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BREAKTHROUGH REPORT14 Aug 2026

Retroactive Breakthrough: AI-assisted refutation of the Jacobian Conjecture for n≥3

A missed historical breakthrough has been corrected into the observatory: Levent Alpöge’s explicit counterexample, credited in part to Claude Fable 5, refutes the Jacobian Conjecture in every dimension n≥3. Independent exact-arithmetic and formal verification support the claim; the n=2 case remains open.

ANALYSIS

Methodology note: Historical index values on this Pulse are normalized to the current methodology for comparability. The original as-published record remains unchanged in the canonical archive.

Levent Alpöge announced an explicit polynomial map in C^3 with constant nonzero Jacobian determinant that is not injective, refuting the Jacobian Conjecture for n≥3. He explicitly credited Claude Fable 5 with work leading to the construction. Independent exact-arithmetic checks and formal verification followed rapidly.

Scope

This does not settle the two-dimensional Jacobian Conjecture, which remains open.

Compass assessment

Evidence: independently verified. Relevance: 10/10. AI role: co-developed/documented assistance. This is strong evidence that frontier AI can materially contribute to novel mathematical discovery on long-standing open problems, but it does not demonstrate an autonomous end-to-end research loop.