The Origins Arc: A Coordinating Formalism for the Passage from the Substituted Chrysene Cluster to the Genetic Code

Abstract

The carrier treated here is a pi-stacked cluster of six substituted chrysene units, each set in one of two orientations, whose orientation pattern is its only datum. This paper coordinates, in one formalism, a continuous derivation from that carrier to a genetic code: to a reading length, to a bounded product repertoire, to a residue rooted at a named framework carbon and formed in a single configuration, to an exogenous element required at exactly two structures, and to punctuation. The derivation invokes no adaptive selection, no antecedent code and no external template, and the capacity argument uses no codon table at any step. The same architecture is specified in the filings as concurrently integrating and synthesising encoded nucleic-acid and phospholipid material, so that membrane, nucleic acid and residue are referred to one object; this paper records that specification and develops the coding branch.

Two cardinals are obtained from carrier structure rather than from the code. The reading length is three because a six-unit cluster carries three catalyst-substrate interfaces, each producing one nucleobase, so the codon follows from the cluster; the conventional derivation, from alphabet size to codon length, is shown to be unavailable within the formalism at every cluster length, the class count lying below the codon supply throughout. The product repertoire is bounded by 2^(2N-2) + 2^(N-1), which is twenty at three vessels, this being the dimension of the algebra of observables the carrier admits. The result is a ceiling; that it is attained requires totality and injectivity, which are stated separately and are not established.

The account is expressed as a mathematical formalism, which is the paper's apparatus rather than its presentation. The registry sequence is the coboundary of the orientation marker, so capacity is the dimension of an invariant algebra; localisation is support on the fixed locus of the reversal, a vertex at odd vessel count and an edge at even; admissibility of a slot set is independence in a transversal matroid; and the consistency relation is a monotonicity together with a factorisation. So expressed, the account admits proof. Two consequences follow that the corpus did not contain: a criterion deciding which reassignments preserve admissibility, and the result that every assignment admitting the consistency relation is regular, which places the observed mapping outside admissibility at two blocks, one of them the block carrying the start codon.

Every quantity is traceable to an explicit structure. Seventeen verification blocks recompute each numerical claim from a graph, a finite enumeration or an incidence; the atom provenance of the residue is complete; the decoration of all twenty structures is printed and audited by molecular formula and by valence at each retained vertex; and a hypothesis register states, for each of the account's eighteen premises, what rests on it and what survives its failure. The coordination yields ten findings across the corpus, four of them corrections, and two conventions are ruled.

What is not claimed is stated before the constructions. Twenty is a ceiling. The architecture yields a single configuration rather than a mixture, and identifying that configuration with the L series is a labelling supplied outside the architecture. The account of initiation holds in the assignment structural consistency selects, not in the observed mapping, and rests on a junction that no article establishes. One intended result, a redox demand functional over the structure series, is demoted to a target: two of its three inputs are now computed, and the parity of the partial balance is the evidence that the third is absent.