Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Quadratic Control Hamiltonian: Legendre Formula, Bounds, Differences in the Law and Admissibility
lemmalem:nc-unitary-quadratic-hamiltonian-2026aProbabilityPDEThe quadratic control Hamiltonian equals the limit of half the sum of the law's values on the squares of the truncated cyclic gradients, lies between 0 and kappa_d^2 |p|^2 / 2, has the corresponding limit formula for differences in the law, and is an admissible Hamiltonian.The Quadratic Control Hamiltonian on Unitary Laws
definitiondef:nc-unitary-quadratic-hamiltonian-2026aProbabilityPDEThe quadratic control Hamiltonian at a unitary law and a momentum is the supremum, over polynomial feedback controls, of minus the pairing of the momentum with the feedback drift minus half the control energy.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.Admissible Hamiltonians on Unitary Laws
definitiondef:nc-unitary-admissible-hamiltonian-2026bAnalysisPDEA Hamiltonian on unitary laws times the gauge space is admissible if it is jointly continuous and depends on the momentum only through its pairing with two-sided tangent vectors.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.Reduced and Cyclically Reduced Words in Unitary Letters
definitiondef:nc-unitary-reduced-words-2026aAlgebraDefines reduced words (no adjacent inverse letters) and cyclically reduced words (also first and last letters not inverse to each other).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.