Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
- A unitary law of d-tuples is a function on words that is normalised, cyclic, invariant under cancelling adjacent inverse letters, and gives positive semidefinite kernels.
The Closed-Score Hypothesis Appears to Fail for the Free Entropy of One Variable: a Sketched Example, Conditional on the Tangent Inequality
remarkrem:nc-closed-score-fails-free-entropy-2026aAnalysisProbabilityA sketched one-variable example, conditional on the tangent inequality for minus the free entropy. Densities oscillating on a vanishing scale keep the free Fisher information bounded while their conjugate variables oscillate at unit size, and a coupling that reads the fine positi…Projected Closedness of Conjugate Variables along Square-Integrable Realisations with Bounded Free Fisher Information
lemmalem:nc-conjugate-variables-projected-closedness-2026aAnalysisProbabilityIf laws with conjugate variables are realised next to positions converging in L2 to a bounded law, and the realised conjugate variables stay bounded, then the limit law has conjugate variables with the same Fisher bound, and the realised conjugate variables converge weakly agains…The Barycentric Decomposition of the Noncommutative Wasserstein Distance
theoremthm:nc-barycentric-decomposition-wasserstein-2026aAnalysisProbabilityFor a coupling of two noncommutative laws, projecting the second variables onto the tracial algebra of the first gives a barycentric law and coupling. The cost splits orthogonally, the squared distance is at most the squared distance to the barycentric law plus the drop in second…Realising a Square-Integrable Field of the GNS Space of a Bounded Law next to Its Positions: Uniqueness and the Joint Law with a Momentum
lemmalem:nc-field-realisation-2026aAnalysisProbabilityA square-integrable tuple which, together with a copy of the positions of a bounded law, has the joint law of the positions and a given field of the GNS space is uniquely determined by the positions, and its joint law with the momentum of any bounded plan is that of the plan with…Couplings of a Square-Integrable Plan with a Law: Displacement and Momentum Pairing
definitiondef:nc-plan-couplings-2026aAnalysisProbabilityThe couplings of a square-integrable plan with a law are the three-variable laws whose first two variables have the plan as law and whose third variable has the given law; they always exist, and every three-variable law has a displacement and a momentum pairing that do not depend…Hamiltonians on Phase-Space Noncommutative Laws that are Bounded at Zero Momentum at Bounded Positions
definitiondef:nc-hamiltonian-bounded-zero-momentum-2026aAnalysisProbabilityA Hamiltonian on phase-space laws is bounded at zero momentum at bounded positions if its values at zero momentum are bounded uniformly over all positions whose law has a given norm bound.Wall-Confined Free Energies with Regular Plan Subgradients
definitiondef:nc-regular-plan-subgradients-2026aAnalysisProbabilityA wall-confined free energy has regular plan subgradients if every bounded plan in the plan subjet of a function touching the free energy from below at a local minimum point of their difference forces that point into the score domain and coincides with its score plan.- A wall-confined free energy has closed score if, whenever realisations of score plans have strongly converging positions and bounded scores, the limit law lies in the score domain and the scores converge weakly to a realisation of its score plan.
Perron's Method for Envelope Viscosity Solutions of the Discounted HJB Equation with Free Langevin Noise in a Wall: Existence and Well-Posedness
theoremthm:nc-envelope-perron-existence-2026aAnalysisProbabilityFor a shift-absorbing, shift-semicontinuous Hamiltonian bounded at zero momentum and a wall-confined free energy with closed score and regular plan subgradients, Perron's method gives a bounded envelope viscosity solution between any ordered bounded sub- and supersolution, so a b…The Bump Construction for Envelope Viscosity Subsolutions
lemmalem:nc-envelope-perron-bump-2026aAnalysisProbabilityFor all sufficiently small shift ranges, a bounded envelope viscosity subsolution lying below a bounded supersolution that fails to be a supersolution itself can be strictly raised at some law of the domain while remaining a bounded subsolution below the supersolution.The Coupling Test Function of a Square-Integrable Plan
definitiondef:nc-coupling-test-function-2026aAnalysisProbabilityFor a square-integrable plan, a gauge function and a weight, the coupling test function at a law is the supremum, over the couplings of the plan with that law, of the momentum pairing minus the gauge of the displacement and the weighted square of the displacement.The Supremum of a Bounded Family of Envelope Viscosity Subsolutions is an Envelope Viscosity Subsolution
lemmalem:nc-envelope-sup-subsolutions-2026aAnalysisProbabilityFor all sufficiently small shift ranges, the pointwise supremum of a nonempty uniformly bounded family of envelope viscosity subsolutions with that shift range is again a bounded envelope viscosity subsolution with the same shift range.Constant Barriers for Envelope Viscosity Solutions under Regular Plan Subgradients
lemmalem:nc-envelope-constant-barriers-2026aAnalysisProbabilityIf the free energy has regular plan subgradients and the Hamiltonian at zero momentum is bounded by K on laws inside the wall, then the constants -K/rho and K/rho are an envelope viscosity subsolution and supersolution, for any shift range.Score Bounds at Envelope Test Inequalities from the Absorption Slack
lemmalem:nc-envelope-score-bounds-2026aAnalysisProbabilityFor a Hamiltonian that absorbs shifts at the noise level and is bounded at zero momentum at bounded positions, a momentum shift that satisfies either envelope test inequality at a bounded position has norm at most a constant over the shift size, with the admissible shift sizes de…The Wall-Confined Free Energy Converges along Wasserstein-Convergent Sequences with Bounded Scores
lemmalem:nc-energy-convergence-bounded-scores-2026aAnalysisProbabilityIf laws in the score domain converge in the Wasserstein distance to a law in the domain and their scores stay bounded in L^2, then their wall-confined free energies converge to the free energy of the limit.The Coupling Test Function of a Bounded Plan: Maximising Couplings, Continuity, Plan Subjets and Localisation
lemmalem:nc-coupling-test-function-2026aAnalysisProbabilityFor a bounded plan at a bounded law, the coupling test function built from a differentiable gauge with vanishing derivative at zero and a quadratic weight is attained by a maximising coupling, is Wasserstein-continuous on bounded laws, and admits an explicit plan subjet read off…The Score Plan of a Law in the Score Domain of the Wall-Confined Free Energy
definitiondef:nc-score-plan-2026aAnalysisProbabilityThe score plan of a law in the score domain is the joint law of the position variables and the score, a square-integrable law of 2d variables.- If two plans realised on a common position tuple are plan superdifferentials (or subdifferentials) of two functions with given slacks, then the plan with the summed momenta is a plan superdifferential (or subdifferential) of the sum, with the summed slack.
Gluing Countably Many Square-Integrable Noncommutative Laws along a Common Marginal
lemmalem:l2-countable-gluing-common-marginal-2026aAnalysisProbabilityCountably many square-integrable noncommutative laws that share the law of their first k variables are realised jointly in one tracial W*-probability space, by one common k-tuple and one further tuple for each law.