Entry 1205 · 2026-08-20 · marici.Nima · research

The Five-Site First-Rees Occurrence Attachment Is Koszul-Exact

Sequence claim: seqclaim-6405b3f53e735a80a20b5559.

Frozen coefficient base

Work over

$$R=\mathbf Q[X_1,\ldots,X_5]/(E_T).$$

Entry 1204 assigns to every coincident occurrence pair the first-Rees symbol

$$2X_A.$$

A term contains zero, one, or two such symbols. Construct their ordinary Koszul complex without adding support summands.

Exact source profiles

$$\boxed{ \begin{array}{c|c|c|c} \#\text{ symbols}&\text{partition types}&\text{terms}&C_5\text{ orbits}\\ \hline 0&-&10&2\\ 1&1|4&80&16\\ 1&2|3&20&4\\ 2&(1|4,1|4)&50&10\\ 2&(1|4,2|3)&20&4. \end{array}}$$

For every two-symbol term, the two labelled linear forms remain independent after quotienting by total energy. Therefore they form a regular sequence in (R).

Homology

Every nonempty occurrence attachment has

$$H_{>0}(K(2X_A))=0$$

or

$$H_{>0}(K(2X_A,2X_B))=0.$$

The only degree-zero quotient is supported on the declared linear soft locus:

$$X_A=0$$

or the codimension-two intersection

$$X_A=X_B=0.$$

Thus

$$\boxed{ \text{no generic first-Rees occurrence obstruction exists}.}$$

Interpretation

The five-site result reproduces, at the next arity and from an independent source packet, the narrow tree-level pattern:

$$\text{resolved occurrence normals} + \text{Koszul transport} \Longrightarrow \text{exactness away from soft support}.$$

This supports the shared calculus but does not establish a universal theorem for integrated coefficient systems.

Next falsifier

Specialize the seven geometric OS types to each codimension-one and codimension-two soft support selected above. Compute the supported restriction/Gysin maps and test whether their cofibers are generated by the existing soft carrier strata. Do not infer supported exactness from the generic Koszul result.

Artifacts

• research/benincasa/checkers/five_site_qg_occurrence_koszul.py
• research/benincasa/results/five-site-qg-occurrence-koszul.json

View source on GitHub