Exact Finite-Sample Permutation Moments for Walsh Interaction-Order Energies
Abstract
WHAT THIS PACKAGE DOES This package goes with the revised Paper-1 manuscript "Exact Finite-Sample Permutation Moments for Walsh Interaction-Order Energies: A Projector Specialization of Quadratic-Form Randomization Theory" (v1.1). It contains the full open-source code, exhaustive verification, benchmarks, and environment details so that anyone can reproduce every number in the paper. WHY IT MATTERS Calculating exact permutation moments for quadratic forms is a known topic. But applying it to Walsh interaction-order projectors on the Boolean cube has not been packaged as a ready-to-use toolkit. This matters because people usually estimate these null moments with large Monte Carlo runs, which are slow and noisy. This package shows that the same mean, variance, and covariance can be computed exactly in microseconds — about 10,000 to 100,000 times faster than Monte Carlo — with agreement to machine precision. WHAT THE PAPER PROVES (SHORT) For a centered finite population placed on the N = 2^n vertices of the Boolean cube, the degree-k interaction-order energy is: E_k = sum_{|A|=k} g_hat_pi(A)^2 By specializing known quadratic-form randomization theory to Walsh projectors, the paper gives exact formulas for the permutation mean, variance, and cross-level covariance. The formulas are verified exhaustively at N = 4 (all 24 permutations) and N = 8 (all 40,320 permutations), and benchmarked up to n = 18. WHAT IS INSIDE - Reference code for the exact null-moment formulas (exact_null_moments).- Two independent energy calculators: dense Fast Walsh-Hadamard Transform (FWHT) and sparse Hamming/Krawtchouk aggregation. They are cross-checked.- Monte Carlo benchmark using 100,000 permutations for runtime comparison.- Exhaustive permutation check at N = 4 (24 permutations) and N = 8 (40,320 permutations).- Edge-case check: B = 0 case (g = [1, 1, 1, -3]), and a grid check showing no valid centered N = 4 field has B < 0.- Timing CSVs with median, interquartile range, minimum, and maximum for each size.- Environment details: Python 3.13.5, NumPy 2.3.5, Intel Xeon Platinum 8573C, 5-vCPU container, kernel 6.18.44.- LaTeX benchmark table matching Table 3 of the paper.- SHA256 integrity manifest for all scripts, CSVs, and the LaTeX table.- README with step-by-step instructions and a fixed random seed (20260917). MAIN COMPUTATIONAL RESULT On the recorded machine, the exact formula is about 10^4 to 10^5 times faster than a 100,000-permutation Monte Carlo estimate of the same first- and second-order moments. Dense and sparse evaluators agree to a maximum relative error below 2.5 x 10^-16 from n = 4 to 18. Timings are hardware-specific; all scripts and seeds are fixed so you can rerun the benchmarks on your own machine. WHO THIS IS FOR Statisticians, applied probabilists, and computational scientists who work with permutation inference, Boolean Fourier analysis, high-order contingency tables, or log-linear interaction models. If you need fast, exact null moments for Walsh-degree energies, you can use this package directly. For the full theorem, see the companion paper. SCOPE AND HONEST LIMITS This is a computational and reproducibility package. It does not make new physical claims, does not imply causality or retrocausality, and does not establish any microscopic Hamiltonian. The theorem is a static finite-sample exchangeability result. It is not a temporal, interventional, or physical law. Timings are machine-specific and are given as benchmarks, not universal performance claims. Full permutation distribution and tail probabilities are handled in the companion Paper-2 preprint, not here. CITATION If you use this package, please cite the paper and this record. Paper-1 v1.1 preprint: 10.5281/zenodo.22958349Paper-2 preprint: 10.5281/zenodo.22955773QDL Research Suite v1.0.0: 10.5281/zenodo.22952902Historical J601 reproducibility package: 10.5281/zenodo.22811634 LICENSE Source code — MIT. Data and benchmark outputs — CC BY 4.0. See the LICENSE file for full text.