Torsion Cell Area from the Prismatic Hilbert Action: A Derivation of Acell = 4G₄ ln 9 under Isolated Horizon Boundary Conditions

Gap G3 of the Event-Horizon-State Computer Memory (EHSCM) architecture (PTRH Paper 11) calls for a derivation of the torsion cell area Acell = 4G₄ ln 9 ≈ 8. 79 ℓ²Pl from the prismatic Hilbert action of PTRH Paper 1, rather than as a self-consistency ansatz. We close this gap under one clearly stated hypothesis. The Z9 prismatic F-crystal (M, φM) with φM (e) = ζ₉ e, established in Paper 1, defines nine Frobenius eigenspaces in the bulk labelled by the ninth roots of unity ζ₉ⁿ: n = 0,. . . , 8, which project to the horizon under the Isolated Horizon Conjecture (IHC). Two results follow. (1) Uniform area V: The Z9 cyclic symmetry of the Frobenius acts transitively on the nine eigenspaces. Because the prismatic Hilbert action is Z9-equivariant, any Z9-equivariant area operator assigns equal area Acell to every eigenspace. (2) Main theorem C conditional on IHC: Under the conjecture that the nine bulk F-crystal eigenspaces project to independent degrees of freedom on the horizon, each torsion cell carries degeneracy d = 9. Combined with the Bekenstein–Hawking entropy bound SBH = A/4G₄, this uniquely fixes Acell = 4G₄ ln 9 ≈ 8. 79 ℓ²Pl. This upgrades the EHSCM cell area claim from model-level ansatz M to derived result conditional on IHC C. The derivation has no free parameter analogous to the Barbero–Immirzi parameter of loop quantum gravity. A direct AdS₃ analogue is established via the Z9 parafermion CFT (Zamolodchikov–Fateev 1987; Gepner–Qiu 1987). The remaining open problem — proving IHC by establishing prismatic isolated horizon boundary conditions for the Z9 F-crystal on the bifurcation surface — is identified and a three-step proof programme is outlined.