Informational Condensation in the Rank-1 Limit: Emergence of Causal Order from a Pre-Geometric Substrate

This work originated from a simple, free-form exploration. Science, while fundamentally built upon method, is first and foremost curiosity taking shape. Although it may initially appear excessively technical, science is merely curiosity disciplined by method. But it is always, at its core, curiosity. This paper may seem densely technical, but it is simply the way I found to give directionto what appears to be a trivial idea, yet is the driving force behind many searches: to understand the Universe. That being said, this toy model investigates the spectral dynamics of high-dimensional correlation systems and numerically demonstrates the emergence of a stable dominant mode in pre-geometric regimes. A combined analysis of the spectrum of the informational metric (g = − log ρ), spectral entropy, and the projected Hessian reveals a robust phenomenon: Informational Causal Condensation. Computational experiments at scales N = 512 and N = 1024 show that: (i) the entropy of the collapsed spectrum converges to S ≈ 0, indicating dimensional complexity collapse; (ii) no geometric subspace emerges spontaneously; (iii) the condensed mode remains stable under infinitesimal perturbations (H ≥ 0). These results provide a mathematical description of a pre-geometric state in which spatial structure is absent and only a single ordering axis becomes dominant. We propose this informational ordering as a structural precursor to physical time, distinct from the dynamical time dimension of relativistic cosmology. This work establishes a minimal numerical model describing how a purely informational system can exhibit causal dominance without generating geometry. A critical methodological consideration in this study is the potential criticism regarding the inevitability of dominance. In correlation-based systems with strictly positive entries, the existence of a single dominant eigenvalue (λ1) is algebraically guaranteed by the Perron-Frobenius Theorem. This raises the risk that the observed Rank-1 dominance might be interpreted not as new physics, but rather as a trivial algebraic artifact inherent to the matrix choice. To address this distinction, we introduce the analysis of the Phase Behavior vs. Correlation Parameter (β). Figure [Phase Behavior] illustrates the system’s dynamic response: instead of exhibiting a static dominance, the system displays a dynamic crossover. We observe a reciprocal evolution: the dominance ratio (λ1/λ2) remains trivial (≈ 1) in the disordered regime but exhibits exponential growth exactly as the Spectral Entropy collapses from high disorder to zero. This sharp, parameterdependent behavior demonstrates that the condensation is a physical regime driven by correlation intensity, rather than a static property of the matrix. We thus characterize a numerical relationship governing the emergence of order. This approach also defines the precision of our terminology. The data reveals a continuous trajectory from a high-entropy state to a zero-entropy condensed state, rather than a discontinuous jump. Therefore, we adopt the cautious and robust description of a "phase crossover" or "condensation regime", adjusting our scientific tone to the numerical evidence. This distinction is crucial for rigorously positioning our work within the context of pre-geometric physics. Using large-scale numerical diagonalization of the informational distance matrix g = − log ρ constructed from N = 1024 random points, we identify a remarkably clear crossover — controlled by the inverse correlation length β — from a disordered spectral phase (high entropy) to a fully condensed rank-1 phase. The core finding is that the dominance ratio λ1/λ2 grows exponentiallywithin the same interval where the spectral entropy vanishes (S → 0), confirming the stability of the emergent order. The spectral analysis demonstrates that this condensed phase possesses only a single ordering axis and actively suppresses the emergence of geometric spatial structure. We conclude that the phenomenon is a non-trivial, parameter-driven precursor to causality in pre-geometric system <p><strong>GitHub repository</strong>: <a href="https://github.com/fonteleslrivka/Emergence-of-Causal-Order-from-a-Pre-Geometric-Substrate">github.com/fonteleslrivka/Emergence-of-Causal-Order-from-a-Pre-Geometric-Substrate</a><br><strong>License</strong>: CC-BY-4.0 (paper) – MIT (code)</p>