{"assumptions": ["The state space is countable, carries a measurable space, and every singleton is measurable.", "\\(\\mu\\) is a nonnegative measure and \\(K\\) is a measurable kernel; neither is assumed finite or normalized.", "For every pair \\(x,y\\), \\(\\mu(\\{x\\})K(x,\\{y\\})=\\mu(\\{y\\})K(y,\\{x\\})\\) in the extended nonnegative reals."], "binding_sha256": "ac978a2c81f53d2c0ce45bb253fdafd93de190bb9ff64e1eab954ea9e29249b8", "domain": "sampling", "family": "atomic-detailed-balance", "id": "sogb-001", "level": "L3", "library_domains": ["mcmc", "discrete-sampling"], "question": "Prove that for all measurable sets A,B, \\(\\int_A K(x,B)\\,d\\mu(x)=\\int_B K(y,A)\\,d\\mu(y)\\). Explain why infinite total mass and zero-mass states cause no cancellation problem. Does this imply convergence to a probability law?", "skills": ["proof", "assumption-audit", "semantic-boundary"], "split": "public-dev", "title": "From atomic balance to reversibility", "track": "natural-language"}
{"assumptions": ["The state space is countable, carries a measurable space, and every singleton is measurable.", "\\(\\mu\\) is a nonnegative measure and \\(K\\) is a measurable kernel; neither is assumed finite or normalized.", "For every pair \\(x,y\\), \\(\\mu(\\{x\\})K(x,\\{y\\})=\\mu(\\{y\\})K(y,\\{x\\})\\) in the extended nonnegative reals."], "binding_sha256": "ac978a2c81f53d2c0ce45bb253fdafd93de190bb9ff64e1eab954ea9e29249b8", "domain": "sampling", "family": "atomic-detailed-balance", "id": "sogb-001", "level": "L3", "library_domains": ["mcmc", "discrete-sampling"], "question": "Complete the fixed Lean target below. Keep its imports, context and proposition unchanged. Use the repository-pinned Lean/Mathlib. This is open-library reconstruction; the target module is not imported directly.\n\n/- SamplingOptimisationGeometryBench public-dev reference.\n   Reuses the existing project proof; retain LICENSE and NOTICE. -/\nimport Mathlib.Probability.Kernel.Invariance\nimport Mathlib.MeasureTheory.Integral.Lebesgue.Countable\n\nopen MeasureTheory ProbabilityTheory\nopen scoped ENNReal\n\nexample {α : Type*} [MeasurableSpace α]\n    [Countable α] [MeasurableSingletonClass α] (κ : Kernel α α) (μ : Measure α)\n    (h : ∀ x y, μ {x} * κ x {y} = μ {y} * κ y {x}) :\n    Kernel.IsReversible κ μ := by\n  -- Replace this comment with your proof.\n", "skills": ["proof", "assumption-audit", "semantic-boundary"], "split": "public-dev", "title": "From atomic balance to reversibility", "track": "lean"}
{"assumptions": ["\\(E\\) is a complete real inner-product space, \\(f:E\\to\\mathbb R\\), and \\(z\\) is an attained global minimizer.", "The real parameters \\(\\alpha,\\beta,h\\) satisfy \\(h\\ge0\\) and \\(\\beta h\\le1\\). For all \\(x,y\\), the quadratic upper model holds: \\[f(y)\\le f(x)+\\langle\\nabla f(x),y-x\\rangle+\\tfrac\\beta2\\|y-x\\|^2.\\]", "For every \\(x\\), \\(2\\alpha(f(x)-f(z))\\le\\|\\nabla f(x)\\|^2\\). Here \\(\\nabla f\\) is the representing vector of the derivative where \\(f\\) is Fréchet differentiable, and zero otherwise. The upper-model and PL inequalities are supplied assumptions; global differentiability is not inferred from the notation."], "binding_sha256": "f5d34b22cd89382ed252b2d1ed63a8b0763a005dbe5940fd24174b14710d19e8", "domain": "optimisation", "family": "pl-gradient-iteration", "id": "sogb-002", "level": "L3", "library_domains": ["optimisation"], "question": "For the actual iterates \\(x_{n+1}=x_n-h\\nabla f(x_n)\\), prove \\(f(x_N)-f(z)\\le(1-\\alpha h)^N(f(x_0)-f(z))\\). Do not add convexity or assume \\(1-\\alpha h\\ge0\\). Explain separately what happens when this factor is negative, and when the bound describes genuine geometric decay.", "skills": ["calculation", "proof", "assumption-audit"], "split": "public-dev", "title": "PL descent: the sign of the contraction factor", "track": "natural-language"}
{"assumptions": ["\\(E\\) is a complete real inner-product space, \\(f:E\\to\\mathbb R\\), and \\(z\\) is an attained global minimizer.", "The real parameters \\(\\alpha,\\beta,h\\) satisfy \\(h\\ge0\\) and \\(\\beta h\\le1\\). For all \\(x,y\\), the quadratic upper model holds: \\[f(y)\\le f(x)+\\langle\\nabla f(x),y-x\\rangle+\\tfrac\\beta2\\|y-x\\|^2.\\]", "For every \\(x\\), \\(2\\alpha(f(x)-f(z))\\le\\|\\nabla f(x)\\|^2\\). Here \\(\\nabla f\\) is the representing vector of the derivative where \\(f\\) is Fréchet differentiable, and zero otherwise. The upper-model and PL inequalities are supplied assumptions; global differentiability is not inferred from the notation."], "binding_sha256": "f5d34b22cd89382ed252b2d1ed63a8b0763a005dbe5940fd24174b14710d19e8", "domain": "optimisation", "family": "pl-gradient-iteration", "id": "sogb-002", "level": "L3", "library_domains": ["optimisation"], "question": "Complete the fixed Lean target below. Keep its imports, context and proposition unchanged. Use the repository-pinned Lean/Mathlib. This is open-library reconstruction; the target module is not imported directly.\n\n/- SamplingOptimisationGeometryBench public-dev reference.\n   Reuses the existing project proof; retain LICENSE and NOTICE. -/\nimport AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentBasic\nimport Mathlib.Analysis.SpecificLimits.Basic\n\nopen Set\nopen scoped RealInnerProductSpace\nopen AutoSamplingTheory.TechnicalLemmas.Analysis\nvariable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]\n\nexample {f : E → ℝ} {α β h : ℝ} {z : E}\n    (hz : IsMinOn f univ z) (hh : 0 ≤ h) (hstep : β * h ≤ 1)\n    (hu : ∀ x y, f y ≤ f x + inner ℝ (gradient f x) (y - x) + β / 2 * ‖y - x‖ ^ 2)\n    (hpl : ∀ x, 2 * α * (f x - f z) ≤ ‖gradient f x‖ ^ 2)\n    (x₀ : E) (N : ℕ) :\n    f ((fun x => x - h • gradient f x)^[N] x₀) - f z ≤\n      (1 - α * h) ^ N * (f x₀ - f z) := by\n  -- Replace this comment with your proof.\n", "skills": ["calculation", "proof", "assumption-audit"], "split": "public-dev", "title": "PL descent: the sign of the contraction factor", "track": "lean"}
{"assumptions": ["\\(E\\) is a real normed vector space; \\(V:E\\to\\mathbb R\\); \\(x_0\\) is an attained global minimizer; \\(k\\) is real.", "Use the following strong-convexity convention (also used by Mathlib): on all of \\(E\\), for \\(a,b\\ge0\\) with \\(a+b=1\\), \\[V(ax+by)\\le aV(x)+bV(y)-\\tfrac k2 ab\\|x-y\\|^2.\\]", "No differentiability or finite-dimensionality assumption is made."], "binding_sha256": "b65dd9a70ee7e52100bc85f865904dc278535225ecc2eaa01929897f888543c2", "domain": "geometry", "family": "strong-convex-midpoint-envelope", "id": "sogb-003", "level": "L2", "library_domains": ["log-concave-sampling", "optimisation"], "question": "Use only the midpoint instance and global minimality to prove \\(V(x)\\ge V(x_0)+\\tfrac k4\\|x-x_0\\|^2\\) for every x. Track the constant. Does this proof establish the sharper k/2 envelope, or finiteness of a Gibbs normalizer?", "skills": ["calculation", "assumption-audit", "semantic-boundary"], "split": "public-dev", "title": "A quadratic envelope from one midpoint", "track": "natural-language"}
{"assumptions": ["\\(E\\) is a real normed vector space; \\(V:E\\to\\mathbb R\\); \\(x_0\\) is an attained global minimizer; \\(k\\) is real.", "Use the following strong-convexity convention (also used by Mathlib): on all of \\(E\\), for \\(a,b\\ge0\\) with \\(a+b=1\\), \\[V(ax+by)\\le aV(x)+bV(y)-\\tfrac k2 ab\\|x-y\\|^2.\\]", "No differentiability or finite-dimensionality assumption is made."], "binding_sha256": "b65dd9a70ee7e52100bc85f865904dc278535225ecc2eaa01929897f888543c2", "domain": "geometry", "family": "strong-convex-midpoint-envelope", "id": "sogb-003", "level": "L2", "library_domains": ["log-concave-sampling", "optimisation"], "question": "Complete the fixed Lean target below. Keep its imports, context and proposition unchanged. Use the repository-pinned Lean/Mathlib. This is open-library reconstruction; the target module is not imported directly.\n\n/- SamplingOptimisationGeometryBench public-dev reference.\n   Reuses the existing project proof; retain LICENSE and NOTICE. -/\nimport AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity\nimport Mathlib.Analysis.Convex.Strong\n\nopen Set\n\nexample\n    {E : Type*} [NormedAddCommGroup E] [NormedSpace ℝ E]\n    {V : E → ℝ} {k : ℝ} (hV : StrongConvexOn (Set.univ : Set E) k V)\n    (x₀ : E) (hx₀ : IsMinOn V (Set.univ : Set E) x₀) :\n    ∀ x : E, V x₀ + (k / 4) * ‖x - x₀‖ ^ 2 ≤ V x := by\n  -- Replace this comment with your proof.\n", "skills": ["calculation", "assumption-audit", "semantic-boundary"], "split": "public-dev", "title": "A quadratic envelope from one midpoint", "track": "lean"}
