Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Perron's Method for Envelope Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall: Existence and Well-Posedness
theoremthm:nc-envelope-perron-existence-2026aAnalysisProbabilityFor a shift-absorbing, shift-semicontinuous Hamiltonian bounded at zero momentum and a wall-confined free energy with closed score and regular plan subgradients, Perron's method gives a bounded envelope viscosity solution between any ordered bounded sub- and supersolution, so a b…The Bump Construction for Envelope Viscosity Subsolutions
lemmalem:nc-envelope-perron-bump-2026aAnalysisProbabilityFor all sufficiently small shift ranges, a bounded envelope viscosity subsolution lying below a bounded supersolution that fails to be a supersolution itself can be strictly raised at some law of the domain while remaining a bounded subsolution below the supersolution.The Coupling Test Function of a Square-Integrable Plan
definitiondef:nc-coupling-test-function-2026aAnalysisProbabilityFor a square-integrable plan, a gauge function and a weight, the coupling test function at a law is the supremum, over the couplings of the plan with that law, of the momentum pairing minus the gauge of the displacement and the weighted square of the displacement.The Supremum of a Bounded Family of Envelope Viscosity Subsolutions is an Envelope Viscosity Subsolution
lemmalem:nc-envelope-sup-subsolutions-2026aAnalysisProbabilityFor all sufficiently small shift ranges, the pointwise supremum of a nonempty uniformly bounded family of envelope viscosity subsolutions with that shift range is again a bounded envelope viscosity subsolution with the same shift range.Constant Barriers for Envelope Viscosity Solutions under Regular Plan Subgradients
lemmalem:nc-envelope-constant-barriers-2026aAnalysisProbabilityIf the free energy has regular plan subgradients and the Hamiltonian at zero momentum is bounded by K on laws inside the wall, then the constants -K/rho and K/rho are an envelope viscosity subsolution and supersolution, for any shift range.Score Bounds at Envelope Test Inequalities from the Absorption Slack
lemmalem:nc-envelope-score-bounds-2026aAnalysisProbabilityFor a Hamiltonian that absorbs shifts at the noise level and is bounded at zero momentum at bounded positions, a momentum shift that satisfies either envelope test inequality at a bounded position has norm at most a constant over the shift size, with the admissible shift sizes de…The Wall-Confined Free Energy Converges along Wasserstein-Convergent Sequences with Bounded Scores
lemmalem:nc-energy-convergence-bounded-scores-2026aAnalysisProbabilityIf laws in the score domain converge in the Wasserstein distance to a law in the domain and their scores stay bounded in L^2, then their wall-confined free energies converge to the free energy of the limit.The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation
lemmalem:nc-coupling-test-function-2026aAnalysisProbabilityFor a bounded plan at a bounded law, the coupling test function built from a differentiable gauge with vanishing derivative at zero and a quadratic weight is attained by a maximising coupling, is Wasserstein-continuous on bounded laws, and admits an explicit plan subjet read off…The Score Plan of a Law in the Score Domain of the Wall-Confined Free Energy
definitiondef:nc-score-plan-2026aAnalysisProbabilityThe score plan of a law in the score domain is the joint law of the position variables and the score, a square-integrable law of 2d variables.- If two plans realised on a common position tuple are plan superdifferentials (or subdifferentials) of two functions with given slacks, then the plan with the summed momenta is a plan superdifferential (or subdifferential) of the sum, with the summed slack.
Gluing Countably Many Square-Integrable Noncommutative Laws along a Common Marginal
lemmalem:l2-countable-gluing-common-marginal-2026aAnalysisProbabilityCountably many square-integrable noncommutative laws that share the law of their first k variables are realised jointly in one tracial W*-probability space, by one common k-tuple and one further tuple for each law.A Nondecreasing Differentiable Majorant with Vanishing Derivative at Zero for a Little-o Modulus
lemmalem:little-o-differentiable-majorant-2026aAnalysisA bounded nonnegative function on an interval [0,r] that vanishes faster than linearly at zero lies below a nondecreasing, differentiable function on the half-line that vanishes at zero, with bounded derivative tending to zero at zero.Envelope Viscosity Solutions versus Free-Energy-Penalised Viscosity Solutions: Envelopes of Semicontinuous Functions, and Semicontinuous Envelope Solutions are Penalised Solutions under Absorption
lemmalem:nc-envelope-penalised-bridge-2026aAnalysisPDEFor a Wasserstein upper semicontinuous function the upper penalty envelope is the function minus delta times the free energy (and dually); consequently, when the Hamiltonian absorbs shifts, a semicontinuous envelope subsolution or supersolution is also a free-energy-penalised one…Shift-Convergent Sequences of Positions, Momenta and Shifts in a Tracial W*-Probability Space
definitiondef:nc-shift-convergent-sequence-2026aAnalysisProbabilityIn one tracial W*-probability space: positions with laws of norm bound r converging in L^2, momenta converging in L^2, and bounded shifts converging weakly.Comparison, Uniqueness and Continuity on Energy Sublevel Sets for Envelope Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall
theoremthm:nc-envelope-comparison-2026aAnalysisPDEWithout any semicontinuity assumption, a bounded envelope subsolution lies below a bounded envelope supersolution on the domain of the free energy; bounded envelope solutions are unique and automatically Wasserstein-continuous on every sublevel set of the free energy.Comparison at a Fixed Level for Envelope Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall
lemmalem:nc-envelope-fixed-level-comparison-2026aAnalysisPDEAt each fixed level delta, the upper penalty envelope of a bounded envelope subsolution lies below the lower penalty envelope of a bounded envelope supersolution up to an error e(delta) that tends to zero with delta and depends only on the sup-norm bound and the data.Envelope Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall
definitiondef:nc-envelope-viscosity-solution-2026aAnalysisPDEAt each level delta up to a shift range, test the penalty envelope of u by strict extrema against arbitrary test functions with plan jets; at such a point the law must have a score, and the equation must hold with the momentum shifted by delta times the score, keeping the good te…The Discounted HJB Equation with Free Langevin Noise in a Wall, Envelope Form: Standing Notation
settingset:nc-envelope-hjb-2026aAnalysisPDEStanding notation for the envelope theory of the free Langevin HJB equation in a wall: the data of the penalised setting, the Wasserstein metric on the domain, self-adjoint scores and shifted plans, and the penalty envelopes of bounded functions.Shift Semicontinuity of a Hamiltonian on Phase-Space Noncommutative Laws
definitiondef:nc-hamiltonian-shift-semicontinuity-2026aAnalysisPDEAlong strongly convergent bounded positions and momenta and weakly convergent bounded shifts in one tracial W*-probability space, the Hamiltonian with an upward shift plus the good quadratic term is lower semicontinuous, and with a downward shift minus that term is upper semicont…Absorption of Shifts by a Hamiltonian on Phase-Space Noncommutative Laws
definitiondef:nc-hamiltonian-shift-absorption-2026aAnalysisPDEA Hamiltonian absorbs shifts at noise level sigma if, at bounded positions, shifting the momentum by delta Q changes it by at most delta times a function of the momentum norm plus a quarter of sigma squared delta times the squared norm of Q, for all small delta.