Exact Monotonicity, Unimodality and Hyperbolicity of the Jonah Multipliers: The Spectral Log-Ratio, an Area-Preservation Threshold, and the Absence of Local Bifurcations

We prove exact monotonicity and unimodality properties of the Jonah saddle multipliers. The transverse multiplier λsλ_sλs is strictly increasing from −1-1−1 to 111 with derivative λs′=3(2t−1)(4t−1)/(2t4)λ_s' = 3(2t-1)(4t-1)/(2t^4)λs′=3(2t−1)(4t−1)/(2t4). The longitudinal multiplier λuλ_uλu is unimodal with unique maximum (17+77)/27≈1.3156(17+7√7)/27 ≈ 1.3156(17+77)/27≈1.3156 at t=(7+7)/14t = (7+√7)/14t=(7+7)/14. The spectral log-ratio S=ln⁡(λu/∣λs∣)S = ln(λ_u/|λ_s|)S=ln(λu/∣λs∣) is strictly increasing on (1/2,t0)(1/2, t_0)(1/2,t0) and strictly decreasing on (t0,1)(t_0, 1)(t0,1), diverging to +∞+∞+∞ at t0=(3+3)/6t_0 = (3+√3)/6t0=(3+3)/6. This monotonicity rigorously excludes any local bifurcation (fold, transcritical, pitchfork, flip, or Neimark-Sacker) on the entire branch. We identify an area-preservation threshold tV≈0.6083t_V ≈ 0.6083tV≈0.6083 as the unique root in (1/2,1)(1/2, 1)(1/2,1) of an irreducible quintic, where ∣det⁡DFt(Jt)∣=1| DF_t(J_t)| = 1∣detDFt(Jt)∣=1 and λs=−1/λuλ_s = -1/λ_uλs=−1/λu. The formula for λuλ_uλu is derived directly from the isosceles reduction, and both multiplier formulas are confirmed by exact computation of the characteristic polynomial of the full 3×33 × 33×3 Jacobian at 30 rational values of ttt. All results are exact and reproducible with the supplied Python code.