[
  {
    "aligned_fixed_lean_task": true,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "ac978a2c81f53d2c0ce45bb253fdafd93de190bb9ff64e1eab954ea9e29249b8",
    "boundary": "The Lean target covers the flux identity. The explanatory non-implications are assessed in the natural-language track, not certified by that Lean target.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Probability.KernelReversibility.isReversible_of_singleton_balance",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Probability.KernelReversibility.isReversible_of_singleton_balance",
    "id": "sogb-001",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Probability.KernelReversibility.isReversible_of_singleton_balance",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Probability/KernelReversibility.lean"
  },
  {
    "aligned_fixed_lean_task": true,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "f5d34b22cd89382ed252b2d1ed63a8b0763a005dbe5940fd24174b14710d19e8",
    "boundary": "This exercises the supplied upper-model/PL formulation. It does not derive those hypotheses from smoothness or infer differentiability from a totalized gradient. The broader natural-language discussion is not an extra Lean theorem.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentPL.gradient_descent_pl_value_bound",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentPL.gradient_descent_pl_value_bound",
    "id": "sogb-002",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentPL.gradient_descent_pl_value_bound",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentPL.lean"
  },
  {
    "aligned_fixed_lean_task": true,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "b65dd9a70ee7e52100bc85f865904dc278535225ecc2eaa01929897f888543c2",
    "boundary": "This is a normed-vector-space statement, not a Riemannian or geodesic-convexity theorem. The Lean target proves the k/4 envelope only; neither the sharper bound nor Gibbs normalization is supplied.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity.centered_quadratic_lower_bound_of_strongConvexOn_minimizer",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity.centered_quadratic_lower_bound_of_strongConvexOn_minimizer",
    "id": "sogb-003",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Geometry.StrongConvexity.centered_quadratic_lower_bound_of_strongConvexOn_minimizer",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Geometry/StrongConvexity.lean"
  },
  {
    "aligned_fixed_lean_task": false,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "2274ba27f6d700779d832ff90a2b26d881f2bf5f23c3dfc78689f556020559cf",
    "boundary": "A supporting formal symmetry theorem is not a verified Lean translation of every sentence of this QA. This question neither proves an optimal coupling exists nor derives the triangle inequality.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSymmetry.wassersteinDistance_comm",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSymmetry.wassersteinDistance_comm",
    "id": "sogb-004",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinSymmetry.wassersteinDistance_comm",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinSymmetry.lean"
  },
  {
    "aligned_fixed_lean_task": false,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "ce37724e97b715e2330e57378f5795759da21f91c6f477b863d897b48fe93f74",
    "boundary": "Only the unit sphere with its induced metric is treated. No general Riemannian gradient theorem, retraction guarantee or optimization convergence result is asserted.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Geometry.QuadraticFeatureTangentGradient.quadratic_feature_potential_tangent_gradient",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Geometry.QuadraticFeatureTangentGradient.quadratic_feature_potential_tangent_gradient",
    "id": "sogb-005",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Geometry.QuadraticFeatureTangentGradient.quadratic_feature_potential_tangent_gradient",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Geometry/QuadraticFeatureTangentGradient.lean"
  },
  {
    "aligned_fixed_lean_task": false,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "74f08b42811ce2c35e4669f9e631246262c6038119faa990aea98efb42431cb3",
    "boundary": "No mixing time, spectral independence, full-support theorem or hard-core ergodicity claim follows from this balance argument.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath.randomScan_isReversible",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath.randomScan_isReversible",
    "id": "sogb-006",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath.randomScan_isReversible",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Probability/RandomScanHeatBathReversibility.lean"
  },
  {
    "aligned_fixed_lean_task": false,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "a4e2d5d2fd1273d0cadc2054858c9597b4e090dd0fec5fda5ecdfd5c7d206b78",
