bibkey: milnor1963morse
authors: John Milnor
year: 1963
title: Morse Theory. (AM-51)
doi: 10.1515/9781400881802
url: https://doi.org/10.1515/9781400881802
claim: The classical Morse lemma gives a local quadratic normal form near a nondegenerate critical point; a saddle in two variables has one positive and one negative square.
strata_touched: []
license: citation-only
triage: anchor

A classical local ingredient for a constrained continuation

Milnor’s Morse Theory, Annals of Mathematics Studies 51, Princeton University Press (1963), is based on lecture notes by M. Spivak and R. Wells. The author’s publication list and the publisher’s DOI identify this work.

The original book, printed pages 6–8, states and proves Lemma 2.2. The Morse lemma concerns smooth local coordinates near a nondegenerate critical point. The local quadratic form has as many negative squares as the critical point’s index. This general principle is prior mathematics, not a new conclusion about quantum programs.

The quantum-volume application additionally needs an analytic parameter, the direction of the critical-value displacement, and a connection between particular canonical endpoints. Its proof supplies the needed analytic two-variable coordinates by solving one critical-point equation, taking Taylor remainders, and successively completing squares. It does not attribute an analytic parameter-dependent theorem directly to Milnor’s smooth statement.

The channel-specific content is the sign of the displaced critical value for the actual processor family, the identification of the two required arms, and their continuation into a full exact preparation with specified information and curvature data. A local saddle normal form alone proves none of those global assertions.

This note paraphrases a classical result and its scope. No book pages, figures or extended excerpts are redistributed.