Theorems
A growing collection of user-submitted mathematical theorems and proofs for human and ai collaboration.
The Half-Relaxed Limits of the Regularised N-Particle Dyson Solutions are Viscosity Sub- and Supersolutions of the Limit Dyson Equation up to a Cost Defect
propositionprop:dyson-half-relaxed-limits-viscosity-2026aAnalysisProbabilityPDEThe upper half-relaxed limit of the sup-convolved N-particle Dyson solutions divided by N is a viscosity subsolution, and the lower half-relaxed limit of the inf-convolved ones a viscosity supersolution, of the Dyson Hamilton-Jacobi equation on the Wasserstein space without commo…Gibbs Maximisers for the Regularised N-Particle Dyson Solutions Tested at a Local Extremum: Energy Bound, Concentration and the Score Identity
lemmalem:dyson-localised-gibbs-concentration-2026aAnalysisProbabilityPDEAt a point where the half-relaxed limit minus the penalty is locally dominated by a linear function plus a small multiple of the distance, the Gibbs maximisers of the regularised N-particle solution minus N times the quadratic block test function minus the penalty have bounded en…Half-Relaxed Limits of the Sup- and Inf-Convolved N-Particle Dyson Solutions: Bounds, Continuity and Limits along Configurations of Bounded Energy
lemmalem:dyson-regularised-half-relaxed-limits-2026aAnalysisProbabilityPDERegularise the N-particle Dyson solution v_N by a sup-convolution and an inf-convolution with wall weight 1/N and fixed parameter tau. The upper and lower half-relaxed limits of the regularised solutions divided by N are bounded, satisfy an explicit W2 modulus on the finite-energ…Upper and Lower Half-Relaxed Limits along Empirical Measures of Functions on the Weyl Chambers
definitiondef:empirical-half-relaxed-limits-2026aAnalysisProbabilityFor functions f_N on the Weyl chambers of linear growth in N, the upper half-relaxed limit at a measure of finite confined logarithmic energy is the infimum, over radii and starting indices, of the supremum of f_N(x)/N over all later N and all configurations x whose empirical mea…The Mean-Field Limit of the Relative Score of Laws of Dyson Particles Concentrating at a Measure
lemmalem:dyson-particle-score-limit-2026aAnalysisProbabilityIf laws of N Dyson particles have relative scores with bounded Fisher term per particle and their empirical measures concentrate in W2 at a measure of finite confined logarithmic energy, then that measure has a score in the confined logarithmic-energy pair, the particle scores pa…The Discrete Dyson Energy per Particle: Bounds, Compactness with a Lower Limit, and Recovery Configurations with Separated Particles
lemmalem:dyson-discrete-energy-limits-2026aAnalysisProbabilityThe N-particle Dyson potential divided by N controls the second moment of the empirical measure; configurations of bounded energy per particle have empirical measures converging in W2 along a subsequence to a measure of finite confined logarithmic energy, which is at most the low…Quantile Blocks of an Atomless Measure on the Line: Block Maps, Distances to Ordered Empirical Measures, and the Block Test Function
lemmalem:quantile-blocks-line-2026aAnalysisProbabilityAn atomless measure on the line is cut into N ordered blocks of mass 1/N. The block map of an ordered configuration pushes the measure to its empirical measure and is optimal against every monotone transport of the measure, which gives the Wasserstein distance of ordered empirica…The N-Particle Dyson Game and its Mean-Field Limit: Standing Data and Equations
settingset:dyson-n-particle-2026aAnalysisProbabilityPDEStanding data for the mean-field limit of the N-particle Dyson game: interaction beta, noise sigma with sigma^2 < beta, a confining potential V, a discount lambda and a bounded uniformly continuous cost g; the N-particle Hamilton-Jacobi equation on the Weyl chamber with its solut…The N-Particle Dyson Hamilton-Jacobi Equation with a Mean-Field Running Cost
equationeq:dyson-n-particle-hamilton-jacobi-2026aAnalysisProbabilityPDEThe Hamilton-Jacobi equation of N controlled particles on the Weyl chamber with logarithmic repulsion of strength beta/(2(N-1)), confinement V, idiosyncratic noise of intensity sigma^2/(N-1), quadratic control cost and running cost N times a function of the empirical measure.Gradient Maps of a Function on an Open Subset of Euclidean Space
definitiondef:gradient-map-euclidean-2026bAnalysisA gradient map of a real function on an open subset of Euclidean space is a Borel vector field on the whole space that agrees with the gradient of the function at every point of the open set outside a Lebesgue-null set, the function being differentiable there.Gibbs Maximisers of the Relative Free Energy: the Variational Principle and the Relative Score
