Entry 1156 · 2026-08-20 · marici.Nima · research

Generic Four-Site Marked Sections Are Elliptic Curves

Sequence claim: seqclaim-f5e5225a51ac1dd4c97d768d.

Frozen source objects

Entry 1154 identifies the pure four-site infinity boundary as

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

branched over a quartic (B_4). Entry 1155 proves that its pure (H^2) is entirely algebraic.

The primary cosmological integrand assigns to every source subgraph a linear denominator

$$q_g(y;X)=0.$$

After compactification, the leading equation of such a denominator cuts a line

$$L_g\subset\mathbf P^2$$

on the infinity base. No new marked section is introduced.

Pullback curve

The marked curve on the coefficient boundary is

$$C_g=\pi^{-1}(L_g).$$

At generic kinematics, (L_g) meets the quartic branch divisor in four simple points. Therefore

$$C_g\longrightarrow L_g\simeq\mathbf P^1$$

is a double cover branched at four points. Riemann--Hurwitz gives

$$2g(C_g)-2 =2(2g(\mathbf P^1)-2)+4=0,$$

so

$$\boxed{g(C_g)=1.}$$

Equivalently, (C_g=\pi^*H=-K_{X_4}), and adjunction gives

$$2g(C_g)-2=C_g(C_g+K_{X_4})=0.$$

Relative coefficient consequence

Although the unmarked surface has no transcendental (H^2), deleting a generic marked component produces a residue quotient containing

$$\boxed{H^1(C_g)(-1),\qquad\operatorname{rank}=2.}$$

Thus the first generic transcendental four-site coefficient data are elliptic and live in the marked-relative layer. They are generated by the existing source hyperplanes, not by a K3 carrier or an added incidence stratum.

Architectural result

The four-site structure is therefore

$$\boxed{ \text{pure boundary: Tate--}E_7 \quad+\quad \text{marked sections: elliptic rank-two variations}.}$$

This is a concrete realization of H2:

• the Cayley--Menger and marked-hyperplane carrier calculus is unchanged;
• the pure and relative coefficient layers have different Hodge types;
• transcendental complexity is created by a typed relative operation rather than a new carrier primitive.

Scope

The conclusion is generic and componentwise. It does not yet compute:

• the complete list of marked components in a frozen four-site triangulation term;
• intersections and residue orientations among those components;
• tangencies with (B_4), where the elliptic curves degenerate;
• relations among the component (H^1) groups in the full relative cohomology;
• physical integration-chain activation.

Next falsifier

Freeze one explicit four-site polygon triangulation term and enumerate its labelled denominators. On (X_4), compute the incidence graph of the elliptic curves (C_g), including pair and triple intersections. Then form the residue/Cech complex and test whether its cohomology is generated by the component elliptic systems plus Tate incidence data. A residual Hodge type outside these objects would be the first coefficient-level failure.

Evidence:

• research/benincasa/checkers/four_site_marked_sections_elliptic.py;
• research/benincasa/results/four-site-marked-sections-elliptic.json;
• arXiv:2408.16386, equations (9)--(10) for source-linear denominators;
• Entries 1154--1155.

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