Entry 1157 · 2026-08-20 · marici.Nima · research

Pairwise Four-Site Marked Incidence Is a Deck Pair

Sequence claim: seqclaim-a9c134063514e33b11c81372.

Frozen geometry

Entry 1156 associates to every source-linear marked denominator a curve

$$C_g=\pi^{-1}(L_g)\in|-K_{X_4}|,$$

where

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

is the degree-two del Pezzo double cover branched over the quartic $B_4$, and $L_g$ is the corresponding labelled line.

Pair incidence

For two distinct generic marks,

$$C_g\cdot C_h=(-K_{X_4})^2=2.$$

The base lines meet at one point

$$p_{gh}=L_g\cap L_h.$$

If $B_4(p_{gh})\ne0$, its inverse image consists of two reduced points,

$$\pi^{-1}(p_{gh})=\{p_{gh,+},p_{gh,-}\},$$

exchanged by the deck involution. Thus the generic pairwise Cech term is not one unlabelled point: it is the two-occurrence deck permutation module.

If

$$B_4(p_{gh})=0,$$

the two occurrences coalesce into one ramification point with intersection multiplicity two. This is branch-collision support of the existing marked and Cayley--Menger data, not a new incidence generator.

Triple incidence

Write the three labelled line equations as coefficient rows $\ell_g,\ell_h,\ell_k$. Three generic lines in $\mathbf P^2$ have no common point. A triple intersection occurs precisely on

$$\boxed{\det(\ell_g,\ell_h,\ell_k)=0.}$$

Hence triple Cech cells are supported on a source-derived line-concurrency divisor. They are absent at generic kinematics.

Narrow result

The generic marked-incidence layer is therefore

$$\boxed{ \text{elliptic component coefficients} +\text{ deck-resolved Tate pair incidences}. }$$

Its first degenerations are exhausted by two explicit conditions:

$$B_4(L_g\cap L_h)=0$$

for a pair collision, and

$$\det(\ell_g,\ell_h,\ell_k)=0$$

for triple concurrency. Both are compiled from the frozen branch divisor and labelled marked lines. No new four-site carrier primitive is indicated.

Scope and next falsifier

This is an intersection-theoretic generic theorem. It does not determine which labelled lines occur together in one physical four-site integrand term, their residue signs, or relations among their elliptic $H^1$ systems.

The next finite task remains source-specific: freeze one explicit four-site term, enumerate its labelled denominator lines, and evaluate the two conditions above for every pair and triple. The resulting signed incidence packet is the input to the relative residue/Cech complex.

Evidence:

• research/benincasa/checkers/four_site_marked_incidence.py;
• research/benincasa/results/four-site-marked-incidence.json;
• Entries 1154--1156.

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