Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Wall-Confined Free Energy and Its Score
definitiondef:nc-wall-confined-free-energy-2026aAnalysisProbabilityThe wall-confined free energy is a free entropy penalty plus the wall energy of radius R; its score, defined where the conjugate variables and the wall force exist, is the wall force minus the conjugate variables.The Conjugate Variables of a Free Entropy Penalty are Displacement Monotone along Optimal Couplings
lemmalem:nc-free-entropy-penalty-monotone-2026aAnalysisProbabilityAdding the tangent inequalities of a free entropy penalty at the two ends of an optimal coupling shows that the conjugate variables are displacement monotone (decreasing) along it.Free Entropy Penalties: Displacement Convexity with Minus the Conjugate Variables as Gradient
definitiondef:nc-free-entropy-penalty-2026aAnalysisProbabilityA free entropy penalty is a real function on a set of noncommutative laws, bounded below on norm balls, with weak-star closed sublevel sets, satisfying the tangent inequality along optimal couplings with gradient minus the conjugate variables.The Wall Energy: Finiteness Forces the Norm Bound, Closed Sublevel Sets, and Monotonicity of the Wall Force along Couplings
lemmalem:nc-wall-energy-2026aAnalysisProbabilityLaws of finite wall energy have norm bound R and nonnegative energy; sublevel sets of the wall energy are weak-star closed; and the wall force is monotone along every coupling.The Wall Energy of a Given Radius and the Wall Force of a Noncommutative Law
definitiondef:nc-wall-energy-2026aAnalysisProbabilityDefines the wall energy of radius R of a noncommutative law as the series of its even moments scaled by powers of R, its finiteness domain, and the wall force as the limit in the GNS space of the differentiated partial sums.Monotonicity of Odd Powers under a Tracial State
lemmalem:nc-odd-power-trace-monotone-2026aAnalysisProbabilityFor self-adjoint noncommutative polynomials a and b and an odd power n, a tracial state takes a real nonnegative value on the product of a^n - b^n with a - b.Marginal Isometries, Bounded Plans and Displacement Pairings for Noncommutative Laws
definitiondef:nc-marginal-pairings-2026aAnalysisProbabilityFixes the isometries from the GNS spaces of the two marginals into that of a joint law, bounded plans with their momentum norm, and the pairings of GNS vectors with the momentum of a plan and with the displacement of a coupling.The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law
definitiondef:nc-conjugate-variables-2026aAnalysisProbabilityDefines the free difference quotient of a noncommutative law as a linear functional on polynomials, and the conjugate variables as the vectors of the GNS space that represent these functionals.Uniqueness of a Continuous Viscosity Solution of the Mean-Field Control Equation on the Torus
corollarycor:torus-control-uniqueness-2026aProbabilityPDEFor the control Hamiltonian of a control datum, a continuous viscosity subsolution lies below a continuous viscosity supersolution, so the mean-field control equation has at most one continuous viscosity solution.The Discounted Hamilton-Jacobi-Bellman Equation of Mean-Field Control with Idiosyncratic Noise on the Torus
equationeq:torus-control-hjb-2026aProbabilityPDEThe discounted Hamilton-Jacobi-Bellman equation on the torus Wasserstein space with the control Hamiltonian of a control datum.The Control Hamiltonian is Continuous Along Couplings and Satisfies the Score-Perturbation Bound and the Structure Condition
lemmalem:torus-control-hamiltonian-conditions-2026aProbabilityPDEThe control Hamiltonian of a control datum satisfies all three conditions of the torus comparison theory, with explicit constants in the score-perturbation bound.Control Data and the Control Hamiltonian on the Torus Wasserstein Space
definitiondef:torus-control-hamiltonian-2026aProbabilityPDEA control datum is a bounded drift and a bounded running cost on the torus that are Lipschitz in the position and the measure. Its control Hamiltonian integrates half the squared momentum minus the drift times the momentum minus the running cost.Comparison and Uniqueness for Continuous Viscosity Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space
corollarycor:torus-comparison-uniqueness-2026aProbabilityPDEUnder continuity along couplings, the score-perturbation bound and the structure condition, a continuous viscosity subsolution lies below a continuous viscosity supersolution, so the equation has at most one continuous viscosity solution.Viscosity Sub- and Supersolutions on the Torus Wasserstein Space are Entropy-Penalised Viscosity Sub- and Supersolutions
theoremthm:torus-viscosity-penalised-bridge-2026aProbabilityPDEIf the Hamiltonian is continuous along couplings and satisfies the score-perturbation bound, every upper semicontinuous viscosity subsolution is an entropy-penalised viscosity subsolution, and likewise for lower semicontinuous supersolutions. The proof regularises the test functi…Comparison Principle for Entropy-Penalised Viscosity Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space
theoremthm:torus-penalised-comparison-2026aProbabilityPDEIf the Hamiltonian satisfies the structure condition, an upper semicontinuous entropy-penalised subsolution lies below a lower semicontinuous entropy-penalised supersolution at every measure of finite entropy.Entropy-Penalised Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space
definitiondef:torus-penalised-viscosity-solution-2026aProbabilityPDESub- and supersolutions tested at strict extrema of the function minus an intrinsic test function and a small multiple of the entropy. The extremal measure must have finite Fisher information, and the score form of the equation must hold there up to a tolerance proportional to th…Viscosity Subsolutions, Supersolutions and Solutions of the Discounted Hamilton-Jacobi-Bellman Equation on the Torus Wasserstein Space
definitiondef:torus-viscosity-solution-2026aProbabilityPDEViscosity sub- and supersolutions tested by Laplacian test functions at global maxima and minima, with the Laplacian form of the equation holding as an inequality at the touching point.The Discounted Hamilton-Jacobi-Bellman Equation with Idiosyncratic Noise on the Torus Wasserstein Space
equationeq:torus-hjb-2026aProbabilityPDEThe discounted equation: discount rate times U, plus the Hamiltonian at the gradient, minus half the squared noise intensity times the Laplacian, equals zero. It is also recorded in score form, where the Laplacian is replaced by minus the pairing of the gradient with the score.The Structure Condition Along Optimal Maps for a Hamiltonian on the Torus Wasserstein Space
definitiondef:torus-hamiltonian-structure-2026aProbabilityPDEA Hamiltonian satisfies the structure condition if, at two absolutely continuous measures joined by optimal maps, its values at the scaled displacement momenta differ by an amount that is small when the scaled squared distance plus the distance is small.The Score-Perturbation Bound for a Hamiltonian on the Torus Wasserstein Space
definitiondef:torus-hamiltonian-score-perturbation-2026aProbabilityPDEA Hamiltonian satisfies the score-perturbation bound if a small multiple of a tangent field changes it by at most a quarter of the noise intensity squared times that multiple times the squared norm of the field, plus a term linear in the multiple that is uniform over bounded mome…