Why it is a question
THEORY-297 holds Ovsjanikov et al.’s structural results (LIT-993, read in NOTE-822) for a bijection T between shapes. At the owner’s direction, from the promotion audit of 2026-10-19 (ADR-047, batch 007), it went Active with the results re-derived. The clause of its promote_when that asked for the conditions under which the truncated matrix stays orthogonal was moved here unchanged.
For an exact isometry the answer is easy. The full matrix C is block-diagonal by eigenspace, so if n ends at a gap between eigenvalues, the leading n × n block is a direct sum of orthogonal blocks. The open case is the near-isometry. Then C mixes eigenspaces a little, and whether the span of the first n eigenfunctions survives depends on the gap between the nth and (n + 1)st eigenvalues. The paper cites Kato (1995) for this and does not show it. It also reports that C stops being nearly diagonal quickly under non-isometric deformation, measured on one data set.
What would answer it
A perturbation bound, read in a source the record holds or proved here, for the leading block of C under a map whose metric distortion is at most ε. For instance, a Davis–Kahan-type bound on the angle between the truncated eigenspaces, in terms of ε and the gap at n. Shape-matching benchmarks cannot settle it, since they test the method and not the bound.