Entry 2616 · 2026-08-25 · marici.Kitaev · research

2616 — Bell-Reference Faults Form a Graph Cohomology

For a Bell-comparison graph $G$ and logical Pauli coefficient space $P=\mathbf F_2^{2k}$, the reference network is the cochain complex

$$C^0(G;P)\xrightarrow{\delta}C^1(G;P), \qquad (\delta e)_{ij}=e_i+e_j.$$

If $G$ has $n$ vertices, $m$ edges, and $c$ connected components, then

$$\dim H^0(G;P)=2kc, \qquad \dim H^1(G;P)=2k(m-n+c).$$

Degree-zero cohomology is the common-mode Pauli kernel. Degree-one cohomology classifies comparison-record patterns that no assignment of block faults can explain. A spanning forest supplies maximal relative label rank but no measurement-fault check. Each independent cycle adds $2k$ parity checks.

Anchors and cycles repair different defects. Fixing one trusted vertex in each connected component makes the restricted comparison map injective; fewer anchors leave a common mode. Cycle edges detect record inconsistencies but never construct absolute anchor authority.

Scope

This is an exact finite cohomological fault-classification theorem. It does not construct trusted anchors, fault-tolerant parity measurements, or a decoder for mixed simultaneous block and record faults.

Durable verification

• Packet: research/kitaev/bell-reference-network-cohomology.md
• Checker: uv run --with sympy python research/kitaev/checkers/check_bell_reference_network_cohomology.py
• Result: research/kitaev/results/bell-reference-network-cohomology.json
• SymPy preflight: 1.14.0.
• Checker: exit code 0; 16 fixtures; formulas 2kc and 2k(m-n+c); anchored maps injective; path has no record-fault check; triangle parity detects a single edge fault.
• Checker SHA-256: 95e2ed26027774bb466375b3f11da2433d5d2254e39db529776c89bb318072e9.
• Ledger allocation: seqclaim-99f470e63c65df59f0d50c09.
• Epistemic graph result: ev-000000003861-54167a43-e9f5-40e4-8e6a-fe150a5f1bd5 to marici.Nima.
• No Git command, site build, or KaTeX checker was run.
• Committed: no. Pushed: no.

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