{"binding_sha256": "ac978a2c81f53d2c0ce45bb253fdafd93de190bb9ff64e1eab954ea9e29249b8", "family": "atomic-detailed-balance", "id": "sogb-001", "messages": [{"content": "{\n  \"assumptions\": [\n    \"The state space is countable, carries a measurable space, and every singleton is measurable.\",\n    \"\\\\(\\\\mu\\\\) is a nonnegative measure and \\\\(K\\\\) is a measurable kernel; neither is assumed finite or normalized.\",\n    \"For every pair \\\\(x,y\\\\), \\\\(\\\\mu(\\\\{x\\\\})K(x,\\\\{y\\\\})=\\\\mu(\\\\{y\\\\})K(y,\\\\{x\\\\})\\\\) in the extended nonnegative reals.\"\n  ],\n  \"binding_sha256\": \"ac978a2c81f53d2c0ce45bb253fdafd93de190bb9ff64e1eab954ea9e29249b8\",\n  \"domain\": \"sampling\",\n  \"family\": \"atomic-detailed-balance\",\n  \"id\": \"sogb-001\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"mcmc\",\n    \"discrete-sampling\"\n  ],\n  \"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?\",\n  \"skills\": [\n    \"proof\",\n    \"assumption-audit\",\n    \"semantic-boundary\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"From atomic balance to reversibility\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "On a countable measurable-singleton space, expand each restricted integral and each kernel mass into singleton masses. The left side becomes \\[\\sum_{x\\in A}\\sum_{y\\in B}K(x,\\{y\\})\\mu(\\{x\\}).\\] These sums take values in \\([0,\\infty]\\).\n\nApply the assumed balance identity to each summand. Nonnegativity permits exchanging the two sums even when the result is infinite: \\[\\sum_{x\\in A}\\sum_{y\\in B}\\mu(\\{y\\})K(y,\\{x\\})=\\sum_{y\\in B}K(y,A)\\mu(\\{y\\}).\\]\n\nRecognize the last expression as the right restricted integral. No division by a singleton mass, subtraction of infinities, or signed Fubini argument occurred. A zero atom needs no special conditional version for this flux identity.\n\nReversibility alone proves neither normalization, irreducibility, convergence, nor a mixing or query-cost bound. Those require separate assumptions and arguments.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "ac978a2c81f53d2c0ce45bb253fdafd93de190bb9ff64e1eab954ea9e29249b8", "family": "atomic-detailed-balance", "id": "sogb-001", "messages": [{"content": "{\n  \"assumptions\": [\n    \"The state space is countable, carries a measurable space, and every singleton is measurable.\",\n    \"\\\\(\\\\mu\\\\) is a nonnegative measure and \\\\(K\\\\) is a measurable kernel; neither is assumed finite or normalized.\",\n    \"For every pair \\\\(x,y\\\\), \\\\(\\\\mu(\\\\{x\\\\})K(x,\\\\{y\\\\})=\\\\mu(\\\\{y\\\\})K(y,\\\\{x\\\\})\\\\) in the extended nonnegative reals.\"\n  ],\n  \"binding_sha256\": \"ac978a2c81f53d2c0ce45bb253fdafd93de190bb9ff64e1eab954ea9e29249b8\",\n  \"domain\": \"sampling\",\n  \"family\": \"atomic-detailed-balance\",\n  \"id\": \"sogb-001\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"mcmc\",\n    \"discrete-sampling\"\n  ],\n  \"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\",\n  \"skills\": [\n    \"proof\",\n    \"assumption-audit\",\n    \"semantic-boundary\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"From atomic balance to reversibility\",\n  \"track\": \"lean\"\n}\n", "role": "user"}, {"content": "/- 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  intro A B _ _\n  have hm (ν : Measure α) (s : Set α) : ν s = ∑' y : s, ν {(y : α)} := by\n    simpa using (lintegral_countable (μ := ν) (fun _ => 1) (Set.to_countable s))\n  rw [lintegral_countable _ (Set.to_countable A),\n    lintegral_countable _ (Set.to_countable B)]\n  conv_lhs => enter [1, x]; rw [hm (κ x) B, ← ENNReal.tsum_mul_right]\n  conv_rhs => enter [1, y]; rw [hm (κ y) A, ← ENNReal.tsum_mul_right]\n  rw [ENNReal.tsum_comm]\n  apply tsum_congr\n  intro y\n  apply tsum_congr\n  intro x\n  simpa only [mul_comm] using h (x : α) (y : α)\n", "role": "assistant"}], "split": "public-dev", "track": "lean", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "f5d34b22cd89382ed252b2d1ed63a8b0763a005dbe5940fd24174b14710d19e8", "family": "pl-gradient-iteration", "id": "sogb-002", "messages": [{"content": "{\n  \"assumptions\": [\n    \"\\\\(E\\\\) is a complete real inner-product space, \\\\(f:E\\\\to\\\\mathbb R\\\\), and \\\\(z\\\\) is an attained global minimizer.\",\n    \"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.\\\\]\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"f5d34b22cd89382ed252b2d1ed63a8b0763a005dbe5940fd24174b14710d19e8\",\n  \"domain\": \"optimisation\",\n  \"family\": \"pl-gradient-iteration\",\n  \"id\": \"sogb-002\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"optimisation\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"calculation\",\n    \"proof\",\n    \"assumption-audit\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"PL descent: the sign of the contraction factor\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "Substitute the actual gradient step into the quadratic model. Using \\(h\\ge0\\) and \\(\\beta h\\le1\\) yields \\[f(x-h\\nabla f(x))\\le f(x)-\\tfrac h2\\|\\nabla f(x)\\|^2.\\]\n\nMultiply the PL bound by h/2≥0. With \\(g(x)=f(x)-f(z)\\ge0\\) and \\(q=1-\\alpha h\\), obtain \\(0\\le g(x-h\\nabla f(x))\\le qg(x)\\).\n\nIf q≥0, multiplication preserves order. Induction along the actual iterates gives \\(g(x_N)\\le q^N g(x_0)\\), including N=0.\n\nIf q<0, the same one-step inequality holds for every x and forces g(x)=0: otherwise qg(x)<0 contradicts nonnegativity of the next gap. Hence every gap is zero and the displayed bound still holds. Nontrivial geometric decay requires \\(0\\le q<1\\), rather than merely the existence of a real power q^N.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "f5d34b22cd89382ed252b2d1ed63a8b0763a005dbe5940fd24174b14710d19e8", "family": "pl-gradient-iteration", "id": "sogb-002", "messages": [{"content": "{\n  \"assumptions\": [\n    \"\\\\(E\\\\) is a complete real inner-product space, \\\\(f:E\\\\to\\\\mathbb R\\\\), and \\\\(z\\\\) is an attained global minimizer.\",\n    \"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.\\\\]\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"f5d34b22cd89382ed252b2d1ed63a8b0763a005dbe5940fd24174b14710d19e8\",\n  \"domain\": \"optimisation\",\n  \"family\": \"pl-gradient-iteration\",\n  \"id\": \"sogb-002\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"optimisation\"\n  ],\n  \"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\",\n  \"skills\": [\n    \"calculation\",\n    \"proof\",\n    \"assumption-audit\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"PL descent: the sign of the contraction factor\",\n  \"track\": \"lean\"\n}\n", "role": "user"}, {"content": "/- 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  let T : E → E := fun x => x - h • gradient f x\n  let q : ℝ := 1 - α * h\n  have hnonneg (x : E) : 0 ≤ f x - f z := sub_nonneg.mpr (hz (mem_univ x))\n  have hrec (x : E) : f (T x) - f z ≤ q * (f x - f z) := by\n    have hd := GradientDescentBasic.gradient_step_descent_of_quadratic_upper_bound hh hstep hu x\n    have hp := mul_le_mul_of_nonneg_left (hpl x) (show 0 ≤ h / 2 by positivity)\n    dsimp [T, q]\n    nlinarith\n  by_cases hq : 0 ≤ q\n  · have hr := le_geom (u := fun n => f (T^[n] x₀) - f z) hq N (by\n      intro k _\n      rw [Function.iterate_succ_apply']\n      exact hrec _)\n    exact hr\n  · have hzero (x : E) : f x - f z = 0 := by\n      have hc : q * (f x - f z) ≥ 0 := (hnonneg (T x)).trans (hrec x)\n      have hneg : q < 0 := lt_of_not_ge hq\n      have hx := hnonneg x\n      nlinarith\n    change f (T^[N] x₀) - f z ≤ q ^ N * (f x₀ - f z)\n    rw [hzero, hzero, mul_zero]\n", "role": "assistant"}], "split": "public-dev", "track": "lean", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "b65dd9a70ee7e52100bc85f865904dc278535225ecc2eaa01929897f888543c2", "family": "strong-convex-midpoint-envelope", "id": "sogb-003", "messages": [{"content": "{\n  \"assumptions\": [\n    \"\\\\(E\\\\) is a real normed vector space; \\\\(V:E\\\\to\\\\mathbb R\\\\); \\\\(x_0\\\\) is an attained global minimizer; \\\\(k\\\\) is real.\",\n    \"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.\\\\]\",\n    \"No differentiability or finite-dimensionality assumption is made.\"\n  ],\n  \"binding_sha256\": \"b65dd9a70ee7e52100bc85f865904dc278535225ecc2eaa01929897f888543c2\",\n  \"domain\": \"geometry\",\n  \"family\": \"strong-convex-midpoint-envelope\",\n  \"id\": \"sogb-003\",\n  \"level\": \"L2\",\n  \"library_domains\": [\n    \"log-concave-sampling\",\n    \"optimisation\"\n  ],\n  \"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?\",\n  \"skills\": [\n    \"calculation\",\n    \"assumption-audit\",\n    \"semantic-boundary\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"A quadratic envelope from one midpoint\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "At \\(m=(x_0+x)/2\\), strong convexity gives \\[V(m)\\le\\tfrac12V(x_0)+\\tfrac12V(x)-\\tfrac k8\\|x-x_0\\|^2.\\] The factor \\(k/8\\) comes from \\((k/2)(1/2)(1/2)\\).\n\nGlobal minimality gives \\(V(x_0)\\le V(m)\\). Combine the two inequalities and multiply by 2 to obtain \\(V(x_0)+\\tfrac k4\\|x-x_0\\|^2\\le V(x)\\). No derivative or limit was used.\n\nThis midpoint calculation proves the k/4 envelope, not the sharper k/2 constant. Obtaining a sharper result needs another argument; it must not be credited to this calculation.\n\nThe envelope alone is not a normalization theorem. One sufficient route assumes k>0, a finite-dimensional Euclidean space with Lebesgue reference measure and measurable V, then compares exp(−V) with an integrable Gaussian and separately checks positivity of the normalizer. These are conditions for that route, not necessary conditions for every Gibbs normalization: on ℝ, V(x)=|x| is convex with k=0 and exp(−|x|) has integral 2.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "b65dd9a70ee7e52100bc85f865904dc278535225ecc2eaa01929897f888543c2", "family": "strong-convex-midpoint-envelope", "id": "sogb-003", "messages": [{"content": "{\n  \"assumptions\": [\n    \"\\\\(E\\\\) is a real normed vector space; \\\\(V:E\\\\to\\\\mathbb R\\\\); \\\\(x_0\\\\) is an attained global minimizer; \\\\(k\\\\) is real.\",\n    \"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.\\\\]\",\n    \"No differentiability or finite-dimensionality assumption is made.\"\n  ],\n  \"binding_sha256\": \"b65dd9a70ee7e52100bc85f865904dc278535225ecc2eaa01929897f888543c2\",\n  \"domain\": \"geometry\",\n  \"family\": \"strong-convex-midpoint-envelope\",\n  \"id\": \"sogb-003\",\n  \"level\": \"L2\",\n  \"library_domains\": [\n    \"log-concave-sampling\",\n    \"optimisation\"\n  ],\n  \"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\",\n  \"skills\": [\n    \"calculation\",\n    \"assumption-audit\",\n    \"semantic-boundary\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"A quadratic envelope from one midpoint\",\n  \"track\": \"lean\"\n}\n", "role": "user"}, {"content": "/- 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  intro x\n  have hx₀_min : ∀ y : E, V x₀ ≤ V y := isMinOn_univ_iff.mp hx₀\n  have hmid_min : V x₀ ≤ V ((1 / 2 : ℝ) • x₀ + (1 / 2 : ℝ) • x) := hx₀_min _\n  have hstrong :=\n    hV.2 (by simp : x₀ ∈ (Set.univ : Set E)) (by simp : x ∈ (Set.univ : Set E))\n      (by norm_num : 0 ≤ (1 / 2 : ℝ)) (by norm_num : 0 ≤ (1 / 2 : ℝ))\n      (by norm_num : (1 / 2 : ℝ) + (1 / 2 : ℝ) = 1)\n  have hstrong' :\n      V ((1 / 2 : ℝ) • x₀ + (1 / 2 : ℝ) • x) ≤\n        (1 / 2 : ℝ) * V x₀ + (1 / 2 : ℝ) * V x - (k / 8) * ‖x - x₀‖ ^ 2 := by\n    rw [norm_sub_rev x₀ x] at hstrong\n    convert hstrong using 1\n    · simp [smul_eq_mul]\n      ring_nf\n  have hle := hmid_min.trans hstrong'\n  nlinarith\n", "role": "assistant"}], "split": "public-dev", "track": "lean", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "2274ba27f6d700779d832ff90a2b26d881f2bf5f23c3dfc78689f556020559cf", "family": "coupling-swap-near-optimality", "id": "sogb-004", "messages": [{"content": "{\n  \"assumptions\": [\n    \"Let E be a finite-dimensional real Euclidean space with its Borel sigma-algebra. Let μ,ν be probability measures with finite second moments.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"2274ba27f6d700779d832ff90a2b26d881f2bf5f23c3dfc78689f556020559cf\",\n  \"domain\": \"transport\",\n  \"family\": \"coupling-swap-near-optimality\",\n  \"id\": \"sogb-004\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"statistical-optimal-transport\"\n  ],\n  \"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?\",\n  \"skills\": [\n    \"proof\",\n    \"assumption-audit\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Wasserstein symmetry without an optimal plan\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "The coordinate swap S(x,y)=(y,x) is continuous and hence Borel measurable. For γ′=S#γ, the identities π₁∘S=π₂ and π₂∘S=π₁ give marginals ν,μ by composition of pushforwards.\n\nNonnegative change of variables and symmetry of the norm give \\[\\int\\|x-y\\|^2\\,d(S_\\#\\gamma)=\\int\\|y-x\\|^2\\,d\\gamma=\\int\\|x-y\\|^2\\,d\\gamma.