This short report investigates order-dependent geometric structurearising from noncommutative histories in a classical statistical system.Using the two-dimensional Ising model as a concrete physical setting,we show that history-induced structure is not directly observablein conventional scalar projections, but can be detected througha gauge-invariant low-order complex-mode representation. The analysis demonstrates that noncommutative update historiesgive rise to a reproducible low-dimensional geometry,which remains hidden under phase-dependent scalar observables.This work provides a minimal physical realization of theorder-projection framework introduced in earlier theoretical studies,without invoking quantum dynamics or wavefunction-based assumptions.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 (Tue,) studied this question.