A Symmetry-Induced Spectral Collapse in a Torus-Parameterized Non-Hermitian Matrix Family: With an Application to Kerr Ringdown Decay-Rate Scaling

We give a complete spectral characterization of a 6×6 torus-parameterized non-Hermitian matrix family A(x,t₁,t₂) = −iI + B(x,t₁,t₂). Our main result is a symmetry-induced collapse: at the unique coupling x = 1, the shifted matrix B becomes real symmetric for every (t₁,t₂) ∈ T², forcing all eigenvalues of A to lie on the horizontal line Im(λ) = −1. We quantify the transition by the imaginary spread Δ(x), proving Δ(1) = 0 and Δ(x) > 0 for x ≠ 1, and we identify parameter locations attaining spectral-radius extrema on the torus. We further show the global spectral radius ρ(x) is non-monotone for small x, providing a concrete counterexample to monotonic expansion intuition in non-Hermitian coupling. We prove robustness: for x = 1 + ε with small |ε|, the imaginary spread satisfies Δ(1+ε) = O(|ε|). Beyond this specific family, we establish a mechanism theorem: any family A(x) = −iI + B(x) where B(x_c) is real symmetric at a unique x_c exhibits the same collapse phenomenon. We interpret the collapse as decay-rate synchronization in a damped-oscillator network and note a structural parallel with the suppression of Kerr quasinormal-mode damping as a/M → 1. Using 15 binary black hole mergers from GWTC-3 and GW250114, we find strong correlation (r = 0.905, ρ_s = 0.886) between remnant spin and ringdown lifetime—consistent with the spectral-flow interpretation of decay-rate compression near critical regimes.