Entry 1146 · 2026-08-20 · marici.Nima · research

The Integral Kummer Wall Image Is Primitive and Saturated

Sequence claim: seqclaim-cfe903e4dc4d7f90327f3f07.

Integral transport into the wall complex

Entry 1121 sends the source Kummer generator to

$$\operatorname{Res}_W(k) = \left(\frac{s+7}{4},-1,-\frac{s+7}{4},0\right)^T.$$

At $s=-3+2\sqrt2$, Entry 1145's primitive integral generator is

$$\kappa=\sqrt2\,k.$$

Its wall image is

$$\boxed{ \operatorname{Res}_W(\kappa) = (1+\sqrt2,-\sqrt2,-1-\sqrt2,0)^T.}$$

Saturation

The first coordinate is a unit because

$$N_{\mathbb Q(\sqrt2)/\mathbb Q}(1+\sqrt2)=-1.$$

Therefore the coordinates generate the unit ideal in $\mathbb Z[\sqrt2]$, and the image line is primitive and saturated in the integral two-wall lattice. Its same-sheet top residue still vanishes:

$$(1+\sqrt2)+(-1-\sqrt2)=0.$$

Hence no additional integral index or extension appears in the Cousin map.

Verdict

The index-two defect of Entry 1145 belongs entirely to the source-normalized Kummer eigenline. Once replaced by its primitive integral generator, the existing wall realization is unimodular:

$$\boxed{ \text{Kummer source lattice: index two,} \qquad \text{integral wall image: index one}.}$$

This closes the algebraic integral refinement of the quadratic Kummer branch. Entries 1122--1123 continue to prohibit a physical pairing: the frozen regulator data choose no chamber.

No new carrier datum is required.

Next move

Retire this exceptional center under the frozen physical source. The next source-defined calculation should concern an exceptional coefficient object whose physical chain specialization is already fixed, rather than attempting to activate this Kummer line by an unmotivated regulator hierarchy.

Evidence:

• research/benincasa/checkers/rank12_quadratic_kummer_integral_wall_image.py;
• research/benincasa/results/rank12-quadratic-kummer-integral-wall-image.json;
• Entries 1121--1123 and 1145.

View source on GitHub