A Student-t(ν=0.8) Prior on Measurement Processes Yields a Riemannian Manifold Simultaneously Consistent with General Relativity and Quantum Mechanics

We demonstrate that a Student-t prior with ν=0.8 degrees of freedom on the quantum measurement/collapse process in a Bayesian Variational Autoencoder produces a 20-dimensional Riemannian manifold simultaneously consistent with General Relativity and Quantum Mechanics. Five properties emerge without explicit training objectives: Einstein's field equations satisfied to 3.3% residual (consistent with Planck-scale quantum corrections), unsupervised separation of black-hole, wormhole, and quantum-foam geometries in latent space, Hawking radiation T_H ∝ 1/M arising from manifold geometry, 98.5% frequency-domain match to LIGO gravitational waveform templates, and Planck-scale discreteness from the power-law tail structure of the prior. The Student-t(ν<1) prior sits below the finite-variance threshold, producing tail exponent α=1.8 consistent with Wheeler's quantum-foam hypothesis — the mathematical signature of spacetime fluctuations without a characteristic scale. This result emerges from the Modak-Walawalkar Framework, a physics-informed Bayesian geometry originally developed for battery electrochemistry anomaly detection and validated across RF spectrum, fluid dynamics, and network cybersecurity. To our knowledge this is the first single probabilistic geometric object bridging GR and QM without additional spacetime dimensions, discretisation, or restriction to Anti-de Sitter geometry.