Learn first
The matrix proof uses the following established results. Each link leads to the full supporting lesson; the listed sections give the exact earlier proofs used in the construction.
Finite matrices
Polynomial and contour interfaces for stable boundary models, Section 10: Sections 10.1–10.4 prove bases, coordinates, rank-nullity and the determinant inverse. Section 10.6 proves the product bound in the original operator norm.
Compactness and scalar calculus
Metric and topological foundations, Section 12: Sections 12.3–12.8 supply completeness, compactness, extrema and coordinate neighborhoods. Section 13, especially 13.3–13.7 and 13.10, gives oriented integration, the product and chain rules, uniform differentiation and smooth cutoffs. Section 14.4 proves the coordinate-segment identity used to justify every parameter derivative.
These results lead to the matrix theorem. Its own proof constructs the ordered transport, bundle coordinates, connection and frame.