Entry 1168 · 2026-08-20 · marici.Nima · research

The Physical Four-Site Total-Energy Point Is a Threefold A1 Node

Sequence claim: seqclaim-81fcc26c9ccfdba43932b895.

Frozen local problem

Entry 1167 identifies the physical $q_G$-residue infinity coefficient as a quartic double solid. Entry 1166 fixes the point reached by radial loop infinity:

$$p_\infty=[1:1:1:1].$$

Use the chart $y_4=1$, with $u_i=y_i-1$ for $i=1,2,3$. For

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

the linearization at $p_\infty$ is

$$J= \begin{pmatrix} -2&2&0\\ 0&-2&2\\ 0&0&-2 \end{pmatrix}, \qquad \det J=-8.$$

Thus the difference coordinates are an étale coordinate system at the physical projective point.

Exact quadratic form

Writing $H=\operatorname{adj}(G)$, the leading branch term is

$$B_2=-\frac14u^TJ^THJu.$$

Since $\det(\operatorname{adj}G)=\det(G)^2$, its Hessian determinant is

$$\boxed{ \det\operatorname{Hess}(B_2)=-8\det(G)^2. }$$

Away from the existing Gram support $\det(G)=0$, the quadratic form is nondegenerate. The branch surface has an ordinary double point, and the double solid has the étale/Morse local form

$$\boxed{ W^2=u_1^2+u_2^2+u_3^2. }$$

This is an $A_1$ hypersurface singularity of complex dimension three.

Vanishing-cycle data

The Milnor number is

$$\mu=1.$$

Its vanishing homology is a rank-one $H_3$, topologically generated by an $S^3$. The local monodromy acts as $+1$ on the vanishing line itself. This does not make the global degeneration trivial: Picard--Lefschetz transport may act by a nontrivial transvection on the ambient $H_3$ local system.

Narrow result

The physically reached four-site total-energy singularity carries one canonical local vanishing-cycle direction. It is generated by the existing total-energy and Cayley--Menger geometry; no new carrier divisor is needed. Unlike the forced marked nodes of Entry 1166, this singularity lies on the literal physical infinity closure.

The calculation establishes the local coefficient object, not its global embedding or the numerical pairing with the physical chain.

Next falsifier

Compute the strict transform of the source radial Cayley--Menger chain in this ordinary blowup, including the measure valuation and orientation. Determine whether its exceptional boundary is the vanishing $S^3$, zero, or a proper multiple. Then compute the resulting Picard--Lefschetz extension inside the global quartic-double-solid $H_3$.

Evidence

• research/benincasa/checkers/four_site_qg_physical_node.py
• research/benincasa/results/four-site-qg-physical-node.json
• Entries 1166--1167.

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