Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Gaussian Free-Energy Pair is Uniformly Displacement Convex, with Closed Score Along Couplings and Regular Penalised Maxima
lemmalem:gaussian-free-energy-pair-regularity-euclidean-2026aAnalysisProbabilityThe Gaussian free-energy pair is (a/c_max)-displacement convex, has closed score along couplings, and has regular penalised maxima.The Gaussian Free-Energy Pair is a Penalty Pair: Translations, the First Variation, Lower Semicontinuity, the Moment Bound and the Map Property
lemmalem:gaussian-free-energy-penalty-pair-euclidean-2026aAnalysisProbabilityThe Gaussian free-energy pair is a penalty pair: translations change the penalty by an explicit quadratic, so its translation Hessian is diagonal with entries a/c_i; its first variation along gradients of test functions is minus the temperature times the Ornstein-Uhlenbeck functi…The Gaussian Free-Energy Pair: Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure
definitiondef:gaussian-free-energy-pair-euclidean-2026aAnalysisProbabilityThe Gaussian free-energy pair has as penalty the temperature times the relative entropy with respect to a diagonal Gaussian, on measures of finite second moment and finite relative entropy, and as score the temperature times the relative score, on those of finite relative Fisher…The Entropy Inequality for Nonnegative Functions, and Quadratic Moments Bounded by the Relative Entropy with Respect to a Diagonal Gaussian Measure
lemmalem:entropy-quadratic-moment-diagonal-gaussian-euclidean-2026aAnalysisProbabilityThe integral of a nonnegative function is at most the relative entropy plus the logarithm of its exponential moment under the reference measure; relative to a diagonal Gaussian this bounds diagonal quadratic moments, and the second moment, linearly by the relative entropy with ex…Relative Entropy and Relative Score with Respect to a Diagonal Gaussian Measure: Comparison with the Entropy, the Score and the Relative Free Energy of a Quadratic Potential
lemmalem:diagonal-gaussian-relative-entropy-score-euclidean-2026aAnalysisProbabilityFor measures with finite second moment, relative entropy with respect to a diagonal Gaussian is finite exactly when the entropy is, and equals entropy plus half the weighted second moment plus the log-normalizer; the relative score is the score plus the scaling map; consequently…Finite Fisher Information Relative to a Diagonal Gaussian Measure: the Ornstein-Uhlenbeck Functional, the Relative Score and the Relative Fisher Information
definitiondef:relative-score-diagonal-gaussian-euclidean-2026aAnalysisProbabilityFor a measure with finite second moment, the Ornstein-Uhlenbeck functional integrates Laplacian minus scaling-map drift of test functions; the measure has finite Fisher information relative to the diagonal Gaussian when this functional is bounded by the L^2 norm of gradients, its…Diagonal Gaussian Measures on Euclidean Space
definitiondef:diagonal-gaussian-measure-euclidean-2026aAnalysisProbabilityThe diagonal Gaussian measure with given variances is the probability measure on R^d whose density with respect to Lebesgue measure is the diagonal Gaussian density.The Diagonal Gaussian Density on Euclidean Space: Regularity, Gradient, Normalization, Second Moments and Exponential Moments of Diagonal Quadratic Forms
lemmalem:diagonal-gaussian-density-euclidean-2026aAnalysisProbabilityThe diagonal Gaussian density is smooth, positive and Borel, its gradient is minus the density times the scaling map, it integrates to one, its i-th coordinate has second moment c_i, and exponential moments of diagonal quadratic forms are explicit products.The Galerkin Problems of the Wick-Square Problem: Standing Notation for Cubes of Modes, Coordinates, the Free Galerkin Quadratic and the Riccati Potential
settingset:wick-square-galerkin-2026aProbabilityPDEStanding notation for the Galerkin problems of the Wick-square problem: for each cutoff an enumeration of the cube of modes, the coordinate map and projection, the free Galerkin quadratic and the Riccati potential.The Counterterm Is Forced at the Level of Galerkin Solutions: Other Counterterms Shift the Solutions by a Constant, and the Bare Solutions Diverge
corollarycor:wick-square-galerkin-counterterm-2026aProbabilityPDEChanging the counterterm shifts each Galerkin solution by a constant, so the Galerkin solutions converge exactly when the counterterm differs from the free-field variances by a cube-summable family; in dimension at least two the bare Galerkin solutions are unbounded and converge…The Galerkin Solutions of the Wick-Square Problem Converge to the Renormalised Viscosity Solution for Every Bounded Lipschitz Running Cost
