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This statement fixes a parameter value for the documented model version. Its effect depends on the equation or update rule that consumes that parameter; it is retained separately from a governing law.
Terms & definitions
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How it works
Read J as the quantity being defined or updated by this relation.
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Assumptions & limits
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Original context
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\subsection*{Execution and an inspectable example}
The 60 instance computations totaled 87.24 seconds, excluding initial validation/JIT warm-up. Execution was serial with one BLAS thread on a Linux x86-64 environment reporting an AMD EPYC 9V74 CPU; Python 3.13.5, NumPy 2.3.5, SciPy 1.17.0 and Numba 0.65.1. Timings are environment-specific.
For the first fixed instance, \texttt{n08\_dense\_r00}, the exact partition is $\{1,2,3,7\}\mid\{4,5,6,8\}$: $J^*=0.2578092851$, $\Phi_{\rm SI}=2.1867917397$ nats and $K=8.4822070674$ nats. Queyranne instead returns $\{1,2,3,6\}\mid\{4,5,7,8\}$ with $J=0.2788420313$. Every other instance, including unsuccessful searches, is retained in the companion data.
\newpage
\section*{6. Interpretation and limits}
\textbf{Normalization materially changes this benchmark.} A separately labeled \emph{post-run diagnostic} found that Queyranne recovered all 60 \emph{unnormalized} stochastic-interaction minima, versus 11 normalized minima. Normalization changed the exact minimizing partition in 37/60 networks. An exhaustive post-run check of 6,804,480 elementary submodularity inequalities found violations in all 60 normalized objective tables. Thus the direct normalized search cannot inherit an exactness guarantee merely from success on a related objective. These post-run diagnostics leave the primary results unchanged.
[Context excerpt; full source linked.]Exact original notation
J=0.2788420313