Entry 1192 · 2026-08-20 · marici.Nima · research

Higher Concurrence Cells Reduce the Four-Site Kummer Sector to Rank Eight

Sequence claim: seqclaim-f0c87d100b20f375b9084880.

Hard-to-vary claim

For the frozen 28-term four-site (q_{\mathcal G})-residue source packet, the complete pair--triple--quadruple (H^0)-Čech complex has deck-odd second cohomology of rank eight:

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

Thus Entry 1190's rank-forty deck-odd pair-to-triple cokernel was not a cohomology group. Fourfold concurrence boundaries kill 32 of those 40 directions. The surviving eight classes require no new carrier stratum.

Missing cells exposed by the tetrahedra

Entry 1191 found alternating tetrahedral vectors in the left cokernel of the pair-to-triple map. That shape is also the signed boundary of a fourfold concurrence cell, so all source subsets of size at least four were audited before retaining the interpretation.

Every term has genuine rank-three fourfold subsets:

$$\begin{array}{c|c|c} \text{geometric marks}&\text{fourfold cells}&\text{term count}\\ \hline 5&1&8\\ 6&2&8\\ 6&3&12. \end{array}$$

There are no nonempty source concurrences of size five or greater.

Deck-resolved full differential

Let (C^1,C^2,C^3) be the (H^0) contributions of pair, triple, and fourfold strata. The maps are the source-labelled signed simplicial coboundaries. In the deck-even sector every nonempty stratum contributes one generator. In the deck-odd sector:

• a split pair contributes its component difference;
• an off-branch triple or fourfold point contributes its deck difference;
• a ramification point contributes no odd generator.

Exact rational row reduction gives

$$\begin{array}{c|c|c|c|c|c} m&\#C^3&\#C^3_-&H^2_+&H^2_-&\text{terms}\\ \hline 5&1&0&3&0&8\\ 6&2&1&8&1&8\\ 6&3&2&7&0&12. \end{array}$$

Both compositions were checked termwise:

$$d_2d_1=0$$

in the even and odd complexes. Summing the source terms gives

$$\boxed{(\dim H^2_+,\dim H^2_-)=(172,8).}$$

Interpretation

The fourfold cells are ordinary incidences of the already frozen marked hyperplanes. Their correction therefore strengthens the carrier/coefficient typing discipline: the apparent rank-forty Kummer packet was partly an artifact of truncating the carrier incidence complex one degree too early.

The eight residual odd classes occur in exactly eight six-mark terms, one per term. They are genuine cohomology of this finite (H^0)-Čech row, but they are not yet physical periods or a global theorem. Their coefficient type, cyclic assembly, higher-row differentials, and pairing with the physical relative chain remain uncomputed.

The exact term indices split into the two source-derived free (C_4) orbits

$$\{2,9,12,22\}, \qquad \{5,10,13,20\}.$$

Hence the residual support has the rational occurrence character of two regular modules,

$$H^2_-\simeq \mathbf Q[C_4]^{\oplus2}$$

as a labelled vector space. The serialized one-triple representatives are quotient-basis choices; they are not asserted to transport literally without pair-boundary corrections.

Falsifier and next calculation

Freeze the two labelled (C_4)-orbit representatives and derive their cyclic transport including pair-boundary corrections. Then test the next available source residue/Gysin differential into or out of them. If it kills them, the Kummer sector closes. If they survive, test physical-chain activation without adding support cells.

Artifacts

• research/benincasa/checkers/four_site_qg_higher_concurrence_audit.py
• research/benincasa/results/four-site-qg-higher-concurrence-audit.json
• research/benincasa/checkers/four_site_qg_full_cech_h2.py
• research/benincasa/results/four-site-qg-full-cech-h2.json

Entries 1189--1191 are superseded only where they identify the truncated pair-to-triple cokernel with the final top-row object.

View source on GitHub