Cone Diameter Growth and the Mixing-Boundary Obstruction for Twisted Bird–Map Operators (Paper 41 in the Non-Holomorphic Fractal Series)

Paper 41 in the Non-Holomorphic Fractal Series. Paper 40 established positive BV cones for twisted Bird-map transfer operators on the central window I_core = [0.84, 0.90] and proved uniform Hilbert projective diameter bounds there, together with a conditional projective contraction theorem and a qualitative boundary obstruction on the enlarged rigidity interval I_rig = [0.70, 1.10]. In this paper we convert that obstruction into a numerical and geometric diagnostic for cone diameter and invariant-density oscillation near the mixing boundary, using only the existing certified grid and transfer-operator infrastructure. We define a projective-diameter proxy D(h, ε) and an invariant-density oscillation functional osc(φ_{h,ε}) on the certified cells, and we report systematic numerical evidence that D(h, ε) remains uniformly larger than the analytic central-window bound on I_rig \ I_core and exhibits nontrivial growth as ε approaches the boundary values 0.70 and 1.10 for at least one kernel. We formulate Conjecture 4.1 (numerical blow-up of the diameter proxy) and Proposition 4.2 (a coarse oscillation lower bound for D(h, ε)), both explicitly labeled as numerical and diagnostic rather than asymptotic theorems.