Exact Binary Projections for Array-RQMC: Joint Laws and Pathwise-Preserving Execution

Exact Binary Projections for Array-RQMC: Joint Laws and Pathwise-Preserving Execution Aoi Kawasaki. Preprint; not peer reviewed. Contribution For a specified two-dimensional Sobol net with linear matrix scrambling and a digital shift, the complete rank-ordered binary innovation vector is uniform over exactly N outcomes at N = 2^m paths. An odd mask and a fair bit describe the law, requiring m independent fair bits per step under the stated ideal randomization model. Independent adjacent antithetic pairs have the same covariance but need not have the same higher-order law; exact nonlinear witnesses give variances 0 versus 2 and 4 versus 2. This characterization enables direct generation without generating unused coordinates or sorting the point set. A second rewrite maintains exactly the reference state order for stopped scalar binary chains with monotone branches, including absorption and floating-point ties. Measured results and limits Across eight fixed benchmark cases in two input encodings, state-order maintenance reduced warm execution time by 37.2% at N = 512 and 67.5% at N = 4,096 relative to direct binary generation with full state sorting. The encodings do not create sixteen independent model laws. The measurements are from one CPU/software stack, not a held-out hardware or dynamics study. The joint-law equivalence of the library and direct generators is an ideal-law statement, not equality under the same finite integer seed. Direct and order-maintained execution do preserve the same finite-seed trajectories and outputs under the stated arithmetic conditions. No new convergence rate or universal variance advantage is claimed. Comparisons with the measured CRN and antithetic implementations remain workload- and reuse-dependent. Avoidable mask initialization was included in the measured baselines; it is not an intrinsic CRN cost. Startup and calibration can outweigh warm-time savings. The publication retains those recorded costs. Included materials and reproducibility scope The record contains the manuscript PDF and a self-contained reproducibility capsule with manuscript/build sources, three array kernels, 19 tests, saved table/figure inputs, two figures, dependency versions, and SHA-256 inventories. The tests include exact finite enumeration, within-project separate reference implementations, absorption and floating-point tie checks, and saved-data arithmetic. They are not third-party peer review. The capsule does not include every raw calibration/validation observation, bootstrap sample, timing round, or baseline/calibration driver. Regenerating saved tables and figures is not a rerun of the full original experiment. No GPU, pretrained model, private workspace, or external dataset is required. Rights and disclosure Publication materials and saved data: CC BY 4.0. Code: MPL-2.0, with per-file scope in LICENSES.txt. Generative-AI assistance is disclosed in the manuscript. The version-specific archival identifier is doi:10.5281/zenodo.22728405.