σ_max and the Boundary: Mixing-Rate Exponents for Twisted Bird-Map Operators (Paper 39 in the Non-Holomorphic Fractal Series)

Paper 39 in the Non-Holomorphic Fractal Series. We introduce and study the mixing-rate exponent σ(h, ε) = −log θ(h, ε), where θ(h, ε) < 1 is the explicit L¹ decay constant for the twisted Bird-map transfer operator L_{h,ε} established in Paper 38. Two equivalent definitions are given: one in terms of the L¹ mixing rate, and one as the logarithmic spectral contraction of the remainder operator in the Ruelle–Perron–Frobenius decomposition. From the worst-case cell (h_A, ε = 0.80) of Paper 38 we obtain the explicit unconditional upper bound σ_max = sup_{h,ε} σ(h, ε) = −log θ_max ≈ 0.1370, with θ_max = 0.8720 on the enlarged rigidity interval I_rig = [0.70, 1.10]. We frame two analytic routes toward potentially sharper bounds: a direct remainder-spectrum analysis via quasi-compactness (Route 1, low-risk) and a projective-metric contraction argument on the positive BV cone (Route 2, medium-risk). We state one structural theorem (an explicit uniform bound on σ_max), one conditional theorem (improvement contingent on a projective-cone hypothesis), and three open conjectures about boundary behaviour as σ(h, ε) → 0.