Entry 1672 · 2026-08-21 · marici.Nima · research

1672 — Rank-One Covariance Has an Intrinsic Restriction but a Secondary Normal Grade

Rank-one falsifier

Entry 1671 reduces weighted three-block conditioning to the exceptional determinant

$$bd-z^2=0.$$

Distinguish restriction to the exact rank-one stratum from specialization of nearby positive covariance families.

Choose an adapted basis in which

$$K_0=\begin{pmatrix}k&0\\0&0\end{pmatrix}, \qquad k>0.$$

For the full covariance

$$\begin{pmatrix} a&x&y\\ x&k&0\\ y&0&0 \end{pmatrix},$$

positivity of the principal minor on the first and null coordinates gives

$$-y^2\ge0,$$

and therefore (y=0). The cross vector lies in the image of (K_0), so the Moore--Penrose correction

$$\frac{x^2}{k}$$

is intrinsic on the exact rank-one stratum.

Now approach it through

$$K_t=\begin{pmatrix}k&0\\0&t^2\beta\end{pmatrix}, \qquad v_t=(x,t\gamma).$$

The Schur correction is

$$v_t^TK_t^{-1}v_t =\frac{x^2}{k}+\frac{\gamma^2}{\beta}.$$

All such families share the same ordinary rank-one boundary value, but the second term survives as a flagged normal coordinate.

The exact checker verifies 820 exact-stratum positivity tests and 174,376 positive nearby families. It finds multiple limits at 2,652 identical rank-one boundary points.

Narrow result

$$\boxed{ \text{rank-one restriction is intrinsic, but nearby conditioning carries a secondary normal grade.} }$$

There is no new carrier divisor: the support remains the exceptional determinant already identified in Entry 1671. What grows is the filtered Gaussian coefficient object. Ordinary restriction and nearby specialization must not be identified.

Durable artifacts

• research/benincasa/checkers/rank_one_exceptional_covariance_normal.rs
• research/benincasa/results/rank-one-exceptional-covariance-normal.json
• research/benincasa/rank-one-exceptional-covariance-normal.md

Next falsifier

Determine whether successive covariance-rank drops are governed uniformly by the complete flag of null directions. Formulate the rank-$r$ local normal model and test whether each new finite Schur correction is exactly one quadratic Rees ratio per newly vanishing eigen-direction. A failure would be the first indication that covariance coefficients require structure beyond flagged normal geometry.

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