    "boundary": "No recursive algorithm, stopping theorem, uncapped/capped comparison, expected oracle cost theorem or full paper result is claimed.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Probability.KernelHybridTelescope.abs_real_comp_commonSuffix_sub_le_integral_eventBound",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Probability.KernelHybridTelescope.abs_real_comp_commonSuffix_sub_le_integral_eventBound",
    "id": "sogb-007",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Probability.KernelHybridTelescope.abs_real_comp_commonSuffix_sub_le_integral_eventBound",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Probability/KernelHybridTelescope.lean"
  },
  {
    "aligned_fixed_lean_task": false,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "ae0a2d0d8f65fbe39a5c2ed55cbe7b93bde710026ec00ed08132abeedefb2794",
    "boundary": "No Gibbs normalization or tail-integrability theorem is supplied by the log-concavity leaf.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_exp_neg_of_convexOn",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_exp_neg_of_convexOn",
    "id": "sogb-008",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Geometry.LogConcavity.logConcaveOn_exp_neg_of_convexOn",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Geometry/LogConcavity.lean"
  },
  {
    "aligned_fixed_lean_task": false,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "a01b6366bfff653a1e2d2d7304b4e284d28669bfebdcfb82e663f26765274315",
    "boundary": "No actual algorithm law, invariant distribution or query-cost result is constructed here.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Probability.KernelTotalVariation.abs_real_comp_sub_le",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Probability.KernelTotalVariation.abs_real_comp_sub_le",
    "id": "sogb-009",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Probability.KernelTotalVariation.abs_real_comp_sub_le",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Probability/KernelTotalVariation.lean"
  },
  {
    "aligned_fixed_lean_task": false,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "7e2364e8fcf6653f355ba88b88d71d81b69d2e3959e42df0e523a13352a1b94d",
    "boundary": "No convexity equivalence, error bound in distance, or convergence rate is asserted by this terminology question.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentPL.gradient_descent_pl_value_bound",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentPL.gradient_descent_pl_value_bound",
    "id": "sogb-010",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentPL.gradient_descent_pl_value_bound",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentPL.lean"
  },
  {
    "aligned_fixed_lean_task": false,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "a5cda7d4b9f7caaf24bf4dbab8205dae1f0a82e2ebb6b9fdc92d81c748863408",
    "boundary": "No general manifold gradient API, retraction theorem, curvature comparison or optimization rate is claimed.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Geometry.QuadraticFeatureTangentGradient.quadratic_feature_potential_tangent_gradient",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Geometry.QuadraticFeatureTangentGradient.quadratic_feature_potential_tangent_gradient",
    "id": "sogb-011",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Geometry.QuadraticFeatureTangentGradient.quadratic_feature_potential_tangent_gradient",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Geometry/QuadraticFeatureTangentGradient.lean"
  },
  {
    "aligned_fixed_lean_task": false,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "d61f3a0414d8d767ef90fd31733e4fb71d9b47d0f6a62b5844c939a7ee915c7f",
    "boundary": "No existence of an optimal plan, uniqueness, or Wasserstein metric theorem is asserted.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Measure.Transport.isCoupling_prod",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Measure.Transport.isCoupling_prod",
    "id": "sogb-012",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Measure.Transport.isCoupling_prod",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean"
  },
  {
    "aligned_fixed_lean_task": false,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "84145f82f459d1cb080b714a85ebad040ad8fd4d034c441b45d77e02c77830ca",
    "boundary": "No efficient conditional-sampling implementation, ergodicity, mixing time or sweep complexity is asserted.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath.randomScan",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath.randomScan",
    "id": "sogb-013",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Probability.RandomScanHeatBath.randomScan",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Probability/RandomScanHeatBath.lean"
