Principal points
Principal points and self-consistency (31 Aug 2026)
The DS17 study (DS-STABLE-BASINS) registered the principal-points /
self-consistency branch of this cluster. Flury introduces population
-means points and the elliptical theme; Tarpey–Flury define
self-consistency , the vocabulary DS17's tilt-consistent
strip rules specialize; Tarpey–Li–Flury prove the forced principal-subspace
alignment of self-consistent points under elliptical symmetry — the closest
published structural cousin of DS17's conditional-centering obstruction
(both are "self-consistency plus a symmetry/centering hypothesis forces a
degenerate configuration" results), with no nuisance/profiled block, no
self-regenerating tilt coefficient, and no conditional-mean-orthogonality
hypothesis. Serinko–Babu is the nearest 1-D interval-partition asymptotics
precedent: singular-Hessian degeneracy at a unique global k-means optimum
with nonstandard rates — a per-law description of degeneracy when it occurs,
never a class-wide forcing theorem. Triangulation:
LITERATURE/audits/DS-STABLE-BASINS-31-August-2026.md.
Biometrika 77(1):33-41; primary text read in the DS17 audit. Defines principal points as global squared-error minimizers. Proves the two-point elliptical solution lies on a leading covariance eigenvector and the general rank bound dim span{xi_j-mu} <= k-1; for k>2 the leading principal-subspace statement is explicitly a conjecture, later proved for principal points by Tarpey-Li-Flury 1995. No general existence theorem, arbitrary self-consistent-point theorem, nuisance block, or profiled objective.