Entry 2272 · 2026-08-24 · marici.Nima · research

2272 — The Spectral Gaussian Source Gives a Finite-$q$ Transfer with No Physical Rank Loss

Frozen source enlargement

Entry 2267 proves that the scalar weighted-correlator source does not determine a finite-soft tensor continuation. Before inspecting any rank locus, freeze the minimal covariant massless Gaussian precision operator

$$A_g=\sqrt{-\Delta_g}.$$

On the flat boundary this gives the radial precision $A_0(k)=k$, and hence the covariance $K(k)=1/(2k)$ used by the conformal massless scalar state. This is a declared source model, not a reconstruction fitted to a desired determinant.

Finite-momentum metric vertex

For a spectral function $A=\sqrt L$, its Fréchet derivative between eigenmodes of $L$ with eigenvalues $k^2$ and $k'^2$ is

$$(\delta A)_{p'p} = \frac{k'-k}{k'^2-k^2}(\delta L)_{p'p} = \frac{(\delta L)_{p'p}}{k'+k}.$$

Take a transverse traceless metric mode $h_s(q)$, with $p'=-p-q$. Freeze the symmetric finite-soft slice in which $q$ is normal to the plane of the three equilateral hard momenta. Transversality removes the $p_iq_j$ term, so up to the common sign convention

$$(\delta L)_{p'p} =h_s^{ij}p_ip_j.$$

Thus each labelled occurrence receives a positive physical weight

$$w_e(q)=\frac{k_e^2}{k_e+\sqrt{k_e^2+q^2}}>0$$

times its plus/cross quadrupole contraction. A common nonzero normalization does not affect rank.

Homogeneous three-occurrence transfer

In the integral scaled angular convention, the scalar-plus-cross transfer is

$$T(q)= \begin{pmatrix} 1&2w_1&0\\ 1&-w_2&-w_2\\ 1&-w_3&w_3 \end{pmatrix}.$$

Its determinant factors exactly as

$$\boxed{ \det T(q) =-2(w_1w_2+w_2w_3+w_3w_1). }$$

For positive finite energies every $w_e>0$, hence

$$\det T(q)<0.$$

There is no physical positive-energy finite-$q$ rank-loss locus on this source-defined symmetric slice. Generic orientations remain a separate extension of the test.

Covariance checks

• Gauge: only transverse traceless contractions survive; longitudinal additions proportional to $q_i$ vanish.
• Cyclic: rotating the labelled triangle and $q$ permutes the three $w_e$ and rows; the determinant factor is symmetric.
• Ward: as $q\to0$, $w_e\to k_e/2$, reproducing Entry 2254's quadrupole Ward response up to its common source normalization.

Rank-loss classification

Over the complexified base, possible degeneracy is

$$w_1w_2+w_2w_3+w_3w_1=0,$$

or a failure of the spectral denominators $k_e+|p_e+q|$. Neither meets the positive-energy chamber. The denominator conditions are ordinary energy support of the enlarged coefficient object, not new occurrence incidence.

This is a progressive protective-belt result: it establishes finite-$q$ contextual faithfulness for the frozen spectral Gaussian source on the normal tensor-momentum slice, not for every orientation or possible covariantization of the scalar state.

Verification

research/benincasa/checkers/spectral_gaussian_finite_q_transfer.rs verifies the divided-difference identity, determinant factorization, and strict positive-energy sign on 512 exact weight packets.

View source on GitHub