Poisson Scale Dipole

Abstract

The off-line curvature dipole is the scale derivative of the Poisson kernel.

Theorem 1.1 (The off-line curvature dipole is a Poisson scale derivative).

$$\begin{gathered} \forall delta,gamma\in \mathrm{Real}\left(\right), \\ 0<delta\Rightarrow \\ \mathrm{let}poissonKernel:=(scale,x\mapsto \frac{scale}{pi\times ((scale{)}^2+(x{)}^2)}), \\ \mathrm{let}curvatureDipole:=(t\mapsto 2\times \frac{(t-gamma{)}^2-(delta{)}^2}{((t-gamma{)}^2+(delta{)}^2{)}^2}), \\ \left(\forall t\in \mathrm{Real}\left(\right),curvatureDipole\left(t\right)=2\times pi\times \mathrm{deriv}\left((scale\mapsto poissonKernel\left(scale,t-gamma\right)),delta\right)\right)\wedge \left(\mathrm{Integrable}\left(curvatureDipole\right)\wedge \mathrm{integral}\left(t,\mathrm{Real}\left(\right),curvatureDipole\left(t\right),\mathrm{volume}\left(\right)\right)=0\right). \end{gathered}$$

Proof. Machine-checked in Lean as D5/S3/Analytic/Curvature/PoissonScaleDipole.poisson_scale_dipole (✓ std3). ∎

Source. Repository-derived.

Commentary.

The pointwise identity differentiates the actual real Poisson kernel in its positive scale parameter. Integrability and zero total mass are transported from the frozen off-line curvature theorem, so this is a representation bridge and introduces no RH premise.

References

• Truth anchor: D5/S3/Analytic/Curvature/PoissonScaleDipole.poisson_scale_dipole
• Dependency: D5/S3/Analytic/Adelic/OffLineCurvatureDipole