Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Entropy and Score Relative to the Gibbs Measure Split into Their Gaussian Parts and the Potential, with a Fisher Information Bound
lemmalem:gibbs-entropy-score-splitting-hilbert-2026aAnalysisProbabilityThe Gibbs measure has density proportional to the weight with respect to the Gaussian reference measure, with the same null sets; relative entropy with respect to it is the Gaussian relative entropy plus the averaged potential over the temperature plus the log-normaliser; a measu…The Relative Score with Respect to the Gibbs Measure of an Admissible Cylindrical Potential
definitiondef:relative-score-gibbs-hilbert-2026aAnalysisProbabilityA measure has a relative score with respect to the Gibbs measure of an admissible cylindrical potential if, coordinate by coordinate, integration by parts against bounded cylindrical functions holds with the Gibbs drift; the components are unique.The Gibbs Measure of an Admissible Cylindrical Potential Relative to a Diagonal Gaussian Measure
definitiondef:gibbs-measure-cylindrical-potential-hilbert-2026aAnalysisProbabilityThe Gibbs measure of an admissible cylindrical potential V at temperature beta is the probability measure with density proportional to exp(-V/beta) with respect to the diagonal Gaussian reference measure.Basic Properties of an Admissible Cylindrical Potential: Continuity, Growth under Noise Translations, Integrability, the Tangent Inequality and Tangency of the Noise Gradient
lemmalem:admissible-cylindrical-potential-hilbert-2026aAnalysisProbabilityAn admissible cylindrical potential is continuous with a continuous noise gradient, grows at most exponentially under noise translations, has integrable slope wherever it is integrable, is integrable with its squared slope against the Gaussian reference measure, satisfies the sem…Admissible Cylindrical Potentials on a Hilbert Space in the Noise Geometry
definitiondef:admissible-cylindrical-potential-hilbert-2026aAnalysisProbabilityAn admissible cylindrical potential is a twice continuously differentiable function of finitely many coordinates that is bounded below, semiconvex in the noise norm, has slope controlled by its value and weighted Laplacian controlled by its squared slope; its noise gradient is fi…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.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 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 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.Well-Posedness of the Hamilton-Jacobi Equation with the Free-Field Score Drift for Laws of Distributions on the Torus: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution
corollarycor:free-field-score-drift-hamilton-jacobi-well-posed-torus-hilbert-2026aProbabilityPDEFor laws of distributions on the torus with finite entropy relative to the free field, and a bounded running cost uniformly continuous for the Wasserstein distance with L^2 cost, the Hamilton-Jacobi equation with the free-field score drift has comparison and exactly one bounded v…On Functions of Finitely Many Fourier Modes the Free-Field Score Drift is Minus the Ornstein-Uhlenbeck Generator with Drift the Laplacian Minus One
lemmalem:free-field-score-drift-fourier-torus-hilbert-2026aProbabilityPDEFor functions of finitely many Fourier modes, the score drift of the free field equals minus the integral of the Ornstein-Uhlenbeck generator with second-order part the sum of second mode derivatives and drift coefficient minus the Fourier weight, which is the Fourier symbol of t…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…Well-Posedness of the Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space: Comparison, Existence and Uniqueness of a Bounded Viscosity Solution
corollarycor:gaussian-score-drift-hamilton-jacobi-well-posed-hilbert-2026aProbabilityPDEFor a bounded, uniformly continuous running cost, the Hamilton-Jacobi equation with Gaussian score drift on laws of finite relative entropy with respect to a diagonal Gaussian measure satisfies comparison and has exactly one bounded viscosity solution.On Integrals of Cylindrical Functions the Gaussian Score Drift is the Noise Ornstein-Uhlenbeck Functional, and the Score-Drift Hamilton-Jacobi Operator Takes Its Pointwise Form
lemmalem:gaussian-score-drift-cylindrical-integral-hilbert-2026aProbabilityPDEOn integrals of bounded C^2 cylindrical functions, the score drift of the Gaussian entropy pair equals beta times the noise Ornstein-Uhlenbeck functional, so the Hamilton-Jacobi operator with Gaussian score drift takes an explicit pointwise form.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 Hamilton-Jacobi Equation with Gaussian Score Drift on a Hilbert Space
equationeq:gaussian-score-drift-hamilton-jacobi-hilbert-2026aProbabilityPDEThe discounted Hamilton-Jacobi equation on laws of finite relative entropy with respect to a diagonal Gaussian measure on a Hilbert space, whose drift term pairs the momentum with beta times the noise score field.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.