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The in-tree of an n-cycle under cyclic conjugation: depth pᵉ⁻¹ for odd p, and 2ᵉ⁻¹−1 for p=2

Frédéric Lefebvre-Naré

For D(q)=q s q⁻¹ on n-cycles, Lefebvre-Naré’s October note claims depth p^(e−1) for odd prime powers p^e, depth 2^(e−1)−1 for powers of two, and the maximum prime-power depth for general n. The shorter proof uses a defect commutator and triangular Sylow-group filtration.

Sources for this record

What was already known

Lefebvre-Naré had studied the dynamics and enumerated small cases. An earlier Claude upper-bound proof had a blocking error; a July argument gave a longer proof of the same formula.

AI involvement AI-led

The note credits Claude with both proof constructions. Lefebvre-Naré supplied the problem, prompts, occasional redirects and publication work.

The author directly credits Claude with both proof constructions while documenting substantial prompting and occasional human steering.

Importance Modest

The exact formula resolves a focused question about this permutation dynamical system.

Limitations

Finite computations test examples, not the universal proof. Lefebvre-Naré says he cannot certify the argument, and no human mathematical review or proof-assistant formalization is documented.

Technical verification Partial

The October note reports exact computations for its finite examples/statements, and the registry reports finite enumeration and backward-tree checks. These test instances and intermediate algebra but not the all-n proof; the note expressly says no human referee has checked it. AI model rereading does not count as human or machine certificate checking.

Human understanding Few

The human author explicitly says he does not understand much of the mathematics and cannot verify the proof; no human mathematician's explanation of the new central mechanism is documented. This is affirmative limited-understanding evidence.

Correction and publication history