  },
  {
    "aligned_fixed_lean_task": false,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "8b10bef6bc3088ea8188089bb960aed747ddeded576261956464483947b6db38",
    "boundary": "No irreducibility, aperiodicity, mixing-rate theorem or general-state-space ergodic theorem is proved.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Probability.KernelReversibility.isReversible_of_singleton_balance",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Probability.KernelReversibility.isReversible_of_singleton_balance",
    "id": "sogb-014",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Probability.KernelReversibility.isReversible_of_singleton_balance",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Probability/KernelReversibility.lean"
  },
  {
    "aligned_fixed_lean_task": true,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "0bf552e05112c84a48c1bb862da7870035eb715a9cd61d0a95474c6694d37a7f",
    "boundary": "The Lean target proves the joint probability interface under exactly the displayed marginal assumptions. Surrounding nonclaims about optimality, unbounded integrability and moments are not separate Lean results. No new production theorem or Registry leaf is introduced.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Measure.Transport.isProbabilityMeasure_of_isCoupling_left",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Measure.Transport.isProbabilityMeasure_of_isCoupling_left",
    "id": "sogb-015",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Measure.Transport.isProbabilityMeasure_of_isCoupling_left",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Measure/Transport.lean"
  },
  {
    "aligned_fixed_lean_task": true,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "69fee45c9a7a53e9d1a4ec5295d530811736ff3e8dec9fbd435f1293d9bc1cd7",
    "boundary": "The Lean conclusion is the displayed representation equivalence for each tangent vector. Stationarity/eigenvector equivalences, optimization rates and arbitrary-manifold or arbitrary-metric claims are separate results. The surrounding explanation is not an additional formal theorem.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Geometry.SphereRayleigh.sphere_rayleigh_characterization",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Geometry.SphereRayleigh.sphere_rayleigh_characterization",
    "id": "sogb-016",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Geometry.SphereRayleigh.sphere_rayleigh_characterization",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Geometry/SphereRayleigh.lean"
  },
  {
    "aligned_fixed_lean_task": true,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "37f2cef51769619910cd718b9d66acd2d57f6a5a5956a6940a4d4dc2ba60eeba",
    "boundary": "Only the displayed one-step averaged event discrepancy is formally concluded. Common-suffix contraction, finite telescoping, actual recursive-history construction, stopping/cap comparisons and expected query costs remain distinct results. It does not align the full earlier hybrid/cost QA or complete a paper theorem.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Probability.KernelHybridTelescope.abs_real_comp_sub_le_integral_eventBound",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Probability.KernelHybridTelescope.abs_real_comp_sub_le_integral_eventBound",
    "id": "sogb-017",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Probability.KernelHybridTelescope.abs_real_comp_sub_le_integral_eventBound",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Probability/KernelHybridTelescope.lean"
  },
  {
    "aligned_fixed_lean_task": true,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "237b6a126e4672415b58ea7f7621fe69ef5d541dac41fb1b42336f828922a1e2",
    "boundary": "The Lean target concludes only the strict-threshold endpoint bound. It does not separately formalize this exercise's Bernoulli shortcut audit, assert optimal-plan existence, or construct a general metric-space instance. The non-strict triangle consumer exists in WassersteinTriangleExact, but is not imported as a shortcut or counted as another QA. The listed disintegration, pushforward and Minkowski Mathlib facts are used in ancestor proofs; the target join calls their ASTIS interfaces.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangle.wassersteinDistance_lt_add_of_lt",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangle.wassersteinDistance_lt_add_of_lt",
    "id": "sogb-018",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Measure.WassersteinTriangle.wassersteinDistance_lt_add_of_lt",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Measure/WassersteinTriangle.lean"
  },
  {
    "aligned_fixed_lean_task": true,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "b12bf7bd53d4c8c85dd4e1658b22f05ea7752e13bfa48a3bbfba72f620c3d52f",