{"assumptions": ["Let E be a finite-dimensional real Euclidean space with its Borel sigma-algebra. Let μ,ν be probability measures with finite second moments.", "A coupling γ has marginals μ,ν. Define W₂(μ,ν) as the infimum of the square root of ∫‖x−y‖² dγ over such couplings. Do not assume an infimizing coupling is attained."], "binding_sha256": "2274ba27f6d700779d832ff90a2b26d881f2bf5f23c3dfc78689f556020559cf", "domain": "transport", "family": "coupling-swap-near-optimality", "id": "sogb-004", "level": "L3", "library_domains": ["statistical-optimal-transport"], "question": "Prove symmetry of W₂ by swapping coordinates of near-optimal couplings. Which measurability and marginal identities are used, and where does the finite-cost hypothesis prevent an invalid subtraction of infinities?", "skills": ["proof", "assumption-audit"], "split": "public-dev", "title": "Wasserstein symmetry without an optimal plan", "track": "natural-language"}
{"assumptions": ["E,H are real Hilbert spaces. F:E→H is Fréchet differentiable at w, with bounded linear derivative D(w). Fix r∈H and V(w)=2⟨r,F(w)⟩.", "Assume ‖w‖=1 and ‖D(w)‖≤M with M≥0. The tangent space of the unit sphere at w is {v:⟨w,v⟩=0}."], "binding_sha256": "ce37724e97b715e2330e57378f5795759da21f91c6f477b863d897b48fe93f74", "domain": "geometry", "family": "sphere-tangent-projection", "id": "sogb-005", "level": "L3", "library_domains": ["riemannian-optimization"], "question": "Set g=2D(w)*r and G=g−⟨g,w⟩w. Prove tangency, the directional-derivative identity on tangent vectors and ‖G‖≤2M‖r‖. Explain precisely why the same formula need not project onto w-perpendicular when w is not a unit vector.", "skills": ["calculation", "proof", "assumption-audit", "counterexample"], "split": "public-dev", "title": "Why a sphere gradient is a projected ambient gradient", "track": "natural-language"}
{"assumptions": ["μ is any probability measure on the finite Boolean cube with n+1 coordinates. K is its uniform random-site heat-bath kernel, with the project's fixed conditional versions at null fibers.", "At any start y with μ({y})>0, the exact singleton transition formula is the sum over coordinates of the retained-coordinate indicator times μ({x}) divided by the corresponding fiber mass, averaged by 1/(n+1). No such formula is assumed at an unsupported start."], "binding_sha256": "74f08b42811ce2c35e4669f9e631246262c6038119faa990aea98efb42431cb3", "domain": "sampling", "family": "random-scan-null-atoms", "id": "sogb-006", "level": "L3", "library_domains": ["discrete-sampling", "mcmc"], "question": "In the detailed-balance proof, handle a target-null atom x without applying the positive-start singleton formula at x. Why is zero mass alone insufficient for an arbitrary kernel? Give a reversible heat-bath example which is not irreducible.", "skills": ["proof", "assumption-audit", "counterexample"], "split": "public-dev", "title": "Null atoms in a random-scan balance proof", "track": "natural-language"}
{"assumptions": ["There are finitely many n adjacent hybrid pairs. At slot i the pair shares a probability law ρᵢ on histories, conditional Markov kernels Kᵢ,Lᵢ and a common downstream Markov kernel Sᵢ.", "A measurable integrable real εᵢ bounds |Kᵢ(x,A)−Lᵢ(x,A)| for ρᵢ-almost every x and every measurable event A. Let βᵢ=∫εᵢ dρᵢ. Total variation means the supremum of absolute event-probability differences, with no extra factor of two."], "binding_sha256": "a4e2d5d2fd1273d0cadc2054858c9597b4e090dd0fec5fda5ecdfd5c7d206b78", "domain": "sampling", "family": "adaptive-hybrid-cost-boundary", "id": "sogb-007", "level": "L4", "library_domains": ["samplewiki", "mcmc"], "question": "Derive an eventwise discrepancy bound Σᵢβᵢ between the first and last output laws. State why actual history laws and a fixed finite call range matter. Then refute the claim that vanishing TV distance alone transfers an unbounded expected query-cost bound.", "skills": ["composition", "assumption-audit", "counterexample"], "split": "public-dev", "title": "Adaptive hybrid error does not certify expected cost", "track": "natural-language"}
{"assumptions": ["On ℝᵈ with d≥1 and Lebesgue reference measure, V is a finite real-valued Borel measurable convex function. Set ρ(x)=exp(−V(x))."], "binding_sha256": "ae0a2d0d8f65fbe39a5c2ed55cbe7b93bde710026ec00ed08132abeedefb2794", "domain": "sampling", "family": "gibbs-shape-normalization", "id": "sogb-008", "level": "L1", "library_domains": ["log-concave-sampling"], "question": "Identify the log-concave shape and its logarithm. What additional numerical condition on Z=∫ρ(x)dx is required before ρ/Z defines a probability density? State which reference measure the word density refers to.", "skills": ["definition", "semantic-boundary"], "split": "public-dev", "title": "A log-concave shape is not yet a probability density", "track": "natural-language"}
{"assumptions": ["π and ν are probability measures. K is a Markov transition kernel. C is the nonnegative random number of oracle calls in a specified run, under its actual input law."], "binding_sha256": "a01b6366bfff653a1e2d2d7304b4e284d28669bfebdcfb82e663f26765274315", "domain": "sampling", "family": "adaptive-hybrid-cost-boundary", "id": "sogb-009", "level": "L1", "library_domains": ["samplewiki"], "question": "Explain separately the meanings of TV(ν,π)≤ε, πK=π and E[C]≤B. Which objects occur in each claim, and why should a theorem statement name the run/input law when discussing expected cost? No implication among the claims needs to be proved.", "skills": ["definition", "semantic-boundary"], "split": "public-dev", "title": "What kind of guarantee does a sampling claim describe?", "track": "natural-language"}
{"assumptions": ["f is differentiable on a real finite-dimensional Euclidean space and attains its global minimum f* at z. Let α>0. Use the convention 2α(f(x)−f*)≤‖∇f(x)‖² for all x."], "binding_sha256": "7e2364e8fcf6653f355ba88b88d71d81b69d2e3959e42df0e523a13352a1b94d", "domain": "optimisation", "family": "pl-gradient-iteration", "id": "sogb-010", "level": "L1", "library_domains": ["optimisation"], "question": "What is the objective gap, what does the PL inequality compare, and is convexity part of this definition? Identify the roles of α and the gradient. Do not infer a gradient-descent rate without a step rule and a descent estimate.", "skills": ["definition", "semantic-boundary"], "split": "public-dev", "title": "Reading a PL inequality before using an algorithm", "track": "natural-language"}
{"assumptions": ["S is the unit sphere in ℝᵈ with d≥2, equipped with its induced Euclidean metric. A real-valued function f is smooth on a neighborhood of S. Let w∈S."], "binding_sha256": "a5cda7d4b9f7caaf24bf4dbab8205dae1f0a82e2ebb6b9fdc92d81c748863408", "domain": "geometry", "family": "sphere-tangent-projection", "id": "sogb-011", "level": "L1", "library_domains": ["riemannian-optimization"], "question": "Identify the tangent space at w and the defining directional-derivative identity for the gradient of f restricted to S. In what sense does this differ from the ambient Euclidean gradient? Do not prove an optimization convergence theorem.", "skills": ["definition", "semantic-boundary"], "split": "public-dev", "title": "Which space contains the sphere gradient?", "track": "natural-language"}
