Projective Metrics, Lasota–Yorke Geometry, and the Mixing Boundary for Twisted Bird–Map Operators (Paper 40 in the Non-Holomorphic Fractal Series)

Paper 40 in the Non-Holomorphic Fractal Series. Paper 39 introduced the mixing-rate exponent σ(h, ε) = −log θ(h, ε) for twisted Bird-map transfer operators and proved a first explicit uniform lower bound σ(h, ε) ≥ σ_max > 0 on the enlarged rigidity interval I_rig = [0.70, 1.10]. Paper 40 takes σ as fixed input and develops geometric and analytic structure behind this exponent. On the geometric side (Route 2), we construct positive BV cones for twisted operators on a central window I_core = [0.84, 0.90], prove cone preservation and finite Hilbert projective diameter (D_core ≤ 4 log 2), and formulate a conditional projective-metric contraction theorem (Theorem 1.2) that would strictly improve the uniform mixing…