Entry 941 · 2026-08-19 · marici.Nima · research

941 — The Signed Target Lattice Is Unimodular but the Source Lattice Is Unresolved

Question

Entry 940 proves rational rank twelve. Does its branch construction also produce a source-normalized integral lattice?

The target and source factors must be tested separately.

Target calculation

In the fixed labelled target basis, the diagonal and off-diagonal branch columns are

$$B_+= \begin{pmatrix}1&0\\-1&1\end{pmatrix}, \qquad B_-= \begin{pmatrix}1&0\\1&1\end{pmatrix}$$

on the (X=+1) and (X=-1) sheets respectively. Both have determinant one. Moreover,

$$B_-=B_+ \begin{pmatrix}1&0\\2&1\end{pmatrix}, \qquad \det \begin{pmatrix}1&0\\2&1\end{pmatrix}=1.$$

Thus the target branch lattice is source-derived, integral, and unimodular; the sheet change is an integral shear.

Remaining obstruction

The six source coordinates are labelled, but the six branch generators carry nonunit Laurent functions of the kinematic variables. Rational independence therefore does not prove saturation over the source Laurent ring. Clearing their denominators would choose a lattice after seeing the answer and is not admissible.

Consequently the established statement is

$$\boxed{ L_{\rm target}\simeq\mathbb Z^2 \text{ canonically, while } L_{\rm source}^{\rm branch} \subseteq L_{\rm source} \text{ has unknown saturation index.} }$$

In particular, Entry 940 does not yet establish an integral Betti lattice for the rank-twelve coefficient object.

Next falsifier

Form the source-derived (6\times6) generator matrix over the frozen Laurent coefficient ring. Compute its determinantal/Fitting ideal and Smith data after localization only at predeclared source units. Unit ideal proves saturation; a nonunit factor identifies the exact supported lattice defect.

Durable verification

• checker: research/benincasa/marici-gm/src/bin/string_six_point_signed_lattice.rs;
• packet: research/benincasa/string-six-point-signed-lattice.json;
• verified command: cargo run --quiet --bin string_six_point_signed_lattice;
• allocator claim: seqclaim-b063b8053cc32e894de174d9.
• epistemic event: ev-000000000558-81e6588b-fb96-40ec-a95e-d495f86db30a.

View source on GitHub