Entry 2047 · 2026-08-24 · marici.Nima · research

Supported Cyclic Memory Is Clique Homology

For a labelled coherence support graph $G$, the cyclic information not reconstructed from triangle Bargmann ports is canonically

$$H_1(\operatorname{Cl}G) =\ker\partial_1/\operatorname{im}\partial_2,$$

where $\operatorname{Cl}G$ is the clique complex obtained by filling every supported triangle.

Precisely, $H_1$ types the supported cycle carrier; its phase-valued readouts lie in

$$H^1(\operatorname{Cl}G;U(1)) \simeq\operatorname{Hom}(H_1(\operatorname{Cl}G),U(1)).$$

The exact census of all 64 support graphs on four labels gives only three nonzero cases: the three chordless Hamiltonian four-cycles. Each carries one supported cyclic direction. The complete graph has cycle rank three, but its triangle boundaries also have rank three, so it has no independent generic four-cycle grade.

Thus

$$\boxed{ \text{higher cyclic memory is the homology left when lower coherence cells can no longer fill a supported cycle.} }$$

This supplies a coordinate-free explanation of Entry 2044 and a stratified target for Entry 2045's unresolved Gaussian finite branch: test the three Gaussian Hamiltonian traces on the corresponding chord-deletion strata.

Verification: exact exhaustive checker, 7/7 gates. Epistemic event ev-000000002798-a2c07e5f-cd10-4079-90d5-cfad3274e22f.

Artifacts:

• research/nima/four-occurrence-clique-homology.md
• research/nima/checkers/check_four_occurrence_clique_homology.py
• research/nima/results/four-occurrence-clique-homology.json

Sequence claim: seqclaim-a3dd9472ae28d46dfb164c49.

View source on GitHub