Elementary moves do not constitute full Cerf theory here

This page records only the elementary moves: introduction and cancellation of a complementary pair of consecutive indices (birth and death), handle slides (handle additions), and the elementary matrix operations they induce on the attaching-belt intersection matrix. It does not construct a one-parameter family of Morse functions, does not assert that any two Morse functions or handle presentations of a cobordism are connected by finitely many of these moves, and does not develop Cerf theory, pseudo-isotopy or the classification of one-parameter families. The removal of excess geometric intersections by ambient isotopy (the Whitney trick) is likewise not available here: the cancellation theorem is proved only under its exact single-transverse-intersection hypothesis.

This remark fixes the proof boundary of the page and is not an existence or classification statement, with no separate proof. The excess-intersection step is deferred, as recorded in the coverage of this pair, to the Whitney trick and surgery below the middle dimension ↗, because the cancellation theorem and its local model are proved here only under the exact single-transverse-intersection hypothesis.