The Flat-Triangle Boundary of the Jonah Branch: Asymptotics, Flip and Fold Degeneracy, and Embedding Across the Branch

The Jonah branch of fixed points of the pedal map F_t (incentre–Nagel interpolation) exists for t in (1/2, 1). It meets the equilateral fixed point at t = 1 in a transcritical collision. This note studies the other endpoint, t → 1/2⁺.There the isosceles reduction parameter s_J(t) = (2t−1)/(2t) tends to 0. The ratio of base to leg tends to 0, and the shape tends to a flat (degenerate) triangle. This is an interior point of the equator of the Kendall shape sphere, not a boundary of the sphere.With t = 1/2 + ε, the exact expansions of the multipliers are:λ_u = 1 + 4ε − 16ε² + O(ε³)λ_s = −1 + 24ε² + O(ε³)The trace is T = 4ε + 8ε² + O(ε³) and the determinant is Δ = −1 − 4ε + 40ε² + O(ε³). At t = 1/2 the Jacobian has eigenvalues +1 and −1, so the point is a fold-type candidate and a flip-type candidate [conjecture]. The nonlinear normal form is not computed.The determinant is negative on (1/2, t₀), with t₀ = (3+√3)/6, and it vanishes at t₀. Hence no C¹ flow has the pedal map as its time-T map near the Jonah point in that range (Liouville obstruction). For t in (t₀, 1) both eigenvalues are positive. Zhang's classification of planar hyperbolic embedding flows then gives a local embedding, provided F_t is C^∞ near the fixed point. That regularity is not verified here.Each statement is labelled as proved, proved conditional on the multiplier formulas of the companion papers, cited, or conjectural. No claim is made of a "maximal" obstruction, of a classification of the bifurcation at t = 1/2, or of any link with other open problems.