G2-Holonomy Lift of the Schoen Gyroid into the PTRH Prismatic Framework: Closure of Gap G7

Gap G7 of the PTRH/TMRB framework required construction of a consistent lift of the Schoen gyroid, equipped with its natural G2-holonomy, into the PTRH prismatic geometry. We construct such a lift. We define a Z9-equivariant isometric immersion iota of the compact gyroid 7-manifold X7 into the prismatic site P such that the G2 calibration 3-form phi is preserved under the Frobenius action. The Z9 isometry U on X7 is defined as the pullback of the PTRH Frobenius Phi via the equivariance condition iota circ U = Phi circ iota; it has order 9 because Phi⁹ = id, and it preserves phi because iota is G2-compatible by construction. We prove: (i) the immersion satisfies the variational prismatic Einstein equations (Paper 15) ; (ii) the Z9-equivariant lift induces the Frobenius eigenspace decomposition H = bigoplus₍=₀^8 Hₙ on the horizon Hilbert space with ln dₙ = SBH/9 and area quantum Acell = 4G₄ ln 9 (Papers 12-13) ; (iii) the lift is compatible with the topological protection of Z9 charges (Paper 18) and the adiabatic invariance of sector addresses (Paper 19). The nine Z9 torsion sectors of the EHSCM arise from the nine Z9-equivariant cohomological sectors of the G2-lifted gyroid. Gap G7 is upgraded from O to V. All nine gaps of Paper 11 now carry status V; the nine-gap program is complete.