Exact Degrees for a Fixed-Double-Pole Rational Hénon Family

We give an all-parameter degree calculation for the birational maps F(x,y)=(beta/x+gamma/x^2-delta*y,x) over the complex numbers, with gamma*delta nonzero. Outside the nontrivial root-of-unity resonances of -delta^3, the first dynamical degree is 2. At a resonance of order ell, two explicit rational generating functions describe the pure inverse-square and mixed cases, with the usual QRT exception at delta=1. The proof counts all finite zero/pole patterns and the boundary contributions on generic coordinate lines. In the mixed case an analytic product chart justifies repeated cancellations, including linearly growing singularity multiplicities that nevertheless yield the same degree series as constant-multiplicity patterns. Classical degree sequences and delayed-confinement polynomials are explicitly credited. The result concerns this fixed-double-pole family, not the full AIM investigation of quadratic rational Henon maps. This is a self-audited, AI-assisted, unrefereed preprint; no independent review, formal verification or absolute priority is claimed.