PNMS Case Study 9 — Einstein Field Equations: Geometric and Energetic Sides in Signature Space

Case study applying the Property-Neutral Matrixscope (PNMS) to the Einstein field equations by projecting both the geometric side (Gμν) and the energetic side (Tμν) into Planck-normalized signature space using the Φ_R operator. The study analyzes Schwarzschild black holes across three mass scales (10 M☉, 10⁴ M☉, 10⁹ M☉). Geometric quantities (Schwarzschild radius, horizon area, Schwarzschild time) and energetic quantities (rest energy, Hawking temperature, Bekenstein entropy) are projected independently and then compared using PNMS analysis classes. Structural results recovered from the signature tables include the invariant product (E/Eₚ)·(T/Tₚ) = 3.9789 × 10⁻², the thermodynamic Smarr relation T_H·S_BH = E/2, and a constant geometric–energetic scaling relation between r_s and E. The study demonstrates how the two sides of the field equations organize relative to each other in Planck-normalized signature space and evaluates their structural coupling using distance analysis