{"assumptions": ["μ,ν are Borel probability measures on ℝᵈ. A candidate joint measure γ is defined on ℝᵈ×ℝᵈ. No transport cost function or independence assumption is imposed on a general coupling."], "binding_sha256": "d61f3a0414d8d767ef90fd31733e4fb71d9b47d0f6a62b5844c939a7ee915c7f", "domain": "transport", "family": "coupling-swap-near-optimality", "id": "sogb-012", "level": "L1", "library_domains": ["statistical-optimal-transport"], "question": "State the two marginal identities that make γ a coupling of μ and ν. Is independence part of the definition? Identify a canonical feasible coupling and explain why feasibility alone does not mean optimality.", "skills": ["definition", "semantic-boundary"], "split": "public-dev", "title": "What information does a coupling contain?", "track": "natural-language"}
{"assumptions": ["The state space is a Boolean cube with m≥1 sites and target probability μ. For each site i, Kᵢ is the fixed coordinate heat-bath kernel, including its specified version on target-null fibers."], "binding_sha256": "84145f82f459d1cb080b714a85ebad040ad8fd4d034c441b45d77e02c77830ca", "domain": "sampling", "family": "random-scan-null-atoms", "id": "sogb-013", "level": "L1", "library_domains": ["discrete-sampling"], "question": "Describe one uniform random-scan step and its mixture kernel. How does that time unit differ from a sweep of all sites? Does the mixture-law description itself specify a mixing time or an oracle-query cost?", "skills": ["definition", "semantic-boundary"], "split": "public-dev", "title": "One random-site step is not one sweep", "track": "natural-language"}
{"assumptions": ["P is a finite row-stochastic transition matrix: P(x,y)≥0 and Σ_y P(x,y)=1. π and ρ are probability row vectors. Use total variation when discussing convergence."], "binding_sha256": "8b10bef6bc3088ea8188089bb960aed747ddeded576261956464483947b6db38", "domain": "sampling", "family": "atomic-detailed-balance", "id": "sogb-014", "level": "L1", "library_domains": ["mcmc"], "question": "Write the identities defining invariance and detailed balance for π. State what convergence from ρ means in this setting. Which identity controls one step started exactly in π, and why is that not the definition of convergence from an arbitrary start?", "skills": ["definition", "semantic-boundary"], "split": "public-dev", "title": "Invariant, reversible and convergent are different claims", "track": "natural-language"}
{"assumptions": ["The spaces \\(\\alpha,\\beta\\) carry arbitrary measurable structures, with the product measurable structure on \\(\\alpha\\times\\beta\\).", "The measure \\(\\gamma\\) couples \\(\\mu\\) and \\(\\nu\\): its coordinate pushforwards satisfy \\(\\gamma.\\mathrm{fst}=\\mu\\) and \\(\\gamma.\\mathrm{snd}=\\nu\\).", "Only \\(\\mu\\) is assumed to be a probability measure. No finiteness or probability assumption on \\(\\gamma\\) or \\(\\nu\\) is supplied independently."], "binding_sha256": "0bf552e05112c84a48c1bb862da7870035eb715a9cd61d0a95474c6694d37a7f", "domain": "transport", "family": "coupling-swap-near-optimality", "id": "sogb-015", "level": "L2", "library_domains": ["statistical-optimal-transport"], "question": "Prove that γ is a probability measure by calculating its total mass from the coupling identities. Identify which marginal identity this argument uses and explain whether ν needs an additional probability assumption. Do not assume the desired joint normalization at the outset.", "skills": ["calculation", "proof", "assumption-audit"], "split": "public-dev", "title": "Recovering joint probability mass from one marginal", "track": "natural-language"}
{"assumptions": ["The spaces \\(\\alpha,\\beta\\) carry arbitrary measurable structures, with the product measurable structure on \\(\\alpha\\times\\beta\\).", "The measure \\(\\gamma\\) couples \\(\\mu\\) and \\(\\nu\\): its coordinate pushforwards satisfy \\(\\gamma.\\mathrm{fst}=\\mu\\) and \\(\\gamma.\\mathrm{snd}=\\nu\\).", "Only \\(\\mu\\) is assumed to be a probability measure. No finiteness or probability assumption on \\(\\gamma\\) or \\(\\nu\\) is supplied independently."], "binding_sha256": "0bf552e05112c84a48c1bb862da7870035eb715a9cd61d0a95474c6694d37a7f", "domain": "transport", "family": "coupling-swap-near-optimality", "id": "sogb-015", "level": "L2", "library_domains": ["statistical-optimal-transport"], "question": "Complete the fixed Lean target below. Keep its imports, context and proposition unchanged. Use the repository-pinned Lean/Mathlib. This is open-library reconstruction; the target module is not imported directly.\n\n/- SamplingOptimisationGeometryBench public-dev reference.\n   Reuses the existing project proof; retain LICENSE and NOTICE. -/\nimport Mathlib.MeasureTheory.Measure.Prod\n\nopen MeasureTheory\nnamespace AutoSamplingTheory.TechnicalLemmas.Measure.Transport\ndef IsCoupling {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]\n    (γ : Measure (α × β)) (μ : Measure α) (ν : Measure β) : Prop :=\n  γ.fst = μ ∧ γ.snd = ν\nend AutoSamplingTheory.TechnicalLemmas.Measure.Transport\nopen AutoSamplingTheory.TechnicalLemmas.Measure.Transport\n\nexample\n    {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]\n    {γ : Measure (α × β)} {μ : Measure α} {ν : Measure β}\n    [IsProbabilityMeasure μ] (hγ : IsCoupling γ μ ν) :\n    IsProbabilityMeasure γ := by\n  -- Replace this comment with your proof.\n", "skills": ["calculation", "proof", "assumption-audit"], "split": "public-dev", "title": "Recovering joint probability mass from one marginal", "track": "lean"}
{"assumptions": ["\\(E\\) is a finite-dimensional real inner-product space with \\(\\dim E=n+1\\), where \\(n\\in\\mathbb N\\). Let \\(S=\\{x\\in E:\\|x\\|=1\\}\\) have its standard smooth sphere structure.", "\\(A:E\\to E\\) is a symmetric continuous linear map and \\(q(y)=\\langle y,Ay\\rangle\\). Fix \\(x\\in S\\), let \\(i:S\\hookrightarrow E\\) be the actual inclusion, and put \\(K=(\\mathbb R x)^\\perp\\).", "Tangent directions \\(w\\in T_xS\\) are paired through \\(D i_x w\\in K\\) using the ambient inner product. No default inner product on chart coordinates is identified with this induced pairing."], "binding_sha256": "69fee45c9a7a53e9d1a4ec5295d530811736ff3e8dec9fbd435f1293d9bc1cd7", "domain": "geometry", "family": "sphere-tangent-projection", "id": "sogb-016", "level": "L3", "library_domains": ["riemannian-optimization"], "question": "For every z in K, prove that D(q restricted to S)_x(w)=⟨D i_x(w),z⟩ for all w in T_xS if and only if z=2(Ax−⟨x,Ax⟩x). Justify the ambient derivative, the unit-normal projection and the uniqueness direction. Use genuine differentiability, not the totalized derivative's fallback value.", "skills": ["calculation", "proof", "assumption-audit"], "split": "public-dev", "title": "A quadratic gradient through the actual sphere differential", "track": "natural-language"}
{"assumptions": ["\\(E\\) is a finite-dimensional real inner-product space with \\(\\dim E=n+1\\), where \\(n\\in\\mathbb N\\). Let \\(S=\\{x\\in E:\\|x\\|=1\\}\\) have its standard smooth sphere structure.", "\\(A:E\\to E\\) is a symmetric continuous linear map and \\(q(y)=\\langle y,Ay\\rangle\\). Fix \\(x\\in S\\), let \\(i:S\\hookrightarrow E\\) be the actual inclusion, and put \\(K=(\\mathbb R x)^\\perp\\).", "Tangent directions \\(w\\in T_xS\\) are paired through \\(D i_x w\\in K\\) using the ambient inner product. No default inner product on chart coordinates is identified with this induced pairing."], "binding_sha256": "69fee45c9a7a53e9d1a4ec5295d530811736ff3e8dec9fbd435f1293d9bc1cd7", "domain": "geometry", "family": "sphere-tangent-projection", "id": "sogb-016", "level": "L3", "library_domains": ["riemannian-optimization"], "question": "Complete the fixed Lean target below. Keep its imports, context and proposition unchanged. Use the repository-pinned Lean/Mathlib. This is open-library reconstruction; the target module is not imported directly.\n\n/- SamplingOptimisationGeometryBench public-dev reference.\n   Reuses the existing project proof; retain LICENSE and NOTICE. -/\nimport AutoSamplingTheory.TechnicalLemmas.Geometry.SphereGradient\nimport Mathlib.Analysis.InnerProductSpace.Rayleigh\n\nnoncomputable section\nopen Metric InnerProductSpace\nopen scoped Manifold InnerProductSpace\nopen AutoSamplingTheory.TechnicalLemmas.Geometry\nvariable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]\n  [FiniteDimensional ℝ E]\n\nexample\n    (n : ℕ) [Fact (Module.finrank ℝ E = n + 1)]\n    (A : E →L[ℝ] E) (hA : A.IsSymmetric) (x : sphere (0 : E) 1) :\n    ∀ z : (ℝ ∙ (x : E))ᗮ,\n      ((∀ w : TangentSpace (𝓡 n) x,\n        mvfderiv (𝓡 n) (fun y : sphere (0 : E) 1 => ⟪(y : E), A (y : E)⟫_ℝ) x w =\n          ⟪mvfderiv (𝓡 n) ((↑) : sphere (0 : E) 1 → E) x w, (z : E)⟫_ℝ) ↔\n        (z : E) = (2 : ℝ) • (A (x : E) - ⟪(x : E), A (x : E)⟫_ℝ • (x : E))) := by\n  -- Replace this comment with your proof.\n", "skills": ["calculation", "proof", "assumption-audit"], "split": "public-dev", "title": "A quadratic gradient through the actual sphere differential", "track": "lean"}