\\] No signed Fubini theorem is required.\n\nThe product coupling has finite quadratic cost since ‖x−y‖²≤2‖x‖²+2‖y‖². Thus the infimum is finite and, for each ε>0, an admissible coupling can be selected with square-root cost less than W₂(μ,ν)+ε. This is near-optimal selection, not an optimal-plan existence theorem.\n\nSwapping this plan gives W₂(ν,μ)≤W₂(μ,ν)+ε. Let ε decrease to zero, then interchange μ and ν to obtain equality. Finiteness justifies the real-valued ε argument; there was no operation ∞−∞. The library also treats an extended-valued generalization, which is not silently identified with this question.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "ce37724e97b715e2330e57378f5795759da21f91c6f477b863d897b48fe93f74", "family": "sphere-tangent-projection", "id": "sogb-005", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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)⟩.\",\n    \"Assume ‖w‖=1 and ‖D(w)‖≤M with M≥0. The tangent space of the unit sphere at w is {v:⟨w,v⟩=0}.\"\n  ],\n  \"binding_sha256\": \"ce37724e97b715e2330e57378f5795759da21f91c6f477b863d897b48fe93f74\",\n  \"domain\": \"geometry\",\n  \"family\": \"sphere-tangent-projection\",\n  \"id\": \"sogb-005\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"riemannian-optimization\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"calculation\",\n    \"proof\",\n    \"assumption-audit\",\n    \"counterexample\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Why a sphere gradient is a projected ambient gradient\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "The chain rule and the adjoint identity give DV(w)[v]=2⟨r,D(w)v⟩=⟨g,v⟩. This uses genuine differentiability, not an assertion about a default derivative value.\n\nSince ‖w‖²=1, \\[\\langle w,G\\rangle=\\langle w,g\\rangle-\\langle g,w\\rangle\\|w\\|^2=0.\\] For v perpendicular to w, ⟨G,v⟩=⟨g,v⟩=DV(w)[v], which identifies the sphere gradient by the induced metric.\n\nThe Pythagorean calculation gives ‖G‖²=‖g‖²−⟨g,w⟩²≤‖g‖². Hence ‖G‖≤‖g‖≤2‖D(w)*‖‖r‖=2‖D(w)‖‖r‖≤2M‖r‖.\n\nWithout normalization, ⟨w,G⟩=⟨g,w⟩(1−‖w‖²). For example, take E=ℝ, w=2 and g=2: G=−6 is not perpendicular to w. For a nonzero general normal, the projector instead divides the coefficient by ‖w‖². This is not a formula for an arbitrary manifold without its own tangent and metric data.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "74f08b42811ce2c35e4669f9e631246262c6038119faa990aea98efb42431cb3", "family": "random-scan-null-atoms", "id": "sogb-006", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"74f08b42811ce2c35e4669f9e631246262c6038119faa990aea98efb42431cb3\",\n  \"domain\": \"sampling\",\n  \"family\": \"random-scan-null-atoms\",\n  \"id\": \"sogb-006\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"discrete-sampling\",\n    \"mcmc\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"proof\",\n    \"assumption-audit\",\n    \"counterexample\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Null atoms in a random-scan balance proof\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "If μ({x})=μ({y})=0, both weighted transition masses vanish: finite Markov transition masses make multiplication by zero harmless.\n\nIf μ({x})=0 but μ({y})>0, use the positive-start formula at y, not x. Each relevant fiber contains y, so its mass is at least μ({y})>0. Every summand for a jump from y to x contains μ({x})=0, hence K(y,{x})=0. Both fluxes are again zero.\n\nFor an arbitrary kernel, a zero target atom does not imply K(y,{x})=0: that conclusion used the actual heat-bath transition law. Target-null starts may have nontrivial transition rules, but their outgoing flux is multiplied by zero.\n\nOn the two-bit cube, let μ give mass 1/2 to each of 00 and 11 and zero to the other states. Updating a single coordinate from either supported state leaves it unchanged. The kernel is reversible for μ, but 00 and 11 cannot communicate. Thus balance supplies neither irreducibility nor convergence from every start.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "a4e2d5d2fd1273d0cadc2054858c9597b4e090dd0fec5fda5ecdfd5c7d206b78", "family": "adaptive-hybrid-cost-boundary", "id": "sogb-007", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"a4e2d5d2fd1273d0cadc2054858c9597b4e090dd0fec5fda5ecdfd5c7d206b78\",\n  \"domain\": \"sampling\",\n  \"family\": \"adaptive-hybrid-cost-boundary\",\n  \"id\": \"sogb-007\",\n  \"level\": \"L4\",\n  \"library_domains\": [\n    \"samplewiki\",\n    \"mcmc\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"composition\",\n    \"assumption-audit\",\n    \"counterexample\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Adaptive hybrid error does not certify expected cost\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "For each slot and fixed event A, Markov event probabilities are measurable and bounded by one, hence integrable under ρᵢ. Integrating the almost-everywhere conditional bound yields an averaged discrepancy at most βᵢ. The error function εᵢ is explicitly integrable; measurability of a supremum over all events was not assumed for free.\n\nA common downstream Markov kernel preserves this uniform event bound by total-variation data processing. The same suffix must be used for the two members of the adjacent pair; otherwise that contraction argument is not applicable.\n\nFor a fixed final event A, telescope the n adjacent probability differences and use the triangle inequality: \\[|P_n(A)-P_0(A)|\\le\\sum_{i=0}^{n-1}\\beta_i.\\] Take the event supremum if desired. Each βᵢ uses its actual shared prefix-history law, not an unrelated ideal or proxy input law.\n\nFor integers N≥2 on the nonnegative reals, take μ_N=δ₀, ν_N=(1−1/N)δ₀+(1/N)δ_{N²} and c(x)=x. Then TV(μ_N,ν_N)=1/N→0, but E_{ν_N}c−E_{μ_N}c=N. All these individual expectations are finite; their gap is unbounded. A uniform cost bound, uniform integrability/tail control, or a separate actual-input cost analysis is needed.\n\nA random recursion depth or stopping cap is not automatically this deterministic finite telescope. Its history construction, stopping/cap comparison, accumulated errors and actual-input expected cost remain separate obligations. This argument alone does not complete the Gaussian Cloud paper.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "ae0a2d0d8f65fbe39a5c2ed55cbe7b93bde710026ec00ed08132abeedefb2794", "family": "gibbs-shape-normalization", "id": "sogb-008", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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)).\"\n  ],\n  \"binding_sha256\": \"ae0a2d0d8f65fbe39a5c2ed55cbe7b93bde710026ec00ed08132abeedefb2794\",\n  \"domain\": \"sampling\",\n  \"family\": \"gibbs-shape-normalization\",\n  \"id\": \"sogb-008\",\n  \"level\": \"L1\",\n  \"library_domains\": [\n    \"log-concave-sampling\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"definition\",\n    \"semantic-boundary\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"A log-concave shape is not yet a probability density\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "The shape ρ is strictly positive and Borel measurable. Its logarithm is −V, a concave function by convexity of V, so this is a positive log-concave shape.\n\nThe normalizing quantity is \\[Z=\\int_{\\mathbb R^d}e^{-V(x)}\\,dx.\\] A probability normalization requires 0<Z<∞, after which p(x)=ρ(x)/Z integrates to one against Lebesgue measure.\n\nLog-concavity of the shape is a pointwise convex-analysis property; the finite integral is a separate analytic condition. This question asks to identify that obligation, not to prove that it holds for every convex potential.\n\nFor example V=0 has the log-concave shape ρ=1 but infinite Lebesgue integral on ℝᵈ. Thus the shape alone is not a normalized sampling target.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "a01b6366bfff653a1e2d2d7304b4e284d28669bfebdcfb82e663f26765274315", "family": "adaptive-hybrid-cost-boundary", "id": "sogb-009", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\"\n  ],\n  \"binding_sha256\": \"a01b6366bfff653a1e2d2d7304b4e284d28669bfebdcfb82e663f26765274315\",\n  \"domain\": \"sampling\",\n  \"family\": \"adaptive-hybrid-cost-boundary\",\n  \"id\": \"sogb-009\",\n  \"level\": \"L1\",\n  \"library_domains\": [\n    \"samplewiki\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"definition\",\n    \"semantic-boundary\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"What kind of guarantee does a sampling claim describe?\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "The first claim compares two probability laws: \\[\\operatorname{TV}(\\nu,\\pi)=\\sup_{A\\text{ measurable}}|\\nu(A)-\\pi(A)|\\le\\varepsilon.\\] Here the convention has no extra factor of two.\n\nThe identity πK=π is invariance of the law π under one application of K. It is a statement about the kernel and its target law, not a bound on ν or a running time.\n\nThe expectation E[C]≤B concerns an algorithmic cost random variable and the probability law governing the run. Different input laws or stopping/cap rules can change that expectation.\n\nThese are different theorem outputs. A proof of one should not be relabelled as a proof of another; mixing, actual-input error and expected query cost require their own links.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "7e2364e8fcf6653f355ba88b88d71d81b69d2e3959e42df0e523a13352a1b94d", "family": "pl-gradient-iteration", "id": "sogb-010", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\"\n  ],\n  \"binding_sha256\": \"7e2364e8fcf6653f355ba88b88d71d81b69d2e3959e42df0e523a13352a1b94d\",\n  \"domain\": \"optimisation\",\n  \"family\": \"pl-gradient-iteration\",\n  \"id\": \"sogb-010\",\n  \"level\": \"L1\",\n  \"library_domains\": [\n    \"optimisation\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"definition\",\n    \"semantic-boundary\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Reading a PL inequality before using an algorithm\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "The objective gap is f(x)−f*, which is nonnegative by global minimality. The parameter α is a positive PL modulus in the stated factor-two convention.\n\nThe inequality \\[2\\alpha(f(x)-f_*)\\le\\|\\nabla f(x)\\|^2\\] compares an objective-value gap with squared gradient norm. It is not a bound on distance to z.\n\nConvexity is not one of the stated PL hypotheses. Do not insert a convexity or strong-convexity premise just because these conditions often appear in nearby optimization theorems.\n\nAn algorithmic rate additionally needs an actual update, a step restriction and a suitable decrease estimate. This question is about the condition itself, not that convergence proof.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "a5cda7d4b9f7caaf24bf4dbab8205dae1f0a82e2ebb6b9fdc92d81c748863408", "family": "sphere-tangent-projection", "id": "sogb-011", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\"\n  ],\n  \"binding_sha256\": \"a5cda7d4b9f7caaf24bf4dbab8205dae1f0a82e2ebb6b9fdc92d81c748863408\",\n  \"domain\": \"geometry\",\n  \"family\": \"sphere-tangent-projection\",\n  \"id\": \"sogb-011\",\n  \"level\": \"L1\",\n  \"library_domains\": [\n    \"riemannian-optimization\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"definition\",\n    \"semantic-boundary\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Which space contains the sphere gradient?\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "The tangent space is \\[T_wS=\\{v\\in\\mathbb R^d:\\langle w,v\\rangle=0\\}.\\] Its vectors are ambient vectors subject to an orthogonality condition.\n\nThe sphere gradient is the tangent vector G satisfying ⟨G,v⟩=Df(w)[v] for every v∈T_wS, using the induced inner product.\n\nThe ambient gradient represents derivatives in all ambient directions; it need not itself be tangent. Its orthogonal projection onto T_wS represents the same derivative on the allowed tangent directions.\n\nThis is the induced-metric sphere setting. Other manifolds and metrics need their own tangent spaces and gradient characterizations, not an automatic reuse of an ambient formula.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "d61f3a0414d8d767ef90fd31733e4fb71d9b47d0f6a62b5844c939a7ee915c7f", "family": "coupling-swap-near-optimality", "id": "sogb-012", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\"\n  ],\n  \"binding_sha256\": \"d61f3a0414d8d767ef90fd31733e4fb71d9b47d0f6a62b5844c939a7ee915c7f\",\n  \"domain\": \"transport\",\n  \"family\": \"coupling-swap-near-optimality\",\n  \"id\": \"sogb-012\",\n  \"level\": \"L1\",\n  \"library_domains\": [\n    \"statistical-optimal-transport\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"definition\",\n    \"semantic-boundary\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"What information does a coupling contain?\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "For every Borel A,B, the marginal constraints are \\[\\gamma(A\\times\\mathbb R^d)=\\mu(A),\\qquad\\gamma(\\mathbb R^d\\times B)=\\nu(B).