Collapse, Convergence, and the Scale of the Prime-Gap Imprint on Polymer Dynamics

A self-avoiding chain whose bending stiffness is programmed by the prime-gap sequence (the prime-driven topological polymer, PDTP) collapses into a state that is measurably less radially sorted than a chain programmed by a random permutation of the same gaps. This paper settles two questions about that imprint. The first question is whether the imprint is a property of the collapsed state or a memory of how the chain was cooled. Two independent relaxation routes at N = 40000, eight-fold slower annealing and a one-million-step isothermal hold that drives the chain to a melt-density globule (radius of gyration 16.81 +/- 0.03, packing fraction 0.489), converge on the same nonzero gap, Delta rho = -0.0269 +/- 0.0033 (8.1 sigma). The gap is present at every size from N = 10000 to 80000 under the converged protocol. The imprint survives collapse, at about 59 percent of the amplitude measured at the original production protocol. The second question is what the size-proportional correlation length of the local-scramble surrogate measures. It measures the readout, not the primes. A retention-scaling theorem for the exactly solvable ideal chain predicts a correlation length proportional to the number of gaps for any readout carried by chain-scale composition. Surrogates that contain no prime-specific arithmetic, the smooth prime-number-theorem envelope and a PNT scale family, reproduce the proportional law of the true primes, with master-curve ratios 0.30, 0.30 and 0.33. The estimator dependence of the correlation length in the Langevin ensembles is reported in full, and the single-point half-crossing is replaced by a master-curve collapse. The inference of extensive arithmetic order in the primes, made in an earlier unpublished draft, is withdrawn. What remains is a smaller, robust statement. The fold registers the composition of the small-prime end of the chain. The smooth PNT envelope supplies a growing fraction of the effect (0.25, 0.34 and 0.57 at N = 20000, 40000 and 80000), and the residual is significant at the sequence level: the prime chain lies below all ten fixed PNT-envelope surrogate sequences (Z_seq = -3.75). This record merges and supersedes two unpublished drafts. It contains the manuscript (LaTeX source and PDF), the exact ideal-chain solver and Langevin engine, the scripts and original drivers behind every table and figure, and the data. Source repository: https://github.com/Ruqing1963/prime-gap-imprint-collapse-and-scale