The (2, 3) -torus knot---the trefoil---lives on the very torus that carries thefinite residue dynamics of the Syracuse map: its two meridional windings encodethe powers of 2, its three longitudinal windings the powers of 3, and itsrational slope 3/2 is a continued-fraction convergent of the irrational slope 3 that governs the dynamics. We use this geometry as an organizing lensfor a set of recently proved results about the Syracuse valuationcocycle JanikFSS: the parity--residue coupling₂ (S (x) ) = (-1) ^ (3x+1), which makes the quadratic character of (/3ʳ) ^ an exact coboundary and biases the mod-3 residues in ratio1\!: \!2; the classification of coboundaries and the uniform spectral gap ofthe synchronized transfer operator (the finite ``H23a'' theorem) ; the keystonearrow (every divergent or cycling orbit casts a finite non-coboundary charactershadow) ; mean-criticality (any counterexample's valuation word must track theslope 3 exactly, on pain of descent) ; the unconditional exclusion ofall Sturmian and subexponential-complexity valuation words by a repetitionrigidity argument whose elementary core is machine-verified in Lean~4; and theeffective exclusion of cycles via linear forms in logarithms. On the torusthese results say: a counterexample must live inside the empirically observedbranch-locus band along the irrational foliation, may not follow anylow-complexity path within it, and cannot close up. We also revisit theempirical k=729 branch locus from 10^11 trajectories and reinterpret its``ghost island'' of 251 cells: it is the geometric shadow of the scale-inducednear-relation 2^729 3^460 (deficit -0. 083), which explains why itappears at 3⁶ and at no smaller power of 3.
John Janik (Fri,) studied this question.