The Jonah Edge State: Edge Tracking in a Two-Dimensional Shape Map with an Exactly Known Saddle
Year: 2026
Type: article
Abstract: The pedal map F_t of the Nagel line acts on triangle shapes. At t = 7/10 it has an isosceles saddle, the Jonah triangle 4:7:7, with exact multipliers 451/343 and −143/343. Trajectories on one side of its stable manifold end at the equilateral triangle, on the other side they flatten, so the stable manifold plays the role of the "edge" of subcritical transition. We run bisection edge tracking on this map and check three predictions against the exact saddle: Bisection on the apex angle converges to 2·arcsin(2/7) = 33.20309919804047° to 7×10⁻¹⁵ degrees in 50 cuts. The number of steps a tracked trajectory lingers near the saddle grows by 8.4 per decade of precision, as predicted by ln 10 / ln(451/343). A tracked point off the symmetry axis approaches the saddle along the edge at 0.4169 per step, equal to |−143/343|, alternating sides. The map can serve as a small test case with known answers for edge-tracking codes. Scope: the map is discrete and two-dimensional; it is not a model of the Navier–Stokes equations and no quantitative link with fluid flows is claimed. Everything is computed for t = 7/10 in double precision; the global separation of the basins is shown numerically, not proved. Script included (NumPy only).
Author: Schloss, Jonah
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225591994