Relational Coherence Dynamics and Emergent Geometric Structure: An Exploratory Pre-Geometric Model Toward Spacetime Emergence

This preprint presents an exploratory pre-geometric framework for examining how effective geometric structure may arise from relational coherence dynamics on a fully connected network of informational degrees of freedom.

The model assigns a scalar coherence variable to each node and investigates how nonlinear dynamics on a fixed symmetric relational substrate can generate structured configurations, spontaneous symmetry breaking, stable attractor-like patterns, and phase-transition-like growth and saturation of coherence-field variance.

Spatial distance, metric structure, and global temporal ordering are not assumed a priori. Instead, an effective relational distance matrix is constructed from the interaction weights as

d_ij ∝ 1 / (K_ij + ε),

where ε is a small regularization constant. Multidimensional scaling (MDS) is then used to obtain an exploratory two-dimensional representation of the relational distance structure.

The numerical simulations use N = 100 nodes, fixed symmetric relational weights K_ij, and coherence-field dynamics with nonlinear amplification, saturation, and network-mediated coupling. The accompanying simulation code reproduces the representative figures discussed in the manuscript, including coherence-field evolution, variance evolution, an exploratory MDS embedding, and a shuffled-coherence control.

This work is intended as an exploratory theoretical and computational framework rather than a complete theory of quantum gravity or spacetime reconstruction. The model does not reconstruct Lorentzian spacetime, establish a relativistic causal structure, or determine a fixed emergent dimensionality.

This Zenodo record includes the preprint manuscript, simulation code, and README file for reproducibility and scholarly discussion.