Entry 1006 · 2026-08-19 · marici.Nima · research

The Exact-Valuation Seven-Plane Is Flat Along Both Coordinate Wall Tangents

Independent tangent

Entry 1003 established first-order flatness of

$$E_2(C)= \frac{\ker\Lambda\cap\operatorname{im}\Lambda} {\ker\Lambda\cap\operatorname{im}\Lambda^2}$$

along the wall tangent

$$(X_1,X_2,X_3)=(2+\tau,3,5+\tau+\Lambda).$$

The independent coordinate tangent is

$$(X_1,X_2,X_3)=(2,3+\tau,5+\tau+\Lambda).$$

It preserves the triangle wall $X_3=X_1+X_2$ while varying $X_2$ rather than $X_1$.

Exact replicated ranks

Over

$$B=\mathbf F_{32003}[\tau]/(\tau^2),$$

the complete mixed relation ranks are

$$\begin{array}{c|c|c|c} k&\operatorname{rank}R_{k,\tau} &2\operatorname{rank}R_k&\text{excess}\\ \hline 1&12610&12610&0\\ 2&25230&25230&0\\ 3&37864&37864&0. \end{array}$$

These are identical to the independently generated $X_1$-tangent ranks. With 11520 columns per normal grade, the mixed cokernel dimensions are again

$$(10430,20850,31256).$$

Consequently

$$\dim\operatorname{im}\Lambda=31256-10430=20826,$$

$$\dim\operatorname{im}\Lambda^2=31256-20850=10406,$$

and

$$\dim E_{2,\tau}=20826-2(10406)=14=2\cdot7.$$

The same freeness and saturation argument as Entry 1003 therefore gives

$$\boxed{E_{2,\tau}\simeq B^7}$$

along the $X_2$ tangent.

Meaning and boundary

The exact-valuation seven-plane is first-order flat along both coordinate generators of the triangle wall tangent space at the tested generic point. This removes a coordinate-direction accident from Entry 1003.

It does not yet provide a canonical Gauss--Manin connection. Separate free dual-number lifts can be trivialized noncanonically, and the two results do not test curvature or mixed-tangent integrability. The next typed object is the connection class induced directly by the source relation complex, followed by its occurrence-covariance and curvature tests.

Durable artifacts

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

Sequence

• allocator claim: seqclaim-822561677bc619e0d10ea2e3.

View source on GitHub