The Heartbeat of Black Holes: Black Hole Ringdown and Gravitational Wave Echoes Under a Dimensionless Geometric Model

Based on a dimensionless geometric model, this paper proves that at the foundational level of geometric source code, all massive systems possess a unified and constant intrinsic angular frequency of ω = 2 (corresponding to a dimensionless period of T_dimless = π). By mapping the characteristic length scale to the black hole's tension diameter R_0 = 2 R_s = 4GM/c^2, we derive the fundamental scalar pulsation frequency formula for a black hole: f = c^3 / (4π GM). For the remnant black hole of mass M_f = 62 M_⊙ produced in the GW150914 merger event, our model predicts a dominant ringdown frequency of 260.61 Hz, which aligns remarkably well with LIGO's observational data. Compared to the quasi-normal mode (QNM) value of 272.60 Hz calculated under mainstream Kerr geometry, a mass-independent, algebraically exact constant deviation of 4.40% is identified. Furthermore, we extend this geometric boundary to the mechanism of gravitational wave (GW) echoes, demonstrating how the scalar breathing mode provides a coordinate-free, spin-axis-independent geometric explanation for horizon boundary pulsations and the echo time delay (Δt_echo ≈ 0.11 s).