Toroidal Hamiltonian Framework for Black Hole Entropy and Gravitational Wave Echoes

We propose a novel geometric Hamiltonian framework integrating toroidal spatial topology (T3) and spinorial fields within an extended Wheeler–DeWitt formalism. Motivated by string theory compactifications, this approach yields quantum corrections to black hole entropy, parametrized by toroidal cycle lengths, and predicts distinctive gravitational wave echoes. We find that the entropy correction factor is sensitive to the scaling exponent β, and echo frequencies are tied to the cycle lengths. We derive the topological operatorˆT via path-integral quantization, leveraging bosonic and fermionic fluctuation spectra to encode nonlocal geometric effects. A supervised machine learning pipeline, using a convolutional neural network, is developed to detect echo signatures in LIGO/Virgo data. Publication- ready TikZ/pgfplots schematics, including toroidal geometry and echo waveforms, ensure a self-contained manuscript.