Entry 2288 · 2026-08-24 · marici.Nima · research

2288 — The Gram-Wall Transfer Has One Cartier Layer and the Completed Nearby Cone Is Exact

Local normal form

Let

$$c=q\cdot\widehat n$$

be the resolved normal to the existing Gram wall. Entries 2276 and 2281 show that the finite-$q$ tensor determinant is

$$\det T=c\,U, \qquad U|_{c=0}\ne0,$$

throughout the generic positive-energy wall. Entry 2283 shows that $T|_{c=0}$ has rank exactly two, so a $2\times2$ minor is a local unit.

Over the local DVR $\mathbb Q[[c]]$, these two facts force Smith form

$$\boxed{ T\sim\operatorname{diag}(1,1,c). }$$

Thus the tensor-only comparison cone has exactly one Cartier-supported coefficient:

$$H^0\operatorname{Cone}(T) \simeq \mathbb Q[[c]]/(c).$$

There is no higher Cartier length, hidden first-normal extension, or extra nearby rank at this wall.

Completion by the direct score port

The direct scalar-plus-quadrupole Gaussian score matrix has determinant $-6$, a unit in $\mathbb Q[[c]]$. Entry 2283 computes its nonzero restriction to the tensor kernel. Therefore adjoining this port contracts the unique Cartier costalk:

$$\boxed{ H^\bullet\operatorname{Cone}(T\oplus Q)=0 }$$

over the entire local Gram neighborhood, including $c=0$.

Nearby-cycle interpretation

The tensor channel alone has one ordinary linear-normal supported class. It is not a new nearby coefficient object: it is the cokernel of a transport map whose determinant crosses the existing Gram wall simply. The complete admitted score-plus-tensor observer complex is locally exact, so no residual nearby-cycle class survives.

Classification:

$$\boxed{ \text{existing Gram Cartier layer} +\text{recoverable transport projection} \Longrightarrow \text{no completed-observer obstruction}. }$$

Verification

research/benincasa/checkers/gram_wall_smith_nearby_cone.rs verifies a local representative with determinant $2c$, a unit two-minor, the resulting Smith invariants, and the unit direct-score completion.

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