Observer-Mode Critical Kernel

Abstract

The symmetric completed-zeta digamma difference has its cosine kernel and is strictly positive on the zero-frequency axis for every nonzero shift.

Theorem 1.1 (The polarized digamma kernel and its axis positivity).

$$\forall t\in \mathrm{Real}\left(\right),tau\in \mathrm{Real}\left(\right),\mathrm{let}a:\mathrm{Real}\left(\right)\to \mathrm{Real}\left(\right)=u:\mathrm{Real}\left(\right)\mapsto \mathrm{re}\left(\mathrm{digamma}\left(\frac{1}{4}+I\left(\right)\cdot \frac{u}{2}\right)\right)-\log \left(\mathrm{pi}\left(\right)\right),\left(\forall u\in \mathrm{Real}\left(\right),\mathrm{IntegrableOn}\left(x:\mathrm{Real}\left(\right)\mapsto \mathrm{archimedeanJumpDensity}\left(x\right)\cdot \left(1-\cos \left(u\cdot x\right)\right),\mathrm{Ioi}\left(0\right),\mathrm{volume}\left(\right)\right)\wedge a\left(u\right)-a\left(0\right)=2\cdot \mathrm{setIntegral}\left(x,\mathrm{Real}\left(\right),\mathrm{Ioi}\left(0\right),\mathrm{archimedeanJumpDensity}\left(x\right)\cdot \left(1-\cos \left(u\cdot x\right)\right),\mathrm{volume}\left(\right)\right)\right)\Rightarrow \left(\frac{1}{2}\cdot \left(a\left(t+tau\right)+a\left(t-tau\right)\right)-a\left(t\right)=2\cdot \mathrm{setIntegral}\left(x,\mathrm{Real}\left(\right),\mathrm{Ioi}\left(0\right),\mathrm{archimedeanJumpDensity}\left(x\right)\cdot \cos \left(t\cdot x\right)\cdot \left(1-\cos \left(tau\cdot x\right)\right),\mathrm{volume}\left(\right)\right)\wedge \left(tau\ne 0\Rightarrow 0<\frac{1}{2}\cdot \left(a\left(0+tau\right)+a\left(0-tau\right)\right)-a\left(0\right)\right)\right)$$

Proof. Machine-checked in Lean as D5/S3/Weil/ZetaGamma/ObserverModeCriticalKernel.observer_mode_critical_kernel (✓ std3). ∎

Source. Repository-derived.

Commentary.

The multiplier a is displayed on its concrete real digamma carrier. The public premise is its positive-scale Levy representation, including integrability for every real frequency.

Polarization gives the cosine-modulated symmetric-difference kernel. At zero frequency, the imported Archimedean jump density and the nonzero shift produce a strictly positive integral.

References

• Truth anchor: D5/S3/Weil/ZetaGamma/ObserverModeCriticalKernel.observer_mode_critical_kernel
• Dependency: D5/S3/Weil/ZetaGamma/ArchimedeanJumpDecomposition