Principal points and self-consistent points of elliptical distributions
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.
Ann. Statist. 23(1):103-112. The principal-subspace theorem proves that if n principal points of a centered elliptical law span dimension q, their span is the leading q-eigenspace. Tarpey-Flury 1996 identifies this as the finite-distinct-support special case of its broader self-consistency theorem, whose arbitrary self-consistent conclusion is only an eigenvector span, not necessarily leading. Closest published geometric cousin of DS17, but no nuisance/profiled block, endogenous tilt, conditional-centering class, or nonexistence theorem. Primary proof remains library-access pending.