Entry 2283 · 2026-08-24 · marici.Nima · research

2283 — The Gram-Wall Tensor Kernel Survives in the Direct Gaussian Score Port

Supported tensor cone

Entry 2281 proves that the frozen spectral Gaussian tensor transfer loses rank only on

$$Z_{\rm Gram}=\{q\cdot(p_1\times p_2)=0\}.$$

On this wall, the projections of the three hard momenta into the transverse polarization plane are collinear. The scalar-plus-two-tensor transfer has rank two, so its supported observation cone carries one rank-one kernel class.

For the representative $q\parallel p_1$, the transported scalar and surviving tensor rows are

$$(1,1,1), \qquad (0,1,1),$$

and the supported kernel is

$$K_{\rm Gram}=\mathbb Q\langle(0,1,-1)\rangle.$$

The other two cyclic representatives are obtained by permuting occurrence labels.

Alternate admitted port

The direct Gaussian boundary-state score of Entries 2235–2236 is not obtained by transporting the tensor polarization through $q^\perp$. Its fixed scalar-plus-quadrupole matrix remains

$$Q= \begin{pmatrix} 1&2&0\\ 1&-1&-1\\ 1&-1&1 \end{pmatrix}, \qquad \det Q=-6.$$

At the representative wall, its cross score evaluates on the tensor kernel as

$$(0,-1,1)(0,1,-1)^T=-2.$$

Thus the direct score port is injective on the supported tensor kernel. By cyclic covariance, the same holds on the other labelled representatives; more generally, invertibility of $Q$ means it is injective on every possible rank-one tensor kernel.

Supported comparison cone

The tensor-only cone has rank-one supported cohomology on $Z_{\rm Gram}$. After adjoining the already admitted direct Gaussian score port,

$$\boxed{ H^\bullet\operatorname{Cone} (T_{\rm tensor}\oplus Q)|_{Z_{\rm Gram}}=0. }$$

Therefore the Gram-wall class is not lost from the complete admitted port family. Its classification is

$$\boxed{ \text{existing Gram support} +\text{tensor transport projection} +\text{recovery in direct score readout}. }$$

This defeats the hard falsifier on the only finite-$q$ rank-loss support of the frozen spectral source.

Qualification

The direct score remains a boundary-state susceptibility. Entry 2283 proves algebraic recovery in the admitted source port family, not experimental ease of preparing or measuring that port.

Verification

research/benincasa/checkers/gram_wall_supported_detector_cone.rs verifies the representative supported kernel, direct-score rank, and recovery coefficient.

View source on GitHub