license": "MIT", "milestones": [], "problem_id": "counterexample_131", "prompt": "Let P range over two-dimensional convex lattice polygons contained in [0,8]^2, normalized so that min{x:(x,y) in P}=min{y:(x,y) in P}=0, and such that S(P)=P intersect Z^2 has exactly 31 points. Join two points of S(P) when their Euclidean distance is one, obtaining the unit-grid graph G(P), and require G(P) to be connected. Let L(P) be its combinatorial Laplacian and require the second eigenvalue lambda_2(P) to be simple. Put B(P)={v in S(P):deg_G(P)(v)<4}. If u is a lambda_2-eigenvector orthogonal to the constants, let E(P) be the union of the vertices where u is maximal and where it is minimal; because lambda_2 is simple, E(P) is independent of the sign chosen for u. Define h(P)=min_{v in E(P)} dist_G(P)(v,B(P)) and q(P)=sum_{v in E(P)} dist_G(P)(v,B(P)). Find an admissible P maximizing h(P). Subject to that, maximize q(P), then minimize |B(P)|, then minimize |Aut(G(P))|, and finally choose the canonical representative under lattice isometries. Output the polygon vertices, all 31 lattice points, the graph, the Laplacian characteristic polynomial, a minimal polynomial and rational isolating interval for lambda_2, an eigenvector with coordinates in Q(lambda_2), and exact sign-comparison certificates identifying E(P). Supply an isomorph-free exhaustive certificate for the global optimum. Determine h(P), q(P), the two extremal vertex sets, lambda_2, the spectral gap lambda_3-lambda_2, the graph diameter, the number of boundary vertices, and the automorphism-group order.", "quality_signals": ["counterexample-oriented discrete analogue", "exact algebraic eigenvalue certificate", "finite convex-polygon search"], "rationale": "This is a discrete Neumann hot-spots problem on lattice-convex planar domains. A positive value of h(P) means that both signs of the first nonconstant mode have all their hottest vertices strictly inside the discrete domain; h(P)=0 means that at least one extremum reaches the boundary. It is not a continuum counterexample, because the graph Laplacian is only a finite analogue of the Neumann Laplacian. Verification is exact: convexity and the lattice-point count are elementary, a Sturm sequence isolates lambda_2 and lambda_3, and signs of coordinates in Q(lambda_2) determine the extrema. The class is nonempty. For example, conv{(0,0),(8,0),(8,1),(3,5),(0,1)} has exactly 31 lattice points, a connected unit-grid graph, and an exact square-free nonzero Laplacian characteristic factor, hence simple lambda_2. The difficult step is proving the optimum over all normalized polygons, not checking a proposed winner.", "research_status": "finite_search_task_existence_or_optimum_not_preasserted", "rl_ready": false, "schema_version": "1.0.0", "source_record_index": 131, "stream": "counterexample_variant", "task_type": "find_and_certify_optimal_object", "title": "Interior hot spots on a convex lattice domain", "verification": "Exact witness checks plus a machine-checkable exhaustive certificate."}
{"difficulty": "frontier_computational_search", "domain": "Diophantine approximation and dynamics", "expected_output": "An exact witness with every requested invariant, plus an independently checkable certificate for feasibility and global optimality.", "inspiration": "C134 — Lonely Runner conjecture, first unresolved case of 14 runners", "license": "MIT", "milestones": [], "problem_id": "counterexample_134", "prompt": "Let V={v_1<...<v_13} be a set of thirteen positive integers with v_13<=210 and gcd(v_1,...,v_13)=1. Define F_V(t)=min_{1<=i<=13} ||t v_i|| for t in R/Z, where ||.|| is distance to the nearest integer, and put Delta(V)=max_t F_V(t). Find V minimizing Delta(V). Subject to that, minimize v_13, then maximize the number of maximizing times in [0,1/2], then maximize the minimum number of active speeds at a maximizing time, and finally choose the lexicographically least tuple. Output V, Delta(V) as a reduced rational number, every maximizing time as a reduced fraction, and the set of active speeds at each such time. Give an exact interval-cover certificate proving F_V(t)<=Delta(V) for every t, together with a witnessing time proving equality, and supply a branch-and-bound or SAT certificate proving the global optimum over all primitive tuples in the stated box. Here the affine symmetry group means the group of real affine bijections x -> ax+b with a != 0 that preserve {0} union V