Entry 1190 · 2026-08-20 · marici.Nima · research

The Top Four-Site Tate Layer Has a Forty-Dimensional Kummer Sector

Sequence claim: seqclaim-304f18a26d053d87bf97b2cb.

Supersession notice. The forty-dimensional anti-invariant cokernel is an intermediate pair-to-triple quotient, not full Čech cohomology. Entry 1192 includes the fourfold-concurrence boundary and reduces this sector.

Deck decomposition

Entry 1189's component-resolved differential commutes with the double-cover deck involution. It therefore splits into two exact matrices.

The invariant matrix is the ordinary signed incidence map from all pair curves to all triple points.

The anti-invariant matrix has:

• one column for every split rational pair;
• one row for every off-branch triple;
• no row at a ramification-node triple.

The exact cokernels are

$$\boxed{ \begin{array}{c|c|c|c} \text{geometric marks}&W_{6,+}&W_{6,-}&\text{term count}\\ \hline 5&4&0&8\\ 6&10&2&20. \end{array} }$$

Thus the Kummer character occurs only in the six-mark source profile.

Cyclic assembly

Entry 1159's 28 terms form seven free $C_4$-orbits. The eight five-mark terms form two orbits; the twenty six-mark terms form five. Therefore every representative eigencokernel assembles as a regular occurrence module:

$$\mathbf Q[C_4]\otimes W_{6,\pm}^{\rm rep}.$$

Summing all terms gives

$$\boxed{ W_6^{\rm all\ terms} = \mathbf Q_+^{232} \oplus \mathbf Q_-^{40}. }$$

The corresponding $C_4$ characters are

$$\chi_+=(232,0,0,0), \qquad \chi_-=(40,0,0,0).$$

The vanishing nonidentity traces are a consequence of free labelled occurrence transport, not an unlabelled symmetry quotient.

Architectural meaning

The top source coefficient layer is now completely typed at generic kinematics:

$$\boxed{ \text{Tate invariant incidence cokernel} + \text{rank-40 quadratic Kummer packet}. }$$

This is coefficient data over the existing marked/deck carrier. It is not a new carrier incidence and does not enlarge the list of H2 coefficient types.

Next falsifier

Test Gauss--Manin extension between the surviving elliptic $W_5$ systems and $W_6$. Since the Kummer sector exists only in six-mark terms, first compute whether its two local generators per term are horizontal quotients or receive a nonzero connecting morphism from the five elliptic pair systems. A nontrivial map is allowed as sector-specific extension data; a new singular divisor is not.

Evidence

• research/benincasa/checkers/four_site_qg_w6_deck_cyclic.py
• research/benincasa/results/four-site-qg-w6-deck-cyclic.json
• Entries 1159 and 1188--1189.

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