Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
Limits of a Uniformly Bounded Sequence of Real Functions with a Common Modulus of Continuity: Bounds, Extraction and Uniform Convergence on Sequentially Compact Sets
lemmalem:equicontinuous-sequence-limits-metric-2026aAnalysisTopologyFor a uniformly bounded sequence of real functions on a metric space sharing a modulus of continuity, the pointwise limit superior and inferior keep the bound and the modulus and are attained along subsequences at each point; where they agree on a sequentially compact set, the co…Well-Posedness of the Renormalised Wick-Square Problem: Existence, Comparison and Uniqueness of Renormalised Viscosity Solutions Differing from the Free Solution by a Bounded Lipschitz Function
theoremthm:wick-square-viscosity-well-posed-2026aAnalysisPDEFor a bounded running cost that is Lipschitz in the norm of order -3, the renormalised Wick-square equation has exactly one renormalised viscosity solution differing from the free solution by a bounded Lipschitz function, comparison holds in that class, and for zero cost the solu…Lipschitz Regularity of Bounded Continuous Viscosity Solutions of a Viscous Hamilton-Jacobi Equation with a Monotone Nonlinearity on a Hilbert Triple
theoremthm:lipschitz-viscous-hamilton-jacobi-hilbert-triple-2026aAnalysisPDEA bounded continuous viscosity solution of a discounted viscous Hamilton-Jacobi equation on a Hilbert triple, with a Hamiltonian given by a form between zero and the identity, a monotone nonlinearity and a running cost that is Lipschitz for the norm of the large space, satisfies…Renormalised Viscosity Solutions versus Viscosity Solutions of the Shifted Equation on the Sobolev Triple of Order Two
lemmalem:wick-square-viscosity-bridge-2026aAnalysisPDEIf w is a bounded continuous viscosity sub- or supersolution of the shifted equation, then the free solution plus w is a renormalised viscosity sub- or supersolution; conversely, when w is moreover Lipschitz for the norm of order -3, the renormalised property of the free solution…Renormalised Viscosity Subsolutions, Supersolutions and Solutions of the Wick-Square Problem, Tested by the Gaussian Penalty plus a Regular Function
definitiondef:wick-square-viscosity-solution-2026aAnalysisPDEA function on the Sobolev space of order -1 is a renormalised viscosity subsolution (supersolution) if the renormalised operator is nonpositive (nonnegative) at the Gaussian penalty plus any regular function touching it from above (below), touching being local in the norm of orde…The Riccati Shift: the Renormalised Operator at the Free Solution plus a Regular Function is the Shifted Operator, Which Satisfies the Hilbert-Triple Comparison Hypotheses, and Its Touching Points Lie in the State Space
lemmalem:wick-square-riccati-shift-2026aAnalysisPDESubtracting the free solution turns the renormalised Wick-square operator into the shifted Hilbert-triple operator, which satisfies the hypotheses of the second-order comparison theory; and at every touching point of a continuous viscosity sub- or supersolution of the shifted equ…The Shifted Hamilton-Jacobi-Bellman Equation of the Wick-Square Problem on the Sobolev Triple of Order Two
equationeq:wick-square-hjb-shifted-2026aAnalysisPDENames the Hamilton-Jacobi-Bellman operator obtained from the renormalised Wick-square operator by subtracting the free solution: a discounted viscous equation on the Sobolev triple of order two with white noise, the gradient form as Hamiltonian, the drift one minus the Laplacian…The Wick-Square Problem on the Sobolev Triple of Order Two: Standing Notation
settingset:wick-square-triple-2026aAnalysisPDEStanding notation for the Wick-square problem on the Sobolev triple of orders -3 and -2: the triple and its Hilbert-triple notation, the white-noise sequence and gradient form, the Riccati data and drift, and the free solution of the renormalised equation.The Sobolev Triple of Order Two for the Wick-Square Problem: the Domain Is the Sobolev Space of Order -1, Unit Families Are Square-Summable, the Gradient Form and the Riccati Drift
lemmalem:wick-square-sobolev-triple-2026aAnalysisPDEFor the Sobolev triple of orders -3 and -2 on the torus in dimension at most three: the domain of the form operator is the Sobolev space of order -1, pairings with unit families are explicit, the space is infinite-dimensional, the unit families are square-summable in the form spa…Mode Derivatives of Linear Combinations and of Restrictions of Twice Differentiable Functions on a Negative Sobolev Space
lemmalem:mode-derivative-calculus-torus-2026aAnalysisMode derivatives are linear and vanish on constants, and the restriction to the Sobolev space of order -1 of a twice continuously differentiable function on a negative Sobolev space has mode derivatives given by its gradient and Hessian evaluated on the unit families.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 Equation of the Wick-Square Problem
equationeq:wick-square-hjb-renormalised-2026aPDEThe renormalised equation asks that at every state of the Sobolev space of order -1 the Wick-ordered cutoff operators applied to the unknown converge, with limit zero.The Cutoff Hamilton-Jacobi-Bellman Equations of the Wick-Square Problem, Wick-Ordered and Bare
equationeq:wick-square-hjb-cutoff-2026aPDEAt each cutoff, the Wick-ordered equation sets the Wick-ordered cutoff operator to zero at every state and the bare equation does the same for the bare cutoff operator.