{"assumptions": ["\\(A,B\\) are arbitrary measurable spaces, \\(\\mu\\) is a probability measure on A, and \\(K,L:A\\leadsto B\\) are Markov kernels.", "\\(\\varepsilon:A\\to\\mathbb R\\) is integrable under μ. On one common μ-full-measure set of inputs x, every measurable event s satisfies \\[|K(x,s)-L(x,s)|\\le\\varepsilon(x).\\] Event masses in this inequality are finite real probabilities."], "binding_sha256": "37f2cef51769619910cd718b9d66acd2d57f6a5a5956a6940a4d4dc2ba60eeba", "domain": "sampling", "family": "adaptive-hybrid-cost-boundary", "id": "sogb-017", "level": "L3", "library_domains": ["samplewiki", "mcmc"], "question": "For every measurable event t, prove |(K composed with μ)(t)−(L composed with μ)(t)|≤∫ε dμ. Give the measurability, finiteness and integrability justifications required to subtract the two real integrals and use almost-everywhere domination. Do not assume a measurable total-variation supremum or a query-cost conclusion.", "skills": ["proof", "assumption-audit", "calculation"], "split": "public-dev", "title": "Averaging a conditional event-error bound", "track": "natural-language"}
{"assumptions": ["\\(A,B\\) are arbitrary measurable spaces, \\(\\mu\\) is a probability measure on A, and \\(K,L:A\\leadsto B\\) are Markov kernels.", "\\(\\varepsilon:A\\to\\mathbb R\\) is integrable under μ. On one common μ-full-measure set of inputs x, every measurable event s satisfies \\[|K(x,s)-L(x,s)|\\le\\varepsilon(x).\\] Event masses in this inequality are finite real probabilities."], "binding_sha256": "37f2cef51769619910cd718b9d66acd2d57f6a5a5956a6940a4d4dc2ba60eeba", "domain": "sampling", "family": "adaptive-hybrid-cost-boundary", "id": "sogb-017", "level": "L3", "library_domains": ["samplewiki", "mcmc"], "question": "Complete the fixed Lean target below. Keep its imports, context and proposition unchanged. Use the repository-pinned Lean/Mathlib. This is open-library reconstruction; the target module is not imported directly.\n\n/- SamplingOptimisationGeometryBench public-dev reference.\n   Reuses the existing project proof; retain LICENSE and NOTICE. -/\nimport AutoSamplingTheory.TechnicalLemmas.Probability.KernelTotalVariation\n\nopen MeasureTheory ProbabilityTheory Set Finset\nopen scoped ProbabilityTheory BigOperators\n\nexample\n    {A B : Type*} [MeasurableSpace A] [MeasurableSpace B]\n    (μ : Measure A) [IsProbabilityMeasure μ]\n    (K L : Kernel A B) [IsMarkovKernel K] [IsMarkovKernel L]\n    (ε : A → ℝ) (hε : Integrable ε μ)\n    (hbound : ∀ᵐ x ∂μ, ∀ s, MeasurableSet s →\n      |(K x).real s - (L x).real s| ≤ ε x) :\n    ∀ t, MeasurableSet t →\n      |(K ∘ₘ μ).real t - (L ∘ₘ μ).real t| ≤ ∫ x, ε x ∂μ := by\n  -- Replace this comment with your proof.\n", "skills": ["proof", "assumption-audit", "calculation"], "split": "public-dev", "title": "Averaging a conditional event-error bound", "track": "lean"}
{"assumptions": ["\\(E\\) is a nonempty normed additive commutative group, with its Borel measurable structure, second-countable topology and a standard Borel measurable space. No vector-space or finite-dimensional structure is assumed.", "\\(\\mu_1,\\mu_2,\\mu_3\\) are probability measures on E. Couplings have exactly the two prescribed coordinate marginals. Define \\[C(\\gamma)=\\left(\\int\\|x-y\\|^2\\,d\\gamma(x,y)\\right)^{1/2}.\\] Both costs and \\(W_2\\) take values in \\([0,\\infty]\\); \\(W_2\\) is the square root of the infimum of squared coupling costs.", "The external thresholds \\(r_{12},r_{23}\\in[0,\\infty]\\) satisfy \\[W_2(\\mu_1,\\mu_2)<r_{12},\\qquad W_2(\\mu_2,\\mu_3)<r_{23}.\\] Neither optimal-plan existence nor finite individual second moments is given."], "binding_sha256": "237b6a126e4672415b58ea7f7621fe69ef5d541dac41fb1b42336f828922a1e2", "domain": "transport", "family": "coupling-gluing-triangle", "id": "sogb-018", "level": "L4", "library_domains": ["statistical-optimal-transport", "log-concave-sampling"], "question": "Prove \\(W_2(\\mu_1,\\mu_3)<r_{12}+r_{23}\\) by choosing strictly near-optimal adjacent plans and gluing them over their common middle marginal. Identify the measure used in Minkowski, justify all three pair-marginal cost identities and explain the measurability/conditional-version requirements. Audit the shortcut \\(\\Gamma'=\\gamma_{12}\\otimes\\mu_3\\): use identical Bernoulli marginals and diagonal adjacent plans to show exactly which identification fails. Do not assume optimal-plan attainment or convert extended costs to real numbers.", "skills": ["proof", "composition", "assumption-audit", "counterexample"], "split": "public-dev", "title": "Gluing the right joint law before applying Minkowski", "track": "natural-language"}
{"assumptions": ["\\(E\\) is a nonempty normed additive commutative group, with its Borel measurable structure, second-countable topology and a standard Borel measurable space. No vector-space or finite-dimensional structure is assumed.", "\\(\\mu_1,\\mu_2,\\mu_3\\) are probability measures on E. Couplings have exactly the two prescribed coordinate marginals. Define \\[C(\\gamma)=\\left(\\int\\|x-y\\|^2\\,d\\gamma(x,y)\\right)^{1/2}.\\] Both costs and \\(W_2\\) take values in \\([0,\\infty]\\); \\(W_2\\) is the square root of the infimum of squared coupling costs.", "The external thresholds \\(r_{12},r_{23}\\in[0,\\infty]\\) satisfy \\[W_2(\\mu_1,\\mu_2)<r_{12},\\qquad W_2(\\mu_2,\\mu_3)<r_{23}.\\] Neither optimal-plan existence nor finite individual second moments is given."], "binding_sha256": "237b6a126e4672415b58ea7f7621fe69ef5d541dac41fb1b42336f828922a1e2", "domain": "transport", "family": "coupling-gluing-triangle", "id": "sogb-018", "level": "L4", "library_domains": ["statistical-optimal-transport", "log-concave-sampling"], "question": "Complete the fixed Lean target below. Keep its imports, context and proposition unchanged. Use the repository-pinned Lean/Mathlib. This is open-library reconstruction; the target module is not imported directly.\n\n/- SamplingOptimisationGeometryBench public-dev reference.\n   Reuses the existing project proof; retain LICENSE and NOTICE. -/\nimport AutoSamplingTheory.TechnicalLemmas.Measure.TransportGluing\nimport AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals\nimport Mathlib.Tactic\n\nnoncomputable section\nopen MeasureTheory\nopen scoped ENNReal\nopen AutoSamplingTheory.TechnicalLemmas.Measure\nopen AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleCore\nopen AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangleMarginals\nvariable {E : Type*} [NormedAddCommGroup E]\n  [MeasurableSpace E] [BorelSpace E] [SecondCountableTopology E]\n  [StandardBorelSpace E] [Nonempty E]\n\nexample\n    (μ₁ μ₂ μ₃ : Measure E)\n    [IsProbabilityMeasure μ₁] [IsProbabilityMeasure μ₂] [IsProbabilityMeasure μ₃]\n    {r₁₂ r₂₃ : ℝ≥0∞}\n    (hr₁₂ : WassersteinSpace.wassersteinDistance μ₁ μ₂ < r₁₂)\n    (hr₂₃ : WassersteinSpace.wassersteinDistance μ₂ μ₃ < r₂₃) :\n    WassersteinSpace.wassersteinDistance μ₁ μ₃ < r₁₂ + r₂₃ := by\n  -- Replace this comment with your proof.\n", "skills": ["proof", "composition", "assumption-audit", "counterexample"], "split": "public-dev", "title": "Gluing the right joint law before applying Minkowski", "track": "lean"}
