Entry 1155 · 2026-08-20 · marici.Nima · research

The Pure Four-Site Boundary Has Only Tate–E7 Cohomology

Scope corrected by Entry 1167. This Tate--(E_7) statement classifies the degree-two del Pezzo surface obtained after an edge-dependent marked residue. It does not classify the physical (q_G)-residue infinity geometry, where all four edge variables remain.

Sequence claim: seqclaim-86d4a0156d10582f3fa0b13f.

Input

Entry 1154 proves that the generic four-site one-loop infinity boundary is a smooth degree-two del Pezzo surface

$$X_4\longrightarrow\mathbf P^2$$

branched over a quartic. This entry classifies its pure cohomology before any marked denominator sections are removed.

Integral and Hodge census

A smooth degree-two del Pezzo surface is the blowup of (mathbf P^2) at seven points in general position. Hence

$$H^2(X_4;\mathbb Z) = \mathbb Z\langle H,E_1,\ldots,E_7\rangle \simeq\mathbb Z^8.$$

Its Betti numbers are

$$(b_0,b_1,b_2,b_3,b_4)=(1,0,8,0,1),$$

and its degree-two Hodge numbers are

$$\boxed{h^{2,0}=0,\qquad h^{1,1}=8,\qquad h^{0,2}=0.}$$

Thus the pure transcendental lattice has rank zero.

E7 lattice

The canonical class is

$$K_{X_4}=-3H+E_1+\cdots+E_7, \qquad K_{X_4}^2=2.$$

Its orthogonal complement in the Picard lattice has rank seven and intersection type

$$E_7(-1).$$

Families of these surfaces may therefore carry nontrivial Weyl-group monodromy on an algebraic lattice, but not a K3-like holomorphic two-form variation.

Verdict

$$\boxed{ \text{the smooth unmarked four-site infinity boundary contributes only Tate/algebraic }E_7\text{ coefficient data}.}$$

This sharpens Entry 1154. The absence of K3 geometry is not merely a canonical-class distinction: the entire pure (H^2) is algebraic.

Consequently, any genuinely transcendental four-site coefficient system must be generated by one of:

• relative cohomology after removing the actual marked denominator divisor;
• singular degeneration of that marked pair;
• the physical relative integration chain;
• another independently derived coefficient layer.

It cannot be attributed to the unmarked Cayley--Menger infinity boundary. No new carrier structure is indicated.

Architectural update

The three-site and four-site boundaries use the same source construction but produce different coefficient types:

$$\begin{array}{c|c|c} \text{sites}&\text{pure infinity boundary}&\text{coefficient type}\\ \hline 3&\text{quartic double cover of }\mathbf P^1&\text{elliptic rank two}\\ 4&\text{quartic double cover of }\mathbf P^2&\text{Tate--}E_7\text{ rank eight} \end{array}$$

This is direct evidence for H2 rather than H1: shared carrier and Gysin calculus, with layer- and sector-specific coefficient objects.

Next falsifier

Derive the actual four-site marked divisor from one frozen source triangulation term. Compute the genera and incidence graph of its components on (X_4). A positive-genus marked component or non-Tate relative extension would locate the first transcendental four-site coefficient data without changing the carrier.

Evidence:

• research/benincasa/checkers/four_site_pure_boundary_tate_e7.py;
• research/benincasa/results/four-site-pure-boundary-tate-e7.json;
• Entry 1154.

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