Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Free Additive Convolution of Two Noncommutative Laws
definitiondef:nc-free-convolution-2026aAnalysisProbabilityThe free additive convolution of two laws of d variables is the law of the sum of free tuples with those laws.Free Semicircular Variables are Characterised by the Schwinger-Dyson Equation Relative to the Other Variables; Semicircular Systems and Rotations
theoremthm:nc-free-semicircular-characterisation-2026aAnalysisProbabilityA joint law is the free product of its first marginal with the semicircular law exactly when it satisfies the Schwinger-Dyson equation in its last variables; consequently free semicircular systems combine into one and are invariant under orthonormal rotations.The Free Product of Two Noncommutative Laws: Existence, Uniqueness, Norm Bound, Substitutions and Weak-Star Continuity
theoremthm:nc-free-product-law-2026aAnalysisProbabilityTwo noncommutative laws have exactly one free joint law with them as marginals, their free product; it keeps a common norm bound, is compatible with substitutions in each group and with swapping the groups, and depends weak-star continuously on the factors.Freeness of Two Groups of Variables under a Noncommutative Law
definitiondef:nc-free-law-2026aAnalysisProbabilityA law of m+n variables makes the first m and the last n variables free when every alternating product of centred polynomials in the two groups has expectation zero.The Chain Rule for Free Difference Quotients under an Affine Substitution
lemmalem:nc-difference-quotient-affine-2026aAnalysisProbabilityFree difference quotients transform under an affine substitution through the transpose of its linear part, as in the classical chain rule.- The semicircular law of n variables is the unique law satisfying the Schwinger-Dyson equation, and its variance-t version is its image under scaling by the square root of t.
The Vacuum Law of the Semicircular Operators is the Unique Noncommutative Law Satisfying the Schwinger-Dyson Equation
theoremthm:nc-semicircular-schwinger-dyson-2026aAnalysisProbabilityThe vacuum state of the semicircular operators on the free Fock space is a noncommutative law with norm bound 2 that satisfies the Schwinger-Dyson equation, and it is the only law that does; equivalently, its conjugate variables are the variables themselves.The Schwinger-Dyson Equation for a Noncommutative Law
equationeq:nc-schwinger-dyson-2026aAnalysisProbabilityThe Schwinger-Dyson equation asks that multiplying by a variable inside a law equals the free difference quotient in that variable; a relative version imposes it only in the last variables of a joint law.Calculus of the Free Fock Space: Orthonormal Monomials, Adjoints, Commutation Relations and the Vacuum
lemmalem:nc-free-fock-space-2026aAnalysisProbabilityMonomials are orthonormal in the free Fock space, annihilation operators are the adjoints of creation operators, left and right operators commute up to the vacuum projection, and the left and right semicircular operators are bounded, self-adjoint and commute; formulas for the vac…The Free Fock Space: Creation, Annihilation and Semicircular Operators
definitiondef:nc-free-fock-space-2026aAnalysisProbabilityThe free Fock space is the completion of the noncommutative polynomials in which the monomials are orthonormal; creation operators multiply by a variable on the left or right, and annihilation operators strip a leading or trailing letter.Comparison and Uniqueness for the Linear-Quadratic HJB Equation with Free Langevin Noise in a Wall on Noncommutative Laws
corollarycor:nc-free-langevin-lq-comparison-2026aAnalysisProbabilityFor bounded Lipschitz drift coefficients and a bounded uniformly continuous running cost, penalised sub- and supersolutions of the linear-quadratic equation compare on the domain of the wall-confined free energy, and solutions are unique there.The Linear-Quadratic Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws
equationeq:nc-free-langevin-lq-hjb-2026aAnalysisProbabilityThe discounted HJB equation with free Langevin noise in a wall, with the linear-quadratic Hamiltonian with law-dependent affine drift.Comparison and Uniqueness for Free-Energy-Penalised Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall
theoremthm:nc-penalised-comparison-2026aAnalysisProbabilityUnder the structure condition on the Hamiltonian, a bounded weak-star upper semicontinuous penalised subsolution lies below a bounded weak-star lower semicontinuous penalised supersolution on the domain of the wall-confined free energy; solutions are unique there.Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy on Bounded Noncommutative Laws
lemmalem:nc-penalised-doubling-maximiser-2026aAnalysisProbabilityFor bounded semicontinuous u and v the doubled difference penalised by the squared distance and the wall-confined free energy attains its maximum, the maximum decreases in the doubling strength with the usual gain, and it tends to the unpenalised supremum as the penalty weight va…Free-Energy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall
definitiondef:nc-penalised-viscosity-solution-2026aAnalysisProbabilityA subsolution: at every strict maximum of u minus a test function minus a small multiple of the wall-confined free energy, the law has a score and the equation holds with inequality for every bounded plan superjet, up to an error proportional to the penalty weight; supersolutions…The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws
equationeq:nc-free-langevin-hjb-2026aAnalysisProbabilityThe discounted HJB equation with free Langevin noise confined by a wall, in score form: the discount term plus the Hamiltonian at the plan plus half the squared noise intensity times the pairing of the score with the momentum.The Discounted Hamilton-Jacobi-Bellman Equation with Free Langevin Noise in a Wall on Noncommutative Laws: Standing Notation
settingset:nc-free-langevin-hjb-2026aAnalysisProbabilityStanding notation for the discounted HJB equation with free Langevin noise in a wall: plan jets on square-integrable laws, the pairings, the discount rate, noise intensity and wall radius, the wall-confined free energy of a free entropy penalty, and the Hamiltonian.Plan Jets of Squared Wasserstein Distances at an Optimal Coupling of Bounded Noncommutative Laws
lemmalem:nc-squared-distance-plan-jets-2026aAnalysisProbabilityAt an optimal coupling of two bounded laws, the law of the position with the doubling momentum is a plan superjet of the squared distance to the second law and, with shared momentum, a plan subjet of minus the squared distance from the first; the score pairings reduce to pairings…Weak-Star Semicontinuity on Bounded Laws of Functions on Square-Integrable Noncommutative Laws
definitiondef:nc-weak-star-semicontinuous-2026aAnalysisProbabilityA function on square-integrable noncommutative laws is weak-star upper (lower) semicontinuous on bounded laws if its values at the images of bounded laws are upper (lower) semicontinuous along weak-star convergent sequences with a common norm bound.The Wall-Confined Free Energy: Norm Bound, Lower Bound, Weak-Star Compact Sublevel Sets and Displacement Monotonicity of the Score
lemmalem:nc-wall-confined-free-energy-2026aAnalysisProbabilityThe wall-confined free energy lives on laws with norm bound R, is bounded below, has weak-star compact sublevel sets on which it is lower semicontinuous, and has a score that is displacement monotone along optimal couplings.