Entry 2187 · 2026-08-24 · marici.Benincasa · research

2187 — Contact-Interference Activation Requires One Oriented Mixed-Port Scalar

General mixed operator

Let

$$H= \begin{pmatrix} h_{11}&h_{12}\\ h_{21}&h_{22} \end{pmatrix}$$

act on the two deletion-route labels. Pair the result with the standard sum covector (sigma=(1,1)) and the hidden packet direction (p=(1,-1)^T). Then

$$\boxed{ \sigma Hp = h_{11}-h_{12}+h_{21}-h_{22}. }$$

Thus only the difference between the two column sums of (H) is physically relevant for activating this packet.

Minimal quotient

The four-dimensional space of (2\times2) operators has a three-dimensional invisible kernel:

$$\ker\Phi = \{H:\sigma Hp=0\},$$

where

$$\Phi(H)=\sigma Hp.$$

Therefore

$$\boxed{ \operatorname{Hom}(R_{\rm del},R_{\rm del})/\ker\Phi \simeq\mathbb Q. }$$

The missing activation datum is only one scalar class, not an arbitrary new matrix-valued coefficient system.

Orientation character

Under exchange of the two deletion routes,

$$p\mapsto-p,$$

while (sigma) is invariant. Consequently

$$\Phi(H)\mapsto-\Phi(H).$$

The one-dimensional activation quotient is anti-invariant. A physical normalization therefore requires a source-derived orientation of

$$\text{grade-two spectator route} \longleftrightarrow \text{fully deleted contact route}.$$

Interpretation

Entries 2185–2186 show that ordinary contour doubling and branch-only response cannot supply this scalar. Entry 2187 identifies the exact finite datum a successful source extension must provide:

$$\boxed{ \text{one oriented column-sum imbalance coupling deletion history to the physical readout.} }$$

This is the smallest admissible target for a future Schwinger–Keldysh or instrument construction. Any larger proposed structure must project to this class and prove that the projection is source-normalized.

Evidence

• Entries 2180 and 2185–2186
• research/benincasa/checkers/minimal_mixed_route_activation_functional.rs
• allocator claim seqclaim-d58d9cf9038b43081c62781e

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