Mathematical Analysis and Analytical Characterization of Mass-Ratio-Dependent Gravitational Recoil and Remnant Diagnostics in Binary Black-Hole Mergers

This study presents a unified mathematical assessment of binary black-hole dynamics, gravitational-wave diagnostics, and recoil predictions within the vacuum Einstein framework. The formulation connects the 3+1 initial-value decomposition, conformal and generalized harmonic evolution variables, horizon diagnostics, Newman–Penrose waveform extraction, and detector-weighted waveform comparison. Quantitative evaluation combines previously reported remnant and radiation measurements with deterministic calculations from a nonspinning phenomenological recoil relation. The equal-mass nonspinning benchmark gives a remnant mass fraction of 0.95162 ± 0.00002 and dimensionless spin of 0.68646 ± 0.00004. The adopted recoil fit reaches approximately 175.2 km s⁻¹ near mass ratio q ≈ 0.36 and vanishes at equal masses, consistently with exchange symmetry. Published dominant quadrupolar energy fractions exceed 98% for equal-mass nonspinning coalescence and are approximately 68% in the extreme-mass-ratio regime. Analytical differentiation and limiting expansions characterize the fitted recoil response. The assessment establishes a consistent interpretation of heterogeneous benchmark quantities while distinguishing analytical model evaluation from independent numerical-relativity simulation. No new spacetime evolutions, detector reconstructions, or waveform-convergence measurements are claimed.