math.AG — Algebraic Geometry

Counterexamples to Hironaka's Characteristic-$p$ Resolution: A Report On Queries to Frontier Agents

Contributed by Taylor Dupuy

Created on 23 October 2026, this manuscript draws on a frontier-dump investigation of Hironaka's proposed positive-characteristic resolution method. Taylor Dupuy initiated the queries during his talk How to Use AI in Mathematical Research Effectively at the University of Rochester Number Theory Seminar on that date. The findings were synthesized with harvest-dumps and developed into this manuscript with using-latexpowers. We examine the proposed positive-characteristic resolution mechanism of Hironaka (2017). Its negative-degree pieces are the unit ideal at a closed singular point where, for a common multiple $m$ of the generator degrees, $\wp(E,m)$ contains an element of order $m$. For the family $x^p+y^n$, with $n>2p$ and $p\nmid n$, the asserted transformation rule fails under the algebraic description of the characteristic algebra. With core focusing read by (44), the fixed local chain and the $\check E$ or $\hat E$ reading of the invariant, the length-one string agrees before and after the first blowup. With renewed chains and corresponding transported choices, the corresponding strings agree along the resolution sequence under either core-focusing reading. The two formulas for core focusing respectively retain the base-hiked exponent and, when its Inv-max stratum is smooth and closed, forget it. We give characteristic-$2$ limits on tail enclosure and a shade increase at a kangaroo point. These calculations do not supply a decreasing invariant or settle resolution. The mathematical draft of this manuscript was AI-generated by Claude Code from an AI-generated research record; its proofs received only AI review, and no human has checked them.