Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
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.- A family of real numbers indexed by the integer lattice is summed along the cubes of side 2N+1 centred at the origin; it is cube-summable when these finite sums converge, and its lattice sum is then their limit.
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.Rotations and Cyclic Derivatives of Words in Unitary Letters
definitiondef:nc-unitary-cyclic-derivative-2026aAlgebraProbabilityThe i-th cyclic derivative of a word is the word polynomial summing, over the occurrences of the i-th unitary or its inverse, the rotation of the word starting at the insertion point, with coefficient i times the sign of the letter.Evaluation of Word Polynomials by a Unitary Law
definitiondef:nc-unitary-law-evaluation-2026aAlgebraProbabilityA unitary law is extended linearly to word polynomials: lambda(X) is the sum of X(w) lambda(w) over a support set of X.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 Trilinear Estimate for Cyclic Gradients of Differences of Unitary Laws
lemmalem:nc-unitary-cyclic-gradient-trilinear-2026aAnalysisProbabilityFor two unitary laws with gauge difference x, the difference of the laws evaluated on the squared truncated cyclic gradients of x is bounded by a constant times the gauge norm of x times the square of its length-weighted gauge, uniformly in the truncation.