Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
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 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.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 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.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…Polynomial Controls on Unitary Laws and Their Energy
definitiondef:nc-unitary-polynomial-controls-2026aProbabilityPDEA polynomial control is a d-tuple of self-adjoint word polynomials; its energy under a unitary law is the sum of the law's values on the squares of its entries.Well-Posedness for Polynomial Sources in the Quadratic Control Problem on Unitary Laws
corollarycor:nc-unitary-quadratic-polynomial-source-2026aProbabilityPDEA source given by the real part of a finite combination of word moments is Gamma-Lipschitz in the gauge distance with an explicit Gamma, so the quadratic control equation is well posed with a Gamma/rho-Lipschitz solution when Gamma is at most rho beta^2/(4K).Well-Posedness and Lipschitz Regularity for the Quadratic Control Problem on Unitary Laws under Strong Free Noise
theoremthm:nc-unitary-quadratic-well-posedness-2026aProbabilityPDEUnder strong free noise, L_f at most rho beta^2/(4K), the quadratic control equation on unitary laws has exactly one viscosity solution; it is L_f/rho-Lipschitz in the gauge distance and bounded by sup|f|/rho.Lipschitz-Cone Comparison for the Quadratic Control Problem on Unitary Laws under Strong Free Noise
theoremthm:nc-unitary-quadratic-cone-comparison-2026aProbabilityPDEIf the source is L_f-Lipschitz in the gauge distance with L_f at most rho beta^2/(4K), every subsolution minus every supersolution of the quadratic control equation is bounded by L times the gauge distance for L between L_f/rho and beta^2/(4K); hence comparison, uniqueness, and t…The Quadratic Control Hamiltonian: Legendre Formula, Bounds, Differences in the Law and Admissibility
lemmalem:nc-unitary-quadratic-hamiltonian-2026aProbabilityPDEThe quadratic control Hamiltonian equals the limit of half the sum of the law's values on the squares of the truncated cyclic gradients, lies between 0 and kappa_d^2 |p|^2 / 2, has the corresponding limit formula for differences in the law, and is an admissible Hamiltonian.The Quadratic Control Hamiltonian on Unitary Laws
definitiondef:nc-unitary-quadratic-hamiltonian-2026aProbabilityPDEThe quadratic control Hamiltonian at a unitary law and a momentum is the supremum, over polynomial feedback controls, of minus the pairing of the momentum with the feedback drift minus half the control energy.Feedback Drifts: the Energy Bound in the Word Gauge, the Momentum Pairing and Two-Sided Tangency
lemmalem:nc-unitary-feedback-drift-2026aProbabilityPDEFeedback drifts lie in the word gauge space with norm at most kappa_d times the square root of the control energy, pair with a momentum p as the limit of the traces of the control against the truncated cyclic gradients of p, and are two-sided tangent vectors to the space of unita…Feedback Drifts of Polynomial Controls on Unitary Laws
definitiondef:nc-unitary-feedback-drift-2026aProbabilityPDEThe feedback drift of a polynomial control a at a unitary law lambda assigns to each cyclically reduced word w the sum over i of lambda(a^i D^i_w), where D^i_w is the i-th cyclic derivative of w.