\\] Equivalently the coordinate pushforwards are μ and ν.\n\nA coupling specifies a joint law with these marginals; it need not be independent. Correlation is allowed as long as the marginal constraints remain satisfied.\n\nThe independent product is a canonical feasible coupling: \\[(\\mu\\otimes\\nu)(A\\times B)=\\mu(A)\\nu(B).\\] Its marginals use the total masses equal to one.\n\nOptimality requires a specified cost and a comparison of its integral over feasible plans. No optimality follows from the marginal identities alone.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "84145f82f459d1cb080b714a85ebad040ad8fd4d034c441b45d77e02c77830ca", "family": "random-scan-null-atoms", "id": "sogb-013", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\"\n  ],\n  \"binding_sha256\": \"84145f82f459d1cb080b714a85ebad040ad8fd4d034c441b45d77e02c77830ca\",\n  \"domain\": \"sampling\",\n  \"family\": \"random-scan-null-atoms\",\n  \"id\": \"sogb-013\",\n  \"level\": \"L1\",\n  \"library_domains\": [\n    \"discrete-sampling\"\n  ],\n  \"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?\",\n  \"skills\": [\n    \"definition\",\n    \"semantic-boundary\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"One random-site step is not one sweep\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "One step first chooses one site I uniformly among the m sites, then draws the new state with kernel K_I, retaining the other coordinates.\n\nThe transition law is the state-independent mixture \\[K(x,A)=\\frac1m\\sum_{i=1}^{m}K_i(x,A).\\] A coordinate resampling may return its previous value, so an update need not change the state.\n\nA sweep applies coordinate updates across all sites according to a specified schedule; it is not this single random-site kernel step. Comparing bounds requires the same time convention.\n\nA noncomputable transition-law definition alone supplies neither an implemented conditional sampler, its query cost, nor a mixing-time estimate. Those guarantees require additional algorithmic and analytic results.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "8b10bef6bc3088ea8188089bb960aed747ddeded576261956464483947b6db38", "family": "atomic-detailed-balance", "id": "sogb-014", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\"\n  ],\n  \"binding_sha256\": \"8b10bef6bc3088ea8188089bb960aed747ddeded576261956464483947b6db38\",\n  \"domain\": \"sampling\",\n  \"family\": \"atomic-detailed-balance\",\n  \"id\": \"sogb-014\",\n  \"level\": \"L1\",\n  \"library_domains\": [\n    \"mcmc\"\n  ],\n  \"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?\",\n  \"skills\": [\n    \"definition\",\n    \"semantic-boundary\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Invariant, reversible and convergent are different claims\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "Invariance is πP=π, or Σ_x π(x)P(x,y)=π(y) for each y. Starting exactly in π therefore gives the same law after one step.\n\nDetailed balance is π(x)P(x,y)=π(y)P(y,x) for every x,y. Summing over x and using the row sum of P at y yields invariance; the stochastic normalization is part of that deduction.\n\nConvergence from ρ is \\[\\operatorname{TV}(\\rho P^t,\\pi)\\longrightarrow0\\quad(t\\to\\infty).\\] It concerns an entire sequence of laws from the specified start, not just the identity πP=π.\n\nFor the identity matrix P, every π is invariant and reversible, but ρP^t=ρ for all t. If ρ≠π, convergence to π fails. Thus one-step invariance is not a definition of general-start convergence.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "0bf552e05112c84a48c1bb862da7870035eb715a9cd61d0a95474c6694d37a7f", "family": "coupling-swap-near-optimality", "id": "sogb-015", "messages": [{"content": "{\n  \"assumptions\": [\n    \"The spaces \\\\(\\\\alpha,\\\\beta\\\\) carry arbitrary measurable structures, with the product measurable structure on \\\\(\\\\alpha\\\\times\\\\beta\\\\).\",\n    \"The measure \\\\(\\\\gamma\\\\) couples \\\\(\\\\mu\\\\) and \\\\(\\\\nu\\\\): its coordinate pushforwards satisfy \\\\(\\\\gamma.\\\\mathrm{fst}=\\\\mu\\\\) and \\\\(\\\\gamma.\\\\mathrm{snd}=\\\\nu\\\\).\",\n    \"Only \\\\(\\\\mu\\\\) is assumed to be a probability measure. No finiteness or probability assumption on \\\\(\\\\gamma\\\\) or \\\\(\\\\nu\\\\) is supplied independently.\"\n  ],\n  \"binding_sha256\": \"0bf552e05112c84a48c1bb862da7870035eb715a9cd61d0a95474c6694d37a7f\",\n  \"domain\": \"transport\",\n  \"family\": \"coupling-swap-near-optimality\",\n  \"id\": \"sogb-015\",\n  \"level\": \"L2\",\n  \"library_domains\": [\n    \"statistical-optimal-transport\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"calculation\",\n    \"proof\",\n    \"assumption-audit\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Recovering joint probability mass from one marginal\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "Let \\(\\pi_1:\\alpha\\times\\beta\\to\\alpha\\) be the first projection, which is measurable for the product measurable structure. The inverse image of the whole first space is the whole product space: \\[\\pi_1^{-1}(\\alpha)=\\alpha\\times\\beta.\\]\n\nBy the definition of pushforward mass and the prescribed first marginal, \\[\\gamma(\\alpha\\times\\beta)=(\\gamma.\\mathrm{fst})(\\alpha)=\\mu(\\alpha)=1.\\] The last equality is exactly the probability assumption on μ.\n\nA measure of total mass one is a probability measure. Thus γ has the required probability interface; its finiteness is a consequence, not an extra premise used to obtain the mass identity.\n\nThis deduction uses only the first marginal identity. The second marginal remains part of the coupling data but no separate probability assumption on ν is needed. This local mass result asserts no optimality, moment bound or integrability of arbitrary unbounded functions.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "0bf552e05112c84a48c1bb862da7870035eb715a9cd61d0a95474c6694d37a7f", "family": "coupling-swap-near-optimality", "id": "sogb-015", "messages": [{"content": "{\n  \"assumptions\": [\n    \"The spaces \\\\(\\\\alpha,\\\\beta\\\\) carry arbitrary measurable structures, with the product measurable structure on \\\\(\\\\alpha\\\\times\\\\beta\\\\).\",\n    \"The measure \\\\(\\\\gamma\\\\) couples \\\\(\\\\mu\\\\) and \\\\(\\\\nu\\\\): its coordinate pushforwards satisfy \\\\(\\\\gamma.\\\\mathrm{fst}=\\\\mu\\\\) and \\\\(\\\\gamma.\\\\mathrm{snd}=\\\\nu\\\\).\",\n    \"Only \\\\(\\\\mu\\\\) is assumed to be a probability measure. No finiteness or probability assumption on \\\\(\\\\gamma\\\\) or \\\\(\\\\nu\\\\) is supplied independently.\"\n  ],\n  \"binding_sha256\": \"0bf552e05112c84a48c1bb862da7870035eb715a9cd61d0a95474c6694d37a7f\",\n  \"domain\": \"transport\",\n  \"family\": \"coupling-swap-near-optimality\",\n  \"id\": \"sogb-015\",\n  \"level\": \"L2\",\n  \"library_domains\": [\n    \"statistical-optimal-transport\"\n  ],\n  \"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\",\n  \"skills\": [\n    \"calculation\",\n    \"proof\",\n    \"assumption-audit\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Recovering joint probability mass from one marginal\",\n  \"track\": \"lean\"\n}\n", "role": "user"}, {"content": "/- 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  constructor\n  rw [← Measure.fst_univ, hγ.1, measure_univ]\n", "role": "assistant"}], "split": "public-dev", "track": "lean", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "69fee45c9a7a53e9d1a4ec5295d530811736ff3e8dec9fbd435f1293d9bc1cd7", "family": "sphere-tangent-projection", "id": "sogb-016", "messages": [{"content": "{\n  \"assumptions\": [\n    \"\\\\(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.\",\n    \"\\\\(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\\\\).\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"69fee45c9a7a53e9d1a4ec5295d530811736ff3e8dec9fbd435f1293d9bc1cd7\",\n  \"domain\": \"geometry\",\n  \"family\": \"sphere-tangent-projection\",\n  \"id\": \"sogb-016\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"riemannian-optimization\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"calculation\",\n    \"proof\",\n    \"assumption-audit\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"A quadratic gradient through the actual sphere differential\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "Symmetry gives the genuine ambient derivative \\[Dq_x(v)=\\langle v,Ax\\rangle+\\langle x,Av\\rangle=2\\langle v,Ax\\rangle.\\] Hence the real gradient is \\(g=2Ax\\). The bounded linear map and quadratic operations supply differentiability; the equality is not an inference from a totalized zero derivative.\n\nFor the actual inclusion, \\(\\operatorname{range}(D i_x)=K\\). The chain rule therefore gives \\[D(q|_S)_x(w)=\\langle D i_x(w),g\\rangle.\\] This uses the inclusion-induced pairing, not an assumed isometry from an arbitrary chart metric.\n\nBecause \\(\\|x\\|=1\\), orthogonal projection onto K satisfies \\[P_Ku=u-\\langle x,u\\rangle x.\\] Substituting g yields the tangent vector \\[G=P_K(2Ax)=2(Ax-\\langle x,Ax\\rangle x).\\]\n\nEvery \\(D i_x(w)\\) lies in K, so orthogonal projection preserves its pairing with g. Thus G represents the restricted differential. Conversely, if a tangent vector z has the same pairing for every w, surjectivity onto K makes \\(z-G\\) orthogonal to every vector in K. Taking its inner product with itself gives \\(z=G\\), proving the equivalence.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "69fee45c9a7a53e9d1a4ec5295d530811736ff3e8dec9fbd435f1293d9bc1cd7", "family": "sphere-tangent-projection", "id": "sogb-016", "messages": [{"content": "{\n  \"assumptions\": [\n    \"\\\\(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.\",\n    \"\\\\(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\\\\).\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"69fee45c9a7a53e9d1a4ec5295d530811736ff3e8dec9fbd435f1293d9bc1cd7\",\n  \"domain\": \"geometry\",\n  \"family\": \"sphere-tangent-projection\",\n  \"id\": \"sogb-016\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"riemannian-optimization\"\n  ],\n  \"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\",\n  \"skills\": [\n    \"calculation\",\n    \"proof\",\n    \"assumption-audit\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"A quadratic gradient through the actual sphere differential\",\n  \"track\": \"lean\"\n}\n", "role": "user"}, {"content": "/- 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  have hx : ‖(x : E)‖ = 1 := mem_sphere_zero_iff_norm.mp x.property\n  have hg : HasGradientAt (fun y : E => ⟪y, A y⟫_ℝ) ((2 : ℝ) • A (x : E)) (x : E) := by\n    apply hasGradientAt_iff_hasFDerivAt.mpr\n    have hraw := (hA.hasStrictFDerivAt_reApplyInnerSelf (x : E)).hasFDerivAt\n    have hfun : (fun y : E => ⟪y, A y⟫_ℝ) = A.reApplyInnerSelf := by\n      funext y\n      simp [ContinuousLinearMap.reApplyInnerSelf_apply, real_inner_comm]\n    have hlin : toDual ℝ E ((2 : ℝ) • A (x : E)) =\n        (2 : ℕ) • innerSL ℝ (A (x : E)) := by\n      ext v\n      simp [toDual_apply_apply, two_smul]\n    rw [hfun, hlin]\n    exact hraw\n  let K := (ℝ ∙ (x : E))ᗮ\n  have hproj (u : E) : (K.orthogonalProjectionOnto u : E) =\n      u - ⟪(x : E), u⟫_ℝ • (x : E) := by\n    change (ℝ ∙ (x : E))ᗮ.starProjection u = _\n    rw [Submodule.starProjection_orthogonal_val,\n      Submodule.starProjection_unit_singleton ℝ hx]\n  have hp : (K.orthogonalProjectionOnto ((2 : ℝ) • A (x : E)) : E) =\n      (2 : ℝ) • (A (x : E) - ⟪(x : E), A (x : E)⟫_ℝ • (x : E)) := by\n    rw [hproj]\n    simp [inner_smul_right, smul_sub, smul_smul]\n  intro z\n  rw [SphereGradient.sphere_gradient_characterization n x hg z]\n  constructor\n  · intro hz\n    exact (congrArg (fun v : K => (v : E)) hz).trans hp\n  · intro hz\n    apply Subtype.ext\n    exact hz.trans hp.symm\n", "role": "assistant"}], "split": "public-dev", "track": "lean", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "37f2cef51769619910cd718b9d66acd2d57f6a5a5956a6940a4d4dc2ba60eeba", "family": "adaptive-hybrid-cost-boundary", "id": "sogb-017", "messages": [{"content": "{\n  \"assumptions\": [\n    \"\\\\(A,B\\\\) are arbitrary measurable spaces, \\\\(\\\\mu\\\\) is a probability measure on A, and \\\\(K,L:A\\\\leadsto B\\\\) are Markov kernels.\",\n    \"\\\\(\\\\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.\"\n  ],\n  \"binding_sha256\": \"37f2cef51769619910cd718b9d66acd2d57f6a5a5956a6940a4d4dc2ba60eeba\",\n  \"domain\": \"sampling\",\n  \"family\": \"adaptive-hybrid-cost-boundary\",\n  \"id\": \"sogb-017\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"samplewiki\",\n    \"mcmc\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"proof\",\n    \"assumption-audit\",\n    \"calculation\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Averaging a conditional event-error bound\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "Fix a measurable event t. Kernel measurability makes \\(k_t(x)=K(x,t)\\) and \\(l_t(x)=L(x,t)\\) measurable real functions. Markov normalization gives \\(0\\le k_t,l_t\\le1\\); since μ is a probability measure, both are integrable.