{"assumptions": ["E is a real inner-product space, not necessarily complete or finite dimensional. The function f:E→ℝ is globally G-Lipschitz, where G≥0.", "A supplied vector g supports f at x globally: \\[f(x)+\\langle g,y-x\\rangle\\le f(y)\\qquad\\text{for every }y\\in E.\\] Existence of supporting vectors, convexity of f and differentiability are not additional premises."], "binding_sha256": "b12bf7bd53d4c8c85dd4e1658b22f05ea7752e13bfa48a3bbfba72f620c3d52f", "domain": "optimisation", "family": "subgradient-energy-average", "id": "sogb-019", "level": "L3", "library_domains": ["optimisation"], "question": "Prove \\(\\|g\\|\\le G\\), making the test point and zero case explicit. Is the same conclusion valid if the support inequality holds only for y in a constraint set C containing x? Give a counterexample with a globally Lipschitz function and explain exactly which step no longer applies.", "skills": ["proof", "assumption-audit", "counterexample"], "split": "public-dev", "title": "Why a global Lipschitz function bounds its supporting vectors", "track": "natural-language"}
{"assumptions": ["E is a real inner-product space, not necessarily complete or finite dimensional. The function f:E→ℝ is globally G-Lipschitz, where G≥0.", "A supplied vector g supports f at x globally: \\[f(x)+\\langle g,y-x\\rangle\\le f(y)\\qquad\\text{for every }y\\in E.\\] Existence of supporting vectors, convexity of f and differentiability are not additional premises."], "binding_sha256": "b12bf7bd53d4c8c85dd4e1658b22f05ea7752e13bfa48a3bbfba72f620c3d52f", "domain": "optimisation", "family": "subgradient-energy-average", "id": "sogb-019", "level": "L2", "library_domains": ["optimisation"], "question": "Complete the fixed Lean target below. Keep its imports, context and proposition unchanged. Use the repository-pinned Lean/Mathlib. This is open-library reconstruction; the target module is not imported directly.\n\n/- SamplingOptimisationGeometryBench public-dev reference.\n   Reuses the existing project proof; retain LICENSE and NOTICE. -/\nimport AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexSubgradient\nimport Mathlib.Topology.MetricSpace.Lipschitz\nimport Mathlib.Data.Finset.Max\nimport Mathlib.Algebra.Order.BigOperators.Group.Finset\nimport Mathlib.Tactic.Linarith\nimport Mathlib.Tactic.Positivity\nimport Mathlib.Tactic.Ring\n\n/- Copyright (c) 2024 Hongjia Chen, Chenyi Li, Wanyi He, Zaiwen Wen.\nReleased under Apache 2.0; retain LICENSE and NOTICE.\nUpstream Optlib 5da27c5f95aa6a8a45b8c14b968ade4c13ff18c3,\nOptlib/Algorithm/SubgradientMethod.lean:80–118,137–232.\nModified for Samplinglib: native Lean 4.33.0, global SupportsAt and unbundled inputs. -/\nopen AutoSamplingTheory.TechnicalLemmas.Analysis.ConvexSubgradient\nopen scoped NNReal RealInnerProductSpace\nvariable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]\n\nexample {f : E → ℝ} {G : ℝ≥0} {x g : E}\n    (hf : LipschitzWith G f) (hs : SupportsAt f x g) : ‖g‖ ≤ (G : ℝ) := by\n  -- Replace this comment with your proof.\n", "skills": ["proof", "assumption-audit", "counterexample"], "split": "public-dev", "title": "Why a global Lipschitz function bounds its supporting vectors", "track": "lean"}
{"assumptions": ["E is a real inner-product space and f:E→ℝ is globally G-Lipschitz with G≥0. Supplied sequences xₖ,gₖ satisfy the global supporting inequality f(xₖ)+⟨gₖ,y−xₖ⟩≤f(y) for every y∈E and every k≥0.", "The actual recurrence is \\[x_{k+1}=x_k-a_k g_k.\\] For the energy estimate take aₖ≥0; for the normalized best-iterate statement assume aₖ>0 and N>0. The comparator z∈E is arbitrary, not necessarily a minimizer.", "For the positive-parameter rate corollary fix N>0, R>0, G>0 and ‖x₀−z‖≤R, then choose the constant step aₖ=R/(G√N). This schedule is selected for that finite horizon."], "binding_sha256": "15a19933c0cbddea0e5e89962c7375f05183709493c77749fe4f0115c0155931", "domain": "optimisation", "family": "subgradient-energy-average", "id": "sogb-020", "level": "L3", "library_domains": ["optimisation"], "question": "Derive the terminal-distance-retaining weighted energy bound, then show that some i<N satisfies \\[f(x_i)-f(z)\\le\\frac{\\|x_0-z\\|^2+G^2\\sum_{k=0}^{N-1}a_k^2}{2\\sum_{k=0}^{N-1}a_k}.\\] Deduce the exact RG/√N bound for the stated constant step. Explain what happens at N=0, R=0 or G=0 and which existence/domain claims this reasoning does not provide.", "skills": ["proof", "composition", "assumption-audit", "calculation"], "split": "public-dev", "title": "From subgradient energy to a finite-run best iterate", "track": "natural-language"}
{"assumptions": ["α and β are measurable spaces; β is nonempty and standard Borel. μ is any finite nonnegative measure on α×β, including the zero measure. No standard-Borel or nonempty assumption is imposed on α.", "Let μ₁ be the first marginal and κ(a,db) a selected measurable regular conditional Markov kernel for the second coordinate given the first. You may use the existing finite-measure disintegration identity \\[\\mu(da,db)=\\mu_1(da)\\kappa(a,db).\\] Its conditional characterization is only μ₁-almost everywhere; the selected version has mass one at every input.", "The actual update retains a and resamples b: \\[K((a,b),d(a',b'))=\\delta_a(da')\\kappa(a,db').\\] Invariant means the measure equality μK=μ, not convergence from every starting measure."], "binding_sha256": "3d72820e315cae6cebacdbed3bc8475fb4479d4a4a6cc7c302f4d534b54fe173", "domain": "mcmc", "family": "conditional-resampling-invariance", "id": "sogb-021", "level": "L3", "library_domains": ["mcmc", "discrete-sampling"], "question": "Prove μK=μ by identifying the measure after projecting to the retained coordinate and after applying the product kernel. Explain the role of measurability, the finite-measure disintegration, and the chosen conditional version on null fibers. Must μ be a probability measure? Give a finite-state example showing why this invariance result alone does not imply convergence from another initial probability law.", "skills": ["proof", "assumption-audit", "counterexample"], "split": "public-dev", "title": "Why retaining one coordinate and conditionally resampling the other is invariant", "track": "natural-language"}
{"assumptions": ["α and β are measurable spaces; β is nonempty and standard Borel. μ is any finite nonnegative measure on α×β, including the zero measure. No standard-Borel or nonempty assumption is imposed on α.", "Let μ₁ be the first marginal and κ(a,db) a selected measurable regular conditional Markov kernel for the second coordinate given the first. You may use the existing finite-measure disintegration identity \\[\\mu(da,db)=\\mu_1(da)\\kappa(a,db).\\] Its conditional characterization is only μ₁-almost everywhere; the selected version has mass one at every input.", "The actual update retains a and resamples b: \\[K((a,b),d(a',b'))=\\delta_a(da')\\kappa(a,db').\\] Invariant means the measure equality μK=μ, not convergence from every starting measure."], "binding_sha256": "3d72820e315cae6cebacdbed3bc8475fb4479d4a4a6cc7c302f4d534b54fe173", "domain": "mcmc", "family": "conditional-resampling-invariance", "id": "sogb-021", "level": "L2", "library_domains": ["mcmc", "discrete-sampling"], "question": "Complete the fixed Lean target below. Keep its imports, context and proposition unchanged. Use the repository-pinned Lean/Mathlib. This is open-library reconstruction; the target module is not imported directly.\n\n/- SamplingOptimisationGeometryBench public-dev reference.\n   Reuses the existing project proof; retain LICENSE and NOTICE. -/\nimport AutoSamplingTheory.TechnicalLemmas.Probability.ConditionalResampling\nimport Mathlib.Probability.Kernel.Invariance\n\nopen MeasureTheory ProbabilityTheory\nnamespace AutoSamplingTheory.TechnicalLemmas.Probability.HeatBath\nnoncomputable def heatBathSnd {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]\n    [StandardBorelSpace β] [Nonempty β] (μ : Measure (α × β)) [IsFiniteMeasure μ] :\n    Kernel (α × β) (α × β) :=\n  (Kernel.id ×ₖ condDistrib Prod.snd Prod.fst μ) ∘ₖ\n    Kernel.deterministic Prod.fst measurable_fst\nend AutoSamplingTheory.TechnicalLemmas.Probability.HeatBath\nopen AutoSamplingTheory.TechnicalLemmas.Probability.HeatBath\nopen AutoSamplingTheory.TechnicalLemmas.Probability\nvariable {α β : Type*} [MeasurableSpace α] [MeasurableSpace β]\n  [StandardBorelSpace β] [Nonempty β]\n\nexample (μ : Measure (α × β)) [IsFiniteMeasure μ] :\n    (heatBathSnd μ).Invariant μ := by\n  -- Replace this comment with your proof.\n", "skills": ["proof", "assumption-audit", "counterexample"], "split": "public-dev", "title": "Why retaining one coordinate and conditionally resampling the other is invariant", "track": "lean"}
