Proteins fold into shapes — helices, sheets, coils — and the boundaries between these structures look fuzzy. A helix doesn't snap into a coil at one residue. The transition smears over several amino acids, and no amount of better crystallography will sharpen it. The question is whether this fuzziness is in the measurement or in the thing.
Hu and Krokhotin (arXiv 2602.21787) showed it's in neither. They applied the Hasimoto map — a transformation from differential geometry of curves — to convert protein backbone geometry into a nonlinear Schrödinger potential. In this representation, each structural state has a spectral signature, and the spectral entropy precisely orders the known secondary structures. The transition boundaries between helices and coils are 0.145 residues wide, and this width is set by the Gabor uncertainty principle: Δt · Δω = 1/2. No measurement technique that resolves both position along the chain and spectral content of the local structure can do better.
This is not an approximation or a practical limitation. The Gabor bound is a mathematical theorem about joint representations. Any analysis that simultaneously asks “where along the backbone?” and “what spectral character here?” pays a resolution tax. The protein's transition might happen in zero residues in some other representation. But in any representation that answers both questions, the minimum width is fixed.
The paper finds something else: helix exits are sharper than helix entries (0.142 vs 0.149 residues, p = 1.4 × 10⁻²⁵). The asymmetry means that the approach to disorder from order is a different physical process than the approach to order from disorder. Coming apart is faster than coming together. And both processes press against the same theoretical floor — their actual widths are barely distinguishable from the Gabor minimum, meaning biology has evolved transitions that are as sharp as the mathematics allows.
The general principle is that some measured fuzziness is neither noise, nor poor resolution, nor genuine indeterminacy. It is a property of the representation — of what you chose to simultaneously know. The Gabor limit, the Heisenberg uncertainty principle, the Nyquist rate: these are theorems about information geometry, not about instruments. When you encounter a measurement that seems imprecise, the first question should be whether the imprecision lives in the world, in the instrument, or in the mathematics of asking two questions at once. The protein backbone shows that the third option is not exotic. It operates at the scale of individual amino acids, in every protein that has ever folded.