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Super-Resolution

Resolve what one sensor folds into a single blur — and certify where you can trust it.
SEE · signal integrity CERN & JWST validated 35×–700× below the FFT frontier

Two things close together — two satellites in proximity, two spectral lines, two vibration modes — fold into one blur when a sensor's window is too short or its aperture too small. Classical super-resolution (CLEAN, MUSIC, compressed sensing) and its deep-learning cousins give an answer everywhere and a validity boundary nowhere: noise-floored, and silently wrong past their aperture. The AI versions are worse — they hallucinate structure with full confidence.

Noether Super-Resolution unfolds the blur exactly, using the sensor's own calibration as a pin — and ships an aperture certificate: the recovery frontier, computed once from the pin, transported to every frame, with an integrity flag that fires the moment the answer leaves the region where it can be trusted. It is the protection-level artifact a safety case is written around, for resolution.

Everyone else's super-resolution answers everywhere. This one answers, and tells you exactly where to stop believing it.

THE TEST   On real public data, blind, with the industry gold standard as the parity bar: CERN Open Data (93,474 dimuon events, PDG as referee), JWST (a JADES galaxy spectrum, NIST line ratios), and a zero-padded FFT periodogram with known ground truth.
THE RESULT   Pull apart what the detector folded, and certify the boundary:
  • Pulled from inside the detector's own fold. The Υ(2S) resonance, natively merged into one shoulder at the dataset's resolution, recovered to 0.21 MeV of the PDG value — with the gauge-invariant mass ratios agreeing to half a part per thousand.
  • The gold standard's own frontier, measured. The 8×-oversampled FFT resolves down to c=0.7/T and dies below — and at its frontier its peak positions carry 25% bias, with no warning. The pinned unfold recovers 35× to 700× below it, graded by noise tier, every recovery inside a certified frontier.
  • You are told exactly where to stop trusting it. Past the frontier, a fit gets more perfect while the answer gets more wrong — the certificate carries that as its alarm, so you never act on a confident number that has quietly gone wrong. The output states where recovery is trustworthy and declares UNRESOLVED beyond it.
  • Attested on the data it ran on. The warrant carries the public datasets by cryptographic hash; a third party re-derives the synthetic frontiers from the certificate alone and confirms the public claims by the pinned files.
Horizontal bar chart on a log axis of resolving power. The top grey bar is the FFT gold standard at Rayleigh, resolving to c-star equals 0.7 over T. Below it three green bars: the Noether field tier at noise ten-to-the-minus-two reaching fifty times, the field tier at ten-to-the-minus-three reaching two hundred times, and the exact tier reaching one thousand times finer than the FFT frontier. A note reads: the FFT is dead below c equals 0.7 over T, 25 percent position bias at its edge.
Resolve below the display cone. Measured on the same two-tone scenes with known ground truth: the FFT gold standard resolves to c=0.7/T and no further; the pinned unfold recovers 50× at field noise, up to 1000× on exact ledgers — each tier stated on the certificate's face, with the recovery frontier as the protection level.

The CERN unfold — the Υ triplet from one shoulder

StepWhat happenedThe number
The dataCERN Open Data dimuon events, selected to opposite-charge global-muon pairs; the file carried by hash93,474 events · sha-pinned
The foldat this dataset's resolution the Υ(2S) and (3S) are natively merged into one shoulder — the naive census sees two peaks where there are threecensus: 2 of 3
The unfoldthe triplet pinned and separated; scale-corrected positions vs PDG, and the gauge-invariant ratios2S to 0.21 MeV · ratios <5×10−4
The frontiera fold-depth scan finds where recovery holds and where the solution runs to a continuum — predicted by pin-counting, measured at b*=150 MeVfrontier certified · measured

Super-Resolution steel thread — CERN Open Data and JWST/JADES public spectra, sha-pinned; referees PDG masses and NIST line ratios; the FFT gold standard measured on ground truth. A demonstration of the certified-recovery capability, not a detector reconstruction pipeline.

How — the pin, the unfold, the frontier

  1. The scene is a superposition of pinned kernels. The sensor's response — the point-spread function, the spectral line list, the mode comb — pins each component's shape; recovery is an exact unfold of the superposition, not a fit against noise. The engine is sealed; the mathematics is published. the pin is the field's unit law
  2. How many, from the data itself. The number of components is named by the data's own cost saturation — the window tells you what it holds — not assumed and not hand-tuned. blind model selection
  3. The frontier is computed once, carried everywhere. The recovery boundary follows from the pin structure and the window; it transports to every frame the sensor takes, so certification precedes deployment — the protection-level shape. frontier · death-rattle alarm past it
  4. The answer is a certificate. Recovered positions, the frontier, and the tier (exact vs field-pin, with the pin's error budget) ride in a warrant re-derivable from the file. verify · status · mint