{"assumptions": ["M is a metric space, F:M→ℝ, α∈ℝ, and P selects curves γ:ℝ→M. For every selected curve and 0≤t≤1 assume \\[F(\\gamma(t))\\le(1-t)F(\\gamma(0))+tF(\\gamma(1))-\\frac{\\alpha t(1-t)}2d(\\gamma(0),\\gamma(1))^2.\\] The predicate P itself does not certify geodesicity, speed, regularity or existence of curves between all endpoints.", "A particular γ with P(γ) is supplied. The scalar composition φ(t)=F(γ(t)) has the genuine two-sided derivative φ′(0)=g. Neither a Riemannian/Wasserstein gradient nor a logarithmic map is supplied. α may have either sign."], "binding_sha256": "2cfe0ffa4f03a84301a91017152eea5265cf548015c4f28f0d9d9fa9f12817b5", "domain": "geometry", "family": "convex-flow-first-order-rate", "id": "sogb-022", "level": "L3", "library_domains": ["riemannian-optimization", "statistical-optimal-transport"], "question": "Prove \\[F(\\gamma(1))\\ge F(\\gamma(0))+g+\\frac\\alpha2d(\\gamma(0),\\gamma(1))^2.\\] Show the positive-time quotient and its right-hand limit, retaining the exact constant. Explain why this does not itself identify g with a geometric gradient pairing or supply geodesics for arbitrary endpoint pairs. Is an arbitrary value returned by a totalized derivative enough to replace the differentiability hypothesis?", "skills": ["proof", "calculation", "assumption-audit", "semantic-boundary"], "split": "public-dev", "title": "From a selected-curve chord inequality to its first-order bound", "track": "natural-language"}
{"assumptions": ["M is a metric space, F:M→ℝ, α∈ℝ, and P selects curves γ:ℝ→M. For every selected curve and 0≤t≤1 assume \\[F(\\gamma(t))\\le(1-t)F(\\gamma(0))+tF(\\gamma(1))-\\frac{\\alpha t(1-t)}2d(\\gamma(0),\\gamma(1))^2.\\] The predicate P itself does not certify geodesicity, speed, regularity or existence of curves between all endpoints.", "A particular γ with P(γ) is supplied. The scalar composition φ(t)=F(γ(t)) has the genuine two-sided derivative φ′(0)=g. Neither a Riemannian/Wasserstein gradient nor a logarithmic map is supplied. α may have either sign."], "binding_sha256": "2cfe0ffa4f03a84301a91017152eea5265cf548015c4f28f0d9d9fa9f12817b5", "domain": "geometry", "family": "convex-flow-first-order-rate", "id": "sogb-022", "level": "L3", "library_domains": ["riemannian-optimization", "statistical-optimal-transport"], "question": "Complete the fixed Lean target below. Keep its imports, context and proposition unchanged. Use the repository-pinned Lean/Mathlib. This is open-library reconstruction; the target module is not imported directly.\n\n/- SamplingOptimisationGeometryBench public-dev reference.\n   Reuses the existing project proof; retain LICENSE and NOTICE. -/\nimport Mathlib.Analysis.Calculus.Deriv.Slope\nimport Mathlib.Topology.MetricSpace.Basic\n\nopen Filter Set\nopen scoped Topology\nnamespace AutoSamplingTheory.TechnicalLemmas.Geometry.GeodesicConvexity\ndef IsAlphaGeodesicallyConvex\n    {M : Type*} [MetricSpace M]\n    (isGeodesic : (ℝ → M) → Prop) (F : M → ℝ) (alpha : ℝ) : Prop :=\n  ∀ path : ℝ → M, isGeodesic path →\n    ∀ t : ℝ, t ∈ Icc (0 : ℝ) 1 →\n      F (path t) ≤\n        (1 - t) * F (path 0) + t * F (path 1) -\n          (alpha * t * (1 - t) / 2) * dist (path 0) (path 1) ^ 2\nend AutoSamplingTheory.TechnicalLemmas.Geometry.GeodesicConvexity\nopen AutoSamplingTheory.TechnicalLemmas.Geometry.GeodesicConvexity\n\nexample\n    {M : Type*} [MetricSpace M]\n    {isGeodesic : (ℝ → M) → Prop} {F : M → ℝ} {alpha : ℝ}\n    (hconvex : IsAlphaGeodesicallyConvex isGeodesic F alpha)\n    {path : ℝ → M} (hpath : isGeodesic path)\n    {gradientPairing : ℝ}\n    (hderiv : HasDerivAt (fun t => F (path t)) gradientPairing 0) :\n    F (path 1) ≥\n      F (path 0) + gradientPairing +\n        alpha / 2 * dist (path 0) (path 1) ^ 2 := by\n  -- Replace this comment with your proof.\n", "skills": ["proof", "calculation", "assumption-audit", "semantic-boundary"], "split": "public-dev", "title": "From a selected-curve chord inequality to its first-order bound", "track": "lean"}
{"assumptions": ["E is a complete real inner-product space. f:E→ℝ is differentiable everywhere and α-strongly convex on E for α≥0, in the chord convention \\[f((1-s)x+sy)\\le(1-s)f(x)+sf(y)-\\frac\\alpha2s(1-s)\\|y-x\\|^2\\qquad(0\\le s\\le1).\\] The corresponding first-order inequality is a derived adapter, not an extra premise. A global minimizer z is supplied, not constructed.", "T≥0 and X:ℝ→E is continuous on [0,T]. For every u∈[0,T), X has the genuine derivative within the forward ray [u,∞) at u equal to −∇f(X(u)). This is the actual trajectory/regularity premise; a scalar gap recurrence is not given.", "The observation t lies in (0,T]. Gradient notation is the Hilbert/Riesz gradient supplied by differentiability, not an arbitrary value of a totalized API. If T=0 there is no such observation."], "binding_sha256": "4a0607efb0c7bf22a09cc2d1ced0f8033582c57fd04e0225279f2e0782d169dc", "domain": "optimisation", "family": "convex-flow-first-order-rate", "id": "sogb-023", "level": "L4", "library_domains": ["optimisation"], "question": "Starting from the actual flow, prove the nonnegative objective-gap estimate \\[0\\le f(X(t))-f(z)\\le\\begin{cases}\\|X(0)-z\\|^2/(2t),&\\alpha=0,\\\\\\alpha\\|X(0)-z\\|^2/[2(e^{\\alpha t}-1)],&\\alpha>0.\\end{cases}\\] Derive objective monotonicity and the distance inequality needed for the fixed later time t rather than assuming them. Explain the time/curvature/domain conditions, why the theorem constructs no flow, and why directly substituting t=0 in totalized division is invalid; give a valid positive-initial-gap example for that last point.", "skills": ["proof", "composition", "calculation", "assumption-audit", "counterexample"], "split": "public-dev", "title": "An actual gradient flow: objective decay with the zero-curvature branch", "track": "natural-language"}
{"assumptions": ["E is a complete real inner-product space. f:E→ℝ is differentiable everywhere and α-strongly convex on E for α≥0, in the chord convention \\[f((1-s)x+sy)\\le(1-s)f(x)+sf(y)-\\frac\\alpha2s(1-s)\\|y-x\\|^2\\qquad(0\\le s\\le1).\\] The corresponding first-order inequality is a derived adapter, not an extra premise. A global minimizer z is supplied, not constructed.", "T≥0 and X:ℝ→E is continuous on [0,T]. For every u∈[0,T), X has the genuine derivative within the forward ray [u,∞) at u equal to −∇f(X(u)). This is the actual trajectory/regularity premise; a scalar gap recurrence is not given.", "The observation t lies in (0,T]. Gradient notation is the Hilbert/Riesz gradient supplied by differentiability, not an arbitrary value of a totalized API. If T=0 there is no such observation."], "binding_sha256": "4a0607efb0c7bf22a09cc2d1ced0f8033582c57fd04e0225279f2e0782d169dc", "domain": "optimisation", "family": "convex-flow-first-order-rate", "id": "sogb-023", "level": "L4", "library_domains": ["optimisation"], "question": "Complete the fixed Lean target below. Keep its imports, context and proposition unchanged. Use the repository-pinned Lean/Mathlib. This is open-library reconstruction; the target module is not imported directly.\n\n/- SamplingOptimisationGeometryBench public-dev reference.\n   Reuses the existing project proof; retain LICENSE and NOTICE. -/\nimport AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexFirstOrder\nimport Mathlib.Analysis.InnerProductSpace.Calculus\nimport Mathlib.Analysis.ODE.Gronwall\n\nopen Set\nopen scoped RealInnerProductSpace\nopen AutoSamplingTheory.TechnicalLemmas.Analysis\nvariable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]\n\nexample {f : E → ℝ} {X : ℝ → E} {z : E} {α T : ℝ}\n    (hα : 0 ≤ α) (_hT : 0 ≤ T) (hf : Differentiable ℝ f)\n    (hsc : StrongConvexOn univ α f) (hz : IsMinOn f univ z)\n    (hX : ContinuousOn X (Icc 0 T))\n    (hflow : ∀ t ∈ Ico 0 T, HasDerivWithinAt X (-gradient f (X t)) (Ici t) t) :\n    ∀ t ∈ Ioc 0 T, 0 ≤ f (X t) - f z ∧\n      f (X t) - f z ≤ if α = 0 then ‖X 0 - z‖ ^ 2 / (2 * t)\n        else α / (2 * (Real.exp (α * t) - 1)) * ‖X 0 - z‖ ^ 2 := by\n  -- Replace this comment with your proof.\n", "skills": ["proof", "composition", "calculation", "assumption-audit", "counterexample"], "split": "public-dev", "title": "An actual gradient flow: objective decay with the zero-curvature branch", "track": "lean"}
