Remark: Uniqueness is not existence or gauge identification

Full column rank must be checked in the chosen presentation. If redundant coordinates or gauge freedoms remain, the lemma does not assert uniqueness before those freedoms are fixed. It also does not produce an accepted output.

The supplied VIII-A/B records report fixed-presentation matching uniqueness and rejected updates CheungVIIIA,CheungVIIIB. The matrix interface above proves the corresponding elementary criterion, without asserting that an unprinted native matrix has been independently evaluated.