Fit a boost from one record and test unused event differences

1. Define events, frames and conserved quantities

An inverse problem needs a training record, an independent prediction and explicit uncertainty. The original training clock is at rest in S. A measured ratio of its event differences in S′ is Δx′/Δt′=−0.6 with hard bound ±e, so the frame boost lies in β=0.6±e. Only this ratio is supplied; no absolute training time increment is available to impose an additional hidden constraint.

Freeze that boost range before using the separate S event pair (Δt,Δx)=(5,3). The central boost predicts (Δt′,Δx′)=(4,0). The unused record instead reports (4,a), with each coordinate bounded by ±0.05. These are original fictional readings and hard error limits, not actual synchronized-clock data or statistical confidence intervals.

Propagating bounds requires more than evaluating every quantity at the same endpoint. For x′(β)=γ(3−5β), the derivative is γ³(3β−5), negative over this domain. Its extrema are at the boost endpoints. For t′(β)=γ(5−3β), the derivative is γ³(5β−3), so the minimum occurs at the interior value β=0.6 and equals 4; the maximum is the larger endpoint value.

The tool compares coordinate intervals as a necessary compatibility screen. Disjoint intervals reject the training-derived range under the declared bounds; overlap is not unique-model proof and in a general multidimensional problem need not guarantee one common parameter fits every coordinate simultaneously. Reusing the unused record to adjust the fit would change its evidentiary role. A difference between central invariant values must also be interpreted with the measurement bounds, not treated as an exact violation.

2. Reconstruct the original case

For e=0.02, β lies from 0.58 to 0.62. The unused spatial prediction spans about −0.127453 to 0.122757, and time spans 4 to 4.002030. If a=0.2, the measured spatial interval is 0.15 to 0.25, disjoint from the predicted interval, so this declared model/bound combination fails the unused check.

3. Predict and investigate

Why can taking only endpoint values miss the correct lower time bound?

Derive both β derivatives, include the interior stationary point, freeze the training range and compare the unused coordinate intervals. Keep calibration changes, uncertainty revision and new-model proposals distinct from original validation.

Primary mechanism context. Checked 31 October 2026. Original derivations, records and diagrams; no physical apparatus operation or actual independent clock/particle measurement is claimed. Retain first reasoning and public-feedback exposure in the course evidence workspace when registered.