ESampling without restrictions?
A 2018 paper on multiplicative calculus says waves can be modelled "without sampling restrictions", with exp(ikx) needing only two points regardless of k; those two points carry the supplied lifted-phase data, not ordinary samples. Sampled waves alias: two frequencies that differ by a multiple of 2 pi/h produce exactly the same samples. No calculus, multiplicative or ordinary, can separate them from those samples alone.
The paper's result is exact on a different kind of data: points on the Riemann surface of the logarithm, where the continuous phase comes with each sample. Supply the phase and any frequency is recoverable. Remove it and Nyquist is back.
Bars mark where each estimate is exact: |omega h| < pi, |omega h| < pi/2, the band (w0 - pi/h, w0 + pi/h], and everywhere. The lifted estimate is exact only because the phase itself is supplied as an input.
The precise statement
For integers m and n, exp(i (omega + 2 pi m/h) n h) = exp(i omega n h). Any estimator that sees only the samples gives both frequencies the same answer, so for one of them its error is at least pi |m| / h.
The forward principal-branch quotient (z_(n+1)/z_n)^(1/h) returns omega mod 2 pi/h. The centered quotient the paper uses, (z_(n+1)/z_(n-1))^(1/(2h)), returns omega mod pi/h: exact only for |omega h| < pi/2, half the Nyquist range. A known band of width 2 pi/h or the lifted phase makes recovery exact.
Where it breaks
Fed the samples of omega = 1 + 2 pi, the forward quotient returns 1, the alias. Fed omega = 2, the centered quotient returns -1.141592654 = 2 - pi. The impossibility covers ordinary complex samples only; with the lifted phase the scheme is exact for every k, which is the paper's own setting.
The paper's Remark 4 marks the other edge: once log v is not a low-degree polynomial (a carrier plus an offset), the points-per-wavelength limit returns. Its abstract leaves the lifted-phase condition out, hence PARTIAL.
Prior art and sources
Classical results it reduces to
- Shannon 1949 (aliasing)
- Bandpass sampling
- Phase unwrapping (Itoh 1982)
NNC anchors
- Modelling wave propagation without sampling restrictions using the multiplicative calculus I (cubillos_2018_wave_propagation_multiplicative)
- On an alternative view to complex calculus (bashirov_2018_alt_view_complex_calculus)
OpenAI families named (context only)
- Exact Fourier transforms below n log n (family 130)
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Write-up and checks: docs/research/nnc-openai-crossanalysis-2026-10/X3/FINDINGS.md. Every number on this page is computed in your browser by lib/crossovers/, tested against independent closed forms in lib/crossovers.test.ts.