Tight upper bound on $d_{GH}(S^1,S^{2k+1})$: GPT's short proof

What the math gives to ML

The paper's central construction is a normalized odd-frequency Fourier embedding of the circle into a high-dimensional unit sphere. Its transferable asset is not the Gromov-Hausdorff proof itself, but the explicit translation-invariant similarity kernel generated by multiple harmonics while preserving constant feature norm. This suggests a drop-in positional or cyclic-feature encoding for transformers and other networks, with frequency count and spectral weights controlling how sharply the model distinguishes nearby and distant phases. The proposal is most suitable for periodic, algorithmic, or extrapolation-heavy tasks rather than as an unconditional replacement for language-model positional encodings.