On the Diophantine Stability of the Syracuse Map

We present the results of an exhaustive branch locus computation forthe Collatz map on the family of finite tori²ₖ = (/k) ², carried out for alln 10^11 across 27 hierarchically nested grid levels (k \1, 2, 3, , 19683\). The computation consumed61. 5~hours on a 24-core Intel Core Ultra~9 275HX, accumulating25. 1 10^12 Collatz steps. Eight figures document theprincipal findings: cell saturation, torus heatmaps, Diophantinesignatures, shift-of-finite-type (SFT) forbidden transitions, foliation enrichment collapse, cell-level statistics, convergenceover~N, and transition/shadow analysis. A comparison with theearlier 10^10 run reveals new structure invisible at lower~N: a ``Diophantine ghost island'' of 251~cells at perpendiculardistance~370 from the main branch locus, created by the best rationalapproximant 460/729 ₃ 2, together with a sharpenedgap in the shadow offset distribution and a new downward-slopingfeature in the foliation-position shadow profile.