Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
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.The Shift of a Bounded Plan by a Self-Adjoint Field
definitiondef:nc-plan-shift-2026aAnalysisProbabilityShifting a bounded plan (a joint law of positions and momenta) by t times a self-adjoint field on the position law adds t times the field to the momenta in the canonical realisation of the plan; the result is a square-integrable law of 2d variables.Conjugate Variables, Wall Forces and Scores are Square-Integrable Tuples of the GNS Space
lemmalem:nc-score-self-adjoint-2026aAnalysisProbabilityThe conjugate variables of a noncommutative law, its square-integrable wall force and hence the score of the wall-confined free energy consist of vectors fixed by the conjugation, so they are square-integrable tuples of the law's tracial W*-probability space.The Commutator of a Square-Integrable Vector with a Bounded Self-Adjoint Operator: Linearity, the Norm Bound, Vacuum Vectors and Dependence on the Law Only
lemmalem:nc-commutator-norm-law-2026aAnalysisProbabilityThe commutator of a vector of the standard form with a bounded self-adjoint operator is linear in the vector and bounded by twice the operator norm, reduces to the operator commutator on vacuum vectors, and the sum of squared commutator norms of a momentum tuple with a bounded po…Square-Integrable Tuples with a Bounded Law are Vacuum Tuples of Bounded Self-Adjoint Operators
lemmalem:l2-tuple-bounded-law-2026aAnalysisProbabilityIf the law of a square-integrable tuple in a tracial W*-probability space is a noncommutative law with norm bound R, the tuple is the vacuum tuple of a unique self-adjoint operator tuple of the algebra, whose entries have operator norm at most R and whose law is that law.Tangent Inequalities for the Wall Energy along Couplings and for the Wall-Confined Free Energy along Optimal Couplings
lemmalem:nc-wall-confined-free-energy-tangent-2026aAnalysisProbabilityThe wall energy lies above its tangent at a law with square-integrable wall force along every coupling, and the wall-confined free energy lies above its tangent, given by the score, along every optimal coupling.Plan Jets of a Series of Squared Wasserstein Distances at Optimal Couplings of Bounded Noncommutative Laws
lemmalem:nc-squared-distance-series-plan-jets-2026aAnalysisProbabilityFor a summable series of squared Wasserstein distances to bounded laws, realised along optimal couplings in one tracial W*-probability space, the law of the position with the summed displacement momentum is a plan superjet, its negative momentum gives a plan subjet of the negativ…Gluing Countably Many Noncommutative Couplings with a Common First Marginal in One Tracial W*-Probability Space
lemmalem:nc-star-gluing-2026aAnalysisProbabilityCountably many couplings of a bounded law with further bounded laws are realised simultaneously by bounded self-adjoint tuples in a single tracial W*-probability space.Sublevel Sets of the Wall-Confined Free Energy are Closed for the Wasserstein Distance
lemmalem:nc-wall-confined-free-energy-w2-closed-2026aAnalysisProbabilityAlong a Wasserstein-convergent sequence of bounded laws with bounded wall-confined free energy, the limit has finite free energy with the same bound; hence the free energy is lower semicontinuous on its domain for the Wasserstein distance.Conditional Comparison and Uniqueness for the Linear-Quadratic HJB Equation with Free Langevin Noise in a Wall, Penalised by Voiculescu's Free Entropy
corollarycor:nc-chi-free-langevin-lq-uniqueness-2026aAnalysisProbabilityIf minus the free entropy satisfies the conjectured tangent inequality along optimal couplings, then it is a free entropy penalty, and the linear-quadratic HJB equation with free Langevin noise in a wall, penalised by it, satisfies comparison and has at most one bounded weak-star…Displacement Convexity of Minus the Free Entropy: the Tangent Inequality along Optimal Couplings
conjectureconj:nc-free-entropy-displacement-convex-2026aAnalysisProbabilityConjecture: minus the free entropy satisfies the tangent inequality along optimal couplings, with minus the conjugate variables as its gradient.Minus the Free Entropy Is Bounded Below on Bounded Laws and Has Weak-Star Closed Sublevel Sets
corollarycor:nc-free-entropy-penalty-clauses-2026aAnalysisProbabilityMinus the free entropy is bounded below on laws with a norm bound and has weak-star closed sublevel sets there, so it satisfies the lower-bound and closed-sublevel clauses of a free entropy penalty.The Free Entropy: the Semicircular Upper Bound, Semicircular Values, Smoothing, the de Bruijn Identity and Upper Semicontinuity
lemmalem:nc-free-entropy-basic-2026aAnalysisProbabilityThe free entropy is at most that of the semicircular law with the same second moment, equals (d/2)log(2 pi e s) on the semicircular law of variance s, is finite after any positive time of free heat flow, increases along the flow by half the integrated free Fisher information, and…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.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…