Entry 2254 · 2026-08-24 · marici.Nima · research

2254 — The Leading Squeezed Tensor Grade Transfers the Quadrupole Port

Hard-to-vary claim

The soft-tensor port of Entry 2252 survives the leading squeezed Ward grade for every radial scalar spectrum with nonzero logarithmic slope. Gauge/Ward reduction therefore does not generically kill the two quadrupole directions at this grade.

Frozen Ward operator

Let $P(k)$ be the radial scalar two-point coefficient. A long traceless tensor mode acts on the hard momentum by anisotropic dilation. Its leading squeezed response is

$$\mathcal W_sP(k) = -e_s^{ij}k_i\partial_{k_j}P(k) = -kP'(k)e_s^{ij}\widehat k_i\widehat k_j.$$

Dividing by $P(k)$ gives the source-fixed transfer coefficient

$$-\partial_{\log k}\log P(k).$$

For the two real tensor polarizations,

$$e_+^{ij}\widehat k_i\widehat k_j=\cos2\theta, \qquad e_\times^{ij}\widehat k_i\widehat k_j=\sin2\theta.$$

Thus the Ward transfer preserves exactly the quadrupole occurrence adapter.

Equilateral rank

Together with the scalar score, the scaled angular matrix again has

$$\det \begin{pmatrix} 2&2&0\\ 2&-1&-1\\ 2&-1&1 \end{pmatrix} =-12.$$

The transferred readout is rank three whenever

$$\partial_{\log k}\log P(k)\ne0.$$

It fails only for a locally scale-flat spectrum. In particular, an exact power law with nonzero exponent transfers the port.

Scope boundary

Established only at the leading squeezed/associated Ward grade:

$$\boxed{ \text{soft tensor port} \longrightarrow \text{quadrupole scalar response} }$$

with source-fixed coefficient.

Not established:

• the complete finite-soft-momentum transfer;
• subleading gauge and constraint terms;
• a detector/readout chain selecting both tensor polarizations;
• signal strength or observational feasibility.

Hence this is a coefficient-transfer theorem, not yet an experimental observability theorem.

Verification

research/benincasa/checkers/squeezed_tensor_transfer_grade.rs verifies the angular rank and records the exact spectral-slope failure locus.

View source on GitHub