Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
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.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.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.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.