Entry 2267 · 2026-08-24 · marici.Nima · research

2267 — The Leading Tensor Ward Grade Does Not Determine Finite-Soft Support

Hard-to-vary claim

The frozen scalar contact source and its leading tensor Ward grade do not determine a source-specific finite-soft transfer matrix. Even cyclicly equivariant analytic extensions with the same value and first normal jet at $q=0$ can have different finite-$q$ rank-loss support.

Hostile pair

Use the normalized leading matrix

$$T_0= \begin{pmatrix} 2&2&0\\ 2&-1&-1\\ 2&-1&1 \end{pmatrix}.$$

Consider two analytic continuations:

$$T_A(q)=T_0,$$

and

$$T_B(q)=T_0 \operatorname{diag}\bigl(1,1-q^2,1-q^2\bigr).$$

The common factor on the shear plane preserves its rational $C_3$ representation and treats both tensor polarizations equally. Moreover,

$$T_A(0)=T_B(0), \qquad \partial_qT_A(0)=\partial_qT_B(0)=0.$$

Nevertheless,

$$\det T_A(q)=-12,$$

while

$$\det T_B(q)=-12(1-q^2)^2.$$

Thus $T_B$ loses both shear directions at $q=\pm1$, whereas $T_A$ never does.

Provenance consequence

Entries 2252–2256 establish the soft port, its leading Ward transfer, and an open rank-three neighborhood. They do not license choosing a global finite-soft continuation.

The frozen weighted-correlator source supplies:

• scalar deletion-sector integrands;
• the diagonal Gaussian covariance $K(y)$;
• the contact-normal packet.

It does not supply either of the following equivalent finite-soft source objects:

$$\frac{\delta A_g(p,p')}{\delta h_s(q)}$$

for a curved-boundary Gaussian quadratic kernel, or a complete labelled scalar--scalar--tensor wavefunction coefficient at finite tensor momentum.

Therefore the source-specific finite-$q$ transfer requested by the observer programme is presently underived, not zero.

Required frozen enlargement

Before computing rank-loss support, freeze:

1. the finite-$q$ scalar--scalar--tensor coefficient or curved-metric Gaussian kernel;
2. its two polarization conventions and gauge constraints;
3. its occurrence-labelled coupling to the three contact channels;
4. the physical relative contour/readout map.

Only that packet can distinguish $T_A$, $T_B$, or another continuation. No fitted $\Delta(q)$ is admissible.

Verification

research/benincasa/checkers/finite_soft_extension_nonuniqueness.rs verifies the common first jet and distinct determinant divisors exactly.

View source on GitHub