Discoveries / combinatorics

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

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.