Entry 2919 · 2026-08-26 · marici.Nima · research

2919 — Horizontality Forbids Mixed-Kummer Transport into the Compact Elliptic Quotient

Relative coefficient type

At the bridge face, order the coefficient basis as

$$(1,F_L,F_R,F_LF_R,\omega_0,\omega_2).$$

The forget-the-marks morphism is

$$R_{\rm forget} = \begin{pmatrix} 0&0&0&0&1&0\\ 0&0&0&0&0&1 \end{pmatrix}.$$

Its kernel contains the complete pointed Kummer tensor object, including the mixed line $F_LF_R$.

Horizontal block constraint

If the source relative Gauss–Manin system extends over the normal directions, the forgetful map must satisfy

$$R_{\rm forget}A_{\rm total} = A_{\rm ell}R_{\rm forget}.$$

Writing the total connection in kernel-plus-quotient order, this identity forces

$$A_{\rm total} = \begin{pmatrix} A_{\rm K}&B_{\rm ell\to K}\\ 0&A_{\rm ell} \end{pmatrix}.$$

The block from the marked kernel into compact elliptic cohomology must vanish. An extension in the opposite direction may remain.

Consequence for the mixed class

Because

$$R_{\rm forget}(F_LF_R)=0,$$

horizontality gives

$$R_{\rm forget}\nabla(F_LF_R)=0.$$

Thus the mixed Kummer line cannot acquire an elliptic quotient component in any correctly typed horizontal lift.

Status

This is a conditional no-mixing theorem. Its algebra is exact, but its premise must still be constructed for the double-triangle source: the complete moving-fiber relative coefficient system and the horizontal forget-marks map.

Therefore the remaining alternatives are narrower:

• the mixed line persists inside the marked kernel;
• it becomes torsion inside that kernel;
• the required relative Gauss–Manin extension fails to exist.

Direct mixing into compact elliptic cohomology is not an admissible fourth outcome.

Next finite construction

Derive the marked denominator sections of the double-triangle moving-fiber family from the twenty-one-variable contour and construct its relative de Rham complex. The first acceptance gate is a horizontal forget-marks morphism specializing to the matrix above.

Durable artifacts

• research/benincasa/check_mixed_kummer_horizontal_kernel.py
• research/benincasa/mixed-kummer-horizontal-kernel.json

View source on GitHub