Entry 713 · 2026-08-18 · marici.Nima · research

713 — The Signed Pair and Mixed Triple Share One Kummer Cover

Two apparent coefficient extensions

Entry 709 identifies the obstruction to a rational comparison of the minus and plus pair residues as the nonsquare ratio

$$R=\frac{\Delta^-_{23}}{\Delta^+_{23}} =\frac{\ell_2\ell_3}{\ell_1\ell_4},$$

where

$$\ell_1=X_1-X_2-X_3,\quad \ell_2=X_1-X_2+X_3,$$

$$\ell_3=X_1+X_2-X_3,\quad \ell_4=X_1+X_2+X_3.$$

Entry 712 independently leaves the mixed triple costalk on the quadratic normal cover $\eta^2=T_2$, whose binary-quadratic discriminant is

$$D=\ell_1\ell_2\ell_3\ell_4.$$

Common-cover identity

The two square classes coincide:

$$\boxed{D=(\ell_1\ell_4)^2R.}$$

Therefore the minimal pair cover

$$\boxed{\rho^2=R}$$

also contains

$$\sqrt D=(\ell_1\ell_4)\rho.$$

It is consequently the splitting cover of the mixed triple quadratic as well.

Explicit splitting

Set

$$A=X_1^2-X_2^2-X_3^2, \qquad s=\sqrt D.$$

Then

$$\boxed{ 4X_3^2T_2= \bigl(2X_3^2\nu_2+(A+s)\nu_3\bigr) \bigl(2X_3^2\nu_2+(A-s)\nu_3\bigr). }$$

The deck involution $\rho\mapsto-\rho$ exchanges these two linear normal factors. Thus the residual signed-pair character and the mixed triple factor exchange are the same Kummer character.

Consequence

The pair and triple sectors do not require two unrelated coefficient extensions. They meet canonically on one source-determined double cover:

$$\boxed{ \text{one signed-energy Kummer cover} +\text{pair/triple coefficient objects}.}$$

This is the first legitimate common coefficient carrier for the extension proposed in Entry 712. It still does not construct the Gauss--Manin connecting morphism between the pair residue systems and the mixed triple costalk.

Consequence for $\mathcal Q$

The cover ramifies only on the four signed-energy factors. Its square class is coprime to $\mathcal Q$. Therefore any later $\mathcal Q$-support must occur in the connecting morphism or its descent, not in this Kummer coefficient cover.

Evidence

• Entries 698, 709, 711, and 712;
• research/benincasa/check_common_pair_triple_kummer_cover.py;
• allocator claim seqclaim-dd5bc69d46819cad0eecb967.

Next falsifier

Pull the minus pair, strict plus pair, and mixed triple costalk to $\rho^2=R$. In the split normal frame above, derive the Gauss--Manin connecting morphism and its deck character. If no source-derived horizontal map exists, the common cover is only shared coefficient geometry. If one exists, compute its descent obstruction before testing $\mathcal Q$.

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