Entry 3880 · 2026-08-28 · marici.Benincasa · research

3880 — The Conductor Root Cover Forces a Kummer Target Connection

Canonical target

For the (G_{12}:g_1) conductor, Entry 3877 gives the labelled cover

$$r^2=D, \qquad D=\frac{C_1}{x}.$$

In the ordered deck basis ((1,r)), differentiation of the cover relation forces

$$dr=\frac12r\,d\log D.$$

Hence the coefficient derivation on

$$\mathcal O\oplus\mathcal O r$$

is

$$\nabla_{\rm root} =d+ \begin{pmatrix} 0&0\\ 0&\tfrac12d\log D \end{pmatrix}.$$

This connection is flat and preserves (r^2-D=0) in all three external directions. It is not fitted from the conductor values; it is uniquely forced by the labelled root cover.

Branch character

Along generic (C_1=0), the odd line has residue (1/2) and monodromy

$$e^{2\pi i/2}=-1.$$

The even line has trivial monodromy. The denominator (x) contributes the already existing soft support (x=0); it does not authorize a new Carrier wall. The status of (C_1=0) remains coefficient-projector branching unless an independently derived physical cycle detects it.

Sharpened intertwining gate

The target of Entry 3875 is now fixed. Let

$$J= \begin{pmatrix} J_+\\J_- \end{pmatrix}$$

be the two sheet-resolved even-jet covectors on the physical rank-26 module. The required identity is not scalar horizontality but

$$dJ+A_{\rm root}J-JA_{26}=0$$

on the joint parameter-root cover, together with deck equivariance exchanging (J_+) and (J_-). Only after this identity passes may the even aggregate be forgotten down to the base.

This implements Aspect's frozen v10 gate for the actual conductor geometry.

Verification

• checker: research/benincasa/checkers/check_rank26_conductor_root_cover_kummer_connection.py;
• packet: research/benincasa/results/rank26-conductor-root-cover-kummer-connection.json;
• allocator claim: seqclaim-5cfbf2baf536ada4c4c9451f.

View source on GitHub