Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Wick Square of the Free Field on the Torus as Diagonal Couplings: Summability in Dimension at Most Three, the Riccati Condition, and the Divergent Wick Constant
lemmalem:free-field-wick-square-data-torus-hilbert-2026aAnalysisProbabilityOn negative Sobolev spaces of the torus, the Wick square of the free field corresponds to diagonal couplings that are summable Wick couplings in dimension at most three, satisfy the Riccati condition, and have a divergent Wick constant in dimension at least two.Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift and a Score-Paired Wick-Square Cost on a Hilbert Space, by Gaussian Dressing
theoremthm:wick-square-score-drift-hamilton-jacobi-well-posed-hilbert-2026aAnalysisPDEFor the Hamilton-Jacobi equation with Gaussian score drift and a score-paired Wick-square cost on laws over a Hilbert space, viscosity solutions relative to the dressing profile satisfy comparison, exist and are unique among those differing from the profile by a bounded function;…Gaussian Dressing of the Score-Paired Wick-Square Cost on a Hilbert Space: Riccati Coefficients, Dressed Variances, the Quadratic Profile, the Constant, the Dressed Pair and the Shifted Operator
lemmalem:gaussian-dressing-wick-square-hilbert-2026aAnalysisPDEPer-mode Riccati coefficients give a dressed diagonal Gaussian and a quadratic profile; shifting the unknown by the profile turns the Wick-square Hamilton-Jacobi operator into the Gaussian score-drift operator relative to the dressed Gaussian with the running cost shifted by an a…The Score-Paired Wick-Square Cost is the Wick-Ordered Series, and Cutoff Counterterms Are Forced up to a Convergent Constant
propositionprop:wick-square-score-cost-renormalisation-hilbert-2026aAnalysisProbabilityThe score-paired Wick-square cost equals the Wick-ordered series of second moments minus variances; cutoff costs with arbitrary counterterms converge exactly when the counterterms differ from the Wick constants by a convergent sequence, and the bare cutoff costs diverge when the…The Hamilton-Jacobi Equation with Gaussian Score Drift and a Score-Paired Wick-Square Cost on a Hilbert Space
equationeq:wick-square-score-drift-hamilton-jacobi-hilbert-2026aAnalysisPDEThe first-order Hamilton-Jacobi equation on laws relative to a diagonal Gaussian in which the Gaussian score is paired with the momentum minus the gradient of the Wick-square corrector, so that the singular Wick-square cost enters through the score.The Wick-Square Corrector and the Score-Paired Wick-Square Cost Relative to a Diagonal Gaussian Measure on a Hilbert Space
definitiondef:wick-square-score-cost-hilbert-2026aAnalysisProbabilityFor diagonal Wick couplings, the Wick-square corrector is a diagonal quadratic profile, and the Wick-square cost of a measure in the score domain is defined as the pairing of the Gaussian score with the corrector's gradient.Changing the Penalty of a Noise Penalty Pair by a Noise Intrinsic Test Function and a Constant Does Not Change the Viscosity Notion
lemmalem:noise-penalty-change-viscosity-hilbert-2026aAnalysisPDEIf two noise penalty pairs share their domains and their penalties differ by a noise intrinsic test function plus a constant, with scores differing by its gradient, then they define the same viscosity subsolutions and supersolutions.Viscosity Subsolutions, Supersolutions and Solutions on the Noise Wasserstein Space Relative to a Noise Penalty Pair and a Profile
definitiondef:noise-profile-viscosity-solution-hilbert-2026aAnalysisPDEA function is a viscosity sub- or supersolution relative to a noise penalty pair and a profile, a noise intrinsic test function, when its difference with the profile is one for the operator shifted by the profile's value and gradient.Changing the Variances of a Diagonal Gaussian Reference by a Diagonal Quadratic: the Same Measures and Domains, with Penalty and Score Shifted by a Diagonal Quadratic Profile
lemmalem:dressed-gaussian-entropy-pair-hilbert-2026aAnalysisProbabilityChanging the inverse variances of a diagonal Gaussian reference by a summable diagonal perturbation leaves the noise-connected measures, the finite-entropy measures and the score domain unchanged, and shifts the relative entropy by a diagonal quadratic profile plus a constant and…The Gaussian Relative Score Paired with the Gradient Field of a Diagonal Quadratic Profile
