Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Admissible Hamiltonians on Unitary Laws
definitiondef:nc-unitary-admissible-hamiltonian-2026bAnalysisPDEA Hamiltonian on unitary laws times the gauge space is admissible if it is jointly continuous and depends on the momentum only through its pairing with two-sided tangent vectors.Perron's Method on Unitary Laws: the Bump Construction
lemmalem:nc-unitary-perron-bump-2026aProbabilityPDEFor any admissible Hamiltonian: if the lower envelope of a subsolution fails the supersolution test at a law, a local bump gives a strictly larger subsolution below a prescribed lower semicontinuous bound.Perron's Method on Unitary Laws: the Upper Envelope of a Bounded Family of Subsolutions Is a Subsolution
lemmalem:nc-unitary-perron-sup-2026aProbabilityPDEFor any admissible Hamiltonian, the upper semicontinuous envelope of the supremum of a family of subsolutions with a common upper bound is a subsolution.Viscosity Sub- and Supersolutions of the Hamilton-Jacobi-Bellman Equation on Unitary Laws
definitiondef:nc-unitary-viscosity-solution-2026aProbabilityPDEViscosity sub- and supersolutions are bounded semicontinuous functions on unitary laws satisfying the equation's inequalities at relative extrema against C^1 test functions on the gauge space.The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise
equationeq:nc-unitary-hjb-2026aProbabilityPDEFor a Hamiltonian H on unitary laws times the gauge space, the discounted first-order equation rho V + H(lambda,p) - (beta^2/2)<p, Theta lambda> = f(lambda), with the free unitary heat generator as noise direction.- Standing data for equations on unitary laws with free unitary noise: the gauge space and gauge distance, C^1 test functions, the free unitary heat generator as noise direction, discount and noise constants, and a continuous source.
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…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.The Penalty Envelopes of a Bounded Function on the Domain of the Wall-Confined Free Energy
definitiondef:nc-penalty-envelopes-2026aAnalysisPDEFor a bounded function u and a level delta > 0, the penalty envelopes are the Wasserstein upper semicontinuous envelope of u minus delta times the free energy and the lower semicontinuous envelope of u plus delta times the free energy, on the domain of the free energy.Comparison and Uniqueness under Wasserstein Semicontinuity for the Controlled von Neumann HJB Equation with Free Langevin Noise in a Wall, Including the Free Entropy Penalty
corollarycor:nc-von-neumann-comparison-w2-2026aAnalysisPDEFor the controlled von Neumann HJB equation with free Langevin noise in a wall and a uniformly continuous running cost, a bounded Wasserstein-upper-semicontinuous penalised subsolution lies below a bounded Wasserstein-lower-semicontinuous penalised supersolution on the wall domai…The Controlled von Neumann Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws
equationeq:nc-von-neumann-hjb-2026aAnalysisPDEThe discounted HJB equation with free Langevin noise in a wall whose Hamiltonian is the controlled von Neumann Hamiltonian, written in score form.The Controlled von Neumann Hamiltonian at Bounded Positions: Evaluation, the Structure Condition and Uniform Continuity in the Momentum
lemmalem:nc-von-neumann-hamiltonian-conditions-2026aAnalysisPDEAt a square-integrable position tuple with a bounded law, the von Neumann Hamiltonian is computed from the bounded operator tuple of the position; for a uniformly continuous running cost it satisfies the structure condition at bounded positions exactly up to the running cost, and…The Controlled von Neumann Hamiltonian on Phase-Space Noncommutative Laws
definitiondef:nc-von-neumann-hamiltonian-2026aAnalysisPDEThe Hamiltonian of the controlled von Neumann dynamics with quadratic control cost: half the sum of the squared L^2 norms of the commutators of momenta with bounded positions, minus a running cost of the position law; off bounded positions only the running cost term remains.Comparison and Uniqueness under Wasserstein Semicontinuity for the Linear-Quadratic HJB Equation with Free Langevin Noise in a Wall, Including the Free Entropy Penalty
corollarycor:nc-free-langevin-lq-comparison-w2-2026aAnalysisPDEFor the linear-quadratic equation, bounded penalised sub- and supersolutions that are Wasserstein semi-continuous on the domain of the wall-confined free energy are ordered there and continuous solutions are unique; under the conjectured tangent inequality this holds with Voicule…Comparison and Uniqueness under Wasserstein Semicontinuity for Free-Energy-Penalised Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall
theoremthm:nc-penalised-comparison-w2-2026aAnalysisPDEIf the Hamiltonian satisfies the structure condition at bounded positions and is uniformly continuous in the momentum there, a bounded penalised subsolution that is Wasserstein upper semicontinuous on the domain of the wall-confined free energy lies below a bounded Wasserstein lo…