{"assumptions": ["Ω is an uncountable set equipped with the countable/co-countable σ-algebra: a measurable set is countable or has countable complement. Define μ(B)=0 for countable B and μ(B)=1 for co-countable B. Both input and output spaces use this σ-algebra.", "For measurable B define K(x,B)=1_B(x) and L(x,B)=μ(B). A Markov kernel must be a probability measure in B for each x, and measurable in x for every measurable B. The total-variation event supremum is not assumed measurable.", "Compare the statements \\[(\\forall B\\text{ measurable})\\;(K(x,B)=L(x,B)\\text{ for }\\mu\\text{-a.e. }x)\\] and \\[\\mu\\text{-a.e. }x\\;(\\forall B\\text{ measurable})\\;K(x,B)=L(x,B).\\] In the second statement the same exceptional set must work for all events."], "binding_sha256": "a64992f051cc8dbb2603c11d72a720f0121aeb064ca12a66d0efd321c5e014da", "domain": "mcmc", "family": "adaptive-hybrid-cost-boundary", "id": "sogb-024", "level": "L3", "library_domains": ["mcmc", "samplewiki"], "question": "Verify that μ is a probability measure and K,L are measurable Markov kernels. Prove the first quantified statement but refute the second. Compute both output laws after averaging over μ: does this example refute equality of the averaged laws? Explain exactly what it does—and does not—show about the uniform eventwise hypothesis in the local conditional-error bound, and why it is not a standard-Borel counterexample.", "skills": ["proof", "assumption-audit", "counterexample", "semantic-boundary"], "split": "public-dev", "title": "One null set for every event, or a different null set for each event?", "track": "natural-language"}
{"assumptions": ["E is a finite-dimensional complete real inner-product space with its Borel measurable structure and volume measure, including the zero-dimensional case.", "W:E→ℝ is C². The density exp(−W) is integrable and its real normalizer Z=∫exp(−W) is strictly positive. Write μ(dx)=Z⁻¹exp(−W(x))dx.", "Real constants satisfy 0<m≤M and, for every x,a, m‖a‖²≤D²W(x)[a,a]≤M‖a‖². The actual position map id is in L²(μ). Retain both curvature bounds and this moment premise of the current source API.", "The imported Gibbs C¹ Poincaré theorem gives m Varμ(f)≤∫‖∇f‖²dμ for genuinely C¹ observables with f and ∇f in L²(μ), under the displayed Gibbs hypotheses."], "binding_sha256": "9331f23fb1a4e9f66961f15c1233923302255d9b4ff3c953d01a7ad61d76e7b6", "domain": "sampling", "family": "posterior-curvature-smoothing", "id": "sogb-025", "level": "L3", "library_domains": ["log-concave-sampling", "samplewiki"], "question": "Prove Covμ(a,a)≤‖a‖²/m for every a∈E, interpreting Covμ(a,a) as the covariance of the linear observable x↦⟨a,x⟩ with itself. Establish its smoothness and L² properties, evaluate its Dirichlet energy, and justify the covariance/variance identification. Separate premises of this current API from stronger claims about when moments or an upper curvature bound are mathematically necessary.", "skills": ["proof", "calculation", "assumption-audit"], "split": "public-dev", "title": "From Gibbs Poincaré to a directional covariance bound", "track": "natural-language"}
{"assumptions": ["E is a finite-dimensional complete real inner-product space with its Borel measurable structure and volume measure, including the zero-dimensional case.", "W:E→ℝ is C². The density exp(−W) is integrable and its real normalizer Z=∫exp(−W) is strictly positive. Write μ(dx)=Z⁻¹exp(−W(x))dx.", "Real constants satisfy 0<m≤M and, for every x,a, m‖a‖²≤D²W(x)[a,a]≤M‖a‖². The actual position map id is in L²(μ). Retain both curvature bounds and this moment premise of the current source API.", "The imported Gibbs C¹ Poincaré theorem gives m Varμ(f)≤∫‖∇f‖²dμ for genuinely C¹ observables with f and ∇f in L²(μ), under the displayed Gibbs hypotheses."], "binding_sha256": "9331f23fb1a4e9f66961f15c1233923302255d9b4ff3c953d01a7ad61d76e7b6", "domain": "sampling", "family": "posterior-curvature-smoothing", "id": "sogb-025", "level": "L3", "library_domains": ["log-concave-sampling", "samplewiki"], "question": "Complete the fixed Lean target below. Keep its imports, context and proposition unchanged. Use the repository-pinned Lean/Mathlib. This is open-library reconstruction; the target module is not imported directly.\n\n/- SamplingOptimisationGeometryBench public-dev reference.\n   Reuses the existing project proof; retain LICENSE and NOTICE. -/\nimport AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GibbsC1Poincare\nimport Mathlib.Probability.Moments.CovarianceBilin\n\nopen MeasureTheory ProbabilityTheory InnerProductSpace\nopen scoped RealInnerProductSpace\nopen AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities\nnoncomputable section\nset_option autoImplicit false\nset_option backward.isDefEq.respectTransparency false\n\nexample\n    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]\n    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] [CompleteSpace E]\n    (W : E → ℝ) (hW : ContDiff ℝ 2 W)\n    (hI : Integrable (fun x => Real.exp (-W x)))\n    (hZ : 0 < ∫ x, Real.exp (-W x))\n    (m M : ℝ) (hm : 0 < m) (hmM : m ≤ M)\n    (hlower : ∀ x a, m*‖a‖^2 ≤ fderiv ℝ (fderiv ℝ W) x a a)\n    (hupper : ∀ x a, fderiv ℝ (fderiv ℝ W) x a a ≤ M*‖a‖^2)\n    (hX : MemLp id 2 ((volume : Measure E).tilted (fun x => -W x))) :\n    let μ := (volume : Measure E).tilted (fun x => -W x)\n    ∀ a : E, ProbabilityTheory.covarianceBilin μ a a ≤ ‖a‖^2/m := by\n  -- Replace this comment with your proof.\n", "skills": ["proof", "calculation", "assumption-audit"], "split": "public-dev", "title": "From Gibbs Poincaré to a directional covariance bound", "track": "lean"}
{"assumptions": ["Work on ℝᵈ with d≥1 and Lebesgue measure. Let V∈C²(ℝᵈ), with real constants m≤M satisfying mI≼∇²V≼MI everywhere; m and M need not be nonnegative.", "Fix η>0 with c=m+η⁻¹>0. Define Cη=(2πη)⁻ᵈᐟ², qη(y)=Cη∫exp(−V(x)−‖x−y‖²/(2η))dx and Uη(y)=−log qη(y). Do not assume ∫exp(−V)<∞ or a probability prior at the outset.", "The existing Gibbs covariance producers may be used once their actual C², positive lower/finite upper posterior curvature, normalization and moment hypotheses are verified. A Gaussian kernel has integral one; elementary polynomial-Gaussian integrability, dominated parameter differentiation and Tonelli are available with their hypotheses checked.", "This is an original extension/audit question motivated by SPHMC v1 §4.1, equation (4.1) and Lemma 4.1, not a claim that the paper states a weakly convex extension. Its original strong-convexity setting and algorithmic conclusions must not be silently replaced."], "binding_sha256": "fd49b5de54ddc240ba020368b624adc6d081d39a2c3e4378761f243fc0827279", "domain": "sampling", "family": "posterior-curvature-smoothing", "id": "sogb-026", "level": "L4", "library_domains": ["samplewiki", "log-concave-sampling", "optimisation"], "question": "A proposed extension says: ‘Drop positive curvature of V, retain m+η⁻¹>0, and reuse the SPHMC smoothing proof unchanged; the result is a normalized smoothed sampling law with gradient Lipschitz constant M/(1+Mη).’ Audit and repair this claim. Establish the pointwise analytic and posterior interfaces rather than assuming the covariance identity, derive sharp two-sided Hessian bounds and a valid nonnegative gradient Lipschitz constant, and decide which normalization and existing Lean API steps still require new work. Give an exact concave-quadratic counterexample to the overclaims. Do not infer an algorithmic complexity theorem.", "skills": ["proof", "composition", "assumption-audit", "counterexample", "semantic-boundary"], "split": "public-dev", "title": "Can the SPHMC smoothing argument survive weak convexity?", "track": "natural-language"}
