Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Free Products, Semicircular Laws, the Free Heat Flow, Free Fisher Information and Free Entropy: Standing Notation
settingset:nc-free-entropy-2026aAnalysisProbabilityStanding notation for free products, semicircular laws, the free heat flow, free Fisher information and free entropy on noncommutative laws.Voiculescu's Non-Microstates Free Entropy of a Noncommutative Law
definitiondef:nc-free-entropy-2026aAnalysisProbabilityVoiculescu's non-microstates free entropy integrates d/(1+t) minus the free Fisher information along the free heat flow; it is defined on the laws whose Fisher information along the flow is integrable near time zero, where it is a real number.The Free Fisher Information along the Free Heat Flow: Bounds, Lower Semicontinuity in Time and Integrability
lemmalem:nc-free-fisher-along-heat-flow-2026aAnalysisProbabilityAlong the free heat flow the free Fisher information lies between d^2/(M+td) and d/t and is lower semicontinuous in time, and the free entropy integrand is integrable exactly when the Fisher information is integrable near time zero.Logarithmic Integrals of Reciprocal Linear Functions over Bounded and Unbounded Intervals
lemmalem:log-integral-half-line-2026aAnalysisThe reciprocal of a linear function integrates to a difference of logarithms over bounded intervals, and the difference of two such reciprocals is integrable over a half-line with a logarithm of a ratio as its integral.Free Fisher Information: the Cramer-Rao Inequality, the Semicircular Laws, and Closedness under Weak-Star Limits
lemmalem:nc-free-fisher-information-2026aAnalysisProbabilityThe free Fisher information satisfies the Cramer-Rao inequality with the second moment, equals d/s on the semicircular law of variance s, and has weak-star closed sublevel sets.Conjugate Variables along the Free Heat Flow: the Projected Semicircular Increment and the Bound d/t on the Free Fisher Information
theoremthm:nc-free-heat-flow-conjugate-variables-2026aAnalysisProbabilityAt every time t>0 of the free heat flow the law has conjugate variables, obtained by projecting the semicircular increment divided by sqrt(t), so its free Fisher information is at most d/t.The Free Fisher Information of a Noncommutative Law
definitiondef:nc-free-fisher-information-2026aAnalysisProbabilityThe free Fisher information of a law with conjugate variables is the sum of the squared norms of its conjugate variables.The Free Heat Flow: Realisation, Norm Bound, Moments, Distance to the Initial Law, Wasserstein Contraction, Semigroup Property and Continuity
lemmalem:nc-free-heat-flow-2026aAnalysisProbabilityThe free heat flow adds a free semicircular tuple of variance t; it enlarges norm bounds by at most 2 sqrt(t), adds t to the diagonal second moments, moves a law by at most sqrt(td) in Wasserstein distance, contracts Wasserstein distance, is a semigroup, and is continuous in time…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.