Spectral analysis of protein backbone geometry reveals abrupt helix-coil boundaries

The boundaries of cooperative helix-coil transitions influence protein allostery and conformational dynamics, yet the persistent one-to-two-residue ambiguity in their assignment remains poorly characterized. We apply the discrete Hasimoto map to translate three-dimensional C $_{\alpha }^{}$ backbone geometry into a one-dimensional discrete nonlinear Schrödinger effective potential and analyze its spatial-frequency structure. Helical segments appear as near-integrable, low-entropy states whose spectral power concentrates at the zero-frequency mode, whereas coil regions show broadband fluctuations. A pointwise integrability residual and a windowed spectral entropy separate the two phases with ROC AUC values of 0.783 and 0.715, and their combination reaches 0.803, while combining the residual instead with a low-frequency energy ratio reaches 0.815. Across 1986 proteins and 19,148 of 21,107 fitted helix-coil boundaries the transition is abrupt, with a median sigmoid width of 0.145 residues that measures the steepness of a single-step discrete jump rather than a literal sub-residue distance; the transition is directionally asymmetric, with helix exits sharper than entries. Across the full dataset every C $_{\alpha }^{}$ geometry-based assignment, including DSSP-calibrated P-SEA and both spectral probes, loses agreement with the DSSP hydrogen-bond reference most acutely at these boundaries, indicating that the assignment ambiguity is a general feature of C $_{\alpha }^{}$ geometry rather than any single algorithm. The windowed spectral probe is subject to a Gabor resolution limit and is therefore outperformed by the pointwise probe, which attains the lattice-limited resolution.