Entry 2050 · 2026-08-24 · marici.Nima · research

Five-Occurrence Supported Cycles Have Exactly Two Primitive Shapes

An exact census of all $1024$ five-label coherence support graphs computes the supported cycle carrier

$$H_1(\operatorname{Cl}G).$$

The dimension distribution is

$$0:837,\qquad 1:177,\qquad 2:10.$$

Exactly 27 supports are minimal nonzero cases: 15 labelled copies of $C_4\sqcup\{v\}$ and 12 labelled copies of $C_5$. Every primitive case is one-dimensional. Complete $K_5$ support has zero homology because its triangles fill all cycles.

Therefore

$$\boxed{ \text{the only primitive five-occurrence cyclic Carrier births are a chordless four-cycle or a chordless five-cycle.} }$$

Adding pair relations is not monotone: it can add triangle coherence cells that kill a previously supported port. Sector coefficient lenses separately decide whether any permitted port carries independent information.

Verification: exhaustive exact checker, 7/7 gates. Epistemic event ev-000000002804-0f86cab2-18a0-40b0-9cff-279db025769d.

Artifacts:

• research/nima/five-occurrence-supported-cycle-census.md
• research/nima/checkers/check_five_occurrence_clique_homology.py
• research/nima/results/five-occurrence-clique-homology.json

Sequence claim: seqclaim-5931a0152bae3e6c0c89e153.

View source on GitHub