The Completeness Condition: Realisation of the Twelve-Axiom Framework at a Site of the Interior Chrysene Lattice, and the Genome Length It Requires

Article 79 delivers a substituted-chrysene stock to the protocell and Article 78 places an interior lattice of that stock inside the vessel, reporting that the lattice restores a three-dimensional representation lost at origination. Neither paper asks the prior question: at which site of the dividing vessel, if any, do the twelve axioms of Article 40 have a model at all? The axioms are realised by contractions between adjacent cells, so the question is settled by neighbour relations and not by how much chrysene a phase holds. We answer it. A site realises every contraction of Article 40, Definition 19 only if it has both stacking neighbours, both lateral neighbours in each planar direction, and a covalent linker to each of them. No phase of the arc satisfies all three: the stock columns of Article 79 and the strand realise the stacking contraction and have no lateral extent; the bilayer has lateral extent, no covalent lateral bridge, and at two leaflets no interior stacking site. Fibre and base are realised in different phases and at no common site outside the interior lattice, so without it the base–fibre decomposition is realised nowhere in the vessel: the axioms have no geometric realisation at any site, which is what survival through division has to mean for a framework stated over a network. The lattice that carries it has a size. The least occupied box with a complete site is $3 \times 3 \times 3 = 27$ nodes, which is exactly the configuration Article 78 tabulates as $\kappa_{\mathrm{ICL}} = 1$ and defers as a target; under Article 78's midplane cleavage the least such box whose halves are each complete is $3 \times 3 \times 6 = 54$ nodes. Over arbitrary configurations the minima are $7$ and $14$, and we give three reasons for taking the box figures as governing. Article 78's minima of 8 and 16 carry no complete site, so within boxes the heritable minimum rises by $3.38$, which is affordable at $0.128$ per cent of the per-period pool. The binding constraint is geometric, and it is settled exactly: fitting one complete lattice in the lumen is containment of a convex body in a ball, not packing, so no packing fraction enters and the circumradius of the wrapped unit decides it. The least genome length admitting a heritable complete lattice is $L_{\mathrm{crit}} = 222$ bp, $11$ per cent above Article 78's minimum viable length; at $L=200$ none fits. The two inheritance routes then separate — Type A needs $222$ bp and Type B, needing two units in one lumen, at most $637$ — so over some four hundred base pairs the framework survives only by cleavage of a single lattice. What the lattice does not achieve is stated before it is mistaken for a result: it does not make the generated algebra Morita equivalent to $\mathfrak{A}_{\mathfrak{C}}$, since an approximately finite algebra has vanishing $K_1$ and no finite box or limit of boxes is equivalent to a crossed product with $K_1 \neq 0$. Enlarging the lattice changes nothing, so Article 78's corner is not an artefact of its minimality.