Exact Projection Quality of OneTwo Sobol' Sequences at 65,536 Points

Abstract

Low-dimensional projection quality is a practical bottleneck for Sobol' sequences used in quasi-Monte Carlo integration and computer graphics. For a digital net with 2^16 = 65,536 points, the t-value measures exact equidistribution in dyadic boxes, with smaller values indicating stronger uniformity. We determine the exact m = 16 census for all 345 pair-aligned four-dimensional blocks in the released 692-dimensional OneTwo construction: 9 blocks have t = 4, and the remaining 336 have t = 5. For dimensions 25–28, determinant-one minors prove the upper bound, while a unique row dependency with mask 0xD2F yields an explicit empty dyadic box for the lower bound. Exact computations over F_2 agree across three independent rank routes, direct enumeration checks all 65,536 points, and a Lean companion checks the finite worked-block encoding. The worked certificate is self-contained; reproducing the complete census additionally uses the exact upstream direction-number table identified by a pinned public commit and file hashes.