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Entry 1183 · 2026-08-20 · marici.Nima · research

Boundary-Sum Marks Are Four-Node Two-Conic Sections

Sequence claim: seqclaim-8a6035531d909274eb452c6f.

Two source-derived section types

Entry 1182's smooth E7E_7 benchmark cannot be applied uniformly to the seven marks of a source OFPT term. Their infinity normals have two types.

Single-edge normal

A spanning-path denominator has leading mark

ye=0.y_e=0.

It misses all eight sign nodes. Generically its plane quartic branch is smooth, so Entry 1182 applies.

Boundary-sum normal

Every connected proper-subgraph denominator has a two-edge boundary normal. For the representative

y1+y2=0,y_1+y_2=0,

one difference coordinate vanishes identically:

Δ1=y22−y12=0.\Delta_1=y_2^2-y_1^2=0.

Set

A=y32−y12,B=y42−y32.A=y_3^2-y_1^2, \qquad B=y_4^2-y_3^2.

The restricted branch quartic is a binary quadratic in A,BA,B:

aA2+2bAB+cB2.aA^2+2bAB+cB^2.

At generic Gram kinematics it factors over its quadratic splitting field into two distinct diagonal conics. Their four intersections are precisely the four sign nodes satisfying ϵ2=−1\epsilon_2=-1.

Thus

boundary-sum mark=degree-two del Pezzo section with four A1 points\boxed{ \text{boundary-sum mark} = \text{degree-two del Pezzo section with four }A_1\text{ points} }

before resolution.

Complete term census

Among the 28 source terms:

zero-node marksfour-node marksterm count0741624.\begin{array}{c|c|c} \text{zero-node marks}&\text{four-node marks}&\text{term count}\\ \hline 0&7&4\\ 1&6&24. \end{array}

Hence most terms contain one smooth-benchmark mark, while four terms contain only four-node two-conic sections.

Consequence

The actual global Gysin packet is not seven independent copies of the smooth E7E_7 kernel. Each boundary-sum section carries an A14A_1^4 contraction inside its resolved del Pezzo lattice, and Entry 1181 supplies the local threefold-node contraction with which it must be glued.

This is still existing carrier geometry:

source marked plane+existing sign-node support+quadratic splitting field.\boxed{ \text{source marked plane} + \text{existing sign-node support} + \text{quadratic splitting field}. }

No new divisor was introduced.

Next falsifier

For one boundary-sum section, identify the four exceptional (−2)(-2)-roots inside the resolved E7E_7 lattice and compute the contraction quotient and its discriminant form. Then compare its local A14A_1^4 boundary map with Entry 1181's occurrence contraction. This determines whether the global Gysin kernel descends primitively or acquires an index.

Evidence

View source on GitHub