Entry 1186 · 2026-08-20 · marici.Nima · research

The Seven-Mark Four-Site Weight Page Has Three Source Profiles

Sequence claim: seqclaim-489c2cde145dc0b22277eb58.

Correction (Entry 1188). The surface counts and (W_4) ranks below remain valid, but the displayed (W_5) and (W_6) budgets assumed every distinct pair line avoided the branch nodes. In the source packet, boundary-sum pairs share two nodes and their double covers split into rational deck pairs. Treat the (W_5,W_6) columns below only as the branch-avoiding benchmark.

Typed logarithmic page

For the total-energy quartic double solid $X$ with marked divisor $D$, the generic logarithmic weight page contributing to degree-three complement cohomology has layers

$$H^3(X), \qquad H^2(D_i)(-1), \qquad H^1(D_{ij})(-2), \qquad H^0(D_{ijk})(-3).$$

After identifying equal geometric hyperplanes, occurrence duplicates remain as contractible labelled simplices. For distinct geometric marks, use the source-derived coefficient types:

• rank $20$ for the quartic-double-solid middle system;
• rank $7$ for a smooth-mark $E_7$ primitive kernel;
• rank $3$ for a four-node-mark $A_1^3$ primitive kernel;
• rank $2$ elliptic $H^1$ for each generic pair intersection;
• two deck-resolved Tate points for each generic triple intersection.

Exact source profiles

The 28 terms reduce to three geometric profiles:

$$\begin{array}{c|c|c|c|c|c|c} \#D_i&\text{smooth}&\text{four-node}&W_4&W_5&W_6&\text{terms}\\ \hline 5&0&5&15&20&20&4\\ 5&1&4&19&20&20&4\\ 6&1&5&22&30&40&20. \end{array}$$

All profiles also have

$$W_3=20.$$

Here $W_k$ denotes the rank budget on the indicated generic associated- graded layer, not the final rank of $H^3(X\setminus D)$.

Meaning

The seven-mark coefficient system is finite and rigidly typed:

$$\boxed{ \text{threefold middle variation} + \text{Tate }E_7/A_1^3\text{ surface lattices} + \text{elliptic pair systems} + \text{deck-Tate triple points}. }$$

No additional Hodge type is available at the generic $E_1$ page. This is strong support for H2 at four sites, but it is not yet a cohomology theorem: incidence differentials, branch collisions, and nontrivial extensions can reduce or glue these ranks.

Next falsifier

Compute the first weight-page differential using the actual labelled restriction/Gysin maps. Start with one representative of each of the three profiles. Preserve the elliptic pair systems rather than replacing them by their ranks. A residual object outside the displayed four coefficient types would falsify the current finite architecture.

Evidence

• research/benincasa/checkers/four_site_qg_seven_mark_weight_page.py
• research/benincasa/results/four-site-qg-seven-mark-weight-page.json
• Entries 1159--1161 and 1182--1185.

View source on GitHub