\n\nThe bind/composition law gives \\[(K\\circ_m\\mu)(t)=\\int k_t\\,d\\mu,\\qquad(L\\circ_m\\mu)(t)=\\int l_t\\,d\\mu.\\] In Lean the underlying masses are extended nonnegative reals. Their finiteness, supplied by the probability-kernel assumptions, justifies conversion to real integrals; the totalized toReal API is not used as if infinity were finite.\n\nThe two integrable event functions can be subtracted inside the integral. The integral triangle inequality gives \\[\\left|\\int k_t\\,d\\mu-\\int l_t\\,d\\mu\\right|\\le\\int|k_t-l_t|\\,d\\mu.\\] The absolute difference is integrable as well.\n\nThe common almost-everywhere hypothesis supplies \\(|k_t-l_t|\\le\\varepsilon\\) almost everywhere for this event. Monotonicity for integrable real functions yields \\[|(K\\circ_m\\mu)(t)-(L\\circ_m\\mu)(t)|\\le\\int\\varepsilon\\,d\\mu.\\] No measurable supremum over all events is needed, and no new global nonnegativity premise on ε is inserted.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "37f2cef51769619910cd718b9d66acd2d57f6a5a5956a6940a4d4dc2ba60eeba", "family": "adaptive-hybrid-cost-boundary", "id": "sogb-017", "messages": [{"content": "{\n  \"assumptions\": [\n    \"\\\\(A,B\\\\) are arbitrary measurable spaces, \\\\(\\\\mu\\\\) is a probability measure on A, and \\\\(K,L:A\\\\leadsto B\\\\) are Markov kernels.\",\n    \"\\\\(\\\\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.\"\n  ],\n  \"binding_sha256\": \"37f2cef51769619910cd718b9d66acd2d57f6a5a5956a6940a4d4dc2ba60eeba\",\n  \"domain\": \"sampling\",\n  \"family\": \"adaptive-hybrid-cost-boundary\",\n  \"id\": \"sogb-017\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"samplewiki\",\n    \"mcmc\"\n  ],\n  \"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\",\n  \"skills\": [\n    \"proof\",\n    \"assumption-audit\",\n    \"calculation\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Averaging a conditional event-error bound\",\n  \"track\": \"lean\"\n}\n", "role": "user"}, {"content": "/- 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  intro t ht\n  have hint (M : Kernel A B) [IsMarkovKernel M] :\n      Integrable (fun x => (M x).real t) μ := by\n    apply (integrable_const (1 : ℝ)).mono'\n      (M.measurable_coe ht).ennreal_toReal.aestronglyMeasurable\n    exact Filter.Eventually.of_forall fun x => by\n      rw [Real.norm_eq_abs, abs_of_nonneg ENNReal.toReal_nonneg]\n      exact (measureReal_le_one : (M x).real t ≤ 1)\n  have happly (M : Kernel A B) [IsMarkovKernel M] :\n      (M ∘ₘ μ).real t = ∫ x, (M x).real t ∂μ := by\n    change (Measure.bind μ M t).toReal = ∫ x, (M x t).toReal ∂μ\n    rw [Measure.bind_apply ht M.aemeasurable]\n    exact (integral_toReal (M.measurable_coe ht).aemeasurable\n      (Filter.Eventually.of_forall fun x => measure_lt_top (M x) t)).symm\n  rw [happly K, happly L, ← integral_sub (hint K) (hint L)]\n  exact abs_integral_le_integral_abs.trans <|\n    integral_mono_ae ((hint K).sub (hint L)).abs hε <|\n      hbound.mono fun x hx => hx t ht\n", "role": "assistant"}], "split": "public-dev", "track": "lean", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "237b6a126e4672415b58ea7f7621fe69ef5d541dac41fb1b42336f828922a1e2", "family": "coupling-gluing-triangle", "id": "sogb-018", "messages": [{"content": "{\n  \"assumptions\": [\n    \"\\\\(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.\",\n    \"\\\\(\\\\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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"237b6a126e4672415b58ea7f7621fe69ef5d541dac41fb1b42336f828922a1e2\",\n  \"domain\": \"transport\",\n  \"family\": \"coupling-gluing-triangle\",\n  \"id\": \"sogb-018\",\n  \"level\": \"L4\",\n  \"library_domains\": [\n    \"statistical-optimal-transport\",\n    \"log-concave-sampling\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"proof\",\n    \"composition\",\n    \"assumption-audit\",\n    \"counterexample\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Gluing the right joint law before applying Minkowski\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "Strict near-optimal selection gives actual couplings \\(\\gamma_{12},\\gamma_{23}\\) with \\[C(\\gamma_{12})<r_{12},\\qquad C(\\gamma_{23})<r_{23}.\\] These are consequences of the transport infimum and positive square-root order, not an assumption that the infimum is attained. A coupling of a probability marginal has total mass one, so both plans are probability measures.\n\nDisintegrate the second plan with respect to its first coordinate. The finite standard-Borel conditional-kernel theorem supplies a Markov version \\(\\kappa(y,dz)\\) with \\[\\gamma_{23}(dy,dz)=\\mu_2(dy)\\kappa(y,dz).\\] Define \\[\\Gamma(d(x,y),dz)=\\gamma_{12}(dx,dy)\\kappa(y,dz).\\] Markov normalization preserves its first pair marginal. Equality \\(\\gamma_{12}.\\mathrm{snd}=\\mu_2=\\gamma_{23}.\\mathrm{fst}\\), kernel measurability and the disintegration identity preserve the second pair marginal. Conditional representatives need only have the correct \\(\\mu_2\\)-almost-everywhere meaning; an unsupported pointwise conditional formula is not used.\n\nWrite \\(\\pi_{ij}\\) for the measurable pair projections from the same triple space. The identities are \\[(\\pi_{12})_\\#\\Gamma=\\gamma_{12},\\qquad(\\pi_{23})_\\#\\Gamma=\\gamma_{23}.\\] Put \\(\\gamma_{13}=(\\pi_{13})_\\#\\Gamma\\). Composing measurable coordinate projections proves that its marginals are \\(\\mu_1,\\mu_3\\). The Borel norm and the product/projection measurability justify nonnegative pushforward integration; no subtraction of infinite integrals occurs.\n\nFor \\(e_{ij}(x,y,z)=\\|x_i-x_j\\|\\), all edge functions are measurable. Nonnegative pushforward integration gives \\[\\|e_{12}\\|_{L^2(\\Gamma)}=C(\\gamma_{12}),\\qquad\\|e_{23}\\|_{L^2(\\Gamma)}=C(\\gamma_{23}).\\] The endpoint identity is \\[\\|e_{13}\\|_{L^2(\\Gamma)}=C(\\gamma_{13}).\\] Pointwise \\(e_{13}\\le e_{12}+e_{23}\\) and extended-nonnegative L² Minkowski apply on this single law Γ, not on two unrelated pair spaces.\n\nUsing the endpoint coupling to bound the infimum, then the preceding identities, yields \\[\\begin{aligned}W_2(\\mu_1,\\mu_3)&\\le C(\\gamma_{13})\\\\&\\le C(\\gamma_{12})+C(\\gamma_{23})\\\\&<r_{12}+r_{23}.\\end{aligned}\\] The coefficient is exactly one on each cost. Strict extended-order addition handles the thresholds without an infinity-minus-infinity operation. No optimal plan, density or individual second-moment assumption is added.\n\nFor the shortcut audit, take \\(E=\\mathbb R\\) and all three marginals equal to \\(\\frac12\\delta_0+\\frac12\\delta_1\\), with both adjacent plans diagonal. Their costs are zero. Under \\(\\Gamma'=\\gamma_{12}\\otimes\\mu_3\\), Y and Z are independent, so \\[\\mathbb P_{\\Gamma'}(Y\\ne Z)=\\tfrac12,\\qquad\\|e_{23}\\|_{L^2(\\Gamma')}=1/\\sqrt2.\\] Its second pair marginal is the independent product, not the diagonal \\(\\gamma_{23}\\), and cannot have cost zero. This refutes the shortcut's cost identification, not the true Wasserstein inequality. Correct conditional gluing makes X=Y=Z in this example.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "237b6a126e4672415b58ea7f7621fe69ef5d541dac41fb1b42336f828922a1e2", "family": "coupling-gluing-triangle", "id": "sogb-018", "messages": [{"content": "{\n  \"assumptions\": [\n    \"\\\\(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.\",\n    \"\\\\(\\\\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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"237b6a126e4672415b58ea7f7621fe69ef5d541dac41fb1b42336f828922a1e2\",\n  \"domain\": \"transport\",\n  \"family\": \"coupling-gluing-triangle\",\n  \"id\": \"sogb-018\",\n  \"level\": \"L4\",\n  \"library_domains\": [\n    \"statistical-optimal-transport\",\n    \"log-concave-sampling\"\n  ],\n  \"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\",\n  \"skills\": [\n    \"proof\",\n    \"composition\",\n    \"assumption-audit\",\n    \"counterexample\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Gluing the right joint law before applying Minkowski\",\n  \"track\": \"lean\"\n}\n", "role": "user"}, {"content": "/- 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  rcases\n      WassersteinSpace.exists_isCoupling_sqrt_lintegral_lt_of_wassersteinDistance_lt\n        μ₁ μ₂ hr₁₂ with\n    ⟨γ₁₂, hγ₁₂, hcost₁₂⟩\n  rcases\n      WassersteinSpace.exists_isCoupling_sqrt_lintegral_lt_of_wassersteinDistance_lt\n        μ₂ μ₃ hr₂₃ with\n    ⟨γ₂₃, hγ₂₃, hcost₂₃⟩\n  rcases TransportGluing.exists_gluing_of_isCoupling γ₁₂ γ₂₃ hγ₁₂ hγ₂₃ with\n    ⟨γ₁₂₃, hfst, hmap₂₃⟩\n\n  have hmap₁₂ : Measure.map (pair12 (E := E)) γ₁₂₃ = γ₁₂ := by\n    change Measure.map Prod.fst γ₁₂₃ = γ₁₂\n    exact hfst\n  have hmap₂₃' : Measure.map (pair23 (E := E)) γ₁₂₃ = γ₂₃ := by\n    change Measure.map (fun p : ((E × E) × E) => (p.1.2, p.2)) γ₁₂₃ = γ₂₃\n    exact hmap₂₃\n\n  have hpair₁₂ :\n      Transport.IsCoupling (Measure.map (pair12 (E := E)) γ₁₂₃) μ₁ μ₂ := by\n    rw [hmap₁₂]\n    exact hγ₁₂\n  have hpair₂₃ :\n      Transport.IsCoupling (Measure.map (pair23 (E := E)) γ₁₂₃) μ₂ μ₃ := by\n    rw [hmap₂₃']\n    exact hγ₂₃\n  have hpair₁₃ :\n      Transport.IsCoupling (Measure.map (pair13 (E := E)) γ₁₂₃) μ₁ μ₃ :=\n    isCoupling_map_pair13_of_pair12_pair23 γ₁₂₃ hpair₁₂ hpair₂₃\n\n  have hedge₁₂ :\n      l2Seminorm γ₁₂₃ (edgeLength12 (E := E)) =\n        (∫⁻ z, WassersteinSpace.quadraticCost (E := E) z ∂γ₁₂) ^\n          (1 / (2 : ℝ)) := by\n    rw [l2Seminorm_edge12_eq_pairCost, hmap₁₂]\n  have hedge₂₃ :\n      l2Seminorm γ₁₂₃ (edgeLength23 (E := E)) =\n        (∫⁻ z, WassersteinSpace.quadraticCost (E := E) z ∂γ₂₃) ^\n          (1 / (2 : ℝ)) := by\n    rw [l2Seminorm_edge23_eq_pairCost, hmap₂₃']\n  have hendpoint :\n      WassersteinSpace.wassersteinDistance μ₁ μ₃ ≤\n        l2Seminorm γ₁₂₃ (edgeLength13 (E := E)) := by\n    calc\n      WassersteinSpace.wassersteinDistance μ₁ μ₃ ≤\n          (∫⁻ z, WassersteinSpace.quadraticCost (E := E) z\n            ∂Measure.map (pair13 (E := E)) γ₁₂₃) ^ (1 / (2 : ℝ)) :=\n        WassersteinSpace.wassersteinDistance_le_sqrt_lintegral_of_isCoupling\n          μ₁ μ₃ (Measure.map (pair13 (E := E)) γ₁₂₃) hpair₁₃\n      _ = l2Seminorm γ₁₂₃ (edgeLength13 (E := E)) :=\n        (l2Seminorm_edge13_eq_pairCost γ₁₂₃).symm\n\n  calc\n    WassersteinSpace.wassersteinDistance μ₁ μ₃ ≤\n        l2Seminorm γ₁₂₃ (edgeLength13 (E := E)) := hendpoint\n    _ ≤ l2Seminorm γ₁₂₃ (edgeLength12 (E := E)) +\n          l2Seminorm γ₁₂₃ (edgeLength23 (E := E)) :=\n      l2_edge_triangle γ₁₂₃\n    _ = (∫⁻ z, WassersteinSpace.quadraticCost (E := E) z ∂γ₁₂) ^\n          (1 / (2 : ℝ)) +\n        (∫⁻ z, WassersteinSpace.quadraticCost (E := E) z ∂γ₂₃) ^\n          (1 / (2 : ℝ)) := by rw [hedge₁₂, hedge₂₃]\n    _ < r₁₂ + r₂₃ := ENNReal.add_lt_add hcost₁₂ hcost₂₃\n", "role": "assistant"}], "split": "public-dev", "track": "lean", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "b12bf7bd53d4c8c85dd4e1658b22f05ea7752e13bfa48a3bbfba72f620c3d52f", "family": "subgradient-energy-average", "id": "sogb-019", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"b12bf7bd53d4c8c85dd4e1658b22f05ea7752e13bfa48a3bbfba72f620c3d52f\",\n  \"domain\": \"optimisation\",\n  \"family\": \"subgradient-energy-average\",\n  \"id\": \"sogb-019\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"optimisation\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"proof\",\n    \"assumption-audit\",\n    \"counterexample\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Why a global Lipschitz function bounds its supporting vectors\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "Because the support relation is global, its test point may be y=x+g. Thus \\[\\|g\\|^2=\\langle g,g\\rangle\\le f(x+g)-f(x).\\] This tests the supplied relation; it does not use a theorem constructing subgradients.\n\nGlobal Lipschitz growth and the absolute-value bound give \\[f(x+g)-f(x)\\le|f(x+g)-f(x)|\\le G\\|g\\|.\\] Combining the inequalities yields \\(\\|g\\|^2\\le G\\|g\\|\\).\n\nIf g=0 the conclusion is immediate from G≥0. Otherwise \\(\\|g\\|>0\\), so division by this strictly positive number gives \\(\\|g\\|\\le G\\). No completeness, differentiability or finite-dimensional compactness is used.\n\nFor constrained support take E=ℝ, f≡0, G=0, C=[0,∞), x=0 and g=−1. Then \\[f(0)+g(y-0)=-y\\le0=f(y)\\qquad(y\\in C),\\] but \\(\\|g\\|=1>0=G\\). The chosen point x+g=−1 is outside C. This refutes the constrained variant, not the globally quantified theorem.