The Quantum Wavefunction of a Black Hole from Dual Compact Geometry: Standing Membrane Modes, Non-Rotational Parity, and Gravitational Echoes

Traditional quantum mechanics forces Schrödinger and Dirac complex wavefunctions onto flat Euclidean backgrounds, relying on artificial far-field boundary conditions (ψ → 0 as r → ∞) to enforce quantization. This approach leads to an unphysical infinite superposition of tension overtones (n) and mathematical singularities. In parallel, mainstream General Relativity relies heavily on rotational models when analyzing black hole Quasi-Normal Modes (QNMs), dismissing odd-parity modes as unphysical or forbidden by momentum conservation. Here, we present a purely geometric framework for the "black hole quantum wavefunction" constructed on a dual compact space. We derive a dual acceleration field, aₗ(x) = 1+l(l+1)/x² - x², completely independent of Planck's constant and complex operators. Microscopic tensor analysis reveals that the numerator term $1$ originates from the trace of the intrinsic Ricci curvature scalar of the two-dimensional unit sphere S² (1/2R⁽²⁾ = 1), while $l(l+1)$ represents the geometric stiffness of the Laplace–Beltrami shear tensor gradient. The 2D area scale transformation u = x² and dual area scale relations uncover a microscopic-macroscopic dual conjugate horizon, yielding the golden ratio area equations (u² - u - 1 = 0 and u² + u - 1 = 0) governed by 2D area scales u = φ ≈ 1.618 and u = 1/φ ≈ 0.618, which regulate black hole thermodynamic phase transitions and holographic fractals. This geometric field possesses a unique minimum equilibrium point, xeq = [1+l(l+1)]1/4, which strictly locks the dynamical tension degree of freedom, eliminating the redundant radial quantum number n and restoring perfect alignment with the Bekenstein–Hawking area law of the Holographic Principle. Furthermore, odd-parity modes ($l=1,3,5$) are reinterpreted as non-rotational, anti-phase "heartbeat" oscillations of the horizon sphere. Ultimately, we derive a tight drumhead eigenfrequency spectrum, fₗ = f₀ · [1+l(l+1)]1/8, providing an explicit coherent detection template for odd dark-mode echoes following black hole ringdown in LIGO/Virgo gravitational-wave observations.