{"assumptions": ["For each n≥0 let Ωₙ={0,1}ⁿ with the full sigma algebra and uniform probability μₙ, giving every configuration mass 2⁻ⁿ. The space Ω₀ has one configuration, not zero configurations. All functions considered are finite real-valued, with no positivity or normalization assumption on h.", "Write φ(s)=s log s for s>0 and φ(0)=0. Define Entₙ(h²)=Eμₙφ(h²)−φ(Eμₙh²). This convention agrees with h²*Real.log(h²), including zeros.", "For j<n, εʲ is obtained by toggling only coordinate j. Define Dₙh(ε) as the sum over all j<n of (h(ε)−h(εʲ))², not divided by n.", "You may use the existing two-point inequality, valid for all real a,b: [φ(a²)+φ(b²)]/2−φ((a²+b²)/2)≤(a−b)²/2. Do not assume the n-dimensional conclusion or an entropy tensorization theorem."], "binding_sha256": "7da6ccd4e2fbac80e8597c8d5421fb083791810652c7a3838214fa305763cf89", "domain": "discrete sampling", "family": "bernoulli-gaussian-entropy-limit", "id": "sogb-027", "level": "L3", "library_domains": ["discrete-sampling", "log-concave-sampling"], "question": "Prove Entₙ(h²)≤(1/2)EμₙDₙh for all n≥0 and all signed h. Construct the conditional RMS after separating the last coordinate; derive the entropy identity, its energy contraction and the dimension induction with the exact factor. Justify finite-law normalization and all integral domains, including n=0 and zero values. Finally exhibit why replacing that RMS by the ordinary conditional mean can invalidate the entropy identity.", "skills": ["proof", "calculation", "assumption-audit", "counterexample"], "split": "public-dev", "title": "Tensorizing a two-point entropy bound without losing the flip constant", "track": "natural-language"}
{"assumptions": ["Let f:ℝ→ℝ be genuinely C² with compact support, and let γ=N(0,1). The function may be signed or zero. No normalization ∫f²dγ=1 and no positive lower bound on f or that integral are assumed.", "For m≥1, μₘ is uniform probability on {0,1}ᵐ and Sₘ(ε)=m⁻¹ᐟ²Σⱼr(εⱼ), where r(1)=1 and r(0)=−1. The actual pushforward laws of Sₘ converge weakly to γ; this balanced Rademacher CLT is available. Use m=n+1 when taking the limit.", "The finite-product inequality Entμₘ(h²)≤(1/2)EμₘΣⱼ(h(ε)−h(εʲ))² is available for every real h, with finite-law normalization and integrability. Write φ(s)=s log s on s>0 and φ(0)=0.", "Standard weak convergence against bounded continuous observables, finite counting integrals, compact-support derivative bounds, the real mean-value theorem and closedness of real order under limits may be used after checking their hypotheses. Do not assume the entropy or full flip-energy limits as unexplained hypotheses of the NL proof."], "binding_sha256": "1be26d5f3c790d72f77813c7277245953ea18975cdadb455721f7a972b11f302", "domain": "sampling", "family": "bernoulli-gaussian-entropy-limit", "id": "sogb-028", "level": "L4", "library_domains": ["log-concave-sampling", "discrete-sampling"], "question": "Prove the compact scalar Gaussian inequality Entγ(f²)≤2∫(f′)²dγ. Explain why the homogeneous entropy converges even if its squared mass is zero, derive the limit of the sum over all Boolean flips with its exact factor four and a uniform error bound, and justify the final order passage. Distinguish weak-convergence tests from dominated convergence on a changing probability space. Keep the compact C² boundary: do not infer the full noncompact or infinite-dimensional inequality.", "skills": ["proof", "composition", "calculation", "assumption-audit"], "split": "public-dev", "title": "From the Boolean cube to compact Gaussian log-Sobolev", "track": "natural-language"}
{"assumptions": ["Let f:ℝ→ℝ be genuinely C² with compact support, and let γ=N(0,1). The function may be signed or zero. No normalization ∫f²dγ=1 and no positive lower bound on f or that integral are assumed.", "For m≥1, μₘ is uniform probability on {0,1}ᵐ and Sₘ(ε)=m⁻¹ᐟ²Σⱼr(εⱼ), where r(1)=1 and r(0)=−1. The actual pushforward laws of Sₘ converge weakly to γ; this balanced Rademacher CLT is available. Use m=n+1 when taking the limit.", "The finite-product inequality Entμₘ(h²)≤(1/2)EμₘΣⱼ(h(ε)−h(εʲ))² is available for every real h, with finite-law normalization and integrability. Write φ(s)=s log s on s>0 and φ(0)=0.", "Standard weak convergence against bounded continuous observables, finite counting integrals, compact-support derivative bounds, the real mean-value theorem and closedness of real order under limits may be used after checking their hypotheses. Do not assume the entropy or full flip-energy limits as unexplained hypotheses of the NL proof."], "binding_sha256": "1be26d5f3c790d72f77813c7277245953ea18975cdadb455721f7a972b11f302", "domain": "sampling", "family": "bernoulli-gaussian-entropy-limit", "id": "sogb-028", "level": "L3", "library_domains": ["log-concave-sampling", "discrete-sampling"], "question": "Complete the fixed Lean target below. Keep its imports, context and proposition unchanged. Use the repository-pinned Lean/Mathlib. This is open-library reconstruction; the target module is not imported directly.\n\n/- SamplingOptimisationGeometryBench public-dev reference.\n   Reuses the existing project proof; retain LICENSE and NOTICE. -/\nimport AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.BernoulliLogSobolev\nimport AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianCompactEntropy\nimport AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianFlipEnergy\nimport Mathlib.Topology.Order.OrderClosed\n\nopen MeasureTheory ProbabilityTheory Filter\nopen scoped Topology BigOperators ENNReal\nopen AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities\n\nexample\n    (f : ℝ → ℝ) (hf : ContDiff ℝ 2 f) (hs : HasCompactSupport f) :\n    let γ : Measure ℝ := gaussianReal 0 1\n    (∫ x, (f x)^2 * Real.log ((f x)^2) ∂γ) -\n      (∫ x, (f x)^2 ∂γ) * Real.log (∫ x, (f x)^2 ∂γ) ≤\n    2 * ∫ x, (deriv f x)^2 ∂γ := by\n  -- Replace this comment with your proof.\n", "skills": ["proof", "composition", "calculation", "assumption-audit"], "split": "public-dev", "title": "From the Boolean cube to compact Gaussian log-Sobolev", "track": "lean"}
{"assumptions": ["This is an original faithful reformulation of Sinho Chewi, Lectures on Optimization, arXiv:2605.07006v1, Chapter 3, Exercise 3.2, printed/PDF p.23. The chapter explicitly works with C² functions; the condition-number regime uses 0<α≤β.", "Let f:ℝᵈ→ℝ be C² and globally α-convex and β-smooth, with the quadratic-model normalization α/2 and β/2. Equivalently under this regularity, αI≼∇²f(z)≼βI for every z. The order α≤β is explicit rather than inferred in a zero-dimensional space.", "For h≥0 define T_h(x)=x−h∇f(x), and write κ=β/α. No minimizer or iterate convergence hypothesis is supplied. The target is a pairwise one-step bound and the optimal uniform curvature-endpoint envelope."], "binding_sha256": "9d9f361a44b9b4cbd33ff6379d61f64ca7b45805e410c4021af25e64ba6013c8", "domain": "optimisation", "exercise_source": {"authors": ["Sinho Chewi"], "book_page": 23, "edition": "arXiv:2605.07006v1", "exercise": "3.2", "id": "chewi-opt-v1-exercise-3-2", "pdf_page": 23, "source_url": "https://arxiv.org/pdf/2605.07006v1#page=23", "title": "Lectures on Optimization", "wording_status": "original faithful mathematical restatement; not reproduced source prose"}, "family": "chewi-opt-v1-exercise-3-2", "id": "sogb-029", "level": "L3", "library_domains": ["optimisation"], "question": "Solve Exercise 3.2 in the positive-curvature regime. Use the genuine derivative of T_h, bound its eigenvalues and operator norm, then justify the vector fundamental theorem of calculus along the segment from x to y. Find the step minimizing the resulting endpoint bound and obtain the factor (κ−1)/(κ+1). Prove minimality of that bound, not merely the value at a proposed step, and explain what is—and is not—being optimized.", "skills": ["proof", "calculation", "assumption-audit"], "split": "public-dev", "title": "Chewi Optimization Exercise 3.2: the balanced gradient step", "track": "natural-language"}