Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
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.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.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…Perturbed Maximisers of the Doubled Difference Penalised by the Wall-Confined Free Energy
lemmalem:nc-penalised-doubling-perturbed-maximiser-2026bAnalysisPDEFor bounded u and v the penalised doubled difference has the usual bounds and monotonicity; when u minus delta times the free energy is Wasserstein upper semicontinuous and v plus delta times the free energy lower semicontinuous on the domain, after a small series of squared-dist…The Structure Condition and Uniform Continuity on Bounded Sets Imply Their Forms at Bounded Positions
lemmalem:nc-hamiltonian-bounded-conditions-2026aAnalysisPDEA Hamiltonian satisfying the structure condition satisfies it at bounded positions, and a Hamiltonian uniformly continuous on bounded sets is uniformly continuous in the momentum at bounded positions.The Structure Condition at Bounded Positions for a Hamiltonian on Phase-Space Noncommutative Laws
definitiondef:nc-hamiltonian-structure-condition-bounded-2026aAnalysisPDEThe structure condition required only at pairs of positions whose laws have a given norm bound: exchanging the positions under the doubling momentum raises the Hamiltonian by little when the doubling cost is small.Hamiltonians on Phase-Space Noncommutative Laws that are Uniformly Continuous in the Momentum at Bounded Positions
definitiondef:nc-hamiltonian-momentum-continuous-2026aAnalysisPDEA Hamiltonian is uniformly continuous in the momentum at bounded positions if, for positions whose law has a given norm bound and momenta in a given L2 ball, small L2 changes of the momentum change its value uniformly little.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.A Smooth Variational Principle of Borwein-Preiss Type with a Gauge on a Complete Metric Space
theoremthm:borwein-preiss-gauge-metric-2026aAnalysisTopologyAn upper semicontinuous function bounded above on a complete metric space, perturbed by a summable series of gauge terms centred at a convergent sequence of points, attains a strict maximum at a point whose value is at least that of any prescribed starting point.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…Rays, and Integrability and the Integral of a Real Function on a Borel Subset of the Real Line
definitiondef:integral-borel-subset-real-line-2026aAnalysisRays in the real line, and integrability and the integral of a real function on a Borel set, defined through its extension by zero and Lebesgue measure.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…