Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
- 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.Resolvents in a Tracial W*-Probability Space: Product Bounds, Membership, the Lipschitz Bound, and Extension to Self-Adjoint Vectors
lemmalem:resolvent-tracial-l2-2026aAnalysisIn a tracial W*-probability space: product bounds on vacuum vectors, resolvents stay in the algebra, resolvents are L^2-Lipschitz, and the resolvent at 1 extends uniquely to self-adjoint L^2 vectors.Calculus of Resolvents of Bounded Self-Adjoint Operators: Adjoints, Commutation, the Resolvent Identity, Recovery, Stepping, and Uniform Polynomial Approximation
lemmalem:resolvent-calculus-complex-2026aAnalysisAdjoints and commutation of resolvents, the resolvent identity, the recovery identity A - y^2 Re R_y(A) = A^3 R_y R_{-y}, stepping between resolvent parameters, and uniform polynomial approximation of resolvents.Resolvents of Bounded Self-Adjoint Operators and the Resolvent Transform of a Self-Adjoint Tuple
definitiondef:resolvent-transform-complex-2026aAnalysisDefines the resolvent (A - iyI)^{-1} of a bounded self-adjoint operator and the resolvent transform of a self-adjoint tuple: the real and imaginary parts of the resolvents at y = 1.The Neumann Series, Inverses in Double Commutants, and Invertibility of A - iyI for Self-Adjoint A
lemmalem:neumann-series-resolvent-complex-2026aAnalysisNeumann series with a rate, a powers estimate, inverses of invertible elements of a double commutant, and invertibility of A - iyI for bounded self-adjoint A with inverse norm at most 1/|y|.Scaling a Plan-Jet Viscosity Subsolution of an Equation with a Quadratic Hamiltonian Gives a Strict Subsolution
lemmalem:nc-quadratic-hamiltonian-scaling-2026aAnalysisPDEIf u is a plan-jet subsolution for a quadratic Hamiltonian with convex Lipschitz remainder, then theta u with theta<1 satisfies the subsolution inequality with a gain proportional to the squared momentum, up to a small constant.Quadratic Hamiltonians with a Convex Remainder Lipschitz in the Momentum on Phase-Space Noncommutative Laws
definitiondef:nc-quadratic-hamiltonian-2026aAnalysisPDEA Hamiltonian is quadratic with a convex Lipschitz remainder if it is one half the squared momentum plus a remainder that is convex and Lipschitz with linear growth in the momentum and bounded above at zero momentum.Lifts of Functions on Square-Integrable Noncommutative Laws to Square-Integrable Tuples
definitiondef:nc-law-function-lift-2026aAnalysisPDEThe lift of a function on square-integrable noncommutative laws to a tracial W*-probability space: evaluate the function at the law of the tuple.