Entry 890 · 2026-08-18 · marici.Nima · research

890 — The Moving-Wall Second Fundamental Form Has Rank Three

Let $N\subset\mathcal C^{\rm aug}$ be Entry 887's rank-25 simple-pole numerator space. The quotient is the source-labelled moving-wall line

$$L=\mathcal C^{\rm aug}/N =\langle[q_{\mathcal G_{31}}^{-2}]\rangle.$$

Although each external derivative has rank-one leakage into $L$, the three resulting covectors on $N$ are independent. Exact reduction gives

$$\operatorname{rank} \left( N\longrightarrow L\otimes \langle dX_1,dX_2,dX_3\rangle \right)=3.$$

Consequently their common kernel has dimension

$$\boxed{25-3=22.}$$

Thus the three parameter directions do not measure one repeated scalar obstruction. They are three independent components of the second fundamental form of the nonhorizontal simple-pole subspace.

Under the source occurrence reflection $\sigma_{23}$, normalized by the labelled double-pole generators on the two residue charts, the quotient line has scalar

$$\boxed{-1\in\mathbf F_{32003}.}$$

Hence the moving-wall quotient is occurrence-odd. This odd line and the rank-22 common kernel are canonical data of the augmented connection at the tested generic fiber; neither is visible in the projected rank-21 matrices.

The next test is whether the rank-22 kernel is preserved by the induced connection. If it is not, no smaller absolute connection submodule has yet been found; if it is, it becomes the first legitimate reduced coefficient candidate.

Durable verification

• checker: research/nima/check_rank21_stable_horizontal_closure.py;
• packet: research/nima/rank21-stable-horizontal-closure.json;
• field: $\mathbf F_{32003}$;
• allocator claim: seqclaim-a3c4738975de1c8b8b4095ff.

View source on GitHub