Effective Spacetime Geometry from Admissible Non-Injective Projection

Continuing from the gauge structure established in the companion paper Q6a , we derive effective spacetime geometry from the admissibility framework of Foundation M by importing the continuum limit and geometric-extraction theorems of the companion paper Q5b . Under the lifting hypothesis [H-lift] of Q5b and the Mosco hypotheses of Q5a , the admissibility filter Πq converges to an effective operator Leff on R_τ × Heis₃(R) whose principal symbol defines a four-dimensional effective metric tensor gμν(x) ∝ Aμν(x) of Lorentzian signature $(-,+,+,+)$ (Q5b Theorems 5.2 and 6.1). In the presence of a localised stationary obstruction to the admissibility flow with spherical symmetry, flux conservation within this effective geometry forces the Schwarzschild metric as the unique stationary exterior solution, with the horizon appearing as a degeneracy of the principal symbol rather than as a singularity of the underlying admissible structure. The Einstein equations emerge as consistency conditions of the emergent geometry, with the Einstein Hilbert action derived rigorously from the spectral entropy functional in the Gravity paper . The present paper closes the chain Πq → Leff → gμν → Gμν. Hypothesis [H-lift], on which the results were originally conditional, has since been proved in Q9 ; the main geometric results are therefore unconditional with respect to [H-lift]. No dynamical substrate is required at any step.