    "boundary": "This bounds each supplied global supporting vector. It constructs neither a subgradient nor a measurable/random oracle, gives no constrained-normal bound, and proves no optimization rate by itself. The existing sum_weighted_gap_le theorem consumes this estimate; Bench examples add no Registry credit.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Analysis.SubgradientMethod.norm_le_of_supportsAt",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Analysis.SubgradientMethod.norm_le_of_supportsAt",
    "id": "sogb-019",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Analysis.SubgradientMethod.norm_le_of_supportsAt",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Analysis/SubgradientMethod.lean"
  },
  {
    "aligned_fixed_lean_task": false,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "15a19933c0cbddea0e5e89962c7375f05183709493c77749fe4f0115c0155931",
    "boundary": "This is a deterministic comparator bound for already supplied iterates, supports and steps. It does not construct a subgradient, prove a measurable/random algorithm or establish query complexity, constrained/projected convergence, last-iterate convergence or a global minimizer's existence. Strict positive steps are retained for the source best-iterate theorem even though some weaker sufficient conditions could also be mathematically valid. The full Lean module supports the energy/best-iterate portion, not every corollary in this QA.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Analysis.SubgradientMethod.exists_gap_le",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Analysis.SubgradientMethod.exists_gap_le",
    "id": "sogb-020",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Analysis.SubgradientMethod.exists_gap_le",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Analysis/SubgradientMethod.lean"
  },
  {
    "aligned_fixed_lean_task": true,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "3d72820e315cae6cebacdbed3bc8475fb4479d4a4a6cc7c302f4d534b54fe173",
    "boundary": "This proves one actual conditional-resampling kernel preserves its finite joint target. It does not establish reversibility, irreducibility, uniqueness, convergence, a mixing rate, a generator-domain theorem or the Gibbs law of a continuous-time process. Conditional disintegration is allowed background, not reproved by this short task. The NL finite-state counterexample is not a separate Lean conclusion.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Probability.HeatBath.heatBathSnd_invariant",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Probability.HeatBath.heatBathSnd_invariant",
    "id": "sogb-021",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Probability.HeatBath.heatBathSnd_invariant",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Probability/HeatBath.lean"
  },
  {
    "aligned_fixed_lean_task": true,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "2cfe0ffa4f03a84301a91017152eea5265cf548015c4f28f0d9d9fa9f12817b5",
    "boundary": "The supplied selector is not certified geometric data. No metric/geodesic existence, manifold regularity, Wasserstein first variation, subgradient existence, flow, sampling rate or invariance is concluded. HasDerivAt is retained exactly as in the source; no claim that all its hypotheses are mathematically minimal is made. The actual StrongConvexFirstOrder.firstOrder_lower_bound_of_strongConvexOn supplies an affine-segment/gradient adapter and consumes this limiting lemma.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Geometry.GeodesicConvexity.firstOrder_geodesicConvexity",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Geometry.GeodesicConvexity.firstOrder_geodesicConvexity",
    "id": "sogb-022",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Geometry.GeodesicConvexity.firstOrder_geodesicConvexity",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Geometry/GeodesicConvexity.lean"
  },
  {
    "aligned_fixed_lean_task": true,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "4a0607efb0c7bf22a09cc2d1ced0f8033582c57fd04e0225279f2e0782d169dc",
    "boundary": "The Lean conclusion is the nonnegative finite objective-gap bound at positive observation time for a supplied complete-space trajectory and minimizer. It is not flow existence/uniqueness, discrete gradient-descent complexity, stochastic sampling convergence, Riemannian or Wasserstein gradient flow, or a formalization of the explanatory time-zero counterexample. T≥0 is retained from the actual statement; no necessity/minimality claim is made for every source/API premise. This exercise shares a public family with the first-order limiting task, not an independent private split.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowValue.value_le",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowValue.value_le",
