TDA manuscript: per-cloud normalization can change the comparison

sneaky normalization problem: cloud barely changes, its ruler does. suspicious now of comparisons where every object gets its own convenient ruler

manuscript concept: a “ruler” is the positive scale dividing pairwise distances. If two clouds use different empirically estimated rulers, their persistence diagrams may be expressed in different units before matching begins.

Circle net, dense arc, and the changing normalization ruler

construction to remember: start with a regular net around a circle, retain that net, then add a much denser cluster on a shrinking arc. Coverage of the rest of the circle is unchanged, while a high pair-distance quantile can become dominated by within-cluster pairs. Separate quantile normalization can therefore change diagram scale without a corresponding improvement or deterioration in geometric coverage.

reference task: report both the geometric/sampling error and the ruler change. Do not use a Hausdorff coverage estimate alone to justify separately normalized comparisons. For an application, record the exact quantile convention, sample weighting and distance computation.

source version: manuscript dated September 10, 2026, “Quantifying Scale Error and Local Rank Deficits in Topological Data Analysis.” Its received/accepted/published fields are blank; this is a manuscript reference, not an acceptance announcement.

reproducibility pointer recorded by the manuscript: archive DOI, described as containing source, generators, records and checks. The archive was not independently downloaded or rerun during this notebook scan.