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "b12bf7bd53d4c8c85dd4e1658b22f05ea7752e13bfa48a3bbfba72f620c3d52f", "family": "subgradient-energy-average", "id": "sogb-019", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"b12bf7bd53d4c8c85dd4e1658b22f05ea7752e13bfa48a3bbfba72f620c3d52f\",\n  \"domain\": \"optimisation\",\n  \"family\": \"subgradient-energy-average\",\n  \"id\": \"sogb-019\",\n  \"level\": \"L2\",\n  \"library_domains\": [\n    \"optimisation\"\n  ],\n  \"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\",\n  \"skills\": [\n    \"proof\",\n    \"assumption-audit\",\n    \"counterexample\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Why a global Lipschitz function bounds its supporting vectors\",\n  \"track\": \"lean\"\n}\n", "role": "user"}, {"content": "/- 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  have hs' := hs (x + g)\n  change f x + inner ℝ g (x + g - x) ≤ f (x + g) at hs'\n  rw [add_sub_cancel_left, real_inner_self_eq_norm_sq] at hs'\n  have hl : f (x + g) - f x ≤ (G : ℝ) * ‖g‖ := by\n    calc\n      f (x + g) - f x ≤ |f (x + g) - f x| := le_abs_self _\n      _ ≤ (G : ℝ) * ‖g‖ := by\n        simpa [Real.dist_eq, dist_eq_norm] using hf.dist_le_mul (x + g) x\n  by_cases hg : ‖g‖ = 0\n  · simp [hg]\n  · have hp : 0 < ‖g‖ := lt_of_le_of_ne (norm_nonneg _) (Ne.symm hg)\n    nlinarith\n", "role": "assistant"}], "split": "public-dev", "track": "lean", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "15a19933c0cbddea0e5e89962c7375f05183709493c77749fe4f0115c0155931", "family": "subgradient-energy-average", "id": "sogb-020", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\",\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"15a19933c0cbddea0e5e89962c7375f05183709493c77749fe4f0115c0155931\",\n  \"domain\": \"optimisation\",\n  \"family\": \"subgradient-energy-average\",\n  \"id\": \"sogb-020\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"optimisation\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"proof\",\n    \"composition\",\n    \"assumption-audit\",\n    \"calculation\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"From subgradient energy to a finite-run best iterate\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "For each k, test global support at y=xₖ+gₖ and bound the same increment by Lipschitz growth: \\[\\|g_k\\|^2\\le f(x_k+g_k)-f(x_k)\\le G\\|g_k\\|.\\] If gₖ=0 the norm bound is immediate; otherwise divide by its strictly positive norm to get \\(\\|g_k\\|\\le G\\). Test support again at z to obtain \\[f(x_k)-f(z)\\le\\langle g_k,x_k-z\\rangle.\\] No minimizing comparator or differentiability is needed.\n\nExpanding the actual update in the inner-product space gives \\[\\|x_{k+1}-z\\|^2=\\|x_k-z\\|^2-2a_k\\langle g_k,x_k-z\\rangle+a_k^2\\|g_k\\|^2.\\] Since aₖ≥0, multiplication preserves the support inequality; since aₖ²≥0, the norm bound gives \\[2a_k(f(x_k)-f(z))\\le\\|x_k-z\\|^2-\\|x_{k+1}-z\\|^2+G^2a_k^2.\\] A negative step would invalidate the first inference.\n\nSum these finite inequalities; the distance terms telescope exactly: \\[2\\sum_{k<N}a_k(f(x_k)-f(z))\\le\\|x_0-z\\|^2+G^2\\sum_{k<N}a_k^2-\\|x_N-z\\|^2.\\] This statement permits zero steps and N=0. It retains, rather than assumes away, the nonnegative terminal distance.\n\nFor N>0 choose i minimizing f(xₖ) over the finite nonempty index set {0,…,N−1}. This is a finite minimum, not existence of a global minimizer of f. Put \\(A=\\sum_{k=0}^{N-1}a_k>0\\). Positive weights imply \\[A(f(x_i)-f(z))\\le\\sum_{k<N}a_k(f(x_k)-f(z)).\\] Drop the nonpositive terminal term from the upper bound and divide by 2A>0 to obtain the requested ratio.\n\nWhen a=R/(G√N)>0, the ratio is at most \\[\\frac{R^2+G^2Na^2}{2Na}=\\frac{R^2}{2Na}+\\frac{G^2a}{2}=\\frac{RG}{\\sqrt N}.\\] The chosen schedule depends on N; this does not establish that one fixed infinite run uses all these horizon-dependent schedules simultaneously.\n\nAt N=0 the energy inequality reduces to 0≤0, but there is no i<0 and division by a zero sum cannot create one (Lean's totalized 0/0 does not repair existence). If R=0 then z=x₀ and, for N>0, i=0 already has zero gap. If G=0 global Lipschitz growth makes f constant and the norm lemma forces all supplied gₖ=0; the positive-parameter step formula is unnecessary. These degenerate cases are separate arguments, not a substitution into a formula requiring positive parameters.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "3d72820e315cae6cebacdbed3bc8475fb4479d4a4a6cc7c302f4d534b54fe173", "family": "conditional-resampling-invariance", "id": "sogb-021", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\",\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"3d72820e315cae6cebacdbed3bc8475fb4479d4a4a6cc7c302f4d534b54fe173\",\n  \"domain\": \"mcmc\",\n  \"family\": \"conditional-resampling-invariance\",\n  \"id\": \"sogb-021\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"mcmc\",\n    \"discrete-sampling\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"proof\",\n    \"assumption-audit\",\n    \"counterexample\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Why retaining one coordinate and conditionally resampling the other is invariant\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "The deterministic projection (a,b)↦a is measurable. Pushing μ through it gives exactly its first marginal \\(\\mu_1=(\\mathrm{fst})_\\#\\mu\\). This assertion uses the actual input μ, not an arbitrary starting law with a different first marginal.\n\nThe product kernel a↦δₐ⊗κ(a,·) is measurable because it is the product of the identity kernel and the measurable conditional kernel. Composition with the deterministic projection gives K. Associativity of kernel/measure composition yields \\[\\mu K=\\mu_1(\\mathrm{id}\\times\\kappa)=\\mu_1\\otimes\\kappa.\\] The last expression is a measure–kernel composition product, not the independent product of μ₁ and a fixed marginal.\n\nThe supplied finite-measure disintegration identity now gives \\[\\mu K=\\mu_1\\otimes\\kappa=\\mu.\\] It is justified by the standard-Borel, nonempty β and finite μ interface. No density, positivity, normalization to mass one or division by a marginal density is required. For μ=0 both sides are zero; this is invariant as a finite measure, not a probability law.\n\nConditional versions agree in their prescribed conditional meaning only for μ₁-almost every a. Replacing them on a μ₁-null set does not change the integrated identity. The chosen global Markov version still yields a legitimate mass-one update there, but no claim about conditional support or convergence from such exceptional starting states follows.\n\nFor lack of convergence take α={0,1}, β={∗} with discrete measurable structures and \\(\\mu=(\\tfrac12\\delta_0+\\tfrac12\\delta_1)\\otimes\\delta_\\ast\\). The update never changes either coordinate, so K is the identity kernel. Starting from \\(\\rho=\\delta_{(0,\\ast)}\\), \\[\\rho K^n=\\rho,\\qquad d_{\\mathrm{TV}}(\\rho K^n,\\mu)=\\tfrac12\\quad(n\\ge0).\\] Thus the actual invariant target does not imply uniqueness, irreducibility, mixing or convergence from a different first marginal.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "3d72820e315cae6cebacdbed3bc8475fb4479d4a4a6cc7c302f4d534b54fe173", "family": "conditional-resampling-invariance", "id": "sogb-021", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\",\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"3d72820e315cae6cebacdbed3bc8475fb4479d4a4a6cc7c302f4d534b54fe173\",\n  \"domain\": \"mcmc\",\n  \"family\": \"conditional-resampling-invariance\",\n  \"id\": \"sogb-021\",\n  \"level\": \"L2\",\n  \"library_domains\": [\n    \"mcmc\",\n    \"discrete-sampling\"\n  ],\n  \"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\",\n  \"skills\": [\n    \"proof\",\n    \"assumption-audit\",\n    \"counterexample\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Why retaining one coordinate and conditionally resampling the other is invariant\",\n  \"track\": \"lean\"\n}\n", "role": "user"}, {"content": "/- 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  change ((Kernel.id ×ₖ condDistrib Prod.snd Prod.fst μ) ∘ₖ\n    Kernel.deterministic Prod.fst measurable_fst) ∘ₘ μ = μ\n  rw [← Measure.comp_assoc, Measure.deterministic_comp_eq_map,\n    ← Measure.compProd_eq_comp_prod]\n  exact ConditionalResampling.fst_compProd_condDistrib_snd_eq_self\n", "role": "assistant"}], "split": "public-dev", "track": "lean", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "2cfe0ffa4f03a84301a91017152eea5265cf548015c4f28f0d9d9fa9f12817b5", "family": "convex-flow-first-order-rate", "id": "sogb-022", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"2cfe0ffa4f03a84301a91017152eea5265cf548015c4f28f0d9d9fa9f12817b5\",\n  \"domain\": \"geometry\",\n  \"family\": \"convex-flow-first-order-rate\",\n  \"id\": \"sogb-022\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"riemannian-optimization\",\n    \"statistical-optimal-transport\"\n  ],\n  \"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?\",\n  \"skills\": [\n    \"proof\",\n    \"calculation\",\n    \"assumption-audit\",\n    \"semantic-boundary\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"From a selected-curve chord inequality to its first-order bound\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "Put \\(D=d(\\gamma(0),\\gamma(1))^2\\). For 0<t<1, subtract F(γ(0)) from the chord bound and divide by the strictly positive t: \\[\\frac{\\varphi(t)-\\varphi(0)}t\\le F(\\gamma(1))-F(\\gamma(0))-\\frac\\alpha2(1-t)D.\\] No sign restriction on α is used, and division at t=0 is not performed.\n\nThe condition 0<t<1 holds eventually as t approaches 0 from the right. Genuine differentiability gives the restricted limit \\[\\lim_{t\\downarrow0}\\frac{\\varphi(t)-\\varphi(0)}t=g.\\] The formal proof restricts HasDerivAt.tendsto_slope to the right-neighborhood filter; a defined-but-totalized slope value at zero is irrelevant.\n\nThe upper envelope \\(U(t)=F(\\gamma(1))-F(\\gamma(0))-\\frac\\alpha2(1-t)D\\) is continuous at zero. Taking limits in the eventual ordered inequality gives \\[g\\le F(\\gamma(1))-F(\\gamma(0))-\\frac\\alpha2D.\\] Rearranging proves the assertion with the exact coefficient α/2.\n\nThe proof uses the selected curve's chord inequality and scalar derivative only. It does not construct paths, prove that P encodes constant-speed geodesics, identify their velocities or define a gradient. An empty selection would make the global chord condition vacuous, but would not provide the required witness P(γ). To obtain a Riemannian or transport gradient statement one must separately identify the path derivative with the intended geometric pairing under that geometry's regularity hypotheses.\n\nA totalized derivative API returns some value even when a derivative does not exist. Merely assigning that value does not justify the slope limit. The local statement explicitly supplies HasDerivAt; its proof uses only the right restriction of that stronger assumption. A genuinely one-sided variant would need its own correctly stated API, not deletion of the current premise.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "2cfe0ffa4f03a84301a91017152eea5265cf548015c4f28f0d9d9fa9f12817b5", "family": "convex-flow-first-order-rate", "id": "sogb-022", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"2cfe0ffa4f03a84301a91017152eea5265cf548015c4f28f0d9d9fa9f12817b5\",\n  \"domain\": \"geometry\",\n  \"family\": \"convex-flow-first-order-rate\",\n  \"id\": \"sogb-022\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"riemannian-optimization\",\n    \"statistical-optimal-transport\"\n  ],\n  \"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\",\n  \"skills\": [\n    \"proof\",\n    \"calculation\",\n    \"assumption-audit\",\n    \"semantic-boundary\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"From a selected-curve chord inequality to its first-order bound\",\n  \"track\": \"lean\"\n}\n", "role": "user"}, {"content": "/- 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  let phi : ℝ → ℝ := fun t => F (path t)\n  let upper : ℝ → ℝ := fun t =>\n    F (path 1) - F (path 0) -\n      alpha * (1 - t) / 2 * dist (path 0) (path 1) ^ 2\n  have hslope :\n      Tendsto (slope phi 0) (𝓝[>] 0) (𝓝 gradientPairing) := by\n    apply hderiv.tendsto_slope.mono_left\n    apply nhdsWithin_mono\n    intro t ht\n    simpa only [mem_compl_iff, mem_singleton_iff] using ne_of_gt ht\n  have hupper :\n      Tendsto upper (𝓝[>] 0) (𝓝 (upper 0)) := by\n    have hcontinuous : ContinuousAt upper 0 := by\n      dsimp only [upper]\n      fun_prop\n    have hfilter : 𝓝[>] (0 : ℝ) ≤ 𝓝 0 := inf_le_left\n    exact hcontinuous.tendsto.mono_left hfilter\n  have hslope_le : slope phi 0 ≤ᶠ[𝓝[>] 0] upper := by\n    filter_upwards [self_mem_nhdsWithin,\n      (eventually_lt_nhds zero_lt_one).filter_mono inf_le_left]\n      with t ht htle\n    have htpos : 0 < t := ht\n    have hchord := hconvex path hpath t ⟨htpos.le, htle.le⟩\n    have hquotient :\n        (F (path t) - F (path 0)) / t ≤ upper t := by\n      apply (div_le_iff₀ htpos).2\n      dsimp only [upper]\n      nlinarith\n    rw [div_eq_inv_mul] at hquotient\n    simpa [phi, slope, htpos.ne'] using hquotient\n  have hfirst := le_of_tendsto_of_tendsto hslope hupper hslope_le\n  dsimp only [upper, sub_zero, one_mul] at hfirst\n  linarith\n", "role": "assistant"}], "split": "public-dev", "track": "lean", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "4a0607efb0c7bf22a09cc2d1ced0f8033582c57fd04e0225279f2e0782d169dc", "family": "convex-flow-first-order-rate", "id": "sogb-023", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\",\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"4a0607efb0c7bf22a09cc2d1ced0f8033582c57fd04e0225279f2e0782d169dc\",\n  \"domain\": \"optimisation\",\n  \"family\": \"convex-flow-first-order-rate\",\n  \"id\": \"sogb-023\",\n  \"level\": \"L4\",\n  \"library_domains\": [\n    \"optimisation\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"proof\",\n    \"composition\",\n    \"calculation\",\n    \"assumption-audit\",\n    \"counterexample\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"An actual gradient flow: objective decay with the zero-curvature branch\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "The differentiability and actual one-sided flow law give, for u<T, \\[\\frac{d}{du^+}f(X(u))=\\langle\\nabla f(X(u)),-\\nabla f(X(u))\\rangle=-\\|\\nabla f(X(u))\\|^2\\le0.