Keep calibration, source inference and a new clock test in separate stages

Define independent evidence stages

An indicated frequency can contain both multiplicative gain and additive offset. Two independently known anchors at physical frequencies2 and6 give indications y2 and y6. Fit g=(y6−y2)/4 and B=y2−2g. A separately known frequency4 is unused while fitting; its indication tests the affine response. This test does not establish the same response for every source, temperature or later time.

Derive the composed prediction

A receding source has independently known rest frequency5 in the same physical units. Correct its indicated frequency through the fitted scale, form D=fsource/5 and use the longitudinal relativistic relation D=sqrt[(1−beta)/(1+beta)]. Solving gives beta=(1−D²)/(1+D²). An additive offset does not cancel in a raw indicated-frequency ratio; borrowing the generating beta instead of fitting the source record would erase the inference problem.

Retain uncertainty and boundary limits

Treat each anchor and source indication as independently bounded by0.01. Express corrected source frequency directly as f=2+4(ys−y2)/(y6−y2). The denominator remains positive over the finite packet. This linear-fractional function has extrema at the original reading-box corners, so evaluate all eight corners and transform their extrema to beta. Do not propagate fitted gain and offset as if independent: they share the same readings. These are deterministic conditional bounds, not confidence probabilities.

Test a fresh claim without refitting

A separate clock packet has proper duration4 for the same constant speed. Freeze the laboratory event-duration prediction4gamma and its interval before revealing the laboratory duration. A pulse arrival interval instead includes increasing travel distance and is4gamma(1+beta); it is not interchangeable with time dilation. The earlier independent frequency-four record tests calibration form, while this later clock record tests a composed relativistic prediction. Neither exposed synthetic check is actual learner or physical measurement certification.

Worked original case

For generating beta0.6, gain1.1 and offset0.25, the two anchors indicate2.45 and6.85. They recover gain1.1 and offset0.25 and predict4.65 at the unused frequency4. The receding source has D0.5, physical frequency2.5 and indication3.0, which infer beta0.6. A proper duration4 predicts laboratory duration5; pulse arrivals with proper emission spacing4 are instead separated by8.

Investigation and retained defense

Write the exact calibration and source records used by each fit. Freeze the independent-frequency and laboratory-duration predictions before viewing their checks. Propagate original-reading corners, identify shared-parameter correlations, and then complete a newly assigned six-category packet in the retained evidence workspace with prior exposure and assistance declared.

Changed-case practice

Try the changed case before feedback

For beta=0.8 and proper duration3 time units, find the laboratory event duration.

Numerical result in the requested unitsCompare this result

This practice is public and unsaved. A matching number does not verify understanding. Retain your explanation, assumptions and changed-case evidence in the separate course workspace.

Sources and course-exit boundary

Original virtual packets and derivations checked 31 October 2026 against the primary mechanism reference. This is a finite authored calculus-based scope, not every textbook topic, AP authorization, institutional credit or proof of learner mastery. The final evidence workspace retains original first work, actual criterion-linked critique, a reasoned revision and a genuinely new reviewer-frozen six-category assignment. Synthetic examples and test fixtures cannot stand in for real learner responses or supervised practical evidence.

The distinct event-duration relation is checked against time dilation. The bounded affine inverse and corner proof are original derivations applied to the supplied records.