setwise. Determine Delta(V), its comparison with 1/14, the number and denominator distribution of maximizing times, the active-set sizes, the largest gap between consecutive speeds, and the order of this affine symmetry group.", "quality_signals": ["first unresolved runner count", "exact rational verification", "counterexample-or-near-miss outcome"], "rationale": "After moving to the frame of the slowest of fourteen runners, the remaining thirteen relative speeds can be taken to be distinct positive integers, and the conjectural loneliness threshold is 1/14. This variant asks for the most dangerous primitive speed set below a fixed height, rather than for an arbitrary counterexample. If the optimum is below 1/14, the winning tuple is an actual counterexample; otherwise it is an exact finite near-miss benchmark. For fixed V, F_V is the lower envelope of finitely many triangular waves. Its maxima occur at rational breakpoints or intersections, and Delta(V)<=q is equivalent to a finite collection of near-integer intervals covering the unit circle. Hence a candidate and its exact value are easy to verify. The difficult component is excluding the enormous number of competing thirteen-subsets of {1,...,210}.", "research_status": "finite_search_task_existence_or_optimum_not_preasserted", "rl_ready": false, "schema_version": "1.0.0", "source_record_index": 134, "stream": "counterexample_variant", "task_type": "find_and_certify_optimal_object", "title": "Most dangerous bounded fourteen-runner instance", "verification": "Exact witness checks plus a machine-checkable exhaustive certificate."}
{"difficulty": "frontier_computational_search", "domain": "Algebraic combinatorics and representation theory", "expected_output": "An exact witness with every requested invariant, plus an independently checkable certificate for feasibility and global optimality.", "inspiration": "C135 — King–Tollu–Toumazet positivity conjecture", "license": "MIT", "milestones": [], "problem_id": "counterexample_135", "prompt": "Let lambda, mu, and nu be partitions with at most five parts, largest part at most 12, |lambda|+|mu|=|nu|<=36, lambda lexicographically no larger than mu, and gcd of all their positive parts equal to one. Require the Littlewood–Richardson coefficient c_{lambda,mu}^{nu} to exceed one and the stretching polynomial P(t)=c_{t lambda,t mu}^{t nu} to have degree at least two. Write P(t)=a_0+a_1 t+...+a_d t^d over Q and put m(lambda,mu,nu)=min_{1<=i<=d} a_i. Find an admissible triple minimizing m. Subject to that, maximize the number of coefficients equal to m, then maximize d, then minimize |nu|, and finally choose the canonical triple after interchanging lambda and mu. Output the three partitions, the complete polynomial P, exact values P(0),...,P(6), the associated five-hive inequality system, and a rational generating-function or equivalent certificate for every lattice-point count used in the interpolation. Supply an exhaustive certificate over all bounded triples. Determine m, the complete coefficient vector and common denominator, the degree, the dimension and normalized volume of the hive polytope, its number of vertices, and the values P(1) and P(2).", "quality_signals": ["negative optimum would refute a positivity conjecture", "exact hive-polytope enumeration", "bounded exhaustive search"], "rationale": "For partitions of length at most five, the hive polytope has dimension at most six, so seven exact Littlewood–Richardson counts determine the entire stretching polynomial. A negative optimum would be a KTT counterexample inside a small box; a nonnegative optimum would identify the closest coefficient-level near-miss in that finite class. The primitive and degree conditions remove simple rescalings and linear examples. The class is nonempty: for example, lambda=(3,1), mu=(3,2,1), and nu=(4,3,2,1) have P(t)=(t+1)(t+2)/2. Candidate verification reduces to checking integer hives and rational interpolation, while global optimality requires enumerating and comparing all bounded partition triples.", "research_status": "finite_search_task_existence_or_optimum_not_preasserted", "rl_ready": false, "schema_version": "1.0.0", "source_record_index": 135, "stream": "counterexample_variant", "task_type": "find_and_certify_optimal_object", "title": "Bounded King–Tollu–Toumazet stretching-positivity search", "verification": "Exact witness checks plus a machine-checkable exhaustive certificate."}