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

Source abstract

The Chrysene Formalism is axiomatized on a $\mathbb{Z}^3$-fibrated tensor network whose crossed-product sequence algebra is $\mathfrak{A}_{\mathfrak{C}} = \mathfrak{Q}_{\mathfrak{C}} \rtimes_\alpha \mathbb{Z}^3$. The origination described in the corpus transforms the $\mathbb{Z}^2$ base into membrane undifferentiatedly, leaving the protocell with the fiber corner $P\mathfrak{A}_{\mathfrak{C}}P \cong \mathfrak{Q}_{\mathfrak{C}} \rtimes_\alpha \mathbb{Z}$. We prove that $P\mathfrak{A}_{\mathfrak{C}}P$ is not Morita equivalent to $\mathfrak{A}_{\mathfrak{C}}$, 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, $\pi$-stacked 3,6,9,12-tetrasubstituted chrysene cage with phosphodiester linker arms — as the minimum physically realizable structure that restores the $\mathbb{Z}^2$ 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 $2\times2\times2$ 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 $n_A \geq 1$ or ($n_{B,1} \geq 1$ and $n_{B,2} \geq 1$) guarantees both daughters inherit the representation with probability 1. The collection of ICL units forms an approximately finite-dimensional (AF) $C^*$-algebra approximating $\mathfrak{A}_{\mathfrak{C}}$, 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 $8^\infty$ (Theorem 4.4), connecting the cell-division rule to a classical object in operator algebra theory. We extend Article 76's admissible domain $\Omega_{\mathrm{adm}}$ with a fifth constraint functional $C_5$, 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 $\kappa_{\mathrm{ICL}} > 0$) are expected to template secondary structure; the geometric concordance is recorded as a target.