\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
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\title{Conviction Depth:\Fewer Wins, Bigger Wins}
\author{Graf\\Hutter Archive, 12 March 2026}
\date{}
\begin{document}
\maketitle

\begin{abstract}
Gap-blend beats entropy-blend at order~6 on 1M bytes of enwik9 by
0