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Self-consistency: a fundamental concept in statistics

Tarpey, Flury1996Vector quantization and Voronoi theory

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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 kk-means points and the elliptical theme; Tarpey–Flury define self-consistency E[X∣Y]=YE[X\mid Y]=Y, 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.

Statistical Science 11(3):229-243; primary text reviewed in the DS17 audit. Credits Hastie-Stuetzle 1989 for the terminology and defines random-vector self-consistency E[X|Y]=Y. Theorem 4.1: under linear conditional means, if a self-consistent Y is measurable with respect to the orthogonal projection onto its q-dimensional support span, that span is generated by q covariance eigenvectors (not necessarily the leading q). This is the closest LCM structural antecedent, but has no endogenous tilt, profiled information, or nuisance-rank obstruction.

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