Regular Simplex Hierarchical Gravity Part VI: Dimensional Reduction and Spacetime Emergence via Sequential Deployment

Abstract

This study investigates, from the bottom up,whether the structure of gravity can emerge from apurely geometric game possessing no physical parameters whatsoeverno mass, no time, no curvature. The single local rule employed is the following: on the three-sphere S3, locate the fourthpoint equidistant from three existing points, and connect a regular tetrahedron without deforming it. The fact that ve regular tetrahedra sharing a common edge fail to close, leaving a gap of 7.36◦(the defect angle δ = 360◦ − 5 arccos(1/3), an algebraic necessity following from Niven's theorem),accumulates as frustration during growth, and the system undergoes a jamming transition at approx-imately N ≈ 111 elements, rigidifying. From thestatistics of this limiting state, a strong linear proportionality (r = 0.880) was measured between the local rejection density (the discrete counterpart of energymatter) and the detour rate (the discrete counterpart of spacetime distortion), yielding the local gravitational constant κsim = 1.119 as the proportionality coecient. This is the discrete counterpart of the Einstein equation Gμν = κTμν. Furthermore, this spacetime distortion is pronounced only in three dimensions, and we identify its geometric catalyst as the four dimensional 600-cell. All of these are established facts, obtained undera pre-registration (prediction-rst freezing) protocol with a separation between proposer and auditor. Finally, we present the residual of approximately 1.5% that remains when connecting this local constant to the continuum gravitational constant 8πinterpreted as an unavoidable geometric cost, a gluing allowance, incurred when the discrete constructs the continuoustogether with a new picture of the origin of gravity. This connection and picture, however, are described as interpretations that have not undergone independent verication, and are clearly distinguished from established facts.