\\] Continuity on closed subintervals and the right-derivative comparison theorem imply f∘X is nonincreasing on [0,T]. Thus for 0≤u≤t, \\(f(X(u))-f(z)\\ge f(X(t))-f(z)\\). The source does not assume this monotonicity as an input.\n\nWrite \\(D(u)=\\|X(u)-z\\|^2\\) and \\(h_t=f(X(t))-f(z)\\ge0\\), the latter by global minimality of z. The norm-square derivative and first-order strong-convexity bound give \\[D'(u^+)=-2\\langle\\nabla f(X(u)),X(u)-z\\rangle\\le-\\alpha D(u)-2(f(X(u))-f(z))\\le-\\alpha D(u)-2h_t.\\] The last step freezes the objective gap at the later observation time, using the proved monotonicity and the correct negative sign.\n\nOn the fixed interval [0,t], scalar Gronwall/comparison with constant forcing −2hₜ gives for α=0 \\[D(t)\\le D(0)-2t h_t.\\] Since D(t)≥0 and t>0, \\(h_t\\le D(0)/(2t)\\). Division by 2t requires the positive observation time, not just T≥0.\n\nFor α>0, the same comparison gives \\[D(t)\\le e^{-\\alpha t}D(0)-\\frac{2h_t}{\\alpha}(1-e^{-\\alpha t}).\\] Multiply its nonnegative upper bound by αe^{αt}>0 and rearrange: \\[2h_t(e^{\\alpha t}-1)\\le\\alpha D(0).\\] Because αt>0, e^{αt}−1>0, and division yields the exact stated coefficient. No extra smoothness, Lipschitz-gradient constant or finite dimension is introduced.\n\nThe derivative premise is required only before T; continuity at T supports the closed-endpoint comparison and permits observation t=T. The supplied minimizer and trajectory are never obtained from the rate estimate. If T=0 the observation set is empty. At α=0 the zero-curvature comparison is a separate branch, not evaluating α/(e^{αt}−1) as 0/0.\n\nFor failure at time zero take E=ℝ, f(x)=x²/2, z=0, α=1 and X(u)=e^{-u} on [0,1]. This satisfies the actual gradient-flow equation, with \\[f(X(0))-f(0)=\\tfrac12>0,\\qquad D(0)=1.\\] Direct t=0 substitution in a totalized denominator would return zero and falsely assert a zero initial gap. This refutes only the forbidden endpoint substitution. The positive-time theorem is valid. Transferring it to a curved manifold or a law-valued flow requires new geometric/chain-rule/distance adapters; these data are not automatically present.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "4a0607efb0c7bf22a09cc2d1ced0f8033582c57fd04e0225279f2e0782d169dc", "family": "convex-flow-first-order-rate", "id": "sogb-023", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\",\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"4a0607efb0c7bf22a09cc2d1ced0f8033582c57fd04e0225279f2e0782d169dc\",\n  \"domain\": \"optimisation\",\n  \"family\": \"convex-flow-first-order-rate\",\n  \"id\": \"sogb-023\",\n  \"level\": \"L4\",\n  \"library_domains\": [\n    \"optimisation\"\n  ],\n  \"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\",\n  \"skills\": [\n    \"proof\",\n    \"composition\",\n    \"calculation\",\n    \"assumption-audit\",\n    \"counterexample\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"An actual gradient flow: objective decay with the zero-curvature branch\",\n  \"track\": \"lean\"\n}\n", "role": "user"}, {"content": "/- 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  have hd (u : ℝ) (hu : u ∈ Ico 0 T) :\n      HasDerivWithinAt (fun s => f (X s)) (-‖gradient f (X u)‖ ^ 2) (Ici u) u := by\n    have hgrad : HasFDerivAt f (InnerProductSpace.toDual ℝ E (gradient f (X u))) (X u) :=\n      (hf (X u)).hasGradientAt\n    simpa only [Function.comp_def, InnerProductSpace.toDual_apply_apply,\n      inner_neg_right, real_inner_self_eq_norm_sq] using\n      hgrad.comp_hasDerivWithinAt u (hflow u hu)\n  have hmono : AntitoneOn (fun s => f (X s)) (Icc 0 T) := by\n    intro a ha b hb hab\n    have hc := (hf.continuous.comp_continuousOn hX).mono\n      (show Icc a b ⊆ Icc 0 T from fun u hu => ⟨ha.1.trans hu.1, hu.2.trans hb.2⟩)\n    have hg := le_gronwallBound_of_liminf_deriv_right_le\n      (f' := fun u => -‖gradient f (X u)‖ ^ 2) (δ := f (X a)) (K := 0) (ε := 0) hc\n      (fun u hu r hr => by\n        simpa [slope] using (hd u ⟨ha.1.trans hu.1, hu.2.trans_le hb.2⟩).liminf_right_slope_le hr)\n      le_rfl (fun u _ => by nlinarith [sq_nonneg ‖gradient f (X u)‖]) b ⟨hab, le_rfl⟩\n    simpa [gronwallBound_K0] using hg\n  have hdist (u : ℝ) (hu : u ∈ Ico 0 T) :\n      HasDerivWithinAt (fun s => ‖X s - z‖ ^ 2)\n        (-2 * inner ℝ (gradient f (X u)) (X u - z)) (Ici u) u := by\n    have h := ((hflow u hu).sub_const z).norm_sq\n    simp only [inner_neg_right, real_inner_comm] at h\n    convert h using 1; ring\n  intro t ht\n  refine ⟨sub_nonneg.mpr (hz (mem_univ _)), ?_⟩\n  have hbound := le_gronwallBound_of_liminf_deriv_right_le\n    (f := fun s => ‖X s - z‖ ^ 2)\n    (f' := fun u => -2 * inner ℝ (gradient f (X u)) (X u - z))\n    (δ := ‖X 0 - z‖ ^ 2) (K := -α) (ε := -2 * (f (X t) - f z))\n    (((hX.sub continuousOn_const).norm.pow 2).mono\n      (show Icc 0 t ⊆ Icc 0 T from fun u hu => ⟨hu.1, hu.2.trans ht.2⟩))\n    (fun u hu r hr => by\n      simpa [slope] using (hdist u ⟨hu.1, hu.2.trans_le ht.2⟩).liminf_right_slope_le hr)\n    le_rfl (fun u hu => by\n      have hs := StrongConvexFirstOrder.firstOrder_lower_bound_of_strongConvexOn\n        hsc (fun x _ => (hf x).hasGradientAt) (mem_univ (X u)) (mem_univ z)\n      rw [show z - X u = -(X u - z) by abel, inner_neg_right, norm_neg] at hs\n      have hm := hmono ⟨hu.1, hu.2.le.trans ht.2⟩ ⟨ht.1.le, ht.2⟩ hu.2.le\n      nlinarith) t ⟨ht.1.le, le_rfl⟩\n  have hn := (sq_nonneg ‖X t - z‖).trans hbound\n  by_cases ha : α = 0\n  · simp only [ha, neg_zero, gronwallBound_K0, sub_zero] at hn ⊢\n    apply (le_div_iff₀ (mul_pos (by norm_num) ht.1)).mpr\n    nlinarith\n  · rw [if_neg ha]\n    have hap : 0 < α := lt_of_le_of_ne hα (Ne.symm ha)\n    rw [gronwallBound_of_K_ne_0 (neg_ne_zero.mpr ha), sub_zero] at hn\n    have hmul := mul_nonneg hn hap.le\n    have heq : (‖X 0 - z‖ ^ 2 * Real.exp (-α * t) +\n        (-2 * (f (X t) - f z)) / -α * (Real.exp (-α * t) - 1)) * α =\n        α * ‖X 0 - z‖ ^ 2 * Real.exp (-α * t) +\n          2 * (f (X t) - f z) * (Real.exp (-α * t) - 1) := by\n      field_simp\n    rw [heq] at hmul\n    have he : Real.exp (-α * t) * Real.exp (α * t) = 1 := by\n      rw [← Real.exp_add, show -α * t + α * t = 0 by ring, Real.exp_zero]\n    have hp := mul_nonneg hmul (Real.exp_pos (α * t)).le\n    have hcancel : (α * ‖X 0 - z‖ ^ 2 * Real.exp (-α * t) +\n        2 * (f (X t) - f z) * (Real.exp (-α * t) - 1)) * Real.exp (α * t) =\n        α * ‖X 0 - z‖ ^ 2 - 2 * (f (X t) - f z) * (Real.exp (α * t) - 1) := by\n      calc\n        _ = α * ‖X 0 - z‖ ^ 2 * (Real.exp (-α * t) * Real.exp (α * t)) +\n          2 * (f (X t) - f z) * (Real.exp (-α * t) * Real.exp (α * t)) -\n          2 * (f (X t) - f z) * Real.exp (α * t) := by ring\n        _ = _ := by rw [he]; ring\n    rw [hcancel] at hp\n    rw [div_mul_eq_mul_div]\n    apply (le_div_iff₀ (mul_pos (by norm_num) (sub_pos.mpr (Real.one_lt_exp_iff.mpr\n      (mul_pos hap ht.1))))).mpr\n    nlinarith\n", "role": "assistant"}], "split": "public-dev", "track": "lean", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "a64992f051cc8dbb2603c11d72a720f0121aeb064ca12a66d0efd321c5e014da", "family": "adaptive-hybrid-cost-boundary", "id": "sogb-024", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\",\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"a64992f051cc8dbb2603c11d72a720f0121aeb064ca12a66d0efd321c5e014da\",\n  \"domain\": \"mcmc\",\n  \"family\": \"adaptive-hybrid-cost-boundary\",\n  \"id\": \"sogb-024\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"mcmc\",\n    \"samplewiki\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"proof\",\n    \"assumption-audit\",\n    \"counterexample\",\n    \"semantic-boundary\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"One null set for every event, or a different null set for each event?\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "Uncountability ensures a set cannot be both countable and co-countable, so μ is well defined, with μ(Ω)=1 and μ(∅)=0. For a disjoint sequence of measurable sets, if one is co-countable, all the others are countable and their union is co-countable; the sum and union mass are one. If none is co-countable, the countable union is countable and both masses are zero. This proves countable additivity.\n\nFor each x, K(x,·)=δₓ and L(x,·)=μ are probability measures. For each fixed measurable B, x↦K(x,B)=1_B(x) is measurable and x↦L(x,B)=μ(B) is constant. Hence both assignments are Markov kernels; no diagonal-subset measurability in the product space is asserted or needed.\n\nIf B is countable, L(x,B)=0 and K differs only on the μ-null set B. If B is co-countable, L(x,B)=1 and K differs only on its μ-null complement. Thus \\[\\forall B\\text{ measurable},\\qquad K(\\cdot,B)=L(\\cdot,B)\\quad\\mu\\text{-a.e.}\\] The exceptional set depends on B.\n\nFor every x choose the measurable singleton B={x}. Then \\[K(x,\\{x\\})=1,\\qquad L(x,\\{x\\})=0.\\] Consequently no x has equality for all measurable events. There cannot be a common full-measure set. An uncountable union of the individual null exceptions may be all of Ω; countable closure of null sets does not justify that union.\n\nNevertheless, for each measurable B, \\[(\\mu K)(B)=\\int 1_B(x)\\,d\\mu(x)=\\mu(B),\\qquad(\\mu L)(B)=\\int\\mu(B)\\,d\\mu=\\mu(B).\\] Both output measures are μ. This refutes the quantifier interchange and equality of kernel versions almost everywhere, not the true averaged-law equality. For a fixed output event an averaging proof can use that event's own exceptional set; a common null set is a sufficient stronger API premise, not shown mathematically necessary for the averaged conclusion.\n\nThe output σ-algebra here is not countably generated: any countable family of countable/co-countable generators involves a countable union C of their countable sides, and every generator is constant on Ω∖C. Every generated measurable set is therefore constant there, but the full σ-algebra contains a singleton inside Ω∖C and can separate it from another such point. The uncountable input also is not a countable type. Thus this is outside the countable-input-or-countably-generated-output interface used by Mathlib's finite-kernel uniqueness theorem. Standard-Borel outputs have countably generated σ-algebras; this example does not refute that theorem or the selected conditional-distribution interface.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "9331f23fb1a4e9f66961f15c1233923302255d9b4ff3c953d01a7ad61d76e7b6", "family": "posterior-curvature-smoothing", "id": "sogb-025", "messages": [{"content": "{\n  \"assumptions\": [\n    \"E is a finite-dimensional complete real inner-product space with its Borel measurable structure and volume measure, including the zero-dimensional case.\",\n    \"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.\",\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"9331f23fb1a4e9f66961f15c1233923302255d9b4ff3c953d01a7ad61d76e7b6\",\n  \"domain\": \"sampling\",\n  \"family\": \"posterior-curvature-smoothing\",\n  \"id\": \"sogb-025\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"log-concave-sampling\",\n    \"samplewiki\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"proof\",\n    \"calculation\",\n    \"assumption-audit\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"From Gibbs Poincaré to a directional covariance bound\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "The density assumptions make μ the normalized Gibbs probability measure. This is used in the energy calculation; the notation tilted alone is not permission to treat an arbitrary totalized density expression as a probability measure.\n\nFor a fixed a, set f_a(x)=⟨a,x⟩. The continuous linear functional is genuinely C¹ with derivative v↦⟨a,v⟩ and gradient a at every x. These derivative facts are proved from the continuous-linear-map API, not from a fallback value of totalized fderiv.\n\nThe supplied vector L² hypothesis implies f_a∈L²(μ), by continuity of inner product with a. The gradient is the constant vector a; a constant is in L² of a probability measure. Thus both integrability interfaces of Gibbs C¹ Poincaré have been checked.