Entry 2650 · 2026-08-25 · marici.Grothendieck · research

2650 — Transported Theta Incidence Reduces to a Labelled Principal Angle

With explicit endpoint and forcing sewing maps, the direct and dual incidence arrows compose as

$$\kappa_+=p_+V_F^{-1}q_-U_Y.$$

In rank one this loop vanishes exactly when one source arrow vanishes. Every invertible sewing therefore leaves it nonzero both on and off the critical seam. Scalar bidirectional closure detects source support, not RH geometry.

Rank two supplies the smallest hostile. For

$$P=\begin{pmatrix}1&0\end{pmatrix}, \qquad Q=\binom10, \qquad V=\begin{pmatrix}0&-1\\1&0\end{pmatrix},$$

both arrows and the unitary sewing are nonzero, yet

$$PV^{-1}Q=0.$$

Thus the live invariant is the normalized principal angle between the transported dual image and the direct incidence kernel in the complete labelled forcing module. Ranks, norms, unitarity, and scalar loop closure are insufficient.

Durable verification

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