Boundary treatment helps.
It cannot restore excluded content.
Six known synthetic signals compare periodic projection, zero padding, reflection padding and Hann taper/division. The empirical reconstruction gate stays unchanged.
Smooth-signal recovery differs from full-state recovery.
| Method | Synthetic passes | Limit exposed |
|---|---|---|
| Periodic | 2 / 6 | Boundary mismatch affects smooth transients |
| Zero pad | 4 / 6 | Pulse and omitted-sideband failures |
| Reflect pad | 4 / 6 | Pulse and omitted-sideband failures |
| Hann / divide | 4 / 6 | Improves smooth cases, increases pulse error |
These ground-truth-based synthetic gates differ from the empirical residual gate; their pass rates are not interchangeable. Complex RMS and phase both enter. All 24 evaluations are retained.
The omitted sideband has amplitude 0.3. Every method’s full-state relative RMS error is about 0.287, even when its fundamental-only error is small. Recovering a selected carrier is a distinct objective from recovering the full signal.
All synthetic metrics ↓A guard leaves boundary influence.
The explicit DFT projection kernel gives a conditional error bound from outside-sample uncertainty. A unit impulse just outside the measured segment has a nonzero interior effect even after discarding 1024 samples at each end.
For the fixed padded filter, the outside-row absolute sum ranges approximately 1.06749–1.64220 across the interior. Multiplying it by an independently justified uniform continuation-error bound yields a numerical reconstruction budget. No such physical bound is established here.
Padding agreement does not certify the unknown continuation. The Hann method changes the operator and needs its own division-error analysis. Constructor floating-point checks are not formal interval arithmetic.
Kernel and reconstruction results →Next: an estimator using only measured samples.
Freeze a finite-support quadrature kernel, its delay, transfer response and valid interior before another experimental run. This can remove assumed padded samples from the estimate; it does not certify unrestricted dynamics outside the record.
The present synthetic controls distinguish boundary mismatch from excluded content in known cases. They do not apportion the causes of the experimental phase-gate failure. Acquisition gain/skew, full energy, absolute native support and the action anchor remain open.
Reproduction code ↓Preserved reconstruction v0.1 →