What question did this study set out to answer?

This study aims to explore the emergent properties of spacetime topology through a framework based on causal metric networks.

June 17, 2026Open Access

Spectral Analysis and Low-Rank Structural Transitions Emergent from Dimensionless Causal Metric Networks under Reciprocal Measurement Dynamics

Key Points

This study aims to explore the emergent properties of spacetime topology through a framework based on causal metric networks.
Utilized an information-theoretic framework based on Status-Relational Entropy dynamics.
Conducted spectral analysis on 18,396,565 components of causal relations.
Identified empirical phase transitions in dimensionality and information density within the network.
Isolated a dimensional phase transition at critical coordinate z approximately 5.790.
Detected an abrupt information density barrier of 11 causal components during the transition.
Observed intrinsic rank variations and peak magnitude of the dominant eigenvalue at z = 5.720.

Abstract

We present a rigorous information-theoretic framework for emergent spacetime topology. In this model, the macroscopic velocity of light (c) and the dimensionality of spacetime function as emergent phenomenological expressions. These expressions are governed entirely by the local Information Density within a coordinate-free causal relation network. Grounded in Status-Relational Entropy (SRE) dynamics, time is defined as a sequential count of discrete measurement acts. Space serves as a dimensionless, random-access indexing grid. Global network expansion introduces informational vacancies into the adjacency matrix. This dilution triggers an accumulation of bidirectional routing latencies, manifesting macroscopically as variations in light speed. We deploy a high-throughput parameter-centralized spectral pipeline across 18,396,565 high-confidence causal relation components. Through this data-stream framework, we isolate an empirical dimension phase transition. Spectral decomposition localizes the primordial manifold emergence boundary at critical coordinate z approx 5.790. This dimensional transition is accompanied by an abrupt information density barrier of 11 causal components. Between z approx 4.60 and z approx 5.79, the intrinsic rank undergoes discrete, multi-dimensional oscillations. These algebraic rank variations arise from the stability threshold of the reciprocal handshake protocol. The dominant eigenvalue discrete first-order variation registers an absolute peak magnitude of 7.822e-03 at coordinate z = 5.720. This quantitative marker measures the structural restructuring cost as the network transitions from 2D degeneracy into the late-time 5D intrinsic plain. Across the modern lower-depth domain (z less than or equal to 1.60), extensive data crowding forces the underlying causal topology to break out into a rigid 5.00D intrinsic plain. This plain is smoothly collapsed back into a stable 4.00D random-access spatial indexing grid by the holographic observation projection matrix.

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