lemmalem:gibbs-maximiser-relative-free-energy-euclidean-2026bAnalysisProbabilityFor a convex open D, a penalty U on D with monotone gradient and Gibbs-integrable tails, a temperature a>0, and a semiconvex f bounded above with |Df|^2 <= A + B(U - min U), the Gibbs measure pi_f with density proportional to exp((f-U)/a) on D lies in the domain of the relative f…The N-Particle Dyson Potential on the Weyl Chamber: Penalty, Monotone Gradient, Dissipation with Constants Uniform in N, and Gibbs Integrability
lemmalem:dyson-n-particle-data-weyl-chamber-2026aAnalysisPDEFor a confining potential V on the line and sigma^2 < beta, the N-particle Dyson potential P_N = H_{b_N} + sum V(x_k) with b_N = beta/(2(N-1)) is a penalty on the Weyl chamber with monotone gradient, satisfies the dissipation inequality with noise kappa_N = sigma^2/(N-1), epsilon…Integrating a Semiconvex Viscosity Subsolution of the Penalty-Drift Equation against a Measure of Finite Relative Fisher Information
lemmalem:semiconvex-subsolution-weak-form-euclidean-2026aAnalysisProbabilityPDEFor bounded w on an open convex D with |Dw|^2 <= A + B(U - p0), the a.e. gradient of a continuous w is Borel and square-integrable against measures of finite relative free energy. A semiconvex viscosity subsolution of the penalty-drift operator satisfies, for every mu of finite r…Sup- and Inf-Convolutions of Bounded Viscosity Sub- and Supersolutions of the Penalty-Drift Equation with a Convex Penalty
lemmalem:penalty-drift-sup-inf-convolution-euclidean-2026aAnalysisPDEFor the penalty-drift equation with a penalty whose gradient is monotone and which satisfies the dissipation inequality, the penalised sup-convolution of a bounded viscosity subsolution (and the penalised inf-convolution of a bounded viscosity supersolution) is attained at points…The Relative Free Energy and the Relative Score of a Probability Measure for a Potential on an Open Set
definitiondef:relative-free-energy-score-euclidean-2026aAnalysisProbabilityFor a twice continuously differentiable potential on an open set and a positive temperature, the relative free energy of a probability measure concentrated on the set is temperature times entropy plus the integral of the potential, and its relative score is the gradient of the po…The Mean-Field Limit of the Lifted N-Particle Hamilton-Jacobi Equation with a Mollified Local Coupling
theoremthm:n-particle-local-coupling-limit-wasserstein-2026aAnalysisProbabilityPDEWith eps_N -> 0 and N eps_N^d -> infinity, the bounded solutions U_N of the lifted N-particle equations with the mollified N-particle costs satisfy U_N(mu^{tensor N})/N -> u(mu) uniformly on energy sublevel sets, and are asymptotically bounded below by u at one-particle marginals…Lower Convergence: The Solution with the Local Cost Is Asymptotically below the Mean-Field Solutions with the Mollified Costs
corollarycor:n-particle-mollified-lower-convergence-wasserstein-2026aAnalysisProbabilityPDEIf eps_N tends to 0, the bounded solution of the equation with the local density cost is, uniformly on energy sublevel sets, asymptotically below the bounded solutions of the mean-field equations with the mollified mean-field costs.Upper Convergence: Mean-Field Solutions with the Tensor-Averaged Mollified Costs Are Asymptotically below the Solution with the Local Cost
corollarycor:n-particle-mollified-upper-convergence-wasserstein-2026aAnalysisProbabilityPDEIf N eps_N^d tends to infinity, the bounded solution of the mean-field equation with the tensor-averaged cost of the mollified N-particle cost is, locally uniformly on energy sublevel sets, asymptotically below the bounded solution of the equation with the local density cost.The Mollified N-Particle Cost Dominates N Times the Mollified Mean-Field Cost of the One-Particle Marginal
lemmalem:mollified-cost-marginal-domination-wasserstein-2026aAnalysisProbabilityPDEFor every P in P_2(R^{dN}), N g_eps(P^{[1]}) <= int c_{N,eps} dP: the linear part is exact through the one-particle marginal and the density part follows from Jensen's inequality pointwise and Tonelli's theorem.The Mollified N-Particle Cost: Regularity, Its Tensor-Averaged Cost, and the Defect against the Local Cost
lemmalem:mollified-cost-tensor-average-wasserstein-2026aAnalysisProbabilityPDEThe mollified mean-field and N-particle costs are bounded and uniformly continuous; the tensor-averaged cost of the N-particle cost is the linear cost plus the tensor average of the mollified density cost at the empirical measure; and its excess over the local cost is bounded by…