Beyond the marginal distribution: prime-gap ordering imprints on a self-avoiding polymer through the small-prime end of the chain

We study the Prime-Driven Topological Polymer (PDTP): a coarse-grained self-avoiding chain in which each bead's bending stiffness is an exponential function of its local prime gap, so that dense prime clusters become flexible hinges and long gaps become rigid rods. After overdamped Langevin collapse, we compare the true prime chain with surrogates that preserve the gap multiset exactly. The gross conformation is unaffected by ordering. The Spearman correlation ρ between stiffness and radial position, however, is significantly lower for the true sequence (ρ = +0.105 vs. +0.137 at N = 40 000, p = 6.5×10⁻⁵). The difference persists under four-fold deeper annealing and in isothermally relaxed globules. It also survives a sequence-level test: the prime chain lies below all twelve fixed shuffles (Z_seq = −7.0). Version 4.0 identifies the mechanism. 1. Exact theory. For the ideal chain with a quenched bending field, tangent correlations are products of Langevin functions. The thermally averaged radial profile of any sequence then follows in O(N) operations. A retention-scaling theorem shows that any readout carried by chain-scale structure of the stiffness composition gives a local-scramble correlation length ξ ∝ n_gap automatically. With no fitted parameter, the theory reproduces the measured ξ ≈ 3.5×10² gaps (357 predicted). 2. Localization. In the Langevin globules the ordering signature is confined to the first ~10% of the backbone, the small-prime region where the prime-number-theorem drift is steepest. Excluding those beads abolishes it (Δρ = +0.002 at production depth, +0.003 after isothermal relaxation). The two halves of the chain carry opposite signs. 3. Higher-order nulls. A second-order sufficiency theorem shows that surrogates matching the gap multiset and power spectrum (IAAFT) must reproduce translation-invariant observables to third order. In the Langevin model, detrended IAAFT "fake primes" reproduce most of the effect, and the prime chain ranks 3rd of 20 (Z_seq = −2.3). The Z > 3 criterion for structure beyond second order is not met. 4. A pre-registered prediction failed. A pure prime-number-theorem surrogate (S6'') reproduces the chain-scale composition but recovers only 28% of the effect at N = 40 000. The imprint is therefore real, but it is not a scale-free arithmetic correlation. It reflects the composition of the small-prime end of the chain, read through the end-to-middle radial profile of the collapse. The interpretation of ξ as an irreducible macroscopic arithmetic correlation (versions 2.0–3.0) is withdrawn. The falsification of the local-trap hypothesis (version 2.0) stands. The Supplementary Material develops the exact theory and reports five further results:• no universal power law A₃D ~ ξ^α;• a closed-form spectral-transfer kernel K(q, ω), with a no-go result for hyperuniformity;• a data-processing bound on sequence–conformation mutual information;• sign invariance under monotone stiffness maps;• no amplification of the prime imprint at the θ collapse. All results were obtained on a laptop. The code and data needed to reproduce every figure and number are available at https://github.com/Ruqing1963/prime-driven-topological-polymer (tag v4.0). Files:• pdtp_v4.0.pdf — main paper (17 pp.)• pdtp_v4.0_supplementary.pdf — Supplementary Material (12 pp.)