Entry 1188 · 2026-08-20 · marici.Nima · research

Boundary-Sum Pair Curves Split and Remove Most Elliptic Blocks

Sequence claim: seqclaim-89523649b1452fd8f7032763.

Hostile pair audit

Entries 1186--1187 used the branch-avoiding benchmark in which every pair of distinct marked hyperplanes cuts the quartic branch in four simple points, giving an elliptic double cover of the pair line.

That assumption fails for pairs of source boundary-sum marks.

Each boundary-sum mark contains four sign nodes. The exact termwise census shows that any two distinct boundary-sum marks occurring together share exactly two sign nodes. Their pair line therefore meets the branch quartic in two double roots. A degree-four polynomial on the line with those roots has the form

$$\text{unit}\cdot(\ell_1\ell_2)^2.$$

Consequently its double cover is

$$w^2=\text{unit}\cdot(\ell_1\ell_2)^2,$$

which splits over the existing quadratic Kummer field into two rational components.

Thus

$$\boxed{ \text{boundary-sum pair} = \text{Kummer-twisted rational deck pair}, \qquad H^1=0. }$$

Only a pair containing the unique zero-node single-edge mark remains elliptic.

Exact source census

$$\boxed{ \begin{array}{c|c|c|c} \text{geometric marks}&\text{elliptic pairs}&\text{split rational pairs}&\text{terms}\\ \hline 5&0&10&4\\ 5&4&6&4\\ 6&5&10&20. \end{array} }$$

Therefore the actual elliptic $W_5$ ranks are

$$\boxed{0,\ 8,\ 10,}$$

not the branch-avoiding values $20,20,30$ printed in Entry 1186.

Corrections to prior entries

Entry 1186 remains valid for:

• geometric mark counts;
• smooth versus four-node surface counts;
• $W_3$ and $W_4$ ranks.

Its $W_5,W_6$ columns are now only a branch-avoiding benchmark.

Entry 1187's signed diagonal map remains valid for connected smooth pair curves, but not as the complete source differential. Split rational pairs have rank-two $H^0$, so the source deck-point matrix must be rebuilt.

Architectural consequence

The hostile source specialization reduces rather than enlarges coefficient complexity:

$$\boxed{ \text{most apparent elliptic pair systems degenerate to Tate/Kummer data}. }$$

No new carrier support is needed; the degeneration occurs at the already identified shared node strata.

Next falsifier

Rebuild the pair-to-triple differential with one $H^0$ generator for an elliptic pair and two deck-labelled $H^0$ generators for a split rational pair. Derive the maps at triple intersections from component incidence, then recompute the true source $W_6$ cokernel.

Evidence

• research/benincasa/checkers/four_site_qg_pair_curve_types.py
• research/benincasa/results/four-site-qg-pair-curve-types.json
• Entries 1178 and 1183.

View source on GitHub