Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Free-Field Variances of the Fourier Modes
definitiondef:free-field-variances-torus-2026aAnalysisProbabilityThe free-field variance of a mode is the noise intensity divided by twice its Fourier weight.- At cutoff N the gradient energy of a function at a state is the sum over the kept modes of its squared first mode derivatives; the uncut gradient energy is the limit in N where it exists.
How the Wick-Square Problem Is Renormalised: Equipartition, the Corrector Identity and the Gaussian Penalty
remarkrem:wick-square-renormalisation-2026bAnalysisProbabilityPDEExplains the Wick-square results: the counterterm is the equipartition energy of the free field, the singular cost is the dissipation of the Gaussian penalty under the free dynamics, and this is why the renormalised equation is written with the generator acting on the unknown min…The Galerkin Solutions of the Wick-Square Problem Converge to a Classical Solution of the Renormalised Equation, while the Bare Solutions Diverge
theoremthm:wick-square-hjb-solutions-2026aAnalysisPDEWith zero running cost, the explicit quadratic Galerkin functions built from the Riccati coefficients solve the Wick-ordered cutoff equations; shifted by the divergent equipartition sum they solve the bare cutoff equations and blow up for dimension at least two; the Wick-ordered…The Riccati Coefficients of the Wick-Square Problem: Existence, the Gaussian Profile and Summable Constants
lemmalem:wick-square-riccati-2026aAnalysisFor each mode the scalar Riccati equation of the Wick-square problem has exactly one positive root; it lies below the Gaussian profile, the cost strength over twice the weight, by at most a constant times the inverse squared weight, and the renormalised constants are nonpositive,…The Renormalised Operator of the Wick-Square Problem: the Penalised Form on the Whole State Space, the Wick Domain without the Penalty, and the Forced Counterterm
theoremthm:wick-square-renormalised-domain-2026aAnalysisPDEIf a test function is the Gaussian penalty plus a regular function, the Wick-ordered cutoff operators converge at every state of the Sobolev space of order -1, and the limit is the discount term minus the free-field generator of the test function minus the penalty, plus half the…Regular Functions: Mode Derivatives, the Free-Field Generator, the Gradient Energy, the Renormalised Operator at the Gaussian Penalty plus a Regular Function, and Linear Combinations
lemmalem:wick-regular-functions-2026bAnalysisPDEA regular function is twice differentiable along the modes with explicit mode derivatives; its free-field generator converges at every state to an explicit lattice sum; adding any multiple of the Gaussian penalty keeps the gradient energy convergent; the renormalised operator is…The Cutoff Wick Square is Minus the Free-Field Generator of the Gaussian Penalty, and the Penalised Form of the Cutoff Operators
lemmalem:gaussian-penalty-corrector-2026aAnalysisPDEAt every cutoff, the free-field generator applied to the Gaussian penalty is exactly minus the cost strength times the cutoff Wick square; consequently each cutoff operator equals the discount term minus the generator applied to the test function minus the penalty, plus half the…The Renormalised Hamilton-Jacobi-Bellman Operator of the Wick-Square Problem as the Limit of the Wick-Ordered Cutoff Operators
definitiondef:wick-square-renormalised-operator-2026aAnalysisPDEThe renormalised operator is the limit, as the cutoff is removed, of the Wick-ordered cutoff operators; its domain is the set of pairs of a test function and a state at which that limit exists.The Cutoff Hamilton-Jacobi-Bellman Operators of the Wick-Square Problem, with a General Counterterm
definitiondef:wick-square-cutoff-operators-2026aAnalysisPDEAt cutoff N the Hamilton-Jacobi-Bellman operator of the Wick-square problem is the discount term minus the free-field generator plus half the gradient energy, minus the cost strength times the sum over kept modes of the squared coordinate minus a counterterm, minus a running cost…Regular Functions for the Wick-Square Problem: Diagonal Quadratics of Curvature of Order the Inverse Squared Weight, plus Twice Differentiable Functions on the Sobolev Space of Order -3
definitiondef:wick-regular-functions-2026bAnalysisPDEA function on the state space is regular if it is a diagonal quadratic form whose coefficient at each mode is bounded by a constant times the inverse square of the weight, plus the restriction of a twice continuously differentiable function on the Sobolev space of order -3.The Gaussian Penalty of the Wick-Square Problem
definitiondef:gaussian-penalty-torus-2026aAnalysisPDEThe Gaussian penalty is the cost strength times half the squared norm of the Sobolev space of order -1, that is, the lattice sum of the squared coordinates divided by the weights.The Wick Square with a Mode Cutoff and the Wick Domain
definitiondef:wick-square-torus-2026aAnalysisProbabilityThe Wick square at cutoff N subtracts the free-field variance from the square of each kept coordinate and sums; the Wick domain is the set of states at which these cutoff Wick squares converge, and the Wick square is their limit there.The Free-Field Generator with a Mode Cutoff
definitiondef:free-field-generator-torus-2026aAnalysisPDEAt cutoff N the free-field generator applies to each kept mode half the noise intensity times the second mode derivative minus the weight times the coordinate times the first mode derivative; the uncut generator is the limit in N where it exists.Derivatives of a Function on the Sobolev Space of Order -1 along the Fourier Modes
definitiondef:mode-derivatives-torus-2026aAnalysisA function on the state space is twice differentiable along the modes when each of its restrictions to a line in the direction of a unit family is differentiable everywhere with a derivative differentiable at the base point; the first and second mode derivatives are these derivat…The Wick-Square Problem on the Torus: Standing Notation
settingset:wick-square-torus-2026aAnalysisPDEStanding notation for the Wick-square control problem on the torus of dimension at most three: Fourier modes and weights, the cubes of modes used as cutoffs, the Sobolev space of order -1 as state space with its unit families, and the positive parameters for noise, cost and disco…Lattice Sums of the Fourier Weights of the Torus, and the Negative Sobolev Norms as Lattice Sums
lemmalem:fourier-weight-lattice-sums-2026aAnalysisThe negative powers of the Fourier weights of the n-torus are summable over the lattice exactly in the range n < 2s used here, the inverse weights are not summable for n at least 2, and a coefficient family lies in the Sobolev space of order -m exactly when its weighted squares a…- The partial sums of the harmonic series are not bounded above, so the series does not converge.
Lattice Sums along Cubes: Linearity, Nonnegative Families, Absolute Summability, Comparison and Finitely Supported Families
lemmalem:lattice-cube-sums-2026aAnalysisCube-summable families form a vector space on which the lattice sum is linear; a nonnegative family is cube-summable exactly when its cube sums are bounded, its sum is their supremum and agrees with its series along any enumeration; absolute cube-summability and domination imply…The Cubes of the Integer Lattice are Finite, Nested, Exhaust the Lattice and Have (2N+1)^n Points
lemmalem:lattice-cubes-2026bAnalysisThe cubes of side 2N+1 in the integer lattice of dimension n are finite, increase with N, eventually contain every finite set of lattice points, and contain (2N+1)^n points.