What question did this study set out to answer?

The aim is to explore how the topology of discrete networks influences the emergence of classical physical laws.

February 25, 2026Open Access

The Raza Criterion: Topological Constraints on the Emergence of Inverse-Square Laws in Discrete Manifolds

Key Points

The aim is to explore how the topology of discrete networks influences the emergence of classical physical laws.
Simulated large-scale random geometric graphs with N=50,000.
Identified the Raza Point threshold for non-locality.
Analyzed the effects of shortcuts on manifold regularity.
Established the Raza Point at approximately 0.1 for non-local shortcuts.
Showed that exceeding this threshold collapses the metric regularity of the manifold.
Provided a topological explanation for the 3-dimensionality of our universe.

Abstract

This treatise establishes a formal mathematical bridge between discrete network topology and the continuous laws of classical physics. By simulating large-scale Random Geometric Graphs (N=50, 000), I have identified a definitive and universal threshold of non-locality: the Raza Point (R 0. 1). This work provides empirical and theoretical proof that if non-local shortcuts exceed 10% of the system's diameter, the metric regularity of the manifold collapses, rendering stable force fields impossible. Furthermore, I provide a topological justification for the 3-dimensional nature of our universe by demonstrating the inherent fragility of 4D hyperspaces

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The Raza Criterion: Topological Constraints on the Emergence of Inverse-Square Laws in Discrete Manifolds

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Abstract

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