Entry 1003 · 2026-08-19 · marici.Nima · research

The Exact-Valuation Seven-Plane Is First-Order Flat in One Tangent Direction

From ambient flatness to the intrinsic object

Entry 994 proves that the full length-three Rees cokernel has no first-order tangential obstruction along

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

That ambient statement does not by itself prove flatness of

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

The missing data are the mixed ranks at normal lengths one and two.

Complete truncation census

Over

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

exact syzygy-derivative reduction gives

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

Thus all three truncated cokernels are free over $B$. With 11520 ambient columns per normal grade, their dimensions over the ground field are

$$\dim C_{1,\tau}=2(11520)-12610=10430,$$

$$\dim C_{2,\tau}=4(11520)-25230=20850,$$

$$\dim C_{3,\tau}=6(11520)-37864=31256.$$

Exact-valuation dimension

Because $C_{3,\tau}/\Lambda^jC_{3,\tau}\simeq C_{j,\tau}$,

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

and

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

Since $\Lambda^3=0$,

$$\dim E_{2,\tau} = \dim\operatorname{im}\Lambda -2\dim\operatorname{im}\Lambda^2 =14.$$

The special fiber has dimension seven. Moreover, freeness of the three truncated cokernels makes the corresponding image and kernel sequences free over the local principal ring $B$. The inclusion defining the denominator is therefore saturated, and the exact-valuation quotient is free of rank seven:

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

Meaning

The rank-seven object is no longer merely pointwise stable or unobstructed inside its ambient length-three cokernel. It survives the source-derived first tangential deformation as an intrinsic dual-number-flat subquotient.

This still does not choose a basis or produce a connection matrix. A free first-order deformation admits trivializations, but the canonical Gauss--Manin transport must be derived from the source relations and checked against occurrence symmetry.

Only the $X_1$ tangential direction is proved here. The independent $X_2$ mixed calculation remains in progress.

Next gate

1. replicate all three truncation ranks in the $X_2$ direction;
2. derive the connection class on the free rank-seven deformation without choosing elimination witnesses;
3. test occurrence covariance and compare with the generic rank-seven algebraic kernel.

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-f72b9ab610cc6f716b4e8b99.

View source on GitHub