Spatial Distance as a Projection of Noncommutative Operational Histories

⚠️ 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 work examines whether spatial distance can arise as a derivedrepresentation of noncommutative operational histories rather thanbeing assumed as a primitive background structure. Using numerical experiments on two fundamentally different systems—a neural network without intrinsic spatial coordinates and aclassical Ising model analyzed without reference to its lattice—we extract low-dimensional complex cores from order-dependentresponses generated by noncommutative operations. Pairwise distances reconstructed from these cores show high fidelity(R² ≈ 0.96–0.997) using only two complex modes (k = 1,2).Further analysis reveals that distance geometry stabilizes onlywithin a restricted parameter regime, defining a spatialized phase,while outside this regime the reconstructed geometry becomes rigidor collapses. In addition, operational order affects orientation but not distance:reversing the order of operations leaves pairwise distances invariantwhile reversing the sign of phase orientation. These results suggest that spatial distance may arise as a conditionalrepresentation of projected operational history, providing a minimalnumerical investigation of how geometric relations can emerge fromhistory-dependent 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.