\n\nNormalization and the constant gradient give the exact energy \\[\\mathcal E_\\mu(f_a)=\\int\\|\\nabla f_a(x)\\|^2\\,d\\mu(x)=\\|a\\|^2.\\] There is no hidden dimension factor.\n\nWith these L² hypotheses, the actual covariance bilinear form satisfies \\[\\operatorname{Cov}_\\mu(a,a)=\\operatorname{Var}_\\mu(f_a).\\] The proof uses covarianceBilin_self followed by the integral formula for variance. The vector L² hypothesis is important for interpreting this totalized covariance API as the intended second moment.\n\nApply the imported Poincaré estimate and substitute the energy identity: \\[m\\operatorname{Cov}_\\mu(a,a)\\le\\|a\\|^2.\\] Since m>0, division preserves the inequality and gives the required bound. This also covers a=0 and the zero-dimensional space.\n\nBoth curvature bounds and the position L² premise remain part of this exact Lean interface, although the final constant depends only on m. This proof neither establishes their sharp mathematical necessity nor produces the vector moment from scratch. The SPHMC posterior consumer separately supplies that moment before applying this lemma.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "9331f23fb1a4e9f66961f15c1233923302255d9b4ff3c953d01a7ad61d76e7b6", "family": "posterior-curvature-smoothing", "id": "sogb-025", "messages": [{"content": "{\n  \"assumptions\": [\n    \"E is a finite-dimensional complete real inner-product space with its Borel measurable structure and volume measure, including the zero-dimensional case.\",\n    \"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.\",\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"9331f23fb1a4e9f66961f15c1233923302255d9b4ff3c953d01a7ad61d76e7b6\",\n  \"domain\": \"sampling\",\n  \"family\": \"posterior-curvature-smoothing\",\n  \"id\": \"sogb-025\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"log-concave-sampling\",\n    \"samplewiki\"\n  ],\n  \"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\",\n  \"skills\": [\n    \"proof\",\n    \"calculation\",\n    \"assumption-audit\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"From Gibbs Poincaré to a directional covariance bound\",\n  \"track\": \"lean\"\n}\n", "role": "user"}, {"content": "/- 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  let μ := (volume : Measure E).tilted (fun x => -W x)\n  let : IsProbabilityMeasure μ := isProbabilityMeasure_tilted hI\n  dsimp only\n  intro a\n  let f : E → ℝ := fun x => inner ℝ a x\n  have hf : ContDiff ℝ 1 f := (innerSL ℝ a).contDiff\n  have hp : MemLp f 2 μ := hX.const_inner a\n  have hgrad : gradient f = fun _ => a := by\n    funext x\n    apply (toDual ℝ E).injective\n    rw [toDual_gradient]\n    exact (innerSL ℝ a).fderiv\n  have hq : MemLp (gradient f) 2 μ := hgrad ▸ memLp_const a\n  obtain ⟨_, hb⟩ := GibbsC1Poincare.gibbs_c1_variance_poincare\n    W hW hI hZ m M hm hmM hlower hupper f hf hp hq\n  have henergy : Poincare.dirichletEnergy μ f = ‖a‖^2 := by\n    simp only [Poincare.dirichletEnergy, hgrad]\n    simp\n  rw [henergy] at hb\n  have hcov : covarianceBilin μ a a = Poincare.variance μ f := by\n    rw [covarianceBilin_self hX a, variance_eq_integral hp.aemeasurable]\n    rfl\n  rw [hcov]\n  exact (le_div_iff₀ hm).2 (by simpa only [mul_comm] using hb)\n", "role": "assistant"}], "split": "public-dev", "track": "lean", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "fd49b5de54ddc240ba020368b624adc6d081d39a2c3e4378761f243fc0827279", "family": "posterior-curvature-smoothing", "id": "sogb-026", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\",\n    \"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.\",\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"fd49b5de54ddc240ba020368b624adc6d081d39a2c3e4378761f243fc0827279\",\n  \"domain\": \"sampling\",\n  \"family\": \"posterior-curvature-smoothing\",\n  \"id\": \"sogb-026\",\n  \"level\": \"L4\",\n  \"library_domains\": [\n    \"samplewiki\",\n    \"log-concave-sampling\",\n    \"optimisation\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"proof\",\n    \"composition\",\n    \"assumption-audit\",\n    \"counterexample\",\n    \"semantic-boundary\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Can the SPHMC smoothing argument survive weak convexity?\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "Set W_y(x)=V(x)+‖x−y‖²/(2η). Its Hessian in x lies between cI and bI, where c=m+η⁻¹>0 and b=M+η⁻¹≥c. Positive posterior curvature is available even when V itself is concave. In particular both 1+mη and 1+Mη are positive.\n\nThe lower Taylor bound V(x)≥V(0)+⟨∇V(0),x⟩+(m/2)‖x‖² implies \\[W_y(x)\\ge\\frac c2\\|x\\|^2+\\langle\\nabla V(0)-y/\\eta,x\\rangle+V(0)+\\frac{\\|y\\|^2}{2\\eta}.\\] On every ball ‖y‖≤R, Young's inequality absorbs the linear term and gives W_y(x)≥(c/4)‖x‖²−A_R for a finite constant A_R. Thus the posterior normalizer is finite and strictly positive for every y, with all polynomial position moments finite.\n\nThe integrand is positive and continuous. Its first y derivatives are (x_i−y_i)/η times itself; its second derivatives have factors (x_i−y_i)(x_j−y_j)/η²−δ_ij/η. On ‖y‖≤R their absolute values are bounded by B_R(1+‖x‖²)exp(−c‖x‖²/4), an integrable function independent of y. Dominated parameter differentiation, including continuity of these derivatives, gives qη∈C² and qη>0; hence Uη∈C². This establishes an unnormalized analytic interface without first declaring exp(−V) integrable.\n\nNormalize exp(−W_y) to obtain the actual posterior Π_y; its position is in L² by the same Gaussian bound. Differentiating qη and then its logarithm gives \\[\\nabla U_\\eta(y)=\\eta^{-1}(y-\\mathbb E_{\\Pi_y}X),\\qquad\\nabla^2U_\\eta(y)=\\eta^{-1}I-\\eta^{-2}\\operatorname{Cov}_{\\Pi_y}(X).\\] The moments and dominated differentiation justify these identities; they are not merely identities assigned to totalized derivatives.\n\nApply the Gibbs covariance upper producer to W_y, retaining its normalization, vector L² and both curvature hypotheses. Apply the covariance lower producer to the same potential, representing c,b as positive nonnegative-real parameters where its Lean interface requires them. They give \\[b^{-1}I\\preceq\\operatorname{Cov}_{\\Pi_y}(X)\\preceq c^{-1}I.\\] The upper covariance bound is the Poincaré ingredient; the reverse bound is a separate Cramér–Rao ingredient. Neither direction supplies the other.\n\nSubtracting a covariance from η⁻¹I reverses its order. Substitution and exact algebra therefore yield \\[\\frac{m}{1+m\\eta}I\\preceq\\nabla^2U_\\eta(y)\\preceq\\frac{M}{1+M\\eta}I.\\] For example the lower endpoint is η⁻¹−η⁻²/c=m/(1+mη). These signed bounds remain meaningful when m or M is negative; denominators are positive by the first step.\n\nA one-sided Hessian upper bound is not an operator-norm bound in the presence of negative eigenvalues. Since the Hessian is self-adjoint, the two-sided bounds imply \\[\\|\\nabla^2U_\\eta(y)\\|_{\\rm op}\\le L_\\eta:=\\max\\!\\left\\{\\left|\\frac m{1+m\\eta}\\right|,\\left|\\frac M{1+M\\eta}\\right|\\right\\}.\\] Integrating the genuine derivative of ∇Uη along a line segment proves global Lη-Lipschitz continuity. If m≥0 this reduces to M/(1+Mη), but not in general.\n\nFor the exact failure example take V(x)=−a‖x‖²/2, a>0 and 0<η<1/a, so m=M=−a and c>0. Completing the square gives \\[q_\\eta(y)=(1-a\\eta)^{-d/2}\\exp\\!\\left(\\frac{a\\|y\\|^2}{2(1-a\\eta)}\\right).\\] Hence \\[U_\\eta(y)=\\frac d2\\log(1-a\\eta)-\\frac{a\\|y\\|^2}{2(1-a\\eta)},\\qquad\\nabla^2U_\\eta=-\\frac a{1-a\\eta}I.\\] Its gradient has Lipschitz constant a/(1−aη)>0, whereas the proposed M/(1+Mη) is negative. The two-sided Hessian bound is attained exactly.\n\nFor nonnegative densities Tonelli and the unit-mass Gaussian kernel give, as extended nonnegative integrals, \\[\\int q_\\eta(y)\\,dy=\\int e^{-V(x)}\\,dx.\\] Pointwise finiteness of qη therefore does not imply a probability law. In the example both total masses are infinite. To define the normalized smoothed sampling law one separately needs 0<Z=∫e⁻ⱽ<∞ and then uses qη/Z; its potential Uη+log Z has the same derivatives. These extended-integral conclusions must not be read as Lean real Bochner integrals returning infinity: those real integrals are totalized outside integrability.\n\nThe local GaussianConvolutionRegularity.gaussian_convolution_derivatives assumes a probability input μ; it cannot be called on an unnormalizable prior to replace the new domination proof above. SmoothedHessianUpper.smoothed_hessian_upper assumes positive lower curvature of V, and the paper-specialized SmoothedHessianBounds also assumes its κ and η regime; neither is an unchanged weak-convexity interface. Their posterior covariance ingredients can be reused after the new analytic/normalization adapters, whose complete Lean realization is absent here. No SPHMC dynamics, warm-start, error, query-cost or main-theorem claim follows from this QA.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "7da6ccd4e2fbac80e8597c8d5421fb083791810652c7a3838214fa305763cf89", "family": "bernoulli-gaussian-entropy-limit", "id": "sogb-027", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\",\n    \"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.\",\n    \"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.\",\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.\"\n  ],\n  \"binding_sha256\": \"7da6ccd4e2fbac80e8597c8d5421fb083791810652c7a3838214fa305763cf89\",\n  \"domain\": \"discrete sampling\",\n  \"family\": \"bernoulli-gaussian-entropy-limit\",\n  \"id\": \"sogb-027\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"discrete-sampling\",\n    \"log-concave-sampling\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"proof\",\n    \"calculation\",\n    \"assumption-audit\",\n    \"counterexample\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Tensorizing a two-point entropy bound without losing the flip constant\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "The uniform law has total mass 2ⁿ·2⁻ⁿ=1, also for n=0. Every displayed finite real function is measurable and integrable on this finite space. The value φ(0)=0 removes any fictitious strict-positivity requirement on h or its squared mean; no logarithm of a positive normalizer is used.\n\nSeparate a configuration into (x,β)∈Ωₙ×{0,1}. Put a(x)=h(x,0), b(x)=h(x,1) and \\[R(x)=\\sqrt{\\frac{a(x)^2+b(x)^2}{2}},\\qquad\\ell(x)=\\frac{\\phi(a(x)^2)+\\phi(b(x)^2)}2-\\phi(R(x)^2).\\] Uniform counting gives Eμₙ₊₁g=Eμₙ[(g(x,0)+g(x,1))/2].\n\nBecause R²=(a²+b²)/2, the squared mean is preserved: EμₙR²=Eμₙ₊₁h². Add and subtract Eμₙφ(R²) to get the exact identity \\[\\operatorname{Ent}_{n+1}(h^2)=\\mathbb E_{\\mu_n}\\ell+\\operatorname{Ent}_n(R^2).\\] All subtractions concern finite integrals. This is a proved identity for this finite law, not an invocation of general tensorization.\n\nDefine P(x) as the sum over j<n of [(a(x)−a(xʲ))²+(b(x)−b(xʲ))²]/2, and L(x)=(a(x)−b(x))². Toggling a prefix coordinate preserves the last bit; toggling the last bit exchanges the two slices. Therefore \\[\\mathbb E_{\\mu_{n+1}}D_{n+1}h=\\mathbb E_{\\mu_n}(P+L).\\] Averaging the two last-coordinate square differences gives L, not L/2: they are equal after reversing the subtraction.\n\nFor u=(a,b)/√2 and v=(c,d)/√2 in ℝ², the reverse triangle inequality gives \\[(\\sqrt{(a^2+b^2)/2}-\\sqrt{(c^2+d^2)/2})^2\\le\\frac{(a-c)^2+(b-d)^2}{2}.\\] Apply this to each pair of neighboring prefix configurations and sum over coordinates. It yields DₙR≤P pointwise and hence EμₙDₙR≤EμₙP. The existing Lean helper proves this same RMS contraction by Cauchy–Schwarz algebra.\n\nThe supplied two-point inequality gives ℓ(x)≤L(x)/2 for every prefix x, for arbitrary signs and zeros. Integrating preserves that inequality. Assuming the n-dimensional bound for R, combine the three ingredients: \\[\\operatorname{Ent}_{n+1}(h^2)\\le\\frac12\\mathbb E L+\\frac12\\mathbb E D_nR\\le\\frac12\\mathbb E(L+P)=\\frac12\\mathbb E D_{n+1}h.\\] This keeps the same factor at every dimension.\n\nAt n=0, there is one atom, so Eφ(h²)=φ(Eh²) and entropy is zero; the empty coordinate sum is zero as well. This starts induction without dividing by n or assuming a positive squared mean. It also covers the identically zero function in every dimension.\n\nFor a counterexample to replacing RMS by conditional mean, take a prefix x∈{0,1}, set A(0)=1,A(1)=2, and let h(x,0)=A(x), h(x,1)=−A(x). Then the conditional mean M is identically zero and the actual local entropy ℓ is identically zero. But \\[\\operatorname{Ent}_2(h^2)=\\frac{\\phi(1)+\\phi(4)}2-\\phi(5/2)>0,\\] by strict convexity of φ on the positive interval [1,4]. The proposed replacement would give 0+Ent₁(M²)=0, contradicting the identity. The squared mean must be preserved; averaging signs does not do that.\n\nThe energy here is the full sum of coordinate-flip squared differences under a probability law. Replacing it by an average over coordinates changes constants. The result is a finite functional inequality, not by itself a convergence theorem for a chosen discrete-time sampler or a noncompact Gaussian inequality.