theoremthm:wick-square-galerkin-convergence-2026aProbabilityPDEFor a bounded running cost that is Lipschitz in the distance of H^-3, each Wick-ordered Galerkin equation has exactly one solution differing from the free Galerkin quadratic by a bounded function; these differences are bounded and Lipschitz uniformly in the cutoff, and the Galerk…Lifting Shifted Galerkin Solutions of the Wick-Square Problem to Viscosity Solutions of the Cut-Off Shifted Equation on the Sobolev Triple
lemmalem:wick-square-galerkin-lift-2026aAnalysisPDEA viscosity sub- or supersolution of the shifted Galerkin equation, composed with the coordinate projection, is a viscosity sub- or supersolution of the cut-off shifted equation on the Sobolev triple; continuity is preserved.The Cut-Off Shifted Operators of the Wick-Square Problem: the Hypotheses of the Convergence Theorem with Constants Independent of the Cutoff, Convergence on Bounded Test Data, and Compactness of the Triple
lemmalem:wick-square-shifted-cutoff-data-2026aAnalysisPDEThe cut-off shifted operators satisfy the hypotheses of the Hilbert-triple convergence theorem with constants independent of the cutoff and converge to the uncut shifted operator on bounded test data; and bounded sets of the form space of the Sobolev triple are precompact in the…The Riccati Shift of the Galerkin Equations of the Wick-Square Problem: a Penalty-Drift Equation with Bounded Cost, and Well-Posedness at Each Cutoff
lemmalem:wick-square-galerkin-shift-2026aProbabilityPDESubtracting the free Galerkin quadratic turns the Wick-ordered Galerkin equation into a penalty-drift equation with bounded running cost whose quadratic potential meets the hypotheses of the Euclidean well-posedness theorem; so each Galerkin equation has exactly one solution diff…The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem with Noise, Control and Cost Cut Off to a Cube of Modes
equationeq:wick-square-hjb-shifted-cutoff-2026aProbabilityPDEThe shifted Wick-square operator on the Sobolev triple with the noise, the gradient form, the Riccati drift and the running cost cut off to a cube of modes, the full linear drift being kept, and its equation.The Galerkin Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem in the Coordinates of a Cube of Modes
equationeq:wick-square-galerkin-2026aProbabilityPDEThe Galerkin Hamilton-Jacobi-Bellman operators of the Wick-square problem in the coordinates of a cube of modes, with a general counterterm (Wick-ordered and bare as special cases), the shifted Galerkin operator of penalty-drift form, and their equations.Convergence of Solutions of Viscous Hamilton-Jacobi Equations with Monotone Nonlinearities on a Hilbert Triple under Convergence of the Operators
theoremthm:convergence-viscous-hamilton-jacobi-hilbert-triple-2026aAnalysisPDEFor viscous Hamilton-Jacobi equations with monotone nonlinearities on a Hilbert triple whose form-space bounded sets are precompact in the ambient space, bounded continuous solutions are uniformly Lipschitz and converge, pointwise and uniformly on bounded sets of the form space,…Stability of Viscosity Sub- and Supersolutions on a Hilbert Triple: Limits of Uniformly Bounded, Equicontinuous Sequences under Convergence of the Operators on Bounded Test Data
theoremthm:stability-equicontinuous-hilbert-triple-2026aAnalysisPDEOn a Hilbert triple whose bounded sets of the form space are precompact in the ambient space, the pointwise limit superior of uniformly bounded, equicontinuous viscosity subsolutions of operators converging on bounded test data is a viscosity subsolution of the limit operator, an…Convergence of Second-Order Equation Operators on a Hilbert Triple on Bounded Test Data
definitiondef:operator-convergence-bounded-test-data-hilbert-triple-2026aAnalysisPDETest data bounded by R on a Hilbert triple, and convergence of a sequence of second-order equation operators to a limit operator uniformly on such data, for both delta-shifts.In a Diagonal Hilbert Triple whose Weights Tend to Infinity, Bounded Sequences of the Form Space Have Subsequences Converging in the Ambient Space
lemmalem:diagonal-hilbert-triple-compact-2026aAnalysisIn a diagonal Hilbert triple whose weights tend to infinity, every sequence bounded in the form space has a subsequence converging in the ambient space to a point of the form space with the same bound.