The Interior Chrysene Lattice: Mathematical Necessity, Distributed Inheritance, and Catalytic Organization in the Protocell

The Chrysene Formalism is axiomatized on a Z³-fibrated tensor network whose crossed-product sequence algebra is AC = QC _α Z³. The origination described in the corpus transforms the Z² base into membrane undifferentiatedly, leaving the protocell with the fiber corner PACP QC _α Z. We prove that PACP is not Morita equivalent to AC, establishing an algebraic deficit that no amount of chain expression can close. We construct the Interior Chrysene Lattice (ICL, informally the Abiotic Loom Unit, ALU) — a bilayer-enclosed, π-stacked 3,6,9,12-tetrasubstituted chrysene cage with phosphodiester linker arms — as the minimum physically realizable structure that restores the Z² base representation inside the protocell. We classify ICL units by attachment: Type A (dual-strand tethered), Type B (single-strand tethered), and Type C (untethered). We prove a three-dimensionality lower bound: a Type A unit must contain at least 16 chrysene units (two 2×2×2 cubes) for each half to carry a 3D representation after cleavage; Type B requires at least 8 units. We prove the Redundancy Theorem: any combination with nA ≥ 1 or (nB,1 ≥ 1 and nB,2 ≥ 1) guarantees both daughters inherit the representation with probability 1. The collection of ICL units forms an approximately finite-dimensional (AF) C^*-algebra approximating AC, whose Bratteli diagram is the cell-division morphism. In the minimal viable configuration (one Type A unit), this diagram has uniform multiplicity $(1,1)$ and the AF-algebra is the uniformly hyperfinite algebra of type 8^∞ (Theorem 4.4), connecting the cell-division rule to a classical object in operator algebra theory. We extend Article 76's admissible domain Ωadm with a fifth constraint functional C₅, add the ICL to Article 65's material ledger at 0.05–0.76% of the per-period pool at $L = 200$, and connect to Article 3's tripartite unfolding. Larger ICL units (complexity κICL > 0) are expected to template secondary structure; the geometric concordance is recorded as a target.