The Twenty Registry Classes of a Chrysene Stack: Frame Orbits, Descent of a Letter Transformation, and a Variational Criterion

Abstract

The frame group Fr = <flipm, revm> acts on the binary marker assignments of a chrysene stack of 2N sites. We prove that the orbit map is the registry sequence modulo reversal and that the orbits number 2^(2N-2) + 2^(N-1), which is twenty for 2N=6; that an orbit has size 2 or 4, never 1; that its stabiliser is a subgroup of Fr determined by the class, the two families of size-2 orbits being distinguished by the central entry of the registry sequence; and that by the maximal-frame-invariant theorem of the predecessor every candidate assembly invariant factors through this classification. We then determine when a letterwise transformation into a secondary alphabet transports the classification. Under a single structural clause on the target — a distinguished partition into two two-element classes — transport of the registry invariant, conjugation of the structural group, conjugation of complementation, intertwining of the two frame-group actions, and descent to the orbit sets are equivalent; exactly eight of the twenty-four bijections satisfy them, and a transformation failing them carries a single orbit onto a set meeting four distinct orbits. The eight induce only four maps of orbit sets, so no orbit-level quantity separates more than four; the residual ambiguity is exactly the projection reference the frame supplies. On an ensemble of classes and a weight on the target classes we define a functional, exhibit the exact algebraic locus on which its maximiser fails to be unique, and prove uniqueness off that locus.

Two coincidences of counting are refuted rather than used. Agreement of counts between source and target is evidence of nothing: the target's own classification has twenty classes with the same eight-and-twelve split whatever the transformation. The twenty classes agree with the twenty three-letter multisets only at N=3; at N=4 the counts are seventy-two and thirty-five.

No correspondence with the genetic code is asserted or used: our surjection of the sixty-four words onto the twenty classes has every fibre of size two or four, in the profile eight and twelve, whereas the standard genetic code has fibre sizes one through six over twenty amino acids together with three further stop codons, so the two surjections 64 to 20 are not the same map. That the functional is maximised by any process is a named hypothesis used in no proof, and that its maximisation is a survival property in the sense of the transfer-geometry paper is a Target with its hypotheses named.