Exchange-Memory Criticality as an Origin of Emergent Distance, Time, and Geometry

We propose a framework in which spacetime is not fundamental but emerges from accumulated exchange-memory correlations between underlying quantum objects. When two objects interact, energy is transferred and a pairwise correlation C_ij is updated. Distance is defined as the information cost of maintaining this correlation:
d = −ξ ln(C/C₀)
The memory dilution rate Γᵢ is derived from a maximum entropy principle, yielding a Boltzmann-type equilibrium distribution that connects the framework organically to statistical mechanics. Vacuum correlation C_vac is grounded in zero-point energy, ensuring no two points are ever fully disconnected. Time emerges as the coarse-grained ordering of exchange events. Lorentzian metric signature is derived from the causal asymmetry of correlations, without circular assumption. Three-dimensional space is proposed to emerge as a critical fixed point of a renormalization group flow over the exchange-memory network — a balance between information propagation efficiency and memory storage cost. Gravity is interpreted as a relaxation field arising from local deviations from this critical dimensionality.
A toy numerical study on a random geometric graph indicates that matter sources locally reduce spectral dimension, yielding a potentially distinguishing prediction absent from standard General Relativity. The framework is explicitly incomplete: no microscopic Hamiltonian, Newton constant, graviton, or full Einstein equations are derived. This work is offered as a mathematically structured conceptual foundation inviting rigorous development and computational verification.