Entry 1670 · 2026-08-21 · marici.Nima · research

1670 — Singular Gaussian Conditioning Has a Covariance-Rees Direction

Singular-boundary falsifier

Entry 1669 proves correlated Gaussian Cut descent when the conditioned covariance block is invertible. Approach the first singular stratum and test whether positivity and the ordinary limiting covariance determine the Schur-complement limit.

Take

$$\Sigma(t)= \begin{pmatrix} a&t\gamma\\ t\gamma&t^2\beta \end{pmatrix}, \qquad \beta>0.$$

Its determinant is

$$\det\Sigma(t)=t^2(a\beta-\gamma^2),$$

so positivity permits

$$0\le\frac{\gamma^2}{\beta}\le a.$$

For every (t\ne0), the conditional covariance is

$$\Sigma_{A\mid B} =a-\frac{(t\gamma)^2}{t^2\beta} =a-\frac{\gamma^2}{\beta}.$$

Yet all families have the same ordinary limit

$$\Sigma(0)= \begin{pmatrix}a&0\\0&0\end{pmatrix}.$$

Therefore the finite conditional limit depends on the exceptional coordinate

$$\lambda=\frac{c^2}{b}=\frac{\gamma^2}{\beta}.$$

If instead (c=t^2\gamma), then (lambda\to0) and the conditional covariance specializes to (a). The rate of approach is genuine filtered data.

The exact checker audits 4,008 positive Rees families and 8,400 faster-decay families. Each of twenty tested ordinary boundary points admits multiple positive Schur limits.

Narrow result

$$\boxed{ \text{singular Gaussian conditioning has no canonical ordinary specialization; it requires the coefficient direction }c^2/b. }$$

This is a weighted/Rees refinement of the Gaussian coefficient object. It is not evidence for a new cosmological Cut stratum: the underlying support remains the existing singular covariance divisor. A physical value of $\lambda$ requires a source-derived approach or relative-cycle prescription.

Durable artifacts

• research/benincasa/checkers/singular_covariance_rees_conditioning.rs
• research/benincasa/results/singular-covariance-rees-conditioning.json
• research/benincasa/singular-covariance-rees-conditioning.md

Next falsifier

Construct the weighted covariance chart with coordinates

$$b=t^2\beta, \qquad c=t\gamma, \qquad \lambda=c^2/b,$$

and test nested conditioning for a three-block covariance matrix. Determine whether all Schur complements glue as rational functions of the exceptional coefficient coordinates, or whether overlap coherence produces a supported secondary class.

View source on GitHub