What was already known
Gurvich and Naumova established existence for every n and identified some one- and two-partition cases. They conjectured more than two in the remaining cases.
AI involvement
JD Jones directed the project. The classification and proof arose in consultation among Claude, GPT-6 and Grok, and Claude wrote the new Lean counting development; earlier existence work was retained with credit.
The classification and proof arose in human-directed consultation among AI systems, and Claude wrote the new counting development.
Importance
The classification completes the base-three counting problem and gives infinitely many n=3ᵗ−6 with exactly two partitions, contradicting that conjecture.
Limitations
The result concerns only blocks of size at most three with sums powers of three. It does not address other bases or general m; the source authors have not endorsed the correction.
Technical verification
The source reports kernel-checked exhaustive bases and a sorry-free Lean proof of the four universal classification theorems; independent Python counts supplement finite cases only. This review did not rerun the build.
Independent checking
Lean 4/Mathlib with decide +kernel exact finite search and induction
- Checker
- Lean 4 kernel
- Scope
- GNM.count_eq_one_iff, count_eq_two_iff, three_le_count_iff and count_eq_two_of_add_six
- Basis
- Reported by the linked checker
Human understanding
The detailed proof map is available, but no named human explanation of the central stabilization/frame mechanism is documented; Few is an evidence-gap default.
Correction and publication history
The classification completes the base-three counting problem and gives infinitely many n=3ᵗ−6 with exactly two partitions, contradicting that conjecture.
Read this published revision