Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Compatible Noise Penalty Pairs at a Fine and a Coarse Level
definitiondef:compatible-noise-penalty-pairs-hilbert-2026aAnalysisProbabilityTwo noise penalty pairs at a fine and a coarse level, joined by a mode-restriction link, are compatible with constant c_0 if the push-forward under the mode restriction maps the fine penalty and score domains into the coarse ones and raises the penalty by at most c_0.Two Noise Levels over Hilbert Spaces: a Fine and a Coarse Reading of the Noise Framework and Level-Indexed Notation
settingset:noise-levels-hilbert-2026aAnalysisProbabilityFixes two readings, a fine and a coarse level, of the standing noise Wasserstein and noise penalty framework over two Hilbert spaces with their bases, noise weights and reference measures, together with the level-indexed notation (noise norms, field norms, couplings, distances, t…Comparison, Uniqueness and Convergence of Coarse Solutions Inherited by a Fine First-Order Equation on the Noise Wasserstein Space from a Consistent Family of Coarse Levels
corollarycor:inherited-comparison-first-order-hilbert-2026aAnalysisProbabilityUnder the hypotheses of the cross-level comparison principle, if every coarse equation has a viscosity solution and these are bounded uniformly in the level, then the fine equation satisfies comparison and uniqueness for bounded solutions, and a bounded fine solution is the limit…A Cross-Level Comparison Principle for First-Order Equations on a Fine Noise Wasserstein Space and a Family of Coarse Levels Linked by Mode Restrictions
theoremthm:cross-level-comparison-first-order-hilbert-2026aAnalysisProbabilityIf a family of coarse noise levels, linked to a fine level by mode restrictions, satisfies the hypotheses of the single-level comparison principle with constants uniform in the level, and the consistency defect between the fine and the coarse operators vanishes as the level grows…Bounds, Monotonicity and Borwein-Preiss Perturbed Maximisers of the Cross-Level Doubled Function in the Coarse Noise Wasserstein Distance for Compatible Noise-Closed Penalty Pairs
lemmalem:cross-level-doubling-perturbed-maximiser-hilbert-2026aAnalysisProbabilityFor compatible noise-closed penalty pairs at a fine and a coarse level and upper semicontinuous functions dominated by minus delta times the penalties, the doubled function U(mu)+V(nu)-(alpha/2)W(p_#mu,nu)^2 has a finite supremum, controlled by its diagonal, monotone in the doubl…Cross-Level Data and Eta-Consistency of a Fine and a Coarse Level at (delta, R)
definitiondef:cross-level-consistency-hilbert-2026aAnalysisProbabilityFor compatible noise penalty pairs at a fine and a coarse level linked by a mode restriction, defines cross-level data (a fine measure, a real number and a coarse tangent field) and their R-boundedness, and calls the two levels eta-sub-, eta-super- or eta-consistent at (delta, R)…Noise Intrinsic Test Functions Pulled Back along a Mode Restriction, and the Squared Coarse Noise Wasserstein Distance as a Fine Test Function
lemmalem:restricted-noise-test-function-hilbert-2026aAnalysisProbabilityComposing a coarse-level noise intrinsic test function with push-forward by the mode restriction gives a fine-level test function whose gradient is the pulled-back coarse gradient; in particular the squared coarse distance from the restricted measure to a fixed measure is a fine…Properties of a Mode-Restriction Link: Isometric Embedding, Contraction of Noise Norms and Noise Wasserstein Distances, and Pull-Back of Cylindrical Functions and Tangent Fields
lemmalem:mode-restriction-link-hilbert-2026aAnalysisProbabilityThe mode restriction p of a mode-restriction link is a left inverse of the isometric mode embedding j; both respect the noise norms and are adjoint for them, push-forward by p contracts noise costs and noise Wasserstein distances, and composing with p and j carries cylindrical fu…Mode-Restriction Links between a Fine and a Coarse Noise Level
definitiondef:mode-restriction-link-hilbert-2026aAnalysisProbabilityDefines a mode-restriction link from a fine to a coarse noise level: an injective relabelling of basis modes that preserves the noise weights, together with its mode restriction p and its mode embedding j.The Hubbard-Stratonovich Ising Field on the Lattice Torus: What the Lattice Results Model, the Role of the Choices, and the Temperature Regimes
remarkrem:hubbard-stratonovich-ising-regimes-torus-hilbert-2026aProbabilityExplains why the Ising model enters through the Hubbard-Stratonovich field: spin laws are singular to every Gaussian. It also covers which choices are canonical (the kernels) and which are not (the shift, the split, the lattice-to-torus embedding), and when the convex theory appl…The Gibbs Entropy Pair of the Hubbard-Stratonovich Ising Field is Lambda-Displacement Convex, with Lambda Uniform in the Cutoff when the Shifted Symbol Stays below One
lemmalem:hubbard-stratonovich-ising-convex-torus-hilbert-2026aProbabilityPDEThe Gibbs entropy pair of the Hubbard-Stratonovich Ising field is (1/r-bar_s - K_s)-displacement convex for every admissible split; when the shifted symbol stays below q-bar < 1 it is lambda(q-bar)-displacement convex with lambda(q-bar) = min{1/(1+q-bar), (1-q-bar)/q-bar} > 0, in…Well-Posedness of the Hamilton-Jacobi Equation of the Hubbard-Stratonovich Ising Field on the Lattice Torus at Every Cutoff, Interaction and Shift: Comparison, Existence and Uniqueness
corollarycor:hubbard-stratonovich-ising-hamilton-jacobi-well-posed-torus-hilbert-2026aProbabilityPDEFor every cube cutoff, reflection-symmetric periodic pair interaction and shift, the Hamilton-Jacobi equation driven by the Gibbs score of the Hubbard-Stratonovich Ising field satisfies comparison and has exactly one bounded viscosity solution, for every bounded uniformly continu…The Hamilton-Jacobi Equation of the Hubbard-Stratonovich Ising Field on the Lattice Torus with White Noise
equationeq:hubbard-stratonovich-ising-hamilton-jacobi-torus-hilbert-2026aProbabilityPDEThe Hamilton-Jacobi equation of the Hubbard-Stratonovich Ising field is the Hamilton-Jacobi equation with Gibbs score drift whose Gibbs measure is the Hubbard-Stratonovich field law, written through any admissible split of its Gaussian. Its penalty pair, and hence its operator, i…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…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…