A Geometric Unification of Particles, Spacetime, and Cosmology: From Private Spacetimes to Emergent Gravity and Dark Energy

We present a relativistic and quantizable geometric framework in which each quantum particle possesses an intrinsic spatial geometry governed by a universal stiffness constant A0 and sourced by its wavefunction. The model is defined by a covariant action coupling a complex scalar field to the metric, including a curvature-coupling term ξR|ϕ|2. From this action we derive coupled Einstein–Klein–Gordon equations; repulsive self-gravity emerges naturally, preventing gravitational collapse and stabilizing quantum matter. Interactions stitch together individual world-blocks via junction conditions that enforce continuity of the metric and extrinsic curvature. In the continuum limit, the collective dynamics yield an emergent spacetime satisfying the full Einstein equations, with Newton’s constant G = c4/(8πA0). The harmonic zero-mode of the strain field may acquire a vacuum expectation value set by the Hubble radius, providing a natural explanation for dark energy. Canonical and path-integral quantization are outlined; the problem of time is resolved relationally through stitching constraints, giving rise to an emergent global time operator. Quantum fluctuations of the lapse network can imprint on the primordial power spectrum and generate a stochastic gravitational wave background. The model predicts testable signatures across scales, from matter-wave interferometry to cosmology, offering a unified, falsifiable foundation for quantum gravity.