    "id": "sogb-023",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Analysis.GradientFlowValue.value_le",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Analysis/GradientFlowValue.lean"
  },
  {
    "aligned_fixed_lean_task": false,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "a64992f051cc8dbb2603c11d72a720f0121aeb064ca12a66d0efd321c5e014da",
    "boundary": "The counterexample is on the explicitly non-countably-generated measurable output, not ℝ with its Borel sets or a standard-Borel state space. It does not refute the source averaging theorem, conditional-law uniqueness with the proper countability assumptions, or measurability of any particular supplied bound. The supporting compiled module is not a formal proof of this QA. No claim about joint diagonal measurability, a measurable total-variation supremum, adaptive process construction or expected query cost is made.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Probability.KernelHybridTelescope.abs_real_comp_sub_le_integral_eventBound",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Probability.KernelHybridTelescope.abs_real_comp_sub_le_integral_eventBound",
    "id": "sogb-024",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Probability.KernelHybridTelescope.abs_real_comp_sub_le_integral_eventBound",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Probability/KernelHybridTelescope.lean"
  },
  {
    "aligned_fixed_lean_task": true,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "9331f23fb1a4e9f66961f15c1233923302255d9b4ff3c953d01a7ad61d76e7b6",
    "boundary": "The aligned Lean target is exactly this isotropic directional covariance upper bound with the stated premises. It is not the full anisotropic Brascamp–Lieb inequality, a new proof of position L², an invariant-law or sampling-complexity result. Prose explanations of API necessity and downstream roles are not extra Lean conclusions. No production declaration or Registry credit is added.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GibbsLinearCovarianceUpper.gibbs_linear_covariance_upper",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GibbsLinearCovarianceUpper.gibbs_linear_covariance_upper",
    "id": "sogb-025",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GibbsLinearCovarianceUpper.gibbs_linear_covariance_upper",
    "related_owned_results": [],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/GibbsLinearCovarianceUpper.lean"
  },
  {
    "aligned_fixed_lean_task": false,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "fd49b5de54ddc240ba020368b624adc6d081d39a2c3e4378761f243fc0827279",
    "boundary": "An original, independently reviewable natural-language transfer/audit exercise, not a new production theorem, paper flaw, fully formalized weakly convex sampler or empirical L5 result. The cited local results support only their exact existing strong-convexity/probability interfaces or the positive-curvature posterior substeps. The unnormalized analytic adapter and the entire extended QA have no aligned Lean view. No claim is made at m+η⁻¹≤0, in infinite dimension, or about algorithmic convergence, initialization or cost.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GibbsLinearCovarianceUpper.gibbs_linear_covariance_upper",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GibbsLinearCovarianceUpper.gibbs_linear_covariance_upper",
    "id": "sogb-026",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GibbsLinearCovarianceUpper.gibbs_linear_covariance_upper",
    "related_owned_results": [
      {
        "declaration": "AutoSamplingTheory.TechnicalLemmas.Analysis.GibbsGradientMoment.gibbs_covariance_lower",
        "role": "additional-formal-support-not-aligned-QA"
      },
      {
        "declaration": "AutoSamplingTheory.TechnicalLemmas.Measure.GaussianConvolutionRegularity.gaussian_convolution_derivatives",
        "role": "additional-formal-support-not-aligned-QA"
      },
      {
        "declaration": "AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothedHessianUpper.smoothed_hessian_upper",
        "role": "additional-formal-support-not-aligned-QA"
      },
      {
        "declaration": "AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothedHessianBounds.smoothed_hessian_bounds",
        "role": "additional-formal-support-not-aligned-QA"
      }
    ],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/GibbsLinearCovarianceUpper.lean"
  },
  {
    "aligned_fixed_lean_task": false,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "7da6ccd4e2fbac80e8597c8d5421fb083791810652c7a3838214fa305763cf89",
