Emergent pi and eᵖi from the UD Potential: Singularity Resolution and the Exclusion of Absolute Vacuum

We derive the resolution of black hole singularities and the absence of absolute vacuum directly from the UD field equations. The structural constants π and e^π are obtained by rigorous analysis: π emerges as the first zero of the standing-wave quantization condition in the spherical black hole core, and e^π follows via analytic continuation of this condition across the exact U ↔ D duality boundary. In the black hole interior, confinement of the expansive field DU via local normalization together with the standing-wave condition yields the minimum radius rₘin = π/m₀. The same confinement drives a tachyonic instability diagnosed from the full 4×4 Hessian, generating repulsive quantum vacuum polarization that halts collapse. The analysis extends to rotating black holes via isotropization by DU vacuum polarization. Black hole evaporation follows from the non-vanishing DU energy-momentum tensor. In the exterior, the spatial condensate UD induces metric corrections that provide a testable prediction for the photon sphere radius. The U ↔ D duality proves that absolute vacuum cannot exist. The cosmological constant problem is resolved by distinguishing the fluctuation energy density of DU—confirmed by the Casimir effect—from the dark energy density carried independently by UU. All results follow from the UD action without additional hypotheses or free parameters.