The Z9 Prismatic Stress-Energy Tensor: Tensor-Level Derivation, Conservation, and Closure of Gap G5 in the PTRH Framework

We derive the Z9-decomposed prismatic stress-energy tensor T_prism from first principles within the PTRH framework. Starting from the C-field action S_n = -(1/48) int F_n^2 sqrt(g) d^4x for each Frobenius-eigenspace sector n=0,...,8, the Hilbert prescription T_{n,mu nu} = -(2/sqrt(g)) delta S_n / delta g^{mu nu} yields the explicit form T_n^{mu nu} = (1/6)(F_n^{mu alpha beta gamma} F_n^nu_{alpha beta gamma} - (1/8) g^{mu nu} F_n^2). We prove: (i) symmetry T_n^{mu nu} = T_n^{nu mu} from the symmetry of the metric variation; (ii) conservation nabla_mu T_prism^{mu nu} = 0 on-shell via the diffeomorphism Ward identity and the prismatic C-field equation of motion d(star_prism F_n)=0, the prismatic analogue of the Maxwell equation; (iii) Hodge compatibility of g_prism follows from Nygaard compatibility (Paper 1 Def. 2.4) via the Bhatt-Scholze comparison isomorphism; (iv) sector support: eigenspace orthogonality decouples S_matter = sum S_n, so T_prism = sum T_n with T_n depending only on F_n, and the association T_n with D_n follows from the Paper 14 eigenspace-sector bijection; (v) area consistency with S_BH/9. This closes Gap G5 in the EHSCM gap table, upgrading G5 from [O] to [V]. PTRH Paper 15.