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

Primarily AI-generated textHuman understanding: no partsmath.AG — Algebraic Geometry

Contributed by Taylor Dupuy ↗

Version 1 / Oct 29, 2026 / CC0 1.0

Abstract

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 mm of the generator degrees, ℘(E,m)\wp(E,m) contains an element of order mm. For the family xp+ynx^p+y^n, with n>2pn>2p and p∤np\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 Eˇ\check E or E^\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-22 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.

Provenance statement

This manuscript began with a prompt during Taylor Dupuy's University of Rochester Number Theory Seminar talk, 'How to Use AI in Mathematical Research Effectively,' on 23 October 2026. The demonstration used frontier-dump, followed by harvest-dumps and using-latexpowers, to investigate Hironaka's preprint. Claude Code agents assembled the mathematical draft on 23–24 October. Codex agents revised it on 26–27 October. The source includes a full Software and AI assistance section. The proofs have received AI review only; no human mathematical verification is recorded.
AI provenance filesMANIFEST.txt · 3 KBfrontier-dump--ai-usage.md.txt · 7 KBfrontier-dump--math-audit.md.txt · 39 KBfrontier-dump--report.md.txt · 36 KBfrontier-dump--search_constant_ridge.out.txt · 1 KBfrontier-dump--search_constant_ridge.py.txt · 4 KBfrontier-dump--search_constant_ridge2.out.txt · 1 KBfrontier-dump--search_constant_ridge2.py.txt · 4 KBfrontier-dump--tokens.md.txt · 2 KBfrontier-dump--verify-output.txt.txt · 4 KBfrontier-dump--verify.sage.txt · 13 KBharvest-checks--followup_all_p.out.txt · 5 KBharvest-checks--followup_all_p.sage.txt · 5 KBharvest-checks--integral_closure_check.out.txt · 1 KBharvest-checks--integral_closure_check.sage.txt · 1 KBharvest-checks--kangaroo_narasimhan_check.out.txt · 1 KBharvest-checks--kangaroo_narasimhan_check.sage.txt · 2 KB

References

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  2. Claude Code agent (claude-fable-5-1) for Taylor Dupuy. Adversarial review of H. Hironaka, ``Resolution of Singularities in Positive Characteristics''. 2026. AI-generated research record, not human-reviewed: \textttreport.md, \textttmath-audit.md, \textttverify.sage (sections V1–V4), \textttverify-output.txt, \textttsearch_constant_ridge*.py; directory \textttai/2026-10-23-hironaka-char-p-resolution-review/ of https://github.com/tdupu/dumps (repository may be private)link
  3. Claude Code agent (claude-opus-5-5) for Taylor Dupuy. Coordinator checks for the harvest of the Hironaka review. 2026. SageMath 10.8 scripts \textttintegral_closure_check.sage, \textttfollowup_all_p.sage, \textttkangaroo_narasimhan_check.sage and their outputs; directory \textttlatex/hironaka-char-p-counterexamples/harvest/checks/ of https://github.com/tdupu/dumps (repository may be private)link
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  1. Narasimhan, R. Monomial equimultiple curves in positive characteristic. Proc. Amer. Math. Soc., vol. 89, pp. 402–406. 1983. Bibliographic metadata checked against the journal volume contents and the Clay Mathematics Institute bibliography; primary text not opened
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Version history

  1. v1Submitted by Taylor DupuyInitial depositCurrentOct 28, 2026