C.39.RO:5.1 - Generalize a timing correction and obtain an impossibility result

A recording has picture-to-sound offsets of 40, 42 and 45 milliseconds at three matching markers. The available correction subtracts one constant from the sound times. Centering the extreme offsets gives 42.5 milliseconds and residuals -2.5, -0.5 and +2.5 milliseconds.

The receiving use needs that construction for any nonempty finite set of marker offsets, expressed in one common unit and reference. Let m be the smallest offset and M the largest. Construct:

• correction c = (m + M)/2;
• largest residual magnitude r = (M - m)/2.

Every marker offset lies between m and M, so subtracting c leaves each residual between -r and +r. Any constant correction leaves the two extreme residuals separated by M - m. At least one therefore has magnitude at least (M - m)/2. The proposed correction attains that lower bound: it minimizes the largest residual magnitude at these markers.

The reusable operation takes the marker offsets and returns c and r. For a stipulated tolerance T, r at most T supplies a feasible constant correction at the markers; r greater than T shows that no constant correction can meet that marker tolerance. One marker gives r = 0. An empty set supplies no extremes for this construction.

Now add a marker with offset 48 milliseconds. The operation gives c = 44 and r = 4. For a tolerance of 3 milliseconds, every constant correction fails. The two extreme offsets differ by 8 milliseconds, while two residuals each within 3 could differ by at most 6. This failure witness lets the practitioner stop searching among constant shifts and consider another time correspondence or a changed use.

The derivation concerns the supplied marker offsets. Applying a correction throughout a recording also needs the relation between markers and a qualified editing operation; C.39:5.2 retains that receiving question. Unknown drift between markers remains unresolved by this finite calculation.