Entry 1187 · 2026-08-20 · marici.Nima · research

Only the Deck-Point Layer Has a Nonzero Generic First Differential

Sequence claim: seqclaim-7cf5bda90e6def3cb819919d.

Correction (Entry 1188). This calculation is valid for a branch-avoiding arrangement of five or six smooth connected pair curves. It is not the first differential of the actual source packet: pairs of boundary-sum marks share two branch nodes and split into two rational components. Their extra (H^0) changes the deck-point incidence matrix, while their (H^1) vanishes.

Typed vanishing statements

On the generic normal-crossing locus, Entry 1186's first logarithmic differential is largely forced.

For every marked surface, the pair curve is anticanonical. Hence every primitive $E_7$ or $A_1^3$ class restricts with degree zero:

$$E_7/A_1^3\longrightarrow H^2(C_{ij}) \quad\text{vanishes on the primitive kernel}.$$

Moreover,

$$H^1(D_i)=0$$

for degree-two del Pezzo surfaces and their rational $A_1$ contractions, while

$$H^1(P_{ijk})=0$$

for triple points. Therefore the elliptic pair systems receive and emit no generic first differential.

The only nonzero map relevant to the displayed weight page is

$$\bigoplus_{i<j}H^0(C_{ij}) \longrightarrow \bigoplus_{i<j<k}H^0(P_{ijk,+}\sqcup P_{ijk,-}).$$

Exact deck-point differential

Each connected pair curve maps diagonally to the two deck points:

$$1\longmapsto(1,1),$$

with the standard signed triangle boundary. For $m$ distinct geometric marks, its rank is

$$\binom{m-1}{2}.$$

The deck-anti-invariant line at every triple is untouched. The invariant cokernel has rank $\binom{m-1}{3}$. Thus

$$\operatorname{gr}_6 E_2 \simeq \mathbf Q_-^{\binom m3} \oplus \mathbf Q_+^{\binom{m-1}{3}}.$$

Explicitly:

$$\begin{array}{c|c|c|c|c} m&\operatorname{rank}d_1&\dim(-)&\dim(+)&\dim W_6E_2\\ \hline 5&6&10&4&14\\ 6&10&20&10&30. \end{array}$$

Generic E2 profiles

Entry 1186's three source profiles become

$$\boxed{ \begin{array}{c|c} \text{term count}&(W_3,W_4,W_5,W_6)_{E_2}\\ \hline 4&(20,15,20,14)\\ 4&(20,19,20,14)\\ 20&(20,22,30,30). \end{array} }$$

No new coefficient type appears: the surviving pieces remain threefold, Tate-lattice, elliptic, and deck-Tate data.

Scope and next falsifier

This is the generic branch-avoiding locus. It does not include pair or triple incidences lying on the quartic branch, where deck points coalesce and nearby-cycle maps replace the diagonal restriction.

The next finite falsifier is therefore the branch-collision support of the pair and triple strata in the 28 source terms. Determine whether each collision is generated by the existing node/Gram Kato maps or creates a new coefficient extension. No generic rank census needs to be repeated.

Evidence

• research/benincasa/checkers/four_site_qg_seven_mark_first_differential.py
• research/benincasa/results/four-site-qg-seven-mark-first-differential.json
• Entries 1182--1186.

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