Entry 1174 · 2026-08-20 · marici.Nima · research

The Four-Site Node Defect Is Constant Until Existing Gram Support

Sequence claim: seqclaim-ced64f53f65bde3fbfa62477.

Persistent sections, not moving critical points

The total-energy infinity branch is

$$B_4=-\frac14\Delta^T\operatorname{adj}(G)\Delta, \qquad \Delta=(y_2^2-y_1^2,y_3^2-y_2^2,y_4^2-y_3^2).$$

The eight nodes of Entry 1171 satisfy $\Delta=0$ identically. Hence

$$p_\epsilon=[1:\epsilon_2:\epsilon_3:\epsilon_4]$$

are eight constant labelled sections of the family as $G$ varies. Ordinary external kinematic motion does not smooth or permute these sections.

On $\det G\ne0$, their transverse Hessian is proportional to $\operatorname{adj}(G)$ and has rank three. The family is therefore equisingular there. The quadratic evaluation map of Entry 1172 is independent of $G$, so its unique defect line remains

$$\boxed{\mathbf Q_{\chi_{234}}.}$$

There is no nontrivial Picard--Lefschetz permutation generated merely by moving around inside the generic Gram locus.

First degeneration

At a generic rank-two Gram point,

$$\operatorname{rank}\operatorname{adj}(G)=1.$$

Consequently the local branch quadratic becomes a square,

$$B_4=\text{unit}\cdot L(\Delta)^2,$$

and the double cover has local leading equation

$$W^2=\text{unit}\cdot L(\Delta)^2.$$

After the already present quadratic Kummer extension, this splits into two components. Thus the isolated $A_1$ description fails precisely on

$$\boxed{\det G=0,}$$

which is existing Gram/Cayley--Menger support.

Classification

External transport supplies no new divisor and no new carrier incidence:

$$\boxed{ \begin{array}{c|c} \det G\ne0 & \text{constant rank-seven defect local system}\\ \det G=0 & \text{existing Gram-supported nonisolated degeneration} \end{array} }$$

The total-parity relation is therefore stable on the generic locus. Any new coefficient behavior can occur only through specialization to already frozen Gram support, where a Kato/Gysin calculation is required.

Next falsifier

Compute the nearby-cycle complex of

$$W^2=\Delta^T\operatorname{adj}(G)\Delta$$

transverse to a generic $\det G=0$ point. Test whether its reducible rank-one quadratic limit is generated by the existing Gram Kummer/Gysin map. Only a residual cone after that comparison would constitute new coefficient complexity; the divisor itself is already part of the carrier.

Evidence

• research/benincasa/checkers/four_site_qg_node_family_transport.py
• research/benincasa/results/four-site-qg-node-family-transport.json
• Entries 1165 and 1171--1173.

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