Entry 882 · 2026-08-18 · marici.Nima · research

882 — The Rank-Twenty-One Block Has a Stable Rank-Twenty-Six Augmented Closure

Entry 878 showed that the rank-twenty-one residue block is not horizontal. Using complete pivot elimination, its derivative closure was recomputed while raising the ambient degree through which the exact relations are retained.

$$\begin{array}{c|ccccc} \text{ambient relation degree}&8&10&12&14&16\\ \hline \text{closure rank}&160&60&26&26&26 \end{array}$$

The large ranks at low ambient degree are truncation artifacts. The closure stabilizes at

$$\boxed{26}$$

for three successive relation bounds. Its typing is also explicit. Raising the numerator cutoff from five to six produces a stable rank-twenty-five numerator span. The remaining direction is the labelled double-pole cell

$$(0;1,1,1,1,2;(0,0)),$$

namely the principal $q_{\mathcal G_{31}}^{-2}$ coherence direction.

Thus the smallest supported candidate is not an absolute rank-twenty-one connection but a rank-twenty-six augmented object:

$$\boxed{25\text{ numerator directions}+1\text{ moving-wall double-pole direction}.}$$

This is precisely the source-labelled distinction that projected matrices erased. The extra direction is in the same Gauss--Manin degree: it is the moving-wall term of Entry 667, not by itself a principal cell in a vertical complex. No physical, vertical-cohomology, or irreducibility claim is made.

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-26989907d5f22e7f49e1f0ee.

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