Depth of the in-tree of $s$ under $q \mapsto q s q^{-1}$ on $n$-cycles

Lineage and corrections

Corrects Depth of the in-tree of $s$ under $q \mapsto q s q^{-1}$ on $n$-cycles →

Supersedes Depth of the in-tree of $s$ under $q \mapsto q s q^{-1}$ on $n$-cycles →

Depth of the in-tree of $s$ under $q \mapsto q s q^{-1}$ on $n$-cycles parent of this event

Depth of the in-tree of $s$ under $q \mapsto q s q^{-1}$ on $n$-cycles evidence for this event

The depth theorem, with the full proof (companion paper, 24 July 2026) evidence for this event

The author's 2006 blog on the same question, via the Wayback Machine evidence for this event

The functional graph of the whole map (survey paper, 25 July 2026) evidence for this event

VibeMathed record: Depth of the in-tree of $s$ under $q \mapsto q s q^{-1}$ on $n$-cycles evidence for this event

This event attributed to Claude FABLE 5

Opus 4.8 constructed a branch of the stated depth, giving a lower bound, and believed it had a matching upper bound; that proof was wrong and the statement stayed a conjecture. FABLE 5 later proved it. In the author's summary of the method: "The proof turns conjugation, near $s$, into base-$p$ arithmetic." A cycle near the fixed point splits into a coarse base permutation and a vector of carries in $\mathbb{Z}/p$, $D$ acts on the carries by the carrying of ordinary base-$p$ addition, and the depth comes out as the nilpotency length of a shift difference - exactly that for odd $p$, one less for $p = 2$. The single missing carry that odd primes absorb and $2$ cannot is what produces the two-branch answer. A companion survey paper covers the rest of the graph: the other periodic orbits, congruences on basin sizes, and a cyclic-sieving count. The depth theorem is the substantive part. parent of this event