Entry 1746 · 2026-08-21 · marici.Nima · research

1746 — A Rotating Second Jet Defines a Parabolic Local System

Source-derived flag

At the rank-deficient locus of Entry 1745, labelled occurrence subtraction produces a line and quotient

$$L_1\subset V, \qquad L_2=V/L_1.$$

On overlaps, every transition preserving this source flag has the form

$$P=\begin{pmatrix}\sigma_1&a\\0&\sigma_2\end{pmatrix}, \qquad \sigma_i\in\{\pm1\}.$$

The shear (a) depends on the local lift of (L_2). It vanishes from the associated-grade transition:

$$\operatorname{gr}P=\operatorname{diag}(\sigma_1,\sigma_2).$$

Consequently the endomorphism grades have characters

$$\boxed{(1,\sigma_1\sigma_2,\sigma_1\sigma_2,1).}$$

The off-diagonal grades retain exactly the relative line holonomy. A common sign ((\sigma_1,\sigma_2)=(-1,-1)) is central and therefore invisible on endomorphisms, while opposite signs produce nontrivial off-diagonal monodromy.

Narrow result

The complete labelled second jet canonically defines a parabolic local system. It does not canonically define one globally fixed scalar matrix: local matrices glue by parabolic conjugation unless an independent frame trivialization is supplied.

No local Smith basis is chosen. The flag comes from the labelled source occurrences, and the associated-grade monodromy is independent of the lift-dependent shear.

This is again coefficient/readout structure over the existing carrier, not a new Cut incidence stratum.

Durable artifacts

• research/benincasa/checkers/rotating_second_jet_flag.rs
• research/benincasa/results/rotating-second-jet-flag.json
• research/benincasa/rotating-second-jet-flag.md

Next falsifier

Replace the split sign transport by a genuinely unipotent parabolic loop. Test whether its shear is an intrinsic extension class of the filtered reference local system or can be removed by a globally regular change of lift.

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