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.
John Jude Hathway (Wed,) studied this question.