Allow frequency error.
Keep a positive Gram bound.
The finite-mode leakage witness now covers imperfect orthogonality through an exact rational uncertainty bound. A prospective tighter monitor requirement keeps the tail ceiling below 1 mV.
The full uncertainty family is bounded.
For 100 nominal integer-Hz modes from 100 through 199, per-mode errors ≤5 mHz and relative one-second monitor-duration error ≤10 ppb give a Gram eigenvalue lower bound above 0.9098. Rational off-diagonal row bounds cover every allowed constant-frequency configuration.
A common frequency shift cancels from pair differences. Physical band membership, frequency drift and nonlinear time warping still require separate treatment. The monitor-duration bound is distinct from the acquisition’s 0.2 ns timing requirement.
| Measured finite-mode complex RMS | ≤55 µV |
|---|---|
| Total waveform monitor L2 error | ≤20 µV |
| Outside-model spectral variation | ≤200 µV |
| Conditional tail variation | ≤0.99125 mV |
| Propagated full-state error / phase | <0.008667 V / <0.017509 rad |
Small RMS alone is insufficient.
Two distinct tones separated by 1 microhertz with opposite 1 V coefficients have spectral variation 2 V but complex RMS below 10 µV on the one-second interval. A separation/coercivity premise prevents that cancellation from concealing large coefficients.
The old 60 µV RMS share fails the selected robust sufficient budget. The new 55 µV share is a prospective requirement for this witness; the old orthogonal packet remains intact. A directly certified coefficient-error bound is a different route from the waveform-error bound used here.
Controls and practical limits →Next connect the witness to sampled measurements.
The certificate is exact rational arithmetic. Two floating synthetic eigenvalue diagnostics provide additional checks, not the proof of the uncertainty family. Outside-mode contamination, completeness, normalization and actual monitor accuracy remain independent evidence gates.
Next bound the sampled-monitor reconstruction using actual sample times, quadrature calibration and numerical error. The candidate estimator cannot serve as its own independent reference.
Certificate replay ↓Two synthetic diagnostics →Original preparation protocol →