{"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": ["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 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": ["\\(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": ["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"}