Chaotic Sampling of Spin-Glass Overlap Fields: Arithmetic Selection and Centered Fluctuations

We characterize correlations surviving an independent chaotic probe of frozen overlap fields of two conditionally independent full-Gibbs replicas of the finite zero-field periodic Edwards–Anderson model, sampled by a cat map with Lebesgue initial conditions. The visitation kernel contracts exactly with the squared equilibrium spin correlations. Under homogeneous periodic disorder, gcd(|det(A^k − I)|, L) = 1 makes the cell displacement uniform and forces both the ensemble-averaged, field-centered covariance and its mean-matched shuffle excess to vanish. This sufficient condition excludes lag one for every L ≥ 2 and all positive lags not divisible by three when L is a power of two. Cancellation is not universal: at L = 4 and lag three, exact rational areas and a controlled high-temperature expansion for independent N(0, J₀²) bonds give a shuffle excess 3(βJ₀)²/52 + O((βJ₀)⁴), strictly positive for sufficiently small βJ₀ > 0, while preserving each field's overlap mean. Local gauge symmetry explains the choice of observable: the disorder-averaged law of one full-Gibbs spin snapshot is uniform under gauge-invariant disorder, whereas two-replica overlaps retain squared spin correlations. An existing invariance principle gives a centered CLT for each fixed finite cell field, allowing zero variance. Averaging full-Gibbs overlap fields at finite temperature and L ≥ 2 instead gives a nondegenerate Gaussian variance mixture with an atom at zero. Fieldwise centering separates these fluctuations from the pooled N²χ_SG/L² random-mean contribution. Asymmetric binary readout is only an affine change of drift and scale. The contribution is the explicit correlation-selection criterion and its nonzero matched-control example, together with their fixed-volume statistical interpretation; no joint thermodynamic limit or new general CLT is asserted. This record contains the preprint manuscript and reproducibility materials prepared for initial submission to Physica D: Nonlinear Phenomena. It has not undergone journal peer review. The manuscript PDF is supplied separately for convenient reading. The accompanying archive, chaotic_spin_zenodo_v1.0.0.zip, contains the current manuscript PDF and authoritative LaTeX source, the three external figure PDFs required for compilation, and four original supplementary archives. S1, chaotic_spin_verification_4.zip, contains executed finite-state verification code, explicit fixed bonds, settings, unrounded numerical outputs, and execution records. S2, Figure1_arithmetic_selection_set.zip, contains exact integer arithmetic classifications and Figure 1 reproduction materials. S3, Figure2_displacement_geometry_set.zip, contains exact rational displacement geometry and Figure 2 reproduction materials. S4, Figure3_matched_shuffle_excess_set.zip, contains a redraw of the rounded Table 8 values and Figure 3 reproduction materials. Original thermal enumeration, explicit bonds, and unrounded outputs are provided in S1. The thermal calibration uses one explicitly constructed fixed bond sample and its translation orbit; these translations are not independent Gaussian-disorder samples. No independent Gaussian-disorder numerical average or larger-lattice Monte Carlo study is supplied. The current manuscript and the original four supplementary archives are preserved byte for byte. The manuscript is dated 16 October 2026. Its pre-deposit statement that no public repository or persistent identifier is claimed is retained as a historical snapshot; this Zenodo record provides the public archive. Earlier manuscript versions and preparation histories inside the supplementary archives are retained as provenance and do not supersede chaotic_spin_research_10.tex. The S1 verification driver regenerates its historical revision-4 manuscript; it is not a generator for the current revision-10 manuscript. Follow the top-level README and the individual archive README files for compilation and computational reproduction. Packaging introduces no new scientific calculations. License scope: original software source code, including code inside the supplementary archives, is licensed under the MIT License. The manuscript, numerical data, figures, and documentation are licensed under the Creative Commons Attribution 4.0 International License (CC BY 4.0). These licenses apply to their respective components, not as alternative licenses for the entire collection. Third-party dependencies retain their own licenses. See LICENSES.txt in the archive.