Proof of the Isolated Horizon Conjecture in the PTRH Z₉ Framework:

PTRH Paper 12 derived the torsion cell area Acell = 4G₄ ln 9 ≈ 8. 79 ℓ²Pl conditional on the Isolated Horizon Conjecture (IHC): that the nine bulk Z₉ prismatic F-crystal eigenspaces project to independent degrees of freedom on the bifurcation surface Σ. This paper proves the IHC. Four results establish the proof. (1) Restriction and decomposition V: The closed immersion i: Σ ↪ X induces a Z₉-equivariant restriction map ρ = i* on the prismatic topos. The pullback ρ* (M, φM) decomposes as a direct sum of nine rank-1 F-crystals ⊕Mₙ, one per Frobenius eigenspace, by Maschke's theorem for the Z₉ action on the restricted crystal. (2) Hawking KMS invariance V: Because the prismatic Hamiltonian H is built from the Z₉-invariant metric gₚrism, it commutes with the Z₉ generator U. Z₉ charge is therefore conserved, and the Hawking KMS state ϱ_Ω satisfies U ϱ_Ω U† = ϱ_Ω. (3) Cross-sector independence V: For n ≠ m, a Ward-identity argument using U ϱ_Ω U† = ϱ_Ω and cyclicity of the trace forces all cross-sector correlators tr (ϱ_Ω Oₙ† Oₘ) to vanish exactly. (4) Cell-to-cell independence V: Torsion observables at distinct cells on Σ are spacelike separated; microcausality places them in commuting algebras, ensuring each cell carries algebraically independent degrees of freedom. The IHC V follows from (1) – (4). As a corollary, Theorem G3 of Paper 12 is upgraded from conditional C to fully verified V: the cell area Acell = 4G₄ ln 9 is now a completely derived result of the PTRH framework, with no remaining conjectural steps. The derivation chain runs from the Z₉ prismatic F-crystal of Paper 1 (Bhatt–Scholze prismatic cohomology) through the uniform area proposition of Paper 12 and the IHC proved here. The proof has no free parameter analogous to the Barbero–Immirzi parameter of loop quantum gravity.