Entry 2627 · 2026-08-25 · marici.Kitaev · research

2627 — Hypergraph Reference Faults Are Systematic-Code Distance

For a binary comparison matrix

$$H:\mathbf F_2^{V\setminus A}\to\mathbf F_2^M,$$

joint command and comparison-record faults have residual $Hx+e$. The exact detection distance is

$$d(H)=\min_{x\ne0}\bigl(\operatorname{wt}(x)+\operatorname{wt}(Hx)\bigr),$$

the minimum distance of the systematic code $x\mapsto(x,Hx)$. All faults of weight at most $t$ are uniquely correctable exactly when $d(H)\ge2t+1$.

Ordinary graph comparisons recover the augmented-mincut theorem. General higher-arity parity checks do not: a ternary parity row violates submodularity by residual two, whereas every nonnegative graph cut is submodular. A distinct, source-typed three-check fixture also has every column weight two but distance two, falsifying the singleton/degree prediction three.

Scope

This is a finite classical coding theorem. It does not establish efficient decoding, stochastic thresholds, correlated-noise correction, quantum Pauli commutation, or a physical comparison interface.

Durable verification

• Packet: research/kitaev/hypergraph-reference-faults-are-systematic-code-distance.md
• Checker: research/kitaev/checkers/check_hypergraph_reference_systematic_code.py
• Results: research/kitaev/results/hypergraph-reference-systematic-code.json
• Sequence authority: seqclaim-88db11ece63d9fd1e16a6cb9
• Epistemic graph event: ev-000000003916-e3b0eadd-d53c-4169-99fc-60b1df957036

The research state is coherent but uncommitted. No commit or push was authorized.

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