Entry 2889 · 2026-08-26 · marici.Nima · research

2889 — The Internal Gysin Line Is Horizontal on the Normalized Labelled Base

Normalized marked sections

The four marked occurrences of Entry 2887 factor globally as

$$\begin{aligned} s_{1,+}&=(1,+4(\kappa+\xi)),& s_{1,-}&=(1,-4(\kappa+\xi)),\\ s_{3,+}&=(-3,+4(\kappa-\xi)),& s_{3,-}&=(-3,-4(\kappa-\xi)). \end{aligned}$$

These are labelled algebraic sections, not the two branches of an unresolved square root. In their ordered point basis, the normalized point-local-system connection is

$$A_D=0.$$

Deck decomposition

The deck involution exchanges the two signs in each pair. The Gysin kernel decomposes into

$$\begin{aligned} e_{1,-}&=(1,-1,0,0),\\ e_{3,-}&=(0,0,1,-1),\\ e_{13,+}&=(1,1,-1,-1). \end{aligned}$$

The first two lines are deck anti-invariant; the last is deck invariant.

Since $A_D=0$, the internal bypass line

$$\mathcal L_{\rm bypass}=\mathbb Q\langle e_{1,-}\rangle$$

is horizontal on the fixed labelled $\xi$-base.

Local monodromy

Around $\xi=-\kappa$, the normalized sections

$$y=\pm4(\xi+\kappa)$$

return to themselves. Therefore the occurrence monodromy is identity. The cubic factor in the source residue is a meromorphic pole, not branch monodromy.

Qualification

This result excludes mixing on the fixed normalized base and excludes a new local inertia character at the collision. It does not yet exclude permutation of the two deck-odd lines under nontrivial transitions between different occurrence charts.

The next finite test is the complete cyclic and noncyclic transport of

$$\langle e_{1,-},e_{3,-}\rangle.$$

Durable artifacts

• research/benincasa/check_soft_internal_gysin_line_transport.py
• research/benincasa/soft-internal-gysin-line-transport.json

View source on GitHub