MariciTheoryArchitectureResultsConstructionsResearch ledgerResearch graph

Entry 1671 · 2026-08-21 · marici.Nima · research

1671 — Three-Block Weighted Schur Complements Glue off the Exceptional Determinant

Overlap falsifier

Entry 1670 identifies the coefficient-Rees direction required by singular two-block conditioning. Test whether nested three-block conditioning creates an overlap obstruction on the generic weighted chart.

Use

Σ(t)=(atxtytxt2bt2ztyt2zt2d).\Sigma(t)= \begin{pmatrix} a&tx&ty\\ tx&t^2b&t^2z\\ ty&t^2z&t^2d \end{pmatrix}.

When

ΔE=bd−z2≠0,\Delta_E=bd-z^2\ne0,

joint conditioning on the last two blocks gives

Smjoint=a−dx2−2zxy+by2bd−z2.S_{ m joint} =a-\frac{dx^2-2zxy+by^2}{bd-z^2}.

Eliminating the (b)-block first gives

a−x2b−(y−xz/b)2d−z2/b,a-\frac{x^2}{b} - \frac{(y-xz/b)^2}{d-z^2/b},

while eliminating the (d)-block first gives

a−y2d−(x−yz/d)2b−z2/d.a-\frac{y^2}{d} - \frac{(x-yz/d)^2}{b-z^2/d}.

Exact reduction identifies both expressions with (S_{ m joint}).

The checker verifies 854 invertible exceptional blocks, 93,638 positive covariance families, and 187,276 elimination-order comparisons.

Narrow result

weighted three-block Gaussian conditioning has no generic overlap obstruction.\boxed{ \text{weighted three-block Gaussian conditioning has no generic overlap obstruction.} }

The coefficient-Rees charts glue by ordinary Schur-complement transitivity. No supported secondary class and no new carrier incidence appears away from

ΔE=bd−z2=0.\Delta_E=bd-z^2=0.

This result does not extend through that determinant locus, where the exceptional covariance block itself becomes singular.

Durable artifacts

Next falsifier

Resolve the rank-one exceptional block (bd-z^2=0). Determine whether positivity forces the cross vector ((x,y)) into its image and whether the resulting Moore--Penrose/relative Schur limit is intrinsic on the labelled rank-one stratum. If a further approach ratio is required, identify its weighted coefficient chart without adjoining a carrier cell.

View source on GitHub