Quantized Angular Coupling of Protein Secondary Structures: A Conserved Geometric Distinction Between Alpha-Helices and Antiparallel Beta-Sheets

We report a significant and reproducible geometric distinction between the two primary secondary structure types in proteins. By measuring the angle between the CA(i)–CA(i+4) vector and the N–O hydrogen bond vector in alpha-helices, and the equivalent cross-strand CA–CA vector vs N–O vector in antiparallel beta-sheets, we find that alpha-helices exhibit a conserved angle of 22.9° ± 1.6° while antiparallel beta-sheets exhibit 12.1° ± 2.2° across 69 structurally diverse protein structures (p = 9.06 × 10⁻³⁴, two-sample t-test). Results were obtained from independent discovery (n=27) and validation (n=42) datasets using identical pre-registered analysis code, then confirmed on the combined dataset (n=69). The ∼10.8° separation is highly conserved across proteins of different fold, organism, and function, suggesting it reflects a fundamental constraint of backbone hydrogen bond geometry. The sum of the two means (35.03°) is within 1° of 36°, the pentagonal angle associated with the golden ratio; this passes the pre-registered tolerance window on the combined dataset and is discussed as a candidate geometric relationship requiring further large-scale validation. The angular-linear correspondence between the 23° helix geometry and the 3.6-residue helical pitch is proposed as a candidate physical mechanism.