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 ∇_prism and define prismatic curvature R_prism := ∇_prism². 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 ([BhattScholze], Thm. 6.4 & Cor. 15.4) identifies these equations isomorphically with their classical counterparts. The ℤ₉ prismatic F-crystal, Schoen gyroid compactification, and TMRB recursive branching supply the geometric substrate and the matter source T_prism. This construction is the technical core of the Prismatic Torsional Recursive Holographic (PTRH) framework.