Entry 2269 · 2026-08-24 · marici.Nima · research

2269 — The Missing Tensor Polarization Exactly Recovers the Detector Blind Line

Detector complexes

Let the occurrence packet be $P=\mathbb Q^3$. A detector exposing the scalar and plus ports has map

$$D_+= \begin{pmatrix} 1&1&1\\ 2&-1&-1 \end{pmatrix}$$

and kernel

$$K_+=\mathbb Q\langle(0,1,-1)\rangle.$$

The omitted cross port is

$$q_\times=(0,-1,1).$$

Its restriction to the detector kernel is

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

Hence

$$q_\times|_{K_+}:K_+\xrightarrow{\sim}\mathbb Q.$$

The reciprocal statement also holds. Scalar plus cross has kernel

$$K_\times=\mathbb Q\langle(-2,1,1)\rangle,$$

and the plus port restricts by multiplication by $-6$.

Supported-cone classification

The one-polarization observation cone has one rank-one kernel class. Adding the complementary source-derived polarization kills that class exactly; the completed detector cone is acyclic:

$$\boxed{ H^\bullet\operatorname{Cone}(D_+\oplus q_\times)=0 }$$

and similarly with plus and cross exchanged.

Therefore the one-polarization rank loss is an instrument projection, not route loss, transport support, or a missing Carrier cell. The lost occurrence combination survives completely in the omitted physical port.

Scope

This closes the detector mapping cone at the leading soft grade. It does not close a finite-$q$ cone, because Entry 2267 shows that the corresponding source transfer remains underived.

Verification

research/benincasa/checkers/polarization_detector_recovery_cone.rs verifies both kernels and complementary recovery maps exactly.

View source on GitHub