From UD Theory to Black Holes and Vacuum: A Unified Derivation of the Two Extremes

This paper is Part VI of the UD theory foundational series, which establishes UD theory as a unified metatheoretical framework for physics. Part I derived general relativity from the scalar projection. Part II derived quantum mechanics from the spinor projection. Part III derived electromagnetism from the vector projection. Part IV derived cosmology from the homogeneous and isotropic limit. Part V derived quark properties and gauge symmetries from first principles. This paper examines the two extreme limits of UD theory: the black hole limit (D → 1, U → 0) and the vacuum limit (U → 1, D → 0). These two limits are perfectly symmetric under U ↔ D exchange. In the black hole limit, the U attribute is compressed to its extreme. Its isotropic nature forces it to form a sphere. The critical geometric ratio of circumference to diameter is L/ (2r) = π. This is the Singularity Rejection Index—the critical ratio of expansive effect to condensative effect at which DU (quantum fluctuations) provides repulsive pressure that prevents singularity formation. The irrationality of π is explained by the fact that DU is the source and π is the phenomenon: quantum uncertainty manifests as infinite non-repetition. In the vacuum limit, the U ↔ D symmetry maps the Singularity Rejection Index π to the Vacuum Rejection Index. The expansive attribute U has exponential growth as its natural behavior (base e), while D provides the geometric constraint (π). The unique dimensionless combination respecting both is e^π. This prevents absolute vacuum formation. For a static black hole, we prove that in the exterior region (r > rH), the scalar field φ = C/E is frozen at its cosmic background value φ₀ = 8/17. The field equations reduce to G_μν = 0, yielding the Schwarzschild solution. Consequently, all observable predictions—horizon radius rH = 2GM/c², photon sphere r = 3GM/c², shadow size, and gravitational wave ringdown frequencies—are identical to GR, consistent with all current observations (Event Horizon Telescope, LIGO/Virgo/KAGRA). In the deep interior (r ≪ rH), the scalar field increases toward the limiting value φBH = 4e^π, defined by the equilibrium four-aspect ratio in the extreme collapse regime. The DU field remains non-zero and provides quantum repulsive pressure that halts collapse at a finite radius rₘin > 0. No singularity forms. In the vacuum limit, the total normalization axiom forbids all four aspects from being zero simultaneously. Absolute vacuum does not exist in UD theory. The so-called "vacuum catastrophe"—the 120-order-of-magnitude discrepancy between QFT vacuum energy and observed dark energy—is a pseudo-problem. QFT vacuum energy is a mathematical fiction; dark energy is UU, a physical entity whose value is determined by cosmic evolution. The two extremes exhibit perfect U ↔ D symmetry, with π and e^π as dual structural constants. Their product πe^π is the fundamental structural constant of UD theory. Series DOI links: Part I (GR): doi: 10. 5281/zenodo. 19494052Part II (QM): doi: 10. 5281/zenodo. 19494646Part III (EM): doi: 10. 5281/zenodo. 19497862Part IV (Cosmology): doi: 10. 5281/zenodo. 19498260Part V (Particle Physics): doi: 10. 5281/zenodo. 19498834Part VI (Black Holes and Vacuum): this paper