Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Feedback Drifts: the Energy Bound in the Word Gauge, the Momentum Pairing and Two-Sided Tangency
lemmalem:nc-unitary-feedback-drift-2026aProbabilityPDEFeedback drifts lie in the word gauge space with norm at most kappa_d times the square root of the control energy, pair with a momentum p as the limit of the traces of the control against the truncated cyclic gradients of p, and are two-sided tangent vectors to the space of unita…Feedback Drifts of Polynomial Controls on Unitary Laws
definitiondef:nc-unitary-feedback-drift-2026aProbabilityPDEThe feedback drift of a polynomial control a at a unitary law lambda assigns to each cyclically reduced word w the sum over i of lambda(a^i D^i_w), where D^i_w is the i-th cyclic derivative of w.The Truncated Cyclic Gradient of a Gauge Vector
definitiondef:nc-unitary-cyclic-gradient-2026aAlgebraProbabilityFor a gauge vector p and a length n, the i-th truncated cyclic gradient is the self-adjoint part of the sum of c_w times conj(p(w)) times the i-th cyclic derivative of w, over cyclically reduced words of length at most n.Word Polynomials under a Unitary Law: Linearity, Traciality, Positivity, the Cauchy-Schwarz Inequality and the l1 Bound
lemmalem:nc-unitary-word-polynomials-2026aAlgebraProbabilityEvaluation of word polynomials by a unitary law is linear, tracial and compatible with adjoints; it is positive on X*X, satisfies the Cauchy-Schwarz inequality, and is bounded by the l1 norm, which is submultiplicative.Word Polynomials in Unitary Letters: the Normed *-Algebra of Finitely Supported Word Functions
definitiondef:nc-unitary-word-polynomials-2026aAlgebraProbabilityWord polynomials are finitely supported complex functions on words in unitary letters, with pointwise linear structure, the concatenation product, the adjoint and the l1 norm.The Embedding of Unitary Laws into the Word Gauge Space and the Gauge Distance
definitiondef:nc-unitary-law-embedding-2026aAnalysisProbabilityUnitary laws embed into the gauge space by restriction to cyclically reduced words; this defines the gauge distance between laws.Perron's Method on Unitary Laws: the Bump Construction
lemmalem:nc-unitary-perron-bump-2026aProbabilityPDEFor any admissible Hamiltonian: if the lower envelope of a subsolution fails the supersolution test at a law, a local bump gives a strictly larger subsolution below a prescribed lower semicontinuous bound.Perron's Method on Unitary Laws: the Upper Envelope of a Bounded Family of Subsolutions Is a Subsolution
lemmalem:nc-unitary-perron-sup-2026aProbabilityPDEFor any admissible Hamiltonian, the upper semicontinuous envelope of the supremum of a family of subsolutions with a common upper bound is a subsolution.Viscosity Sub- and Supersolutions of the Hamilton-Jacobi-Bellman Equation on Unitary Laws
definitiondef:nc-unitary-viscosity-solution-2026aProbabilityPDEViscosity sub- and supersolutions are bounded semicontinuous functions on unitary laws satisfying the equation's inequalities at relative extrema against C^1 test functions on the gauge space.The Discounted Hamilton-Jacobi-Bellman Equation on Unitary Laws with Free Unitary Noise
equationeq:nc-unitary-hjb-2026aProbabilityPDEFor a Hamiltonian H on unitary laws times the gauge space, the discounted first-order equation rho V + H(lambda,p) - (beta^2/2)<p, Theta lambda> = f(lambda), with the free unitary heat generator as noise direction.- Standing data for equations on unitary laws with free unitary noise: the gauge space and gauge distance, C^1 test functions, the free unitary heat generator as noise direction, discount and noise constants, and a continuous source.
The Free Unitary Heat Generator Points into the Space of Unitary Laws
lemmalem:nc-free-unitary-heat-inward-2026aAnalysisProbabilityThe free unitary heat direction is an inward tangent vector at every unitary law, realised by a Cayley-transform curve built from a free semicircular family.Gradients at Relative Extrema on the Unitary Law Space: Signs on Inward Tangent Vectors and Vanishing on Two-Sided Ones
lemmalem:nc-unitary-tangent-extrema-2026aAnalysisProbabilityAt a relative minimum on the unitary law space, the gradient of a test function is nonnegative on inward and zero on two-sided tangent vectors; the reverse holds at a relative maximum.Inward and Two-Sided Tangent Vectors to the Space of Unitary Laws
definitiondef:nc-unitary-tangent-vectors-2026aAnalysisProbabilityDefines inward and two-sided tangent vectors to the space of unitary laws as velocities, in the gauge space, of one-sided and two-sided curves of laws.The Free Unitary Heat Generator: Bound, Continuity and Strict Dissipation in the Length-Weighted Gauge
lemmalem:nc-free-unitary-heat-generator-estimate-2026aAnalysisProbabilityThe free unitary heat generator is bounded by |w|^2 on each word, continuous into the gauge space, and strictly dissipative: its differences pair with differences of laws to at most minus one quarter of the squared length-weighted gauge.The Free Unitary Heat Generator on Unitary Laws
definitiondef:nc-free-unitary-heat-generator-2026aAnalysisProbabilityDefines the free unitary heat generator on unitary laws: a diagonal term -|w|/2 lambda(w) plus signed products of lambda over pairs of letters with the same generator.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.