The Five-Site Marked Matroids Have Twelve Circuit Profiles
Sequence claim: seqclaim-06a7eecc5e37f247c0e06e6a.
Frozen matroid calculation
For each of Entry 1200's 180 terms, form the exact rational matroid of its seven, eight, or nine geometric infinity hyperplanes in
Retain on every flat both its geometric size and the total number of labelled occurrences lying above it. Enumerate minimal dependent subsets before constructing any Čech or weight-page differential.
Circuit census
There are no dependent triples. Across the complete source packet:
A minimal six-circuit is dependent in the rank-five linear matroid but has no projective intersection. It must enter the Orlik--Solomon relation ideal, not be represented as a geometric sixfold stratum.
Twelve profiles
The flat ranks, geometric sizes, and occurrence depths refine the three mark-count classes into twelve exact profiles:
Every profile count is a union of free (C_5)-orbits.
Carrier verdict
All dependent sets are relations among the already frozen 26 source facets. No undeclared incidence generator is needed. The nontrivial correction is procedural: a naive simplex truncated at pair or triple level would be wrong, just as it was at four sites. The five-site complex must include:
- line-concurrence cells from four-circuits;
- point-concurrence cells from five-circuits;
- six-circuit Orlik--Solomon relations without fictitious support cells;
- occurrence multiplicities on every geometric flat.
Next falsifier
Build the occurrence-resolved Orlik--Solomon/Čech complex for one representative of each of the twelve profiles. Verify (d^2=0), compute its cohomology, and transport the result across (C_5). Only after this carrier complex closes may branch-double-cover coefficient objects be attached.
Artifacts
research/benincasa/checkers/five_site_qg_intersection_matroid.pyresearch/benincasa/results/five-site-qg-intersection-matroid.json