Fractional Logarithmic Scaling of Gravitational Wave Echoes from Volumetric Planck Cores

Searches for post-merger gravitational wave echoes currently rely on phenomenological exoticcompact object (ECO) models that treat the reflective boundary as an arbitrary free parameter.Typically anchoring this boundary a single Planck length above the event horizon, these models forcepredicted echo delays to adhere to a standard M ln M scaling. In this letter, we transition awayfrom ad-hoc boundary placements by modeling the black hole interior as a discrete thermodynamicsubstrate operating at absolute geometric saturation. By anchoring the physical core radius to thefundamental Planck density limit, we mathematically derive a unique, zero-free-parameter scalinglaw where the logarithmic slope is strictly throttled by a volumetric factor of 2/3. Applying thisformulation to the GW150914 remnant dictates a baseline isotropic delay of 75.9 ms, resolvingto a spin-adjusted observational target of 89.1 ms. This provides a rigorous, highly constrainedcoordinate for current Advanced LIGO and Virgo data pipelines, significantly reducing the trialsfactor and offering a direct empirical test for the volumetric quantization of spacetime.