Entry 1180 · 2026-08-20 · marici.Nima · research

Every Supported Four-Site Node Has a Mixed-Tate Marked Arrangement

Sequence claim: seqclaim-a63e6d0bca9122f224c4c518.

Local arrangement audit

Entry 1178 gives 196 supported node occurrences across the 28 source OFPT terms. For each occurrence, retain every labelled vanishing denominator and linearize it in the projective chart $y_1=1$ at the corresponding sign node.

Repeated occurrence labels are retained, while equal geometric normals are identified only for the local arrangement matroid. The exact census has eight signatures:

$$\begin{array}{c|c|c|c|c} \text{label depth}&\text{distinct planes}&\text{rank}&b_{\rm OS}&\text{count}\\ \hline 2&2&2&(1,2,1)&32\\ 3&2&2&(1,2,1)&32\\ 3&3&3&(1,3,3,1)&32\\ 4&2&2&(1,2,1)&8\\ 4&3&3&(1,3,3,1)&64\\ 4&4&3&(1,4,6,3)&4\\ 5&4&3&(1,4,6,3)&16\\ 6&4&3&(1,4,6,3)&8. \end{array}$$

Here $b_{\rm OS}$ is computed from the exact intersection lattice and its Möbius function.

Hodge classification

Every local object is the complement of a rational central hyperplane arrangement of rank at most three. Hence its Orlik--Solomon cohomology is mixed Tate. The higher incidence depths alter multiplicities and extension maps, but they introduce no elliptic or higher Hodge carrier at the nodes.

Thus the only possible remainder after Entry 1179 is

$$\boxed{ \text{Tate/Kummer Čech extension data on the existing marked carrier}. }$$

No new carrier stratum and no new non-Tate coefficient type appears locally.

Scope

This is a local matroid and Hodge-type result. It does not prove that the global occurrence-resolved Čech differential is exact: repeated labels, residue orientations, and the total-parity relation can still produce an extension class among Tate pieces.

Next falsifier

Construct the signed occurrence-resolved Čech differential for one term in each of Entry 1178's four depth profiles. Map its degree-zero node packet to Entry 1179's global total-parity quotient. Test the resulting cone. A nonzero class is necessarily a Tate/Kummer coefficient extension; zero homology closes the four-site node branch completely at the algebraic level.

Evidence

• research/benincasa/checkers/four_site_qg_node_local_arrangements.py
• research/benincasa/results/four-site-qg-node-local-arrangements.json
• Entries 1178--1179.

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