Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Word Gauge Space Is a Real Hilbert Space, and the Unitary Laws Form a Compact Metric Space in It
lemmalem:nc-unitary-word-gauge-2026aAnalysisProbabilityThe gauge space is a real Hilbert space, and the gauge distance is a metric on unitary laws whose convergence is convergence on every word and for which the unitary laws are compact.The Word Gauge: Weights, the Space of Weighted Square-Summable Word Functions, Its Norm and Inner Product, and the Length-Weighted Gauge
definitiondef:nc-unitary-word-gauge-2026aAnalysisProbabilityDefines the weights theta^|w| with theta = 1/(6144d), summable word functions, the gauge space of weighted square-summable functions on cyclically reduced words with its norm and inner product, and the length-weighted gauge.Unitary Laws Are the Laws of Unitary Tuples, and Are Realised Together with a Free Semicircular Family
lemmalem:nc-unitary-law-realisation-2026aAnalysisProbabilityTraces of words in unitaries of a tracial W*-probability space form a unitary law, and every unitary law is realised this way together with a semicircular family obeying the free second-moment relations.Unitary Laws: Adjoints, the Bound One, and Reduction to Cyclically Reduced Words
lemmalem:nc-unitary-laws-basic-2026aAlgebraProbabilityUnitary laws satisfy lambda(w*) = conj lambda(w) and |lambda| <= 1, reduce to cyclically reduced words, are determined by them, and the constant law 1 exists.- 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.
Counting Cyclically Reduced Words: Finiteness and the Bound (2d)^k
lemmalem:nc-unitary-word-counts-2026aCombinatoricsWords of a given length form a nonempty finite set containing cyclically reduced words, and a sum over words of length k of terms bounded by M is at most M(2d)^k.Words in Unitary Letters: Generators, Signs, Inverse Letters, Lengths and Adjoint Words
definitiondef:nc-unitary-words-2026aAlgebraLetters 1,...,2d encode unitaries and their inverses; defines generators, signs, inverse letters, word lengths and adjoint words.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.