A three-point counterexample to a random lattice discrepancy comparison

Abstract

A full negative answer to the universal expected p-th-power comparison posed by Novak, reporting joint work with Hinrichs, in Oberwolfach Report 13/2004, printed p.698 (problem 30000052; OWR-723-005). For independent continuous uniform generator and shift, three one-dimensional indexed lattice points have expected normalized anchored fourth discrepancy moment 19/1620, exceeding iid 4/405 = 16/1620 by 1/540, with ratio 19/16. The four-page elementary note gives an inclusion-area proof, a separate empty-tail bound 2/[105(n-1)] reversing the iid comparison for every n at least 4, whole-law equality for n at most 2, and second-moment equality for all n and d. Exact overlapping results and broader dimension/exponent results appear in Alper Ferudun, 14 October 2026, DOI 10.5281/zenodo.23372668, prominently credited in the abstract, related-work discussion and bibliography. The source repository PR has GitHub creation record 10 October 2026; this note is a new release on 17 October. Historical repository public visibility and worldwide first discovery/publication are not certified, and no inference about awareness, dependence or copying is made. The result is not claimed new relative to all current literature. A bounded current audit identified no pre-PR full answer; exhaustive priority remains uncertified. Several journal/full bodies and final-version comparisons remain unavailable, and citation coverage is incomplete. The source archive records those limits and preserves every original source-PR file unchanged. The source and verification archive includes three distinct independent mathematical audits, broad and focused priority reviews, a fresh adversarial priority-classification review, seven pinned standard-library exact verification programs with results and a portable ordinary/optimized runner, and dated source/chronology receipts. A separate affine-cell integration verifies the supplementary four-point value 139/17280. A further Fourier argument is a newly supplied application of classical Su 2000 and Anderson et al. 2007 formulas, not evidence of an earlier recorded resolution. Computation supplements standalone analytic proofs. The claim concerns expected p-th powers, not expected norms or the separate existence of good lattice generators. AI tools were used extensively in proof development, exact computation, adversarial verification, literature research and writing. Unrefereed; independent AI audits are not external human peer review, journal acceptance or formal proof-assistant certification.