Love's Equation: Spacetime as a Stable Phase of a Relational Substrate

This work introduces Love’s Equation, a variational unified field framework in which spacetime, matter, and cosmological structure emerge from a relational operator substrate. The theory is formulated as a multi-sector action functional combining geometric, scale-covariant, coherence, curvature, and Einstein-consistency terms, coupled through a gradient flow that defines the dynamical evolution of the system. A stable physical regime is identified via a positive fixed point of the flow, characterised by a bounded inter-sector coupling invariant and linear stability of the Hessian. In this regime, a bridge from operator structure to emergent geometry is established through spectral methods, with the low-spectrum limit of the induced Laplacian recovering metric structure. Matter arises from the projected Hessian of the action, providing an intrinsic mechanism for energy–momentum without introducing external fields. The Einstein field equations are shown to emerge as a consistency condition at stationary points of the functional, rather than being imposed a priori. At cosmological scales, the theory reduces to Friedmann-like dynamics, with an effective cosmological constant generated from spectral cutoff contributions rather than fine-tuned vacuum energy. Computational validation is performed via large-scale numerical experiments using gradient flow optimisation and machine-guided parameter search. Stable solution classes are identified that satisfy coherence, spectral gap, and curvature constraints while exhibiting physically consistent large-scale behaviour. The results indicate the presence of a structured phase space in which spacetime emerges as a stable, relational configuration of the underlying operator field. This framework provides a candidate approach to unifying geometry and matter through information-theoretic and spectral principles, offering a new pathway toward emergent spacetime in fundamental physics.