A primitive-ratio cut and a 7/12 bound for the abc exceptional set
Ruiyi Zhang · Preprint, 2026-10-08 · 16 pages
PDFHTMLDOI 10.5281/zenodo.23247752Source and checks
Abstract
We prove N_1(X) ≪_ε X^(7/12+ε), where N_λ(X) counts the abc triples of exponent λ in [1, X]^3. The previous bound proved in full was X^(13/20+ε), and Bernert (arXiv:2506.13364, Section 4) sketched X^(8/13+ε). We keep the anatomic reduction of Bernert, Browning, Lichtman and Teräväinen. It gives 22 inequalities for the exponents of a box. We add two exact rows per side for the root classes of index at least seven, and a cut. The cut fixes all but two coprime products on two sides. The remaining pairs are primitive points of a lattice. Its determinant is a product of roots on the third side. This gives 1 560 195 further inequalities. An exact Farkas certificate over 8192 cells and 10 820 leaves gives exponent 7/12 for the box count. On each leaf the multipliers sum to at most 18/7. This pays for the losses at λ = 1.
Cite
Ruiyi Zhang, A primitive-ratio cut and a 7/12 bound for the abc exceptional set, preprint, Zenodo, 2026. https://doi.org/10.5281/zenodo.23247752