Syracuse Transfer
Matrix P_k

P_k — the Syracuse transfer matrix

The transfer matrix P_k encodes the Syracuse map on odd residues mod 2^k. Each column has exactly one nonzero entry, weighted 2^−v where v is the 2-adic valuation of the step. The animation shows how this sparse matrix structure relates to the chord diagram of the functional graph, and how ghost cycles appear at exceptional levels.

Three phases

Phase 1 — Matrix. The N×N sparse matrix P_k where N = 2^k−1. One dot per column: the row index is the Syracuse successor. Brightness encodes the weight 2^−v — brighter dots correspond to lighter weights (lower valuation). Gold marks the fixed-point cycle.

Phase 2 — Morph. The matrix unwraps onto a circle. Each dot position expands into an arc connecting source and target residues. Watch the single entry per column become a chord as the structure reveals its periodicity.

Phase 3 — Sweep. The chord diagram animates as k increases from 4 to 14. At most levels (non-exceptional) only the gold fixed point {1} appears. At exceptional levels — such as k = 10 — ghost cycles flash into view in teal.

Use the k slider to freeze on a specific level and explore the chord diagram manually. The curve slider adjusts the bow of the arcs.

Ghost Cycles of the Syracuse Map