Entry 952 · 2026-08-19 · marici.Nima · research

The Exact-Valuation Seven-Plane Is Tangentially Rank-Stable on Generic Wall Samples

Question

Entry 947 identifies the quadratic triangle-wall sector as the fifth cross-effect of the exact-valuation functor,

$$E_2(C)= \frac{\ker(\Lambda:C\to C)\cap \Lambda C} {\ker(\Lambda:C\to C)\cap \Lambda^2C}, \qquad \dim E_2(C)=7.$$

That calculation was made at the wall point $(X_1,X_2,X_3)=(2,3,5)$. Before asking for a Gauss--Manin connection on this object, its rank must at least remain stable under tangential motion along

$$X_3=X_1+X_2.$$

Exact finite test

The source exporter now accepts arbitrary integral tangential coordinates $(X_1,X_2)$ and takes its seven exact normal samples at

$$(X_1,X_2,X_1+X_2+\Lambda), \qquad -3\leq \Lambda\leq3.$$

At ambient relation degree ten, the complete sparse relation packet has 11520 columns and 15256 raw rows at each normal sample. Complete finite-field reduction gives

$$\begin{array}{c|c|c|c} (X_1,X_2,X_3)&\operatorname{rank}R_0&n_1&n_2\\ \hline (2,3,5)&6305&5&7\\ (3,3,6)&6305&5&7\\ (2,4,6)&6305&5&7 \end{array}$$

The two new points move in independent tangential directions from the original one. Their complete cumulative family filtrations also agree:

$$\operatorname{rank}R_0=(264,2021,3989,5117,5757,6117,6305),$$

$$n_1=(0,6,6,6,6,6,5), \qquad n_2=(0,0,0,0,0,0,7).$$

The elimination representatives vary with the point, while these ranks do not. This is the behavior expected of a rank-seven object rather than of a special numerical accident at $(2,3,5)$.

Conclusion and boundary

The intrinsic exact-valuation object is tangentially rank-stable on the tested generic triangle-wall samples:

$$\boxed{\dim E_2(C_X)=7.}$$

This is a sampled rank theorem, not yet a local-system theorem. No canonical comparison

$$E_2(C_X)\longrightarrow E_2(C_{X+dX})$$

or tangential Gauss--Manin operator has been constructed. Consequently the result does not yet identify this seven-plane with the known generic rank-seven algebraic kernel. The next typed calculation is to differentiate the full relation presentation tangentially, derive the induced connection on $E_2$, and test its compatibility with occurrence transport and the infinity-Gysin sequence.

Durable artifacts

• research/nima/export_triangle_wall_dual_rows.py
• research/nima/triangle-wall-dual-relation-rank.json
• research/benincasa/marici-gm/src/bin/triangle_wall_dual_rank.rs

View source on GitHub