Azimuthal finite-difference identities for the slow-spin leading-eikonal Kerr damping rate

The leading-eikonal Kerr quasinormal-mode spectrum is naturally organized at fixed μ = m/L, whereas holding a finite azimuthal number m sends μ to zero as L = ℓ+1/2 increases. Starting from the established leading-eikonal radial system and the leading-L spheroidal expansion through z⁶, we derive the odd-in-μ damping polynomial through seventh order in spin. Its displayed μ³, μ⁵, and μ⁷ terms generate exact azimuthal finite-difference identities after an ordered fixed-m re-expansion. For gravitational s=−2 fundamental modes, the combined Richardson grid contains 64 physical modes at ℓ=384,512,640,768, χ=0.0125,0.025 and m=±1,±3,±4,±5, each evaluated in three continued-fraction configurations (192 rows). Earlier tests at χ up to 0.05 are reported separately. The Richardson-extrapolated seventh-order contrast approaches its exact coefficient with approximately L⁻² behavior; L² times its relative error varies by 6.03% across four multipoles. The nested contrast loses 28–33 decimal digits. A full-grid 80-digit replay with stricter nonlinear termination reproduced the archived contrasts within 1.96×10⁻¹³ relative change and passed the numerical residual criteria. The preceding replay’s single numerical-gate failure remains recorded. These finite-L calculations are consistency tests, not unique coefficient identifications: an unbounded subleading remainder may survive the differences, and all roots share one continued-fraction formulation. The result is a reproducible high-order asymptotic diagnostic rather than a low-multipole waveform model. The first 70-digit calculation retains one residual-gate failure. The amended 80-digit calculation passes all 192 rows, representing 64 physical modes at three numerical configurations. Precision stability does not identify a unique asymptotic coefficient. This is an author preprint and has not been peer reviewed. The supplement preserves nulls, adverse checks and the stated scope. The existing predecessor code and data archive is available at https://doi.org/10.5281/zenodo.22080752.