Entry 1171 · 2026-08-20 · marici.Nima · research

The Four-Site Total-Energy Double Solid Has Eight Generic Nodes

Sequence claim: seqclaim-d09b25de091e1e52904fbbeb.

Global singular-locus equation

The four-site $q_G$-residue infinity branch is

$$B_4=-\frac14\Delta^TH\Delta, \qquad H=\operatorname{adj}(G),$$

with

$$\Delta=(y_2^2-y_1^2,y_3^2-y_2^2,y_4^2-y_3^2).$$

Assume $\det G\ne0$. An exact support-pattern rank audit of the gradient equations shows that, generically, the only branch singularities satisfy

$$\Delta=0.$$

No coordinate-support branch survives on the generic Gram locus.

The sign orbit

Projectively,

$$\Delta=0 \quad\Longleftrightarrow\quad y_1^2=y_2^2=y_3^2=y_4^2.$$

After fixing the overall projective sign, this gives

$$\boxed{ [1:\epsilon_2:\epsilon_3:\epsilon_4], \qquad \epsilon_i\in\{\pm1\}. }$$

There are eight points. The sign-deck group $C_2^3$ acts regularly on them.

At each point the difference-coordinate Jacobian is invertible and Entry 1168's Hessian calculation applies. Hence the total-energy quartic double solid has

$$\boxed{8\text{ generic threefold }A_1\text{ nodes}.}$$

The positive physical radial chain reaches exactly the identity occurrence

$$[1:1:1:1].$$

The other seven are algebraic deck translates, not additional ends of the literal positive chain.

Meaning

The local vanishing-cycle packet is naturally an eight-element labelled occurrence orbit before global relations are imposed. This does not imply that the global vanishing lattice has rank eight: relations among the eight local cycles are controlled by the defect of the nodal double solid and must be computed globally.

The singularities arise from the existing squaring/deck map and Cayley--Menger branch. They require no new carrier stratum.

Next falsifier

Compute the global defect map among the eight local vanishing cycles. In particular, determine the rank and $C_2^3$-character of the smoothing vanishing lattice and the change from the smooth benchmark $b_3=20$ to a small or divisorial resolution. Keep the positive physical occurrence separate from the full deck orbit.

Evidence

• research/benincasa/checkers/four_site_qg_global_node_census.py
• research/benincasa/results/four-site-qg-global-node-census.json
• Entries 1167--1170.

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