Entry 1194 · 2026-08-20 · marici.Nima · research

The Existing Node Kato Cells Do Not Meet the Residual Odd Cech Row

Sequence claim: seqclaim-f3863487e1281579e443758f.

Tested conjecture

Entry 1192's eight deck-odd classes occur only in terms that also contain a fourfold concurrence at a branch node. The nearest predeclared mechanism is the local (A_1) Kato/vanishing-cycle cell of Entries 1178--1181.

The finite test is whether its oriented fourfold boundary defines a nonzero row on the deck-odd triple basis and kills the residual (H^2_-).

Exact nonincidence

For every source branch quadruple, all four of its triple faces are themselves ramification triples. A ramification point has no deck-odd (H^0) generator. Consequently the only variance-compatible Kato row is identically zero:

$$\boxed{d_{\rm Kato}:C^2_-\longrightarrow V_{A_1}=0.}$$

Appending every typed branch-node row therefore leaves

$$\boxed{\dim H^2_-=8.}$$

The augmented matrices still satisfy (d_2d_1=0) termwise.

Meaning

Entry 1181's node-local cone remains acyclic in its own occurrence-resolved support category. It cannot be used to kill Entry 1192's classes: that would transport a node vanishing object across a missing deck-odd restriction map.

This negative result preserves H2. The eight classes remain sector-specific coefficient candidates on the existing carrier; they are not evidence for a new carrier stratum.

Next falsifier

The next admissible mechanism is horizontal rather than node-local: derive the Gauss--Manin connection on the two (\mathbf Q[C_4]) residual modules, including their extension with the smooth elliptic-pair row. Test whether the rank-eight subquotient is horizontal. If it is not, compute its minimal flat closure; if it is, factor its intrinsic singular support before any physical-chain interpretation.

Artifact

• research/benincasa/checkers/four_site_qg_node_kato_completion.py
• research/benincasa/results/four-site-qg-node-kato-completion.json

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