lemmalem:gaussian-score-pairing-diagonal-quadratic-hilbert-2026aAnalysisProbabilityFor a measure with finite weighted Fisher information relative to a diagonal Gaussian, the score paired with the gradient field of a diagonal quadratic profile is the convergent series of normalised second moments minus one, weighted by the coefficients.Diagonal Quadratic Profiles on the Noise Wasserstein Space: Moment Bounds, a Lipschitz Estimate, the Gradient Field, and Noise Intrinsic Test Functions
lemmalem:noise-diagonal-quadratic-profile-hilbert-2026aAnalysisProbabilityIntegrals of diagonal quadratics with suitably summable coefficients are finite on all measures noise-connected to a diagonal Gaussian, depend Lipschitz-continuously on the measure, and are noise intrinsic test functions whose gradient is the corresponding linear field.The Free Field on the Torus as the Gaussian Reference Measure, with Square-Integrable White Noise: Standing Notation
settingset:free-field-white-noise-torus-hilbert-2026aAnalysisProbabilityStanding notation for working on distributions on the torus relative to the free field, with displacements measured in the L^2 norm of white noise: the Gaussian-reference framework instantiated on a negative Sobolev space.White Noise in the Square-Integrable Space and the Free Field on the Torus as Noise Weights and a Variance Sequence on a Negative Sobolev Space
lemmalem:free-field-white-noise-data-torus-hilbert-2026aAnalysisProbabilityOn the negative Sobolev space H^{-m} of the torus with its rescaled trigonometric basis, white noise in L^2 gives a weight sequence and the free field (1 - Laplacian)^{-1} gives a variance sequence; their ratio is the Fourier weight, and the noise space is exactly L^2 of the toru…Integrals of Bounded Cylindrical Functions are Noise Intrinsic Test Functions with the Noise Gradient as Gradient
lemmalem:noise-cylindrical-integral-test-function-hilbert-2026aAnalysisProbabilityThe integral of a bounded C^2 cylindrical function against a measure is a noise intrinsic test function on the noise Wasserstein space, and its gradient along noise couplings is the noise gradient of the function.The Gaussian Entropy Pair is Uniformly Displacement Convex, with Modulus the Temperature over the Variance-to-Noise Bound
lemmalem:noise-gaussian-entropy-pair-convex-hilbert-2026aAnalysisProbabilityWhen the Gaussian variances are at most kappa times the noise weights, the Gaussian entropy pair with temperature beta is beta over kappa displacement convex along noise-optimal couplings, and in particular displacement convex.The Gaussian Entropy Pair Has Regular Penalised Maxima
lemmalem:noise-gaussian-entropy-pair-regular-maxima-hilbert-2026aAnalysisProbabilityEvery local maximum of a noise intrinsic test function minus a positive multiple of the Gaussian entropy penalty has finite weighted Fisher information.The Gaussian Entropy Pair Has Closed Score Along Noise Couplings
lemmalem:noise-gaussian-entropy-pair-closed-score-hilbert-2026aAnalysisProbabilityAlong couplings of vanishing noise cost, measures with uniformly bounded scores converge to a measure that again has finite weighted Fisher information, and the scores converge weakly along the couplings.The Gaussian Entropy Pair is Noise-Closed, and Its Penalty Bounds the Squared Noise Wasserstein Distance to the Reference Measure
lemmalem:noise-gaussian-entropy-pair-closed-hilbert-2026aAnalysisProbabilityThe Gaussian entropy pair has closed and bounded penalty sublevel sets in the noise Wasserstein space, and by Talagrand's inequality the squared distance to the reference measure is at most 2 kappa over beta times the penalty.The Gaussian Entropy Pair is a Noise Penalty Pair, with Nonnegative Penalty and Dense Score Domain
lemmalem:noise-gaussian-entropy-penalty-pair-hilbert-2026aAnalysisProbabilityThe Gaussian entropy pair is a noise penalty pair: its penalty is nonnegative and measures of finite weighted Fisher information are dense among measures of finite relative entropy in the noise Wasserstein distance.A Bound on the Noise Ornstein-Uhlenbeck Functional by Noise Gradients Gives Finite Weighted Fisher Information
lemmalem:ou-functional-fisher-bound-hilbert-2026aAnalysisProbabilityIf the noise Ornstein-Uhlenbeck functional of a measure is bounded by R times the noise-gradient norm on bounded C^2 cylindrical functions, then the measure has a relative score and weighted Fisher information at most R squared.