Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Heat Regularisation of the Entropy: a Laplacian Test Function whose Laplacian is Minus the Fisher Information
lemmalem:torus-heat-regularised-entropy-2026aAnalysisProbabilityThe entropy of the heat-smoothed measure is a Laplacian test function of the measure. Its gradient is the heat average of the score of the smoothed measure, and its Laplacian is minus the Fisher information of the smoothed measure.Heat Regularisation of an Intrinsic Test Function: a Laplacian Test Function whose Laplacian is Minus the Pairing of the Gradient with the Score
lemmalem:torus-heat-regularised-test-function-2026aAnalysisProbabilityComposing an intrinsic test function on the absolutely continuous measures with the heat semigroup gives a Laplacian test function. Its gradient is the heat average of the gradient at the smoothed measure, and its Laplacian is minus the pairing of that gradient with the score. As…Heat Smoothing, Entropy, Fisher Information and Laplacian Test Functions on the Torus Wasserstein Space: Standing Notation
settingset:torus-heat-calculus-2026aAnalysisProbabilityStanding notation for the heat semigroup and heat kernel of the torus, heat averages of vector fields, the entropy and Fisher information, and intrinsic and Laplacian test functions with their gradients, gradient fields and Laplacians.The Laplacian of a Laplacian Test Function on the Torus Wasserstein Space
definitiondef:torus-laplacian-2026aAnalysisProbabilityThe Laplacian of a Laplacian test function at a measure is the integral against that measure of the divergence of its gradient field.Laplacian Test Functions on the Torus: Uniqueness of the Gradient Field, a Criterion, and Linearity of the Test Classes
lemmalem:torus-laplacian-test-function-basic-2026aAnalysisProbabilityA function has at most one gradient field. A continuous function with a tangent-valued gradient field is a Laplacian test function. Intrinsic and Laplacian test functions are closed under linear combinations, with linear gradients and gradient fields.Gradient Fields and Laplacian Test Functions on the Torus Wasserstein Space
definitiondef:torus-laplacian-test-function-2026aAnalysisProbabilityA gradient field of a function on the torus Wasserstein space is a periodic, continuously differentiable representative of its gradient along couplings that depends continuously on the measure together with its derivatives. A Laplacian test function is an intrinsic test function…Lower Semicontinuity of the Torus Fisher Information and Closedness of the Score Along Couplings
lemmalem:torus-fisher-closed-2026aAnalysisProbabilityIf measures of bounded torus Fisher information converge, the limit has Fisher information at most the bound, and pairings of fields converging along couplings with the scores converge to the pairing with the score of the limit.Convergence Along Couplings of Measures Carrying Square-Integrable Vector Fields on the Torus
definitiondef:torus-convergence-along-couplings-2026aAnalysisProbabilityA sequence of measures carrying vector fields converges along couplings to a limit measure and field if some couplings have vanishing torus cost and vanishing mean squared field mismatch.Heat Averages of Vector Fields on the Torus: Regularity, Contraction, Tangency, the Divergence Identity and Synchronous Couplings
lemmalem:torus-heat-average-2026aAnalysisProbabilityAveraging a square-integrable field of a heat-smoothed measure against the torus heat kernel gives a continuously differentiable periodic field. The average does not increase the norm and maps tangent fields to tangent fields; its divergence integrates to minus the pairing with t…The Torus Heat Kernel: Regularity and Bounds, the Density of a Heat-Smoothed Measure, and Its Lipschitz Dependence on the Measure
lemmalem:torus-heat-kernel-2026aAnalysisProbabilityThe torus heat kernel is twice continuously differentiable, even, positive and bounded with bounded derivatives. Convolving it with a measure gives the smooth positive density of the heat-smoothed measure, whose score is the logarithmic gradient of that density, and this density…- The torus heat kernel at time s is the continuous periodic density of the heat semigroup started at the Dirac mass at the origin.
Existence and Uniqueness of a Continuous Periodic Density for the Torus Heat Semigroup Started at the Origin
lemmalem:torus-heat-kernel-unique-2026aAnalysisProbabilityThe heat semigroup on the torus, applied to the Dirac mass at the origin, has exactly one continuous periodic density on the unit cell.Nearly Optimal Composite Displacements on the Torus Converge to the Optimal Displacement
lemmalem:torus-composite-displacement-2026aAnalysisProbabilityIf random points with laws mu and nu are joined by displacements that are lifts of the torus displacement and whose mean squares approach the squared torus Wasserstein distance, then, when mu is absolutely continuous, these displacements converge in mean square to the optimal dis…McCann's Tangent Inequality on the Torus, and Monotonicity of the Score Along Optimal Maps
theoremthm:torus-entropy-tangent-inequality-2026aAnalysisProbabilityOn the torus the entropy lies above its tangent along optimal maps: the entropy of the target is at least that of the source plus the pairing of the source score with the optimal displacement; adding the inequalities in both directions shows the score is monotone along optimal ma…The Entropy of Measures on the Torus: Nonnegativity, Absolute Continuity and Closed Sublevel Sets
lemmalem:torus-entropy-basic-2026aAnalysisProbabilityA probability measure on the torus with finite entropy is absolutely continuous and has nonnegative entropy, and the sets of measures on the torus with entropy at most a given constant are closed for the torus Wasserstein distance.Heat Smoothing on the Torus: Distance to the Identity, Contraction, a Smooth Positive Periodic Density, Finite Entropy, and Finite Fisher Information
lemmalem:torus-heat-semigroup-2026aAnalysisProbabilityHeat smoothing on the torus moves a measure by at most the square root of d times s, contracts the torus Wasserstein distance, and produces a positive periodic density of class C2, finite entropy not exceeding that of the original, and finite torus Fisher information with score t…The Heat Semigroup on the Probability Measures on the Torus
definitiondef:torus-heat-semigroup-2026aAnalysisProbabilityThe heat semigroup at time s sends a probability measure on the torus to the wrap of its Gaussian smoothing at scale s.Half the Squared Torus Wasserstein Distance to a Fixed Measure is an Intrinsic Test Function on the Absolutely Continuous Measures
lemmalem:torus-squared-distance-test-function-2026aAnalysisProbabilityThe optimal displacement fields towards a fixed target vary continuously along couplings of vanishing torus cost between absolutely continuous sources, so half the squared torus Wasserstein distance to a fixed measure is an intrinsic test function on the absolutely continuous mea…Half the Squared Torus Wasserstein Distance to a Fixed Measure: the One-Sided Bound Along Couplings, Differentiability at an Absolutely Continuous Source, and the Tangent Gradient
lemmalem:torus-squared-distance-2026aAnalysisProbabilityHalf the squared torus Wasserstein distance to a fixed measure lies below its first-order expansion plus half the cost along every coupling out of a mapped source; at an absolutely continuous source it is differentiable along couplings with gradient minus the optimal displacement…Intrinsic Test Functions on the Torus Wasserstein Space
definitiondef:torus-intrinsic-test-function-2026aAnalysisProbabilityAn intrinsic test function on a set of probability measures on the torus is a continuous function that is differentiable along couplings at every measure of the set with gradient in the tangent space, the gradients varying continuously along couplings of vanishing torus cost.