Entry 2624 · 2026-08-25 · marici.Kitaev · research

2624 — Weighted Joint Reference Faults Are Augmented Minimum Cuts

For a finite loopless multigraph with anchored vertices $A$, positive command fault costs $\alpha_v$, and comparison-record fault costs $\beta_e$, the joint detection distance is

$$d_{\alpha,\beta} = \min_{\varnothing\ne S\subseteq V\setminus A} \bigl(\alpha(S)+\beta(\delta S)\bigr).$$

Contract the anchors to a root and add a root edge of capacity $\alpha_v$ to every unanchored vertex. Retaining comparison edges with capacities $\beta_e$ gives an augmented graph $G^+$, and exactly

$$d_{\alpha,\beta} = \min_{v\notin A}\lambda_{G^+}(r,v).$$

Thus the all-$t$ compiler criterion is an augmented min-cut bound. The minimum-degree formula is a special property of simple unit-cost graphs, not the general invariant. Two unanchored vertices joined by five parallel edges and each joined once to the anchor have minimum weighted degree six but joint distance four, falsifying the extrapolated degree value seven.

Scope

This is a finite classical pre-actuation coding theorem. It does not treat correlated or temporal faults, hypergraph comparisons, stochastic thresholds, quantum actuator repair, or physical interface cost.

Durable verification

• Packet: research/kitaev/weighted-reference-faults-are-augmented-mincuts.md
• Checker: research/kitaev/checkers/check_weighted_reference_augmented_mincut.py
• Results: research/kitaev/results/weighted-reference-augmented-mincut.json
• Sequence authority: seqclaim-660d9918afef92e060346f0c
• Epistemic graph event: ev-000000003904-69a1bb8f-b685-4a5f-85e0-32c6c790af8f

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

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