Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
A Periodic Potential for a Torus-Cyclically Monotone Set: Lipschitz Periodic Part, Convex Lift and Subgradients
theoremthm:torus-periodic-potential-2026aAnalysisA nonempty torus-cyclically monotone set admits a Lipschitz periodic potential whose difference from half the squared norm is convex, satisfying a one-sided comparison along the set; consequently every nearest lift of the second point of a pair in the set is a subgradient of that…The Support of an Optimal Coupling on the Torus is Torus-Cyclically Monotone
lemmalem:torus-optimal-coupling-cyclically-monotone-2026aAnalysisProbabilityThe support of an optimal coupling of two probability measures on the flat torus is torus-cyclically monotone.Torus-Cyclically Monotone Subset of a Doubled Euclidean Space
definitiondef:torus-cyclically-monotone-2026aAnalysisA set of pairs of points is torus-cyclically monotone if cyclically permuting the second points of finitely many of its pairs never lowers the total squared flat torus distance.The Torus Wasserstein Space is a Sequentially Compact Metric Space in which Optimal Couplings Exist
theoremthm:torus-wasserstein-compact-2026aAnalysisProbabilityAny two probability measures on the flat torus have an optimal coupling; the torus Wasserstein distance is a metric on the probability measures on the torus; and every sequence of such measures has a subsequence converging in that metric.The Torus Wasserstein Distance: Comparison with the Euclidean Distance, Wrapping, and Integrals of Periodic Functions
lemmalem:torus-wasserstein-basic-2026aAnalysisProbabilityMeasures on the flat torus have finite second moment; the torus cost of a coupling is at most its Euclidean quadratic cost; wrapping two measures on Euclidean space does not increase their Wasserstein distance; and integrals of continuous periodic functions depend continuously, a…Probability Measures on the Flat Torus, the Torus Cost of a Coupling, the Torus Wasserstein Distance and Optimal Couplings
definitiondef:torus-wasserstein-distance-2026aAnalysisProbabilityProbability measures on the flat torus are Borel probability measures on Euclidean space carried by the half-open unit cell; the torus cost of a coupling integrates the squared flat torus distance, the torus Wasserstein distance is the square root of the least cost, and a couplin…The Flat Torus Distance: Minimality of the Wrapped Displacement, Periodicity, the Metric on the Unit Cell and the Lipschitz Bound
lemmalem:flat-torus-distance-2026aAnalysisThe wrapped displacement is the shortest integer translate of a point, uniquely so and continuously away from half-integer coordinates; the flat torus distance is lattice-periodic, symmetric, vanishes exactly on lattice differences, satisfies the triangle inequality, is a metric…- Defines the wrapped displacement of a point of Euclidean space, obtained by subtracting the nearest integer from each coordinate, and the flat torus distance between two points as the Euclidean norm of the wrapped displacement between them.
Existence and Uniqueness of a Bounded Continuous Plan-Jet Viscosity Solution of the Linear-Quadratic Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws
corollarycor:nc-lq-existence-2026aAnalysisPDEThe linear-quadratic Hamilton-Jacobi equation on square-integrable noncommutative laws has exactly one bounded continuous plan-jet viscosity solution.The Linear-Quadratic Hamiltonian is Uniformly Continuous on Bounded Sets and Bounded at Zero Momentum
lemmalem:nc-lq-hamiltonian-perron-conditions-2026aAnalysisUnder the standing linear-quadratic hypotheses, the linear-quadratic Hamiltonian is uniformly continuous on bounded sets and bounded at zero momentum by a bound on the running cost.Perron's Method for Plan-Jet Viscosity Solutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws: Existence and Well-Posedness
theoremthm:nc-plan-perron-existence-2026aAnalysisPDEPerron's method: between a bounded usc subsolution and a bounded lsc supersolution there is a bounded continuous plan-jet viscosity solution; if H(X,0) is bounded, the equation has exactly one bounded continuous solution.The Bump Construction for Plan-Jet Viscosity Subsolutions on Square-Integrable Noncommutative Laws
lemmalem:nc-plan-perron-bump-2026aAnalysisIf the lower envelope of a bounded upper semicontinuous subsolution below a supersolution fails to be a supersolution, the subsolution can be raised strictly somewhere while staying an upper semicontinuous subsolution below the supersolution.The Upper Semicontinuous Envelope of a Supremum of Plan-Jet Viscosity Subsolutions is a Plan-Jet Viscosity Subsolution
lemmalem:nc-plan-sup-subsolutions-2026aAnalysisFor a Hamiltonian uniformly continuous on bounded sets, the upper semicontinuous envelope of the supremum of a bounded family of upper semicontinuous plan-jet subsolutions is a subsolution.Constant Plan-Jet Viscosity Subsolutions and Supersolutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws
lemmalem:nc-constant-sub-supersolutions-2026aAnalysisA constant c is a plan-jet viscosity subsolution (supersolution) when rho c + H(X, 0) is nonpositive (nonnegative) for every L^2 tuple X.Hamiltonians on Phase-Space Noncommutative Laws that are Uniformly Continuous on Bounded Sets
definitiondef:nc-hamiltonian-uniformly-continuous-2026aAnalysisDefines when a Hamiltonian on phase-space laws is uniformly continuous on bounded sets, uniformly over realisations of positions and momenta in a common space.Plan Jets at a Common Realisation: Superdifferential and Subdifferential Momenta are Close, and the Plan Jets of a Constant
lemmalem:nc-plan-jets-graph-constants-2026aAnalysisAt a common realisation, the momenta of a plan superjet and a plan subjet differ by at most the sum of the slacks; the plan jets of a constant are exactly the plans whose momentum has L^2 norm at most the slack.Gluing Two Square-Integrable Noncommutative Laws along a Common Marginal
theoremthm:l2-gluing-common-marginal-2026aAnalysisTwo square-integrable noncommutative laws with a common marginal on their first k variables are realised jointly in one tracial W*-probability space, sharing that marginal.The Law of the Resolvent Transform is a Lipschitz Function of the Law, and Reconstruction of a Square-Integrable Tuple from a Bounded Tuple with the Law of Its Transform
lemmalem:l2-resolvent-transform-law-2026aAnalysisThe law of the resolvent transform depends Lipschitz-continuously on the law of the L^2 tuple, and any bounded tuple with the law of the transform reconstructs an L^2 tuple with the original law.The Resolvent Transform of Square-Integrable Tuples: Consistency, Concatenation, the Lipschitz Bound and Universal Polynomial Recovery
lemmalem:l2-resolvent-transform-basic-2026aAnalysisThe resolvent transform of L^2 tuples is consistent with the bounded one, respects concatenation, is 1-Lipschitz in L^2, and a fixed sequence of polynomials recovers every self-adjoint L^2 vector from its transform.The Resolvent of a Self-Adjoint Vector and the Resolvent Transform of a Square-Integrable Tuple
definitiondef:l2-resolvent-transform-2026aAnalysisDefines the resolvent of a self-adjoint L^2 vector and the resolvent transform of an L^2 tuple, a bounded self-adjoint tuple of twice the length.