Projection-Induced Gravity Phase and Metric-Connection Structure without Assuming Spacetime (Projection-Origin Gravity 1)

⚠️ CORRECTION NOTICE (added 2026): The author has since identified serious errors in this work through a systematic adversarial audit. The central claims of this record should not be relied upon. Specific issues found across this research program include implementation errors (e.g., conflation of signed cosine with cos²), circular definitions, proof gaps, results that reduce to known theorems, and effects that vanish under proper null models and direct-influence correction. This record is left online for transparency and as a documented case, not as a valid result. For the author's current, audited methodology, see the later work on classical-baseline auditing of quantum-like contextuality claims: https://doi.org/10.5281/zenodo.20680607 — A. Ogasawara This study investigates whether gravity-like geometric structurecan arise from projection-based order dynamics without assumingspacetime as a primitive background. Building on earlier results showing that spatial distance geometrycan emerge from low-dimensional projection cores, we examine whetherthe same projection dynamics admits a distinct gravity-like phase. Within this phase the system exhibits Poisson-type response togetherwith inverse-square force behavior. Additional diagnostics,including Gauss-law consistency, approximate curl-free structure,action–reaction symmetry, and Kepler-type two-body dynamics,confirm that the interaction behaves consistently with theNewtonian limit. Using projection-generated order fields, we construct an inducedlocal metric and evaluate the associated Levi–Civita connectionand parallel-transport holonomy. The holonomy angle exhibitsarea scaling consistent with curvature, indicating the emergenceof a metric–connection hierarchy within the projection framework. The analysis is performed entirely within a discrete projectionmodel and does not derive Einstein’s field equations. Instead,it provides a structural investigation of how gravitationalgeometry can appear as an ordered phase of projection dynamics. Note: Parts of the manuscript were linguistically and structurally refinedwith the assistance of AI-based tools.All scientific content, analysis, and conclusions are the author's own.