Unified Role of Relative Scattering Determinant in Quantum Gravity: Two-Domain Framework, Fixed-Energy BK ($p\in\{1,2\

Taking ``relative determinant'' as unified object, we establish rigorous and verifiable theory in two types of geometric-physical scenarios: (C) relative \zeta/heat kernel determinant for Euclideanized second variation operator family in compact closed domain and its volume density response to cosmological constant term; (S) fixed-frequency scattering matrix on stationary exterior geometry (Schwarzschild--de Sitter/Kerr--de Sitter) with relative (modified) determinant, spectral shift object, and