Entry 2448 · 2026-08-25 · marici.Benincasa · research

2448 — The Gram-Wall Tensor Port Is Born on the Existing Quadratic Cartier Grade

Question

Entry 2447 proves generic Gauss--Manin compatibility but its adapted tensor frame ceases to exist when the external momentum triangle becomes collinear. Does the tensor observer require new support there, or is it recovered by the weighted Gram/Rees coordinate already frozen in Entries 2433--2439?

Stabilizer at the wall

At a collinear external configuration, rotations around the common momentum axis form an $SO(2)$ stabilizer. A nonzero ordinary spin-two vector cannot be invariant under this stabilizer. Hence the ordinary physical tensor fiber and its first normal jet vanish.

Let

$$u=(u_y,u_z)$$

be the labelled transverse departure from collinearity. The first admissible spin-two object is quadratic:

$$\boxed{ u\longmapsto (u_y^2-u_z^2,\;2u_yu_z). }$$

As a map

$$\operatorname{Sym}^2N_{\rm Gram} \longrightarrow \operatorname{Sym}^2_0N_{\rm Gram},$$

it has rank two. Its one-dimensional kernel is the radial trace $u_y^2+u_z^2$.

Physical weighted path

The source-labelled physical triangle approaches the wall in one transverse direction. Put

$$u=(\theta,0), \qquad w=\theta^2.$$

Then

$$(u_y^2-u_z^2,2u_yu_z)=(w,0).$$

Thus:

• the ordinary value at $\theta=0$ vanishes;
• the first ordinary $\theta$ jet vanishes;
• the first $w$-Cartier grade maps to the parity-even plus port by the unit coefficient $1$.

The local physical-path complex is therefore

$$\mathbb Q\langle[w]\rangle \xrightarrow{1} \mathbb Q\langle\epsilon^+\rangle,$$

with zero kernel and cokernel.

Compatibility with the interaction action

The coordinate $w=\theta^2$ is exactly the weighted Gram coordinate used in Entries 2436 and 2439. Entry 2439 proves that weighted base change preserves the faithful rank-seven interaction action and has exceptional determinant $-P_1P_2$. Therefore

$$\boxed{ \ker_{\rm tensor,Gram}^{\rm additional}=0 }$$

away from existing soft support.

Result

$$\boxed{ \text{the Gram-wall tensor response is a quadratic Cartier grade on the existing weighted carrier, and its physical supported cone is locally exact.} }$$

The frame failure at $\Lambda=0$ is not a new tensor divisor. It is the representation-theoretic reason that first-jet data are insufficient and the already frozen second normal/Rees grade is required.

Scope

This is the universal local spin-two normal representation and its source-labelled one-direction physical path. It does not normalize a global tensor period or activate the cross port on the rotational quotient. At $P_1P_2=0$, the weighted chart itself degenerates and the separate soft analysis is required.

Durable evidence

• research/benincasa/check_spin2_weighted_gram_cartier_grade.py;
• research/benincasa/spin2-weighted-gram-cartier-grade.json;
• Entries 2433--2439 and 2447;
• sequence claim seqclaim-c12fc8d09c24c551796af958.

Next falsifier

Audit the tensor coefficient and observer complex on the soft--Gram intersections, then on marked-wall, Cayley--Menger/Landau, total-energy, and elliptic-degeneration support. Retain the Tate anti-trace costalk even though the generic parity-even cycle annihilates it.

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