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

The Prime-Driven Topological Polymer (PDTP) maps the arithmetic structure of theprime numbers onto a coarse-grained self-avoiding polymer: the local bendingstiffness of each bead is an exponential function of its local prime gap, so denseprime clusters become flexible hinges and long inter-prime runs become rigid rods.Using overdamped Langevin dynamics with simulated annealing, we ask whether thecollapsed conformation is fixed by the composition of this stiffness sequence or byits arithmetic ordering, comparing the true prime chain against surrogate nullmodels that preserve the marginal stiffness distribution exactly while destroyingsequential order. Gross morphology (radius of gyration, tail-weighted core–shell separation) issurrogate-invariant. The finer organization is not: the Spearman rank correlationbetween stiffness and radial position is significantly suppressed for the truesequence (ρ = +0.105 vs. +0.137 at N = 4×10⁴; Welch p = 6.5×10⁻⁵, Cohen d = −0.96),an effect a three-way decomposition attributes to arithmetic ordering rather than tothe gap distribution. Deterministic arithmetic order therefore leaves a small butstatistically robust and size-reproducible imprint on three-dimensional structure. Version 2.0 identifies the mechanism. Two scale-selective "scissors" surrogates — ablock shuffle that destroys long-range order while preserving local order, and itsconverse, a local scramble that preserves the global arrangement while destroyinglocal order — cut the correlation length from opposite sides. Destroying local orderleaves the signal fully intact (indistinguishable from the true chain up to windowsof ~100 consecutive gaps), whereas destroying global order abolishes it at everyscale. Local order is thus neither necessary nor sufficient: the local twin-prime"topological-trap" hypothesis proposed in Version 1.0 is falsified. The signal isinstead driven by an irreducible, macroscopic long-range arithmetic correlation oflength ξ ≈ 3.5×10² gaps (68% CI [1.8, 4.9]×10²), spanning roughly 30% of thesequence studied. Whether ξ grows in proportion to system size (scale-free order) orsaturates at a fixed gap count remains open and is the subject of ongoing runs. This record contains the full manuscript (LaTeX + compiled PDF), the completesimulation and analysis code, the ensemble result files, and all figure-generationscripts, enabling full reproduction of every result. — Changes in Version 2.0 —- New Section 5.2, "Falsification of the local-trap hypothesis and the emergence of macroscopic order," replacing the former "A candidate mechanism."- New scale-selective surrogate experiments S4 (block shuffle) and S5 (local scramble), N = 10⁴, n = 40 per arm.- Measurement of the macroscopic correlation length ξ ≈ 355 gaps.- Updated abstract and conclusion; new Figure 5 (the S4/S5 scissors).- New code (block_shuffle_test.py, block_local_scramble_test.py, make_scissors_figure.py) and data files.- All Version 1.0 data and results are retained unchanged.