Einstein Equations from Prismatic Curvature

We derive the Einstein field equations from a variational principle formulated in prismatic cohomology. Working in the Torsion-Modulated Recursive Branching (TMRB) prismatic framework, we equip the tangent crystal T_X with its canonical prismatic p-connection and define prismatic curvature R_prism. Contractions yield prismatic Ricci and scalar curvatures; the metric g_prism is a polarization of the Nygaard-filtered tangent crystal. The prismatic Hilbert action S = ∫ R_prism vol_prism (hypercohomology over the prismatic site) has first variation δS = ∫ ⟨G_prism, δg_prism⟩ vol_prism + boundary terms. Stationarity implies the prismatic Einstein equations G_prism = 8π T_prism (vacuum: G_prism = 0). The Bhatt–Scholze de Rham comparison theorem (arXiv:1905.08229, Thm. 6.4 & Cor. 15.4) identifies these equations isomorphically with their classical counterparts at the level of cohomology classes; tensor-level descent is flagged as conjectural. An explicit local-coordinate verification on the rank-2 q-de Rham polydisk shows that the ½ coefficient in the Einstein tensor arises from the linearization det(I + λM) = 1 + λ tr(M) + O(λ²) of the volume form, not from analogy with classical general relativity. The Z₉ prismatic F-crystal, Schoen gyroid (conjectured as a calibrated 3-cycle inside a G₂-holonomy 7-manifold — this embedding is open; no G₂-holonomy metric on the Schoen gyroid has been constructed), and TMRB recursive branching supply the geometric substrate and the matter source T_prism. This construction supplies the variational foundation for the Prismatic Torsional Recursive Holographic (PTRH) framework. Deposit contents (v2): compiled PDF (13 pages), LaTeX source, Python verification script for Section 4 (sympy/numpy), BibTeX entry, README. v2 corrects author affiliation (Sarasota, FL), updates all companion DOIs to concept DOIs, labels the G₂-holonomy assumption conjectural, and standardises equation cross-references.