Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Hubbard-Stratonovich Field Law is the Gibbs Measure of an Admissible Cylindrical Potential for Every Admissible Split of Its Gaussian, and the Gibbs Entropy Pair Does Not Depend on the Split
lemmalem:hubbard-stratonovich-gibbs-ising-torus-hilbert-2026aProbabilitySplitting the Hubbard-Stratonovich Gaussian at any level above its Fourier variances writes the field law as the Gibbs measure of an explicit admissible cylindrical potential (quadratic minus log-cosh of the site field) relative to a modified diagonal Gaussian. The resulting Gibb…The Hubbard-Stratonovich Transform Couples the Ising Measure and the Field Law: the Density of the Field Law, the Disintegration of the Joint Law, and Entropy Contraction under Both Kernels
lemmalem:hubbard-stratonovich-transform-ising-torus-hilbert-2026aProbabilityThe Hubbard-Stratonovich field law has an explicit positive continuous density with respect to the Gaussian reference, the joint law disintegrates through the field-to-spin kernel, and the field law is carried back to the Ising measure. Relative entropy does not increase under ei…The Hubbard-Stratonovich Transform of the Ising Measure on the Lattice Torus
definitiondef:hubbard-stratonovich-transform-ising-torus-hilbert-2026aProbabilityCouples the Ising measure on the lattice torus to a Gaussian field on the negative Sobolev space: a spin configuration shifts the Gaussian reference by a vector built from its Fourier coefficients, and a field picks independent spins tilted by its site values. These two kernels d…The Ising Measure on the Lattice Torus with a Reflection-Symmetric Periodic Pair Interaction
definitiondef:ising-measure-lattice-torus-2026aProbabilityThe spin configurations are the maps from the lattice torus to {-1,1}. The Ising measure of an interaction gives each configuration a probability proportional to the exponential of half its interaction energy, normalised by the partition function; the inverse temperature is inclu…The Interaction Matrix of a Reflection-Symmetric Periodic Pair Interaction is Symmetric and Diagonal in the Trigonometric Basis, with Eigenvalues Its Symbol
lemmalem:lattice-pair-interaction-torus-2026aProbabilityFor a periodic pair interaction invariant under coordinate reflections, the interaction matrix on the lattice torus is symmetric and has the trigonometric maps as eigenvectors with eigenvalues given by its cosine symbol; hence the interaction energy is diagonal in the discrete Fo…Reflection-Symmetric Periodic Pair Interactions on the Lattice Torus, Their Interaction Matrices and Their Symbols
definitiondef:lattice-pair-interaction-torus-2026aProbabilityA reflection-symmetric periodic pair interaction is a function on the integer lattice that is periodic with period L and invariant under each coordinate reflection. Its interaction matrix couples two sites through their difference, and its symbol is its cosine transform over one…The White-Noise Embedding of Lattice Fields: Linearity, Inversion by the Site Field, the Site Sum as the White-Noise Norm, and Coordinates
lemmalem:lattice-field-embedding-torus-hilbert-2026aAnalysisThe white-noise embedding of lattice fields into the negative Sobolev space is linear and injective with the site field as left inverse; site evaluations are continuous linear functionals, the white-noise norm of an embedded field is its site sum of squares, the site sum of a noi…The White-Noise Embedding of Lattice Fields into the Negative Sobolev Space of the Torus, and the Site Field of a Distribution
definitiondef:lattice-field-embedding-torus-hilbert-2026aAnalysisThe white-noise embedding sends a lattice field to the distribution on the torus whose Fourier coefficients on the cube of modes are its discrete Fourier coefficients times L^{n/2}, and zero elsewhere. The site field of a distribution is the lattice field obtained by evaluating i…Discrete Fourier Analysis on the Lattice Torus: Counting, Orthogonality, Inversion, Parseval, and the Spin Sum
lemmalem:lattice-torus-discrete-fourier-torus-2026aAnalysisOn the lattice torus of odd side L = 2M+1 the sites and the modes both number L^n, the sampled trigonometric system is orthogonal, the discrete Fourier transform is a linear bijection with an explicit inversion formula and satisfies Parseval's identity, and the sum over all spin…The Lattice Torus of Odd Side, Lattice Fields, and the Discrete Fourier Coefficients of a Lattice Field
