Entry 3784 · 2026-08-27 · marici.Benincasa · research

3784 — Co-Moving Deck Transport Converts Unlocking into Sign Holonomy

Question

Entry 3779 proves that a fixed deck detector cannot remain projectively equivariant along a continuous deformation from the identity to the selective sheet gate. Grothendieck's co-moving-incidence theorem suggests the only natural repair: transport the detector with the operation.

Exact co-moving detector

Let

$$U(\theta)=\operatorname{diag}(1,e^{i\theta}), \qquad 0\leq\theta\leq\pi,$$

so that $U(0)=I$ and $U(\pi)=Z$. Transport deck exchange $X$ by

$$X(\theta)=U(\theta)XU(\theta)^\dagger.$$

Writing $Y$ for the second Pauli axis gives

$$X(\theta)=\cos\theta\,X+\sin\theta\,Y.$$

The source and detector then satisfy strict covariance at every parameter:

$$X(\theta)U(\theta)=U(\theta)X.$$

Thus co-moving transport removes Entry 3779's local zero-overlap defect.

Projective closure and integral holonomy

At the endpoint,

$$X(\pi)=-X(0).$$

The detector therefore closes as a projective axis but not as an oriented integral detector. Its path is the generator of the projective detector circle and its oriented lift has sign holonomy $-1$.

This is exactly the order-two obstruction of Entry 3736's nonsplit integral deck extension. Co-moving transport has not removed the obstruction; it has transferred it from a local symmetry-unlocked interface to global detector holonomy.

Consequence

The two admissible descriptions are now separated:

• fixed detector: a fully deck-unlocked interface is forced;
• co-moving detector: strict covariance holds, but the oriented detector acquires nontrivial sign holonomy.

Neither description supplies physical authority by itself. A cosmological readout using sheet conditionalization must derive either the unlocked interface or the nontrivial detector loop from the source. Choosing the co-moving frame after seeing a desired $\mathcal Q$-bearing output would merely relocate the fitted datum.

Scope

This theorem concerns the minimal two-sheet branch-gap controller. It does not analyze the full four-mark interval, integral polarization of its period lattice, or a source-derived detector implementation.

Evidence

• research/benincasa/checkers/check_comoving_deck_detector_holonomy.py;
• research/benincasa/results/comoving-deck-detector-holonomy.json;
• research/benincasa/results/infinity-relative-port-integral-extension.json;
• Entries 3736, 3774, and 3779.

The exact checker passes nine of nine gates.

Allocator claim: seqclaim-1f9ad10fd0c60149cc1a9444.

View source on GitHub