Twisted Lasota–Yorke Inequalities and Conditional Watson-Band Rigidity for Bird-Map Borel Transforms (Paper 34 in the Non-Holomorphic Fractal Series)

Paper 34 in the Non-Holomorphic Fractal Series. We introduce a family of twisted transfer operators L_{h_W,ε} on BV for Watson-class kernels h_W(y) = y/(1+y²) acting on the autonomous Bird-map y-dynamics, and establish a Lasota–Yorke inequality for each ε ∈ (0,1]. Assuming a spectral gap γ > 0 for the twisted transfer operator at ε = 1 (Hypothesis 3.3), we derive a pressure-type bridge relating the spectral radius of L_{h_W,ε} to the exponential type of the associated Watson-class Borel ladder, and formulate a conditional Watson-band placement statement (Theorem 3.4). Keller–Liverani perturbation theory then yields gap persistence on an open interval I_rig ∋ 1, giving a conditional Watson-band rigidity result (Theorem 4.1). Numerical Ulam-matrix approximations of L_{h_W,ε} on a 2^12-cell partition calibrate the spectral gap at ε ∈ {0.84, 0.90, 1.0}, while independent Watson-ladder fits from Paper 32 place the WBA kernel in the Watson band B_W = [0.79, 0.87] over the same ε-window.