Z₉ Holographic Quantum Error Correction in the PTRH Framework: Operator Algebra Encoding, Spectral Gap Code Distance, and Celestial Finite-N Instance

The holographic quantum error correction (HQEC) program, initiated by Almheiri, Dong, and Harlow and realized concretely by the HaPPY tensor-network code, establishes that AdS/CFT bulk reconstruction is a quantum error correction protocol. All existing holographic codes are constructed by assumption: a tensor network is specified and its error-correction properties derived. We show that the Z9 sector structure of the PTRH framework constitutes a holographic QEC code that was not constructed but derived — emerging from the nine independent gap-closure results of Papers 11–23 of the PTRH series. Three theorems and a proposition are proved. Theorem 1 establishes the Z9 encoding isometry: the sector decomposition H = ⊕H_n with Frobenius-eigenvalue projectors P_n defines a Z9-equivariant isometric embedding V: C^9 → H, with the Z9-symmetric canonical form ΩZ9 supplying the equal-weight sector structure at the encoding locus. Theorem 2 proves that the projectors satisfy the operator algebra quantum error correction (OAQEC) conditions of Beny–Kempf–Kribs for the full class of single-sector errors: P_n E†_a E_b P_m = Cab(n) P_n δₙₘ, a consequence of projector orthogonality alone, with physical content supplied by Zeno sector isolation (Paper 16). Theorem 3 establishes the code distance: the PTRH spectral gap Δ_min = 2sin(π/9) (Paper 18) yields a closed-form correction threshold ½Δ_min = sin(π/9) ≈ 0.342, with d ≥ 3 following from single-sector correctability. To the author's knowledge this is the first closed-form code distance derived from first principles in the holographic QEC literature, without being specified as a construction parameter. The threshold applies to the algebraic noise model of Frobenius operator perturbations. A proposition identifies the non-renormalization result of Paper 23 (Corollary 1) as the statement that the logical observable P_n is preserved under any error-and-correction cycle — a standard consequence of OAQEC that here follows from Zeno sector isolation. An observation establishes that PTRH is the exact finite-N (N=9) instance of the cyclic qudit-chain construction of Guevara and Hu (JHEP 06 (2025) 121), with the relation Δ_min = 2sin(τ/2) where τ = 2π/N = 2π/9 is the GKP phase-space cell size; no continuum limit is required because the Frobenius eigenvalue structure fixes N = 9 exactly. The holographic character is celestial (flat-space), not anti-de Sitter; subregion reconstruction and Ryu–Takayanagi are not claimed. Three open problems are posed: exact code distance d = 3, erasure threshold comparison to HaPPY, and CHY resolution of Paper 22 Conjecture 1.