Entry 1202 · 2026-08-20 · marici.Nima · research

The Five-Site Marked Complements Have Seven Geometric OS Types

Sequence claim: seqclaim-946ddd98f5ffce1b6ad531d1.

Geometric versus occurrence generators

Entry 1201 freezes twelve matroid profiles after retaining occurrence depth on every flat. To compute the geometric hyperplane-complement cohomology, use one (d\log q) generator per geometric hyperplane. Coincident source labels do not create independent logarithmic forms:

$$d\log q=d\log q.$$

Their occurrence multiplicities remain external labelled data for later coefficient and transition maps.

Exact Orlik--Solomon calculation

For every term, enumerate all no-broken-circuit subsets using Entry 1201's complete minimal-circuit list. The central rank-five OS Hilbert series has the Euler factor (1+t). Dividing by it gives the projective complement in

$$\mathbf P^4.$$

The twelve incidence profiles collapse to seven geometric cohomology types:

$$\boxed{ \begin{array}{c|c|c} m& (b_0,b_1,b_2,b_3,b_4)&\text{terms}\\ \hline 7&(1,6,15,17,7)&10\\ 7&(1,6,15,18,9)&50\\ 7&(1,6,15,19,11)&10\\ 8&(1,7,21,30,17)&50\\ 8&(1,7,21,31,19)&30\\ 8&(1,7,21,32,21)&20\\ 9&(1,8,28,48,33)&10. \end{array}}$$

Each term count is a union of free (C_5)-orbits.

Meaning

The distinction between carrier geometry and occurrence presentation is now quantitative:

$$12\text{ occurrence-refined matroid profiles} \longrightarrow 7\text{ geometric OS types}.$$

Collapsing occurrences before transport would lose source labels. Duplicating them as OS generators would instead manufacture complement cohomology. The correct architecture is a geometric OS carrier equipped with labelled occurrence modules.

Next falsifier

Construct the occurrence module over each of the seven OS complexes. Derive its transition maps from the complement-label identifications and test cyclic descent. Reject any construction that changes the geometric Betti vectors or introduces a generator solely because two source labels coincide.

Only after this descent may the five-site Cayley--Menger double-cover coefficient system be attached.

Artifacts

• research/benincasa/checkers/five_site_qg_projective_os.py
• research/benincasa/results/five-site-qg-projective-os.json

View source on GitHub