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "1be26d5f3c790d72f77813c7277245953ea18975cdadb455721f7a972b11f302", "family": "bernoulli-gaussian-entropy-limit", "id": "sogb-028", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\",\n    \"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.\",\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"1be26d5f3c790d72f77813c7277245953ea18975cdadb455721f7a972b11f302\",\n  \"domain\": \"sampling\",\n  \"family\": \"bernoulli-gaussian-entropy-limit\",\n  \"id\": \"sogb-028\",\n  \"level\": \"L4\",\n  \"library_domains\": [\n    \"log-concave-sampling\",\n    \"discrete-sampling\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"proof\",\n    \"composition\",\n    \"calculation\",\n    \"assumption-audit\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"From the Boolean cube to compact Gaussian log-Sobolev\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "Each μₘ is a genuine probability law, and Sₘ is measurable on the finite configuration space. The available CLT identifies its actual pushforward distribution, not just an unrelated Gaussian-looking sequence. Working with m=n+1 avoids using the inverse-square-root scaling at zero in the asymptotic calculation.\n\nSince f is C² and compactly supported, f², φ(f²) and (f′)² are continuous compactly supported functions. In particular f′ is genuinely continuous and supported in the closed support of f. Continuity of s↦s log s at s=0 is understood via its zero extension; continuity of log alone at zero would be false. These observables are bounded, measurable and integrable under γ and every finite law.\n\nWeak convergence may therefore be tested against each of these three bounded continuous observables. The pushforward integral formula converts the resulting expectations to integrals over the actual cube. Put \\[A_m=\\mathbb E f(S_m)^2,\\quad B_m=\\mathbb E\\phi(f(S_m)^2),\\quad C_m=\\mathbb E f'(S_m)^2.\\] They converge to the corresponding Gaussian integrals. This is weak convergence plus admissible test functions, not an application of dominated convergence to functions on different spaces Ωₘ.\n\nAll Aₘ are nonnegative. Continuity of φ on [0,∞), including zero, gives \\[B_m-\\phi(A_m)\\longrightarrow\\int\\phi(f^2)\\,d\\gamma-\\phi\\!\\left(\\int f^2\\,d\\gamma\\right)=\\operatorname{Ent}_\\gamma(f^2).\\] There is no need to take a separate limit of log Aₘ or assume the limiting squared mass is strictly positive. This includes f≡0.\n\nCompact C² regularity supplies finite constants B,K≥0 with |f′|≤B and |f′(y)−f′(x)|≤K|y−x| globally. To obtain K, bound the continuous second derivative on its compact support and use the mean-value theorem. Genuine differentiability is used; the totalized deriv API by itself does not supply this regularity.\n\nFlipping coordinate j changes Sₘ by δⱼ=±2/√m, exactly. For any x,y the mean-value theorem gives a secant a with f(y)−f(x)=a(y−x), |a|≤B and |a−f′(x)|≤K|y−x|; the x=y case is immediate. Thus \\[\\big|(f(y)-f(x))^2-f'(x)^2(y-x)^2\\big|\\le2BK|y-x|^3.\\] This is the squared-increment estimate used by the existing full-flip producer.\n\nLet Gₘ(ε)=Σⱼ(f(Sₘ(εʲ))−f(Sₘ(ε)))². Summing the last estimate over all m coordinates gives the pointwise bound \\[|G_m(\\varepsilon)-4f'(S_m(\\varepsilon))^2|\\le\\frac{16BK}{\\sqrt m}.\\] The leading term is four because m·(2/√m)²=4; it would not be four for an average over coordinates. No independence assumption on the individual flip errors is needed.\n\nFinite-space integrability licenses subtraction inside the expectation, and the probability mass is one. Hence \\[\\left|\\mathbb E G_m-4C_m\\right|\\le\\frac{16BK}{\\sqrt m}\\longrightarrow0.\\] Together with the derivative-observer limit, this proves \\[\\mathbb E G_m\\longrightarrow4\\int(f')^2\\,d\\gamma.\\] This uniform error estimate justifies accumulating m small coordinate errors; a pointwise one-coordinate approximation alone would not.\n\nApply the finite cube inequality to h(ε)=f(Sₘ(ε)). Its energy may use the opposite subtraction order, but squaring makes the two orientations identical. For every m≥1, \\[B_m-\\phi(A_m)\\le\\frac12\\mathbb E G_m.\\] Both sides have finite real limits by the preceding domain and convergence arguments.\n\nPass this family of inequalities through the limits using closedness of ≤ on ℝ. The constant is exactly \\[\\operatorname{Ent}_\\gamma(f^2)\\le\\frac12\\left(4\\int(f')^2\\,d\\gamma\\right)=2\\int(f')^2\\,d\\gamma.\\] This proves the homogeneous compact C² inequality without dividing by ∫f², choosing a representative outside this regular class, or assuming nonzero mass.\n\nThe Lean target invokes the existing entropy and full-flip producers, which internally establish the actual finite/Gaussian L¹ domains and limits described above. It does not take those limits as additional public assumptions. A general Sobolev/noncompact extension would need its own approximation, representative and convergence arguments; this QA supplies none of those conclusions.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "1be26d5f3c790d72f77813c7277245953ea18975cdadb455721f7a972b11f302", "family": "bernoulli-gaussian-entropy-limit", "id": "sogb-028", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\",\n    \"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.\",\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"1be26d5f3c790d72f77813c7277245953ea18975cdadb455721f7a972b11f302\",\n  \"domain\": \"sampling\",\n  \"family\": \"bernoulli-gaussian-entropy-limit\",\n  \"id\": \"sogb-028\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"log-concave-sampling\",\n    \"discrete-sampling\"\n  ],\n  \"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\",\n  \"skills\": [\n    \"proof\",\n    \"composition\",\n    \"calculation\",\n    \"assumption-audit\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"From the Boolean cube to compact Gaussian log-Sobolev\",\n  \"track\": \"lean\"\n}\n", "role": "user"}, {"content": "/- 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  dsimp only\n  let μ : (n : ℕ) → Measure (Fin n → Bool) := fun n =>\n    (Fintype.card (Fin n → Bool) : ℝ≥0∞)⁻¹ • Measure.count\n  let S : (n : ℕ) → (Fin n → Bool) → ℝ := fun n ε =>\n    (Real.sqrt (n : ℝ))⁻¹ * ∑ j : Fin n, if ε j then (1 : ℝ) else -1\n  rcases GaussianCompactEntropy.compact_count_gaussian_entropy_limits f hf hs\n    with ⟨_, _, _, _, _, _, _, tEnt⟩\n  rcases GaussianFlipEnergy.compact_count_gaussian_flip_energy_limit f hf hs\n    with ⟨_, _, tEnergy⟩\n  change Tendsto (fun n : ℕ =>\n    (∫ ε, (f (S (n+1) ε))^2 * Real.log ((f (S (n+1) ε))^2) ∂μ (n+1)) -\n      (∫ ε, (f (S (n+1) ε))^2 ∂μ (n+1)) *\n        Real.log (∫ ε, (f (S (n+1) ε))^2 ∂μ (n+1))) atTop\n    (𝓝 ((∫ x, (f x)^2 * Real.log ((f x)^2) ∂gaussianReal 0 1) -\n      (∫ x, (f x)^2 ∂gaussianReal 0 1) *\n        Real.log (∫ x, (f x)^2 ∂gaussianReal 0 1))) at tEnt\n  change Tendsto (fun n : ℕ => ∫ ε : Fin (n+1) → Bool,\n    (∑ j : Fin (n+1),\n      (f (S (n+1) (Function.update ε j (!ε j))) - f (S (n+1) ε))^2)\n      ∂μ (n+1)) atTop (𝓝 (4 * ∫ x, (deriv f x)^2 ∂gaussianReal 0 1)) at tEnergy\n  have tHalf : Tendsto (fun n : ℕ => (1/2 : ℝ) *\n      ∫ ε : Fin (n+1) → Bool, (∑ j : Fin (n+1),\n        (f (S (n+1) (Function.update ε j (!ε j))) - f (S (n+1) ε))^2)\n        ∂μ (n+1)) atTop (𝓝 (2 * ∫ x, (deriv f x)^2 ∂gaussianReal 0 1)) := by\n    have heq : (1/2 : ℝ) * (4 * ∫ x, (deriv f x)^2 ∂gaussianReal 0 1) =\n        2 * ∫ x, (deriv f x)^2 ∂gaussianReal 0 1 := by ring\n    simpa only [heq] using tEnergy.const_mul (1/2 : ℝ)\n  apply le_of_tendsto_of_tendsto tEnt tHalf\n  exact Eventually.of_forall fun n => by\n    have hn := (BernoulliLogSobolev.bernoulli_function_logSobolev (n+1)\n      (fun ε => f (S (n+1) ε))).2.2.2.2\n    change ((∫ ε, (f (S (n+1) ε))^2 * Real.log ((f (S (n+1) ε))^2) ∂μ (n+1)) -\n      (∫ ε, (f (S (n+1) ε))^2 ∂μ (n+1)) *\n        Real.log (∫ ε, (f (S (n+1) ε))^2 ∂μ (n+1))) ≤\n      (1/2 : ℝ) * ∫ ε : Fin (n+1) → Bool, (∑ j : Fin (n+1),\n        (f (S (n+1) ε) - f (S (n+1) (Function.update ε j (!ε j))))^2)\n        ∂μ (n+1) at hn\n    simpa only [sub_sq_comm] using hn\n", "role": "assistant"}], "split": "public-dev", "track": "lean", "usage": "public curriculum reuse; never an unseen test"}
{"binding_sha256": "9d9f361a44b9b4cbd33ff6379d61f64ca7b45805e410c4021af25e64ba6013c8", "family": "chewi-opt-v1-exercise-3-2", "id": "sogb-029", "messages": [{"content": "{\n  \"assumptions\": [\n    \"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.\",\n    \"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.\",\n    \"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.\"\n  ],\n  \"binding_sha256\": \"9d9f361a44b9b4cbd33ff6379d61f64ca7b45805e410c4021af25e64ba6013c8\",\n  \"domain\": \"optimisation\",\n  \"exercise_source\": {\n    \"authors\": [\n      \"Sinho Chewi\"\n    ],\n    \"book_page\": 23,\n    \"edition\": \"arXiv:2605.07006v1\",\n    \"exercise\": \"3.2\",\n    \"id\": \"chewi-opt-v1-exercise-3-2\",\n    \"pdf_page\": 23,\n    \"source_url\": \"https://arxiv.org/pdf/2605.07006v1#page=23\",\n    \"title\": \"Lectures on Optimization\",\n    \"wording_status\": \"original faithful mathematical restatement; not reproduced source prose\"\n  },\n  \"family\": \"chewi-opt-v1-exercise-3-2\",\n  \"id\": \"sogb-029\",\n  \"level\": \"L3\",\n  \"library_domains\": [\n    \"optimisation\"\n  ],\n  \"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.\",\n  \"skills\": [\n    \"proof\",\n    \"calculation\",\n    \"assumption-audit\"\n  ],\n  \"split\": \"public-dev\",\n  \"title\": \"Chewi Optimization Exercise 3.2: the balanced gradient step\",\n  \"track\": \"natural-language\"\n}\n", "role": "user"}, {"content": "C² regularity makes the actual Hessian H(z)=∇²f(z) symmetric and continuous. The convexity/smoothness models yield αI≼H(z)≼βI. These are genuine derivative statements, not an interpretation of the fallback value of a totalized derivative. They also ensure continuous segment derivatives for the upcoming integral.\n\nThe actual derivative is DT_h(z)=I−hH(z). For h≥0 and any eigenvalue λ of H(z), α≤λ≤β gives \\[1-h\\beta\\le1-h\\lambda\\le1-h\\alpha.\\] Since the derivative is symmetric, its operator norm is bounded by \\[M(h):=\\max\\{|1-h\\alpha|,|1-h\\beta|\\}.\\] In finite dimensions this follows from orthogonal diagonalization. The existing Hilbert-space Lean proof uses the equivalent symmetric Rayleigh-quotient norm bound rather than constructing an eigenvalue list.\n\nPut v=y−x and z(t)=x+tv. The chain rule gives dT_h(z(t))/dt=DT_h(z(t))v. This derivative is continuous on [0,1], so it is Bochner integrable and the vector FTC applies: \\[T_h(y)-T_h(x)=\\int_0^1DT_h(x+tv)v\\,dt.\\] Taking norms and using the uniform operator bound yields \\[\\|T_h(y)-T_h(x)\\|\\le M(h)\\|y-x\\|.\\] The case x=y is included; no division by ‖y−x‖ is needed.\n\nBecause α+β>0, take h★=2/(α+β) and q=(β−α)/(α+β)≥0. Direct algebra gives \\[1-h_\\star\\alpha=q,\\qquad1-h_\\star\\beta=-q.\\] Thus M(h★)=q, and the preceding estimate proves the pairwise bound at that step.\n\nTo show global optimality of the endpoint envelope, fix any real h and let M=M(h). The elementary inequalities 1−hα≤M and hβ−1≤M can be multiplied by the nonnegative weights β and α. Their sum gives \\[\\beta-\\alpha=\\beta(1-h\\alpha)+\\alpha(h\\beta-1)\\le(\\alpha+\\beta)M.\\] Divide by α+β>0 to obtain q≤M(h). Hence h★ achieves the smallest endpoint bound, even when the scalar comparison is made over every real h; in particular it is optimal among nonnegative steps.\n\nSince α>0, κ=β/α is well-defined and κ≥1. Dividing numerator and denominator of q by α gives \\[q=\\frac{\\kappa-1}{\\kappa+1},\\qquad\\|T_{h_\\star}(y)-T_{h_\\star}(x)\\|\\le\\frac{\\kappa-1}{\\kappa+1}\\|y-x\\|.\\] Also q<1, so this is a contraction in the stated source regime. When α=β>0 it gives q=0 and a constant update map, without needing a minimizer premise.\n\nThe optimization concerns the uniform bound derived from the curvature interval, not the actual best step for each individual objective. For example, if f(x)=β‖x‖²/2 and 0<α<β is a valid but loose lower label, the individual map vanishes at h=1/β, whereas the balanced endpoint envelope still has q>0. This does not contradict the minimax result for the supplied class of curvature bounds.\n\nThe owned endpoint theorem permits signed moduli and the optimal-step theorem permits α=0, both on complete real Hilbert spaces. Those are disclosed generalizations. At α=0 the latter is only nonexpansive and κ is not formed. The present exercise uses their positive-curvature Euclidean specialization. Its source's informal large-κ iteration-complexity remark, separate eigenvalue-list output and the explanatory loose-label example are not additional compiled claims of these two declarations.", "role": "assistant"}], "split": "public-dev", "track": "natural-language", "usage": "public curriculum reuse; never an unseen test"}