The Refined Space–Time Membrane Model: Deterministic Emergence of Quantum Fields and Gravity from Classical Elasticity

We present a deterministic elasticity framework—the Space–Time Membrane (STM) model—that unifies quantum‑like phenomena, gauge field emergence, black hole singularity avoidance, and cosmic acceleration within a single high‑order partial differential equation (PDE). By incorporating scale‑dependent elasticity, higher‑order (∇4,∇6 ) derivatives and non‑Markovian decoherence, the STM model replicates key features of quantum field theory while seamlessly introducing gravitational curvature. A bimodal decomposition of the membrane displacement naturally yields spinor fields; enforcing local symmetries on these spinors reproduces gauge bosons (e.g., photon‑like, gluon‑like) as deterministic wave–anti‑wave cycles with zero net energy over each cycle. Multi‑scale expansions reveal that sub‑Planck wave excitations can remain non‑decaying if damping is negligible and the signs of certain couplings (e.g., ΔE and λ) align to stabilise wave amplitudes. Once coarse‑grained, these persistent waves leave a near‑uniform offset in the emergent Einstein‑like field equations, acting as dark energy and driving cosmic acceleration. In addition, black hole interiors are regularised by enhanced stiffness from the higher‑order operators, replacing singularities with solitonic or standing‑wave structures. The model’s non‑Markovian damped PDE also explains wavefunction collapse through deterministic decoherence, reproducing the Born rule and entanglement analogues without intrinsic randomness. Finally, allowing a mild late‑time variation in the leftover vacuum offset addresses the Hubble tension by shifting the expansion rate at low redshifts. Future research will refine numerical PDE simulations, test exact operator self‑adjointness, and compare predictions against high‑precision data to fully assess this deterministic route to reconciling quantum phenomena, black hole physics, and cosmological observations.