Entry 1673 · 2026-08-21 · marici.Nima · research

1673 — Flagged Covariance Degeneration Has One Quadratic Form per Normal Block

Uniform-rank falsifier

Entry 1672 finds a secondary quadratic normal coordinate at the first rank-one degeneration. Test whether successive covariance-rank drops require new structure or follow one invariant rule.

For a vanishing graded covariance block, write

$$K(t)=W(t)BW(t), \qquad v(t)=W(t)\gamma,$$

where

$$W(t)=\operatorname{diag}(t^{w_1},\ldots,t^{w_s})$$

has arbitrary positive integral weights and (B) is an invertible symmetric normal-block covariance.

Then

$$K(t)^{-1}=W(t)^{-1}B^{-1}W(t)^{-1},$$

so

$$\boxed{ v(t)^TK(t)^{-1}v(t)=\gamma^TB^{-1}\gamma. }$$

For every matrix term, the normal exponent cancels:

$$w_i-(w_i+w_j)+w_j=0.$$

Thus the finite nearby correction is invariant under the choice of weighted coordinates inside the graded block.

The exact checker verifies 60 correlated positive-definite normal blocks of ranks one through six, 910 matrix-inverse identities, 60 positive quadratic forms, and 3,640 independent flag-weight cancellations.

Narrow result

$$\boxed{ \text{each covariance normal block contributes one invariant quadratic Schur form.} }$$

The tempting description as one scalar ratio per vanishing eigen-direction is not invariant when the normal block is correlated. It is merely the diagonal basis presentation of

$$\gamma^TB^{-1}\gamma.$$

Successive rank drops are therefore governed by the existing flagged normal calculus with sector-specific symmetric-form coefficients. No additional carrier incidence or higher sewing primitive is indicated.

This theorem assumes that the graded normal block (B) is invertible. Its own rank-drop locus must be handled by the next flag grade, as in Entries 1671--1672.

Durable artifacts

• research/benincasa/checkers/flagged_covariance_quadratic_normal_form.rs
• research/benincasa/results/flagged-covariance-quadratic-normal-form.json
• research/benincasa/flagged-covariance-quadratic-normal-form.md

Next falsifier

Compare this covariance flag with the cosmological energy-normal/Rees flag. Test whether their product degeneration obeys a strict external tensor rule or whether mixed energy--covariance normal terms survive. A nonzero mixed term would be the first coupling not explained by separate carrier flags and coefficient quadratic forms.

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