Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Sums, Real Multiples and the Pairing of Square-Integrable Tuples in a Tracial W*-Probability Space
definitiondef:l2-tuple-pairing-2026aAnalysisDefines sums, real multiples and the real pairing of square-integrable self-adjoint tuples, as the vector operations and inner product of the direct-sum Hilbert space.Comparison and Uniqueness for the Linear-Quadratic Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws
corollarycor:nc-lq-comparison-2026aAnalysisPDEComparison for bounded semicontinuous plan-jet sub- and supersolutions of the linear-quadratic equation, and uniqueness of bounded continuous plan-jet viscosity solutions.The Linear-Quadratic Hamiltonian: Its Lift, the Structure Condition and Its Quadratic Structure
lemmalem:nc-lq-hamiltonian-conditions-2026aAnalysisPDEWith bounded Lipschitz drift coefficients and a bounded uniformly continuous running cost, the linear-quadratic Hamiltonian lifts to the expected formula, satisfies the structure condition, and is quadratic with a convex Lipschitz remainder.The Linear-Quadratic Hamilton-Jacobi Equation with Law-Dependent Affine Drift on Square-Integrable Noncommutative Laws
equationeq:nc-lq-hamilton-jacobi-2026aAnalysisPDEThe linear-quadratic Hamiltonian, one half the squared momentum minus the pairing with a law-dependent affine drift minus a running cost, and its discounted Hamilton-Jacobi equation on noncommutative laws.Comparison Principle for Plan-Jet Viscosity Solutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws
theoremthm:nc-plan-comparison-2026aAnalysisPDEFor a Hamiltonian satisfying the Crandall-Ishii-Lions structure condition that is either Lipschitz in the momentum with linear growth or quadratic with a convex Lipschitz remainder, every bounded usc plan-jet subsolution lies below every bounded lsc plan-jet supersolution.Plan Jets at an Ekeland Point of the Doubled Difference with Logarithmic Confinement on Square-Integrable Noncommutative Laws
lemmalem:nc-plan-doubling-jets-2026aAnalysisPDEAt an Ekeland point of the doubled difference over joint laws, with a logarithmic confinement, the shared doubling momentum gives plan superjets of the first function and plan subjets of the second; the confinement momenta are small, and the doubling distance is controlled by the…Test Functions and Plan Jets: Touching Transfers Plan Jets, and the Jet Form of Plan-Jet Viscosity Solutions
lemmalem:nc-plan-viscosity-jets-2026aAnalysisPDETouching transfers plan jets from the test function to the function, so plan-jet viscosity solutions can be tested equivalently on the plan jets of the function itself.Plan-Jet Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws
definitiondef:nc-plan-viscosity-solution-2026aAnalysisPDEPlan-jet viscosity sub- and supersolutions: whenever a test function touches from above (below) and has a plan super- (sub-)differential with slack, the equation holds as an inequality up to a momentum perturbation of the size of the slack.The Discounted Stationary Hamilton-Jacobi Equation on Square-Integrable Noncommutative Laws
equationeq:nc-discounted-hamilton-jacobi-2026aAnalysisPDEThe discounted stationary Hamilton-Jacobi equation on square-integrable noncommutative laws, with the Hamiltonian evaluated on the joint law of position and gradient.Hamiltonians on Phase-Space Noncommutative Laws that are Lipschitz in the Momentum with Linear Growth
definitiondef:nc-hamiltonian-momentum-lipschitz-2026aAnalysisPDEA Hamiltonian on phase-space laws is Lipschitz in the momentum with linear growth if changing the momentum by Q changes it by at most a constant times one plus the position norm times the norm of Q.The Structure Condition for a Hamiltonian on Phase-Space Noncommutative Laws
definitiondef:nc-hamiltonian-structure-condition-2026aAnalysisPDEThe structure condition of Crandall-Ishii-Lions on phase-space laws, local in the positions: at the doubling momentum alpha(X-Y), moving the base point from X to Y raises the Hamiltonian by little when alpha|X-Y|^2+|X-Y| is small.Plan Superdifferentials, Plan Subdifferentials and Plan Jets with Slack on Square-Integrable Noncommutative Laws
definitiondef:nc-plan-jets-2026aAnalysisPDEA plan is a plan superdifferential (subdifferential) of a function on laws, with slack, if the first-order upper (lower) expansion along the momentum holds uniformly over every tuple jointly realised with the plan; the plan jets are the sets of these.- A plan at a law is a joint law of position and momentum whose first marginal is that law.
Plan Jets and Hamiltonians on Square-Integrable Noncommutative Laws: Standing Notation
settingset:nc-plan-jets-2026aAnalysisPDEStanding notation for plan jets and Hamiltonians on square-integrable noncommutative laws: tuple sums and pairing, lifts of functions on laws to tuples, and the metrics used.Tracial W*-Probability Spaces and Square-Integrable Tuples: Standing Notation
settingset:tracial-w-star-l2-2026aAnalysisProbabilityStanding notation for tracial W*-probability spaces, their traces, conjugations and self-adjoint vectors, trace-preserving embeddings, self-adjoint and L^2 tuples and their laws, with the basic results on them in force.Every Square-Integrable Noncommutative Law is the Law of a Square-Integrable Tuple; Realisation of Couplings and of Almost Optimal Pairs
theoremthm:l2-law-realisation-2026aAnalysisProbabilityEvery L^2 noncommutative law is the law of an L^2 tuple in some tracial W*-probability space; likewise every L^2 law of 2d variables is the law of a pair of L^2 tuples, and any two L^2 laws are the laws of two tuples in a common space whose squared L^2 distance is within any marg…Gluing a Chain of Noncommutative Couplings into Consistent Joint Laws
lemmalem:nc-chain-gluing-2026aAnalysisProbabilityGiven a sequence of noncommutative laws and couplings between consecutive terms, there is a consistent sequence of joint laws of the first k blocks of variables whose consecutive pairs of blocks have the given couplings as laws.Calculus of Laws of Square-Integrable Tuples: Bounded Tuples, the Lipschitz Bound, Moments, Affine Push-Forwards, Couplings and Embeddings
lemmalem:l2-tuple-law-properties-2026aAnalysisProbabilityThe law of an L^2 tuple extends the law of bounded self-adjoint tuples, is 1-Lipschitz for the L^2 norm, has its moments given by inner products, commutes with affine push-forwards, turns pairs into L^2 couplings whose cost is the squared L^2 distance, and is invariant under trac…Square-Integrable Tuples in a Tracial W*-Probability Space: Their Norm, Affine Images, Pairs, Embedded Images and Laws
definitiondef:l2-tuple-law-2026aAnalysisProbabilityAn L^2 tuple of a tracial W*-probability space is a tuple of vectors fixed by its conjugation; the definition fixes their L^2 norm, affine images, pairs and images under embeddings, and defines the law of an L^2 tuple as the L^2 law obtained as the limit of the laws of approximat…Laws of Self-Adjoint Tuples in a Tracial W*-Probability Space: Moments, Affine Images, Couplings, Embeddings and Approximation
lemmalem:bounded-tuple-laws-tracial-2026aAnalysisProbabilityThe law of a self-adjoint tuple in a tracial W*-probability space is a noncommutative law; its moments are inner products of vacuum vectors, affine images and embeddings behave as expected, the joint law of two tuples is a coupling whose cost is the squared L^2 distance, and laws…