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 $\Pi_q$ converges to an effective operator $\mathcal{L}_{\mathrm{eff}}$ on $\mathbb{R}_\tau \times \mathrm{Heis}_3(\mathbb{R})$ whose principal symbol defines a four-dimensional effective metric tensor $g^{\mu\nu}(x) \propto A^{\mu\nu}(x)$ 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 $\Pi_q \to \mathcal{L}_{\mathrm{eff}} \to g^{\mu\nu} \to G_{\mu\nu}$. Q9 proves hypothesis [H-lift], so the main geometric results are unconditional with respect to that hypothesis. No dynamical substrate is required at any step. Interpretive reading. The construction treats spacetime geometry as an effective propagation structure selected by admissibility, rather than as primitive substrate data.