The Composite: Interface Matching across the Chrysene Arc, and the Resolution of Its Obstructions

This paper establishes the formal interface composition of the chrysene protocell arc, verifying how the sequence of transitions from bulk stock separation to vessel division concatenates into a single valid finite execution. Across this unbroken chemical pathway, a substituted chrysene stock separates itself from a polyaromatic bulk, is admitted with its carbon traced atom by atom, draws on a single material pool shared among competing uses, encloses an integrated nucleic-acid strand, builds an ordered chrysene lattice in the lumen where a twelve-axiom framework finds its model, and culminates in a vessel that divides. To connect each stage's output to the next stage's input as a complete atom map, the atom-tracing framework defines an interface as a tuple of fields — atoms with their origin labels, resources, location, charge, orientation, environment, and phase — and establishes that a chain of transitions forms a valid finite execution when adjacent stages match on every shared coordinate. Evaluating these interface equalities across the full sequence yields a rigorous biconditional: the chain concatenates into a valid finite execution precisely when the 17 conditions enumerated here hold, each an explicit equality between the quantities of connected stages. Of these conditions, 11 hold directly and 2 are unified under explicit conventions. The carbon formula of the separation stage reproduces three of the four terms of the 810-carbon assembly on which the atom-by-atom accounting is demonstrated, reflecting their shared foundation in the underlying substitution balances. Polarity descriptions across the corpus align in the same way: the resource-allocation analysis designates a species by its broad class, while the separation analysis specifies its ordering within that class. Two additional interface fields are formalized here to complete the specification, both uniquely determined by the structure of the system. The atom-tracing framework requires an origin label to be carried per site rather than per molecule, fixing the label set at 26 members — 18 core positions and 8 arm positions — across the bulk-core and oxalate origins, and the separation stage's carbon formula reflects this exact partition. Six invariants are verified atom by atom across the five transitions of the separation sequence, and the location field is defined relative to a control volume. Four remaining conditions are harmonized by aligning parameter regimes and notation across the corpus. The material ledger is governed by the integrated-cycle demand table, whose requirement at a genome length of L = 200 base pairs is 15231 units, incorporating a catalyst term without a separate bud-fraction term. This reference frame unifies the resource-pool figure of 42200 and the lumen-lattice figure of 14705: the membrane's share is 86.3 per cent, and the 83.5 per cent figure corresponds to that table's L = 735 row paired with the L = 200 pool. The final three conventions establish a shared protonation state, containment clause, and counting criterion.