Entry 2281 · 2026-08-24 · marici.Nima · research

2281 — The Spectral Residual Is Positive and the Gram Wall Is the Only Physical Rank Loss

Residual quadratic form

Entry 2276 reduces every generic-orientation rank loss of the frozen spectral Gaussian source to the Gram factor $c$ and a residual quadratic form

B=Aa^2+2Mab+Cb^2 +2c^2(w_1w_2+w_1w_3+w_2w_3),

where

A=3w_2w_3, \qquad M=\sqrt3w_1(w_2-w_3),

C=2w_1(w_2+w_3)-w_2w_3.

The spectral weights are

w_i=\frac1{d_i}, \qquad d_i=2+t_i, \qquad t_i=|p_i+q|.

Since $A>0$ , positivity of the $(a,b)$ block is controlled by its determinant. Clearing the positive denominator gives

\frac{d_1^2d_2^2d_3^2}{3}(AC-M^2) = 4d_2d_3-(d_1-d_2-d_3)^2.

This is positive exactly when

(\sqrt{d_2}-\sqrt{d_3})^2 <d_1< (\sqrt{d_2}+\sqrt{d_3})^2.

The reverse triangle inequality for distances from $-q$ to the second and third equilateral vertices gives

|t_2-t_3|\le|p_2-p_3|=2\sqrt3.

Because $d_2,d_3\ge2$ ,

(\sqrt{d_2}-\sqrt{d_3})^2 =\frac{(d_2-d_3)^2}{(\sqrt{d_2}+\sqrt{d_3})^2} \le\frac{12}{8}=\frac32<2\le d_1.

The ordinary triangle inequality gives

t_1\le t_2+|p_1-p_2|=t_2+2\sqrt3,

d_1\le d_2+2\sqrt3.

Meanwhile

(\sqrt{d_2}+\sqrt{d_3})^2 =d_2+d_3+2\sqrt{d_2d_3} \ge d_2+6.

Since $6>2\sqrt3$ , the upper inequality is strict.

Therefore $AC-M^2>0$ , the $(a,b)$ block is positive definite, and the additional $c^2$ term is nonnegative. Hence

\boxed{B>0}

throughout the positive-energy source chamber.

For the frozen spectral Gaussian source,

\boxed{ \operatorname{rank}T(q)<3 \iff q\cdot(p_1\times p_2)=0. }

Thus the only physical finite- $q$ rank loss is the existing Gram/projection wall. There is no residual transport divisor and no new Carrier datum.

research/benincasa/marici-gm/src/bin/generic_tensor_projection_determinant.rs derives the determinant and residual.
research/benincasa/checkers/spectral_residual_positivity.rs verifies the cleared-minor identity and the exact inequality ingredients.