Manuscript XX-E@1 · XX-E.tex · theorem · cf7a8f40
Theorem: Simultaneous exact repair
Post-calculation repair: simultaneous actual-image conditions › A complete conditional affine application · lines 1156–1192 · proof included or linked
XX-E@1#thm:repair, XX-E@1#eq:jointrepair, XX-E@1#eq:jointimage
Under Assumption ass:repair, simultaneous insertion and subsidiary cancellation means
It is possible exactly when
If is one solution, every solution is
If the subsidiary defect is explicitly retained rather than canceled, the insertion-only condition is , with the remaining subsidiary value stated for the chosen correction.
Proof. Equation (eq:jointrepair) is precisely the linear equation . This proves the image criterion. Subtracting two solutions gives the stated common kernel. Dropping the second cancellation requirement gives the insertion-only criterion, without changing the need to report .
Displayed formulas in this passage
XX-E@1#eq:jointrepair · line 1160
XX-E@1#eq:jointimage · line 1165
unlabeled · line 1174