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 Tₚrism from first principles within the PTRH framework. Starting from the C-field action Sₙ = - (1/48) int Fₙ² sqrt (g) d⁴x for each Frobenius-eigenspace sector n=0,. . . , 8, the Hilbert prescription T₍, ₌ₔ ₍ₔ = - (2/sqrt (g) ) delta Sₙ / delta g^mu nu yields the explicit form Tₙ^mu nu = (1/6) (Fₙ^mu alpha beta gamma Fₙⁿu₀₋₏₇₀ ₁₄ₓ₀ ₆₀₌₌₀ - (1/8) g^mu nu Fₙ²). We prove: (i) symmetry Tₙ^mu nu = Tₙ^nu mu from the symmetry of the metric variation; (ii) conservation nablaₘu Tₚrism^mu nu = 0 on-shell via the diffeomorphism Ward identity and the prismatic C-field equation of motion d (starₚrism Fₙ) =0, the prismatic analogue of the Maxwell equation; (iii) Hodge compatibility of gₚrism follows from Nygaard compatibility (Paper 1 Def. 2. 4) via the Bhatt-Scholze comparison isomorphism; (iv) sector support: eigenspace orthogonality decouples Sₘatter = sum Sₙ, so Tₚrism = sum Tₙ with Tₙ depending only on Fₙ, and the association Tₙ with Dₙ follows from the Paper 14 eigenspace-sector bijection; (v) area consistency with SBH/9. This closes Gap G5 in the EHSCM gap table, upgrading G5 from O to V. PTRH Paper 15.