Gravitational Wave Ringdown from the Rendering Algebra R12: The Axis Count from the Modular Spectrum, the Extremal Determination of the Rendering Anchor, and the Nine-Event Confrontation

Abstract

We derive the quasi-normal mode (QNM) spectrum of black hole ringdown from the five axioms {12, π, γ1, ln 2, φ} of Array Cosmology (AC) with zero free parameters.

The Schwarzschild (χ = 0) fundamental mode (l = 2, n = 0) frequency is ω˜R = γ1/(12π) = 0.37494 (GR exact: 0.37367, error 0.34%), and the quality factor is Q = 2 + Znoise = 2.098 (GR: 2.100, error 0.10%). For Kerr black holes,
spin progressively collapses the 12-axis rendering loop to a single Volma stroke (3 axes) via the golden-ratio implosion exponent φ^−2 : the effective axis count is Neff(χ) = 3 + 9(1 − χ) φ−2 , yielding ω˜R(χ) = γ1/[π Neff(χ)] with RMS 2.3% against numerical relativity. The quality factor Q(χ) = (2+Znoise)/(1−χ) 1/2−Znoise achieves RMS 2.2%. The overtone spacing is ∆˜ωI = 2Znoise (0.8% match). Applied to GW150914 (M = 68.5 M⊙, χ = 0.69), the model predicts f = 242 Hz
and τ = 4.41 ms, within 1.1σ and 0.8σ of LIGO observations respectively. All five axioms are employed: 12 (loop topology), π (angular circumference), γ1 (tension anchor), ln 2 (Landauer floor), and φ (spin collapse). This supersedes the previous
AC ringdown paper, correcting the erroneous ×1.699 damping-time scaling