The Placed Source: Exogenous Preparation of the Substituted-Chrysene Assembly, and the Equality of Its Reachable Set with the Endogenous One

This paper compares endogenous self-separation and exogenous site-specific placement of substituted-chrysene stocks, evaluating how transitioning from a bulk-partitioned source to purified, site-directed assembly affects both the set of reachable configurations and the information cost of structural selection. In the endogenous pathway, substitution in a polyaromatic bulk lowers a unit's transfer free energy so that departed units self-assemble into columns; in the placed pathway, the same three substitution families and arm words are delivered as purified material to prescribed sites. Comparing the two regimes establishes that both sources generate the exact same set of reachable structures, while exogenous placement provides direct control over alignment selection. To establish the equivalence of the reachable sets, a configuration is defined as a finite occupied subset of ℤ³ carrying a substitution family and an arm word at each node, together with a binary registry label — Aligned or Unaligned — at each stacking contact. Because the set of such configurations on a column is finite and explicitly enumerable, comparing the two sources reduces to comparing the support of the endogenous distribution with the image of the placement map. Because the endogenous exit distribution is strictly positive at every admitted substitution degree and permits the full arm alphabet, every placed configuration lies within the endogenous support (R_endo = R_exo). Consequently, the reachable end materials coincide under any product map, and for the six-site stack composition their common image is bounded by 20 frame classes. Where the two sources differ is in structural selection, governed by the symmetry properties of the separation filter. A unit's contact pattern is its set of locant–arm pairs, and inversion through the C2h reference frame exchanges positions 3 ↔ 9 and 6 ↔ 12. Any contact containing at least one unit whose pattern is invariant under this inversion is indifferent to its binary alignment state. Because the disubstituted family D = {6, 12} and tetrasubstituted family T = {3, 6, 9, 12} are inversion-invariant and carry arm words constant across inversion orbits, every contact in the delivered stock is alignment-indifferent except the initiator–initiator (I–I) contact, which exhibits a registration number of 3 when aligned and 2 when unaligned. Because the separation filter reads the alignment field solely through the initiator — where I–I contacts occur at a frequency of 2.5 × 10⁻⁷ — an endogenous contact energy without an additional orientation term induces a uniform distribution over the alignment field. Prescribing the alignment state of a stack of length ℓ therefore carries an exact information cost of ℓ − 1 bits: 5 bits for a codon-length stack (corresponding to five binary agreements) and 53 bits at the 54-node heritable lattice minimum. By comparison, the kinetic species-selection cost at the reported demand ratios is 20.0 to 23.0 bits for a six-site alternating column, making the registry contribution approximately one-fifth of the total specification cost. This symmetry analysis also highlights the dual structural role of the trisubstituted initiator. As the sole admitted family that breaks inversion symmetry, the initiator is the unique unit capable of fixing an orientation frame. In the coding-stack architecture, stacks are capped with this trisubstituted unit, and polymerisation terminates there due to the absence of a sidechain at the 9-position — the exact image of the 3-position under C2h inversion. Thus, the chemical feature that terminates chain extension is the same geometric feature that establishes stack orientation. Relative to this asymmetric cap, the two alignments of a symmetric contact become physically distinct states, anchoring the information content of the registry field.