Beyond the marginal distribution: long-range correlations in prime gaps imprint on the 3D conformation of a self-avoiding polymer

Version 3.0 (consistency, protocol-dependence, and reporting revision). No version-2.0 datum is altered and no new production ensemble is introduced. This revision: (i) states the error-bar convention (mean ± s.e.m.) explicitly; (ii) records every ensemble's annealing depth (steps and steps/bead) in the figure captions, resolving the apparent discrepancy between the N = 10⁴ points of Figs. 3 and 4; (iii) identifies the correlation length ξ as a protocol-conditioned observable of the annealed ensemble, and adds a note reporting a deeper-annealed re-measurement from follow-up work (ξ ≈ 4.8×10² gaps, ξ/n_gap ≈ 0.39 at N = 10⁴), which leaves the macroscopic-order conclusion intact; (iv) states the length–stiffness construction coupling and the chain-end boundary treatment explicitly; (v) flags the reduced statistical power of the n = 10 radius-of-gyration comparison; (vi) adds a related-work section positioning the paper against prior art on correlated-sequence random heteropolymers, the conformation-dependent ("protein-like") sequence-design programme and its Lévy-flight sequence correlations (Khokhlov–Khalatur; Govorun et al.), Fibonacci-sequence molecular chains, polymeric compaction as a probe of sequence correlations (Odijk), and the effective limit-periodicity and hyperuniformity of the primes (Torquato et al.); and (vii) fixes the PDF text layer (ligatures now copy and search correctly). The full changelog is in CHANGELOG.md. All version-1.0 and version-2.0 data and results stand unchanged. Abstract. The folding of a heteropolymer is governed by its monomer sequence, yet in the intermediate regime between periodic and random sequences—those that are aperiodic but deterministic—a controlled attribution of three-dimensional structure to sequential ordering, as opposed to composition, has been lacking. We introduce the Prime-Driven Topological Polymer (PDTP), a coarse-grained self-avoiding chain in which the local bending stiffness is set by an exponential map of the local prime gap, so that dense prime clusters act as flexible hinges and the long inter-prime runs as rigid rods. Using overdamped Langevin dynamics with simulated annealing, we ask whether the folded state is fixed by the composition of this sequence or by its arithmetic ordering. To separate the two, we compare the true prime chain against surrogates that preserve the marginal stiffness distribution exactly while destroying sequential order. The gross conformation is surrogate-invariant: neither the radius of gyration nor the tail-weighted core–shell amplitude distinguishes the true chain from a shuffle. The finer organization does: the Spearman correlation between stiffness and radial position is significantly suppressed for the true sequence (ρ = +0.105 vs. +0.137 at N = 4×10⁴; p = 6.5×10⁻⁵, d = −0.96), an effect a decomposition attributes to ordering rather than to the gap distribution. The imprint is quantitatively modest (~1% of the radial-rank variance) but statistically robust and reproducible across system sizes, demonstrating that deterministic arithmetic order leaves a measurable trace on three-dimensional structure—one that survives the entropically favored radial sorting of the collapse. Scale-selective surrogate tests (a block shuffle and its converse, a local scramble) further localize this order: the imprint is destroyed by global rearrangement yet wholly preserved under complete local scrambling within windows of up to ~10² consecutive gaps, revealing that the driver is an irreducible, macroscopic long-range arithmetic correlation of length ξ ≈ 3.5×10² gaps (≈30% of the sequence, under the reference annealing protocol; ξ is a protocol-conditioned observable of the annealed ensemble) and falsifying the local-cluster (topological-trap) hypothesis of version 1. Contents of this record: the version-3.0 paper (PDF and LaTeX source), all simulation code (pre-flight calibration, surrogate/null-model decomposition, finite-size sweep, decisive N = 4×10⁴ experiment, S4 block-shuffle and S5 local-scramble drivers, conformation generator), ensemble result files, figure-generation scripts, and CHANGELOG.md. Code license: MIT.