Arithmetic Orbit Trajectory (AOT) of Continuous Glucose Monitoring: GF(2^8) Sylow-17 Structure Reveals Glycemic Instability Beyond HbA1c

We introduce the Arithmetic Orbit Trajectory (AOT) framework, grounded in the GF(2^8) algebraic structure of AES-128, to characterize continuous glucose monitoring (CGM) time series. CGM values are classified into arithmetic equilibrium states (Z/15Z subgroup, no Sylow-17 component; 80-93 mg/dL in the normoglycemic range) and arithmetic transition states (Sylow-17 component present). The Sylow-17 algebraic critical point 99 mg/dL (0x63=SBOX[0x00], multiplicative order 17) coincides with the ADA prediabetes threshold. In 16 subjects from the BIG IDEAs Lab Glycemic Variability dataset (PhysioNet, 37,080 readings), the AOT Oscillation Index (mean Sylow-17 run length) strongly predicted hyperglycemic area above 100 mg/dL (Spearman rho=0.818, p=0.0001, R^2=0.600), while HbA1c did not (rho=0.208, p=0.44). In the matched HbA1c 5.5-5.7 subgroup (n=9), AOT_OI vs HAA100 rho=0.867 (p=0.003). Subjects in the highest AOT_OI tertile had 2.6x higher HAA100 despite identical TIR (97.7% vs 97.7%), demonstrating AOT captures glycemic stress invisible to standard metrics. These findings constitute a proof-of-concept for the IGCI-AOT framework combining cumulative glycation exposure with algebraic trajectory structure.