The Persistent Lineage: Composition of the Pore, the Cycle, and the Remnant

Abstract

Four earlier articles of this corpus each settled one piece of a composition that none of them performed. The enclosure gave the vessel its membrane; an earlier article showed that vessel dividing under a periodic force and bounded the founder lineage at roughly forty generations; a further paper found the architecture's only candidate osmolyte and left its supply rate as the open question; and the pore paper supplied the conduit and with it continuous satisfaction of the osmolyte-generation condition. This paper asks what the three together imply for the lineage, and finds two answers.

The first answer lifts a bound. The forty-generation bound was conditional on the absence of a particle source; that condition is now false. Under continuous osmolyte supply the reduced volume falls by a computable factor each cycle, and the return time to the division hyperplane after each budding event is bounded by two to three cycles independently of the generation number. The lineage is persistent.

The argument rests on a structural invariance that holds independently of every parameter. The ratio ρ = R/N of remnant count to particle count is preserved exactly by both the per-cycle map and the division reset: it evolves as a Möbius contraction toward unity and cannot be held below that limit by any edge that consumes remnants slower than they are produced. The polypeptide paper must discharge the remnant at the production rate, not merely at a slower one. This rate condition on the next article is derived from the symmetry of the map and from nothing else.

The second answer identifies a new bound, not previously seen, that arises from the same edge that generates the osmolyte. The excision emits one apolar chrysene arm per osmolyte particle, and that arm partitions to the bilayer by the same rule that keeps every non-polar species out of the lumen. The ratio of remnant count to particle count is an invariant of both the per-cycle map and the division reset: it converges monotonically to one and cannot be held below that limit by any edge consuming remnants at a rate below the production rate. The bilayer remnant count therefore grows without bound, and the lineage exits the safe region of bilayer integrity at a cycle number that depends on α and on the assumed tolerance. To survive forty cycles at a tolerance of five remnants per lipid, α must be below 0.013. This is the fourth argument for small α in this corpus and the first that is a necessary condition rather than a cost preference.

A fifth result, stated as an observation rather than a theorem, names for the first time the energetic basis of the whole programme. The chrysene π-system absorbs UV-A photons; intersystem crossing stores 143 kJ/mol in the triplet state; proton-coupled electron transfer along the phosphodiester arms drives all synthesis at a quantum yield of five to ten events per photon. The twelve-hour forcing period of the earlier article is the light phase of a diurnal cycle. Neither a metabolic carrier nor a coding apparatus is required because the molecule is simultaneously antenna, energy carrier, catalyst and substrate. Both features are deferred for the same architectural reason: there is no mediator yet to evolve, because the substrate already is one.