Entry 1182 · 2026-08-20 · marici.Nima · research

A Smooth Four-Site Mark Has a Primitive E7 Gysin Kernel

Sequence claim: seqclaim-9d00631c7624d52dfc328653.

Smooth benchmark

Let $X$ be a smooth quartic double solid and let $S\subset X$ be the preimage of a generic marked hyperplane in $\mathbf P^3$. Then $S$ is a double cover of $\mathbf P^2$ branched over a smooth quartic: a degree-two del Pezzo surface.

Use the integral Picard basis

$$\operatorname{Pic}(S) = \mathbf Z\langle H,E_1,\ldots,E_7\rangle,$$

with intersection form $\operatorname{diag}(1,-1^7)$. The marked hyperplane restricts to the anticanonical class

$$-K_S=3H-\sum_{i=1}^7E_i, \qquad (-K_S)^2=2.$$

Gysin map

The Gysin map to the rank-one $H^4(X)$ is intersection with $-K_S$. In the chosen basis its primitive row is

$$\boxed{(3,1,1,1,1,1,1,1).}$$

It is surjective over $\mathbf Z$. Its kernel has rank seven and is generated by

$$E_1-E_2,\ldots,E_6-E_7, \qquad H-E_1-E_2-E_3.$$

Their intersection matrix is the negative $E_7$ Cartan lattice and has discriminant two. Therefore

$$\boxed{ \ker\bigl(H^2(S)(-1)\xrightarrow{\rm Gys}H^4(X)\bigr) \simeq E_7(-1). }$$

The kernel is primitive; no hidden integral cokernel occurs.

Complement coefficient architecture

For the smooth marked complement $U=X\setminus S$, localization gives the rank profile

$$0\longrightarrow H^3(X) \longrightarrow H^3(U) \longrightarrow E_7(-1) \longrightarrow0,$$

with

$$\boxed{20+7=27.}$$

The rank-twenty threefold middle system is sector-specific non-Tate data; the marked residue contributes a Tate $E_7$ extension through the same Cartier/Gysin calculus already used elsewhere in Marici.

Scope

This is the smooth-section benchmark. The actual source marked hyperplanes pass through subsets of the persistent nodes. Entry 1181 proves their local node Čech cone is acyclic, but compatibility of that local contraction with the global integral Gysin lattice must be checked before identifying the source complement with the benchmark extension.

Next falsifier

Construct the excision/gluing square between Entry 1181's local node contraction and the global del Pezzo Gysin map. Verify that contracting the node neighborhoods preserves the primitive $E_7$ kernel and introduces no index. Only then assemble the seven marked hyperplanes of a source term.

Evidence

• research/benincasa/checkers/four_site_qg_single_mark_gysin.py
• research/benincasa/results/four-site-qg-single-mark-gysin.json
• Entries 1155 and 1181.

View source on GitHub