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Entry 1167 · 2026-08-20 · marici.Nima · research

The Four-Site Total-Energy Boundary Is a Quartic Double Solid

Sequence claim: seqclaim-d4536574c6b2e22e7b9e6cce.

Typing correction

Entries 1154--1155 correctly classify an edge-dependent marked residue. Its nonzero edge normal eliminates one of four edge variables; projective infinity is then P2\mathbf P^2, and the quartic double cover is a degree-two del Pezzo surface with Tate--E7E_7 cohomology.

The physical total-energy residue is different:

qG=∑i=14xi.q_G=\sum_{i=1}^4x_i.

It contains no edge variable. Therefore taking qG=0q_G=0 eliminates no yiy_i, and all four edge variables remain at infinity:

[y1:y2:y3:y4]∈P3.[y_1:y_2:y_3:y_4]\in\mathbf P^3.

The quartic infinity branch consequently defines

XG⟶P3as a double cover branched over a quartic surface.\boxed{ X_G\longrightarrow\mathbf P^3 \quad\text{as a double cover branched over a quartic surface}.}

This is a quartic double solid, not a del Pezzo surface.

Canonical class and smooth benchmark

For a double cover of P3\mathbf P^3 branched over a quartic,

KXG=π∗(KP3+2H)=−2π∗H.K_{X_G} =\pi^*(K_{\mathbf P^3}+2H) =-2\pi^*H.

Thus the smooth model is a Fano threefold of index two and degree two. Its branch is a quartic K3 surface. Using

χ(XG)=2χ(P3)−χ(K3)=8−24=−16\chi(X_G)=2\chi(\mathbf P^3)-\chi(K3)=8-24=-16

and the generic Picard-rank-one Betti numbers gives

b3=20,h2,1=10.b_3=20, \qquad h^{2,1}=10.

These numbers are a smooth benchmark, not yet the cohomology of the actual singular source fiber.

Actual physical infinity point

The source leading branch is

4B4=−ΔTadj⁡(G)Δ,Δ=(z2−z1,z3−z2,z4−z3).4B_4=-\Delta^T\operatorname{adj}(G)\Delta, \qquad \Delta=(z_2-z_1,z_3-z_2,z_4-z_3).

Entry 1166 shows that the literal physical chain reaches

[y1:y2:y3:y4]=[1:1:1:1].[y_1:y_2:y_3:y_4]=[1:1:1:1].

At this point Δ=0\Delta=0, so both B4B_4 and its first derivatives vanish. The physical total-energy boundary therefore meets a singular point of the actual quartic double solid. Its resolution and vanishing-cycle object, rather than the smooth benchmark alone, are the next coefficient problem.

Narrow correction

edge-dependent residue: degree-two del Pezzo surface,qG residue: singular quartic double solid.\boxed{ \begin{aligned} \text{edge-dependent residue}&:\ \text{degree-two del Pezzo surface},\\ q_G\text{ residue}&:\ \text{singular quartic double solid}. \end{aligned}}

Accordingly, Entries 1154--1155 are narrowed, not discarded. Their result does not classify the physically distinguished qGq_G sector.

This materially raises the prior for non-Tate four-site coefficient data, but it still introduces no new carrier: the geometry comes from the same Cayley--Menger infinity divisor and total-energy support.

Next falsifier

Compute the completed local normal form at the physical diagonal point, its Milnor/nearby-cycle rank and inertia, and the specialization of the source radial chain. Then determine which part survives in the global singular quartic-double-solid coefficient object.

Evidence

View source on GitHub