definitiondef:lattice-torus-discrete-fourier-torus-2026aAnalysisThe lattice torus of odd side L = 2M+1 is the set of points w/L of the unit cell with integer coordinates w between 0 and L-1. A lattice field is a real function on it, and its discrete Fourier coefficients are its averages against the real trigonometric maps indexed by the cube…Measures on a Finite Set Given by Point Masses: the Measure, Integrals as Finite Sums, and Relative Entropy
lemmalem:finite-set-weighted-measures-2026aProbabilityOn a finite set with all subsets measurable, nonnegative point masses define a measure (every probability measure arises this way), every real function is integrable with integral the weighted sum of its values, and the relative entropy of two such probability measures (the refer…The Gibbs Dirichlet Form of the Noise Gradient
definitiondef:gibbs-dirichlet-form-hilbert-2026aAnalysisProbabilityDefines the Gibbs Dirichlet form as the inner product, in the square-integrable fields of the Gibbs measure, of Gibbs noise gradients.Cameron-Martin Partial Derivatives on the Gaussian Sobolev Space with the Variances as Weights
definitiondef:cameron-martin-partial-derivatives-hilbert-2026aAnalysisProbabilityReads the Gaussian Sobolev space with the variances as weights, the Cameron-Martin (Malliavin) Sobolev space, and defines the partial derivatives of its elements along the basis vectors.The Potential in Dimension One: Wick Ordering Is a Finite Mass Shift and Nelson's Estimate Is Not Needed
remarkrem:free-field-phi4-one-dimension-torus-hilbert-2026aAnalysisProbabilityExplains that in dimension one Wick ordering is only a finite mass shift and Nelson's estimate is not needed, which techniques could replace the chaos route there, and why the score relative to the Gibbs measure must still be defined intrinsically.The Measure on the Torus of Dimension at Most Two as the Gibbs Measure of a Gibbs-Sobolev Potential, and Its Galerkin Approximation in Total Variation
corollarycor:free-field-phi4-gibbs-measure-torus-hilbert-2026aAnalysisProbabilityIn dimension at most two the phi^4 potential is a Gibbs-Sobolev potential for every exponent, so the phi^4 measure is its Gibbs measure, and the Galerkin Gibbs measures and their normalisers converge to it uniformly over Borel sets.Nelson's Exponential Integrability for the Wick-Ordered Potential on the Torus of Dimension at Most Two
theoremthm:nelson-exponential-integrability-torus-hilbert-2026aAnalysisProbabilityNelson's estimate on the torus of dimension at most two: for every t>0, exp(-tV_N) is bounded in mean under the free field uniformly in the cutoff, exp(-tV) is integrable, and exp(-tV_N) converges to exp(-tV) in every L^p.The Wick-Ordered Potential on the Torus of Dimension at Most Two: Limit of the Galerkin Potentials Almost Everywhere and in Every , Conditional Expectations, Cameron-Martin Derivatives, and the Lower Bound at Each Cutoff
lemmalem:free-field-phi4-potential-torus-hilbert-2026aAnalysisProbabilityThe phi^4 potential is Borel, is the almost-everywhere and L^p limit of the Galerkin potentials with rate N^(-1/8), has them as conditional expectations, and lies in the Cameron-Martin Sobolev space with derivatives in every L^p; each Galerkin potential is bounded below by a cons…The Wick-Ordered Potential of the Free Field on the Torus of Dimension at Most Two
definitiondef:free-field-phi4-potential-torus-hilbert-2026aAnalysisProbabilityDefines the Wick-ordered phi^4 potential of the free field on the torus of dimension at most two as the pointwise limit of the Galerkin potentials along dyadic cutoffs, and zero where that limit does not exist.The Galerkin Wick-Ordered Potentials on the Torus of Dimension at Most Two under the Free Field: Chaos Order, Conditional Expectations, and Increments with Their Cameron-Martin Derivatives
lemmalem:free-field-galerkin-phi4-chaos-torus-hilbert-2026aAnalysisProbabilityIn dimension at most two the Galerkin Wick-ordered phi^4 potentials lie in the chaos up to order four, each is the conditional expectation of a later one given the cube modes, and their increments and Cameron-Martin derivative increments are bounded in square mean by a constant t…Lattice Sums of Powers of the Fourier Weights in Dimension at Most Two: Power Sums, the Growth of the Wick Constants, and the Quartic Convolution Sums
lemmalem:lattice-quartic-sums-torus-2026aAnalysisIn dimension at most two, powers of the Fourier weights are summable above half the dimension, the Wick constants grow slower than any power of the cutoff, and the quartic convolution sums with one weight relaxed by a small power are finite.