The Characteristic Polynomial of the Jonah Saddle

We study the two-dimensional linearisation of the pedal shape map at the Jonah saddle, using the longitudinal and transverse multipliers lambda u(t) and lambda s(t) imported from the companion papers. From them we derive the trace T(t)=(3t minus 1)(2t minus 1)/t squared, the determinant Delta (t)= lambda u lambda s, and the discriminant T squared minus 4 Delta =( lambda u minus lambda s) squared, which is positive on (1/2,1). The determinant changes sign at t 0=(3+ the square root of 3)/6, where the transverse multiplier vanishes and the linearisation has rank one. At t=7/10 the determinant is minus 11 squared times 13 times 41/7 to the 6. The main new result is a Liouville obstruction. For t in (1/2,t 0], the Jacobian determinant at the Jonah point is non-positive, whereas the time-T map of any C to the 1 flow has strictly positive determinant. Hence the pedal map is not the time-T map of a flow near the Jonah point in this range. For t in (t 0,1) the linear part admits a real flow logarithm, so there is no linear obstruction. Whether the nonlinear map embeds in a flow for t in (t 0,1) is left open. Status: the algebraic identities and the Liouville argument are proved, conditional on the imported multiplier formulas. The formula for lambda s is checked numerically against the Jacobian of the two-dimensional map at eight values of t, not proved here.