    "boundary": "An original NL exercise over an existing compiled finite-product theorem, with no equivalent fixed Lean task and no new production declaration. The supporting module does prove the displayed log-Sobolev bound with actual normalization and L¹ conclusions, but not the separate counterexample or every explanatory sentence. General entropy tensorization, noncompact/Hilbert extensions and sampler convergence remain outside this QA.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.BernoulliLogSobolev.bernoulli_function_logSobolev",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.BernoulliLogSobolev.bernoulli_function_logSobolev",
    "id": "sogb-027",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.BernoulliLogSobolev.bernoulli_function_logSobolev",
    "related_owned_results": [
      {
        "declaration": "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.TwoPointEntropy.two_point_squared_entropy_le_half_sq_sub",
        "role": "additional-formal-support-not-aligned-QA"
      }
    ],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/BernoulliLogSobolev.lean"
  },
  {
    "aligned_fixed_lean_task": true,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "1be26d5f3c790d72f77813c7277245953ea18975cdadb455721f7a972b11f302",
    "boundary": "The aligned Lean conclusion is exactly scalar Gaussian homogeneous log-Sobolev for signed compact C² functions, under only those public regularity/support premises. Imported producers close the actual finite-law, moment and convergence obligations; the explanatory constants and warning examples are not new production declarations. No full noncompact, arbitrary Sobolev, finite-Hilbert or infinite-dimensional theorem, sampler rate or SPHMC main-result completion is asserted.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianCompactLogSobolev.compact_gaussian_logSobolev",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianCompactLogSobolev.compact_gaussian_logSobolev",
    "id": "sogb-028",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianCompactLogSobolev.compact_gaussian_logSobolev",
    "related_owned_results": [
      {
        "declaration": "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.BernoulliLogSobolev.bernoulli_function_logSobolev",
        "role": "additional-formal-support-not-aligned-QA"
      },
      {
        "declaration": "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianCompactEntropy.compact_count_gaussian_entropy_limits",
        "role": "additional-formal-support-not-aligned-QA"
      },
      {
        "declaration": "AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianFlipEnergy.compact_count_gaussian_flip_energy_limit",
        "role": "additional-formal-support-not-aligned-QA"
      },
      {
        "declaration": "AutoSamplingTheory.TechnicalLemmas.Probability.BalancedRademacherCLT.balanced_count_sum_tendsto_gaussian",
        "role": "additional-formal-support-not-aligned-QA"
      }
    ],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/GaussianCompactLogSobolev.lean"
  },
  {
    "aligned_fixed_lean_task": false,
    "audience": "expert-review-not-model-input",
    "binding_sha256": "9d9f361a44b9b4cbd33ff6379d61f64ca7b45805e410c4021af25e64ba6013c8",
    "boundary": "A real source-numbered exercise with a complete authored mathematical reference answer and two existing compiled owned core-answer declarations. There is no separately aligned fixed Lean QA here. The Lean APIs are broader than the source domain and use the equivalent Rayleigh formulation; their exact differences and unformalized context remain visible. No new production declaration, wrapper, Registry credit, full textbook completion, standalone eigenvalue-list theorem, iterate complexity or source-commentary formalization is claimed.",
    "declaration": "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentOptimalStep.optimal_gradient_step",
    "edge_semantics": "position only; no inferred proof-dependency or necessity",
    "graph_url": "../../../underlying-lean-graph/index.html?focus=decl%3AAutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentOptimalStep.optimal_gradient_step",
    "id": "sogb-029",
    "node_id": "decl:AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentOptimalStep.optimal_gradient_step",
    "related_owned_results": [
      {
        "declaration": "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentOptimalStep.gradient_step_endpoint_bound",
        "role": "additional-formal-support-not-aligned-QA"
      },
      {
        "declaration": "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentOptimalStep.gradient_step_endpoint_bound",
        "role": "owned-textbook-core-answer",
        "supports": [
          "hessian-segment-bound"
        ]
      },
      {
        "declaration": "AutoSamplingTheory.TechnicalLemmas.Analysis.GradientDescentOptimalStep.optimal_gradient_step",
        "role": "owned-textbook-core-answer",
        "supports": [
          "balanced-minimax-step"
        ]
      }
    ],
    "source_file": "AutoSamplingTheory/TechnicalLemmas/Analysis/GradientDescentOptimalStep.lean"
  }
]
