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 bound. On the analytic side (Route 1), we relate σ(h, ε) to the twisted Lasota–Yorke constants from Paper 34 and the certified spectral gap from Papers 37–38, deriving explicit inequalities (Theorems 1.3 and 1.4). We isolate distortion growth near the boundary of I_rig as the main obstruction to extending projective-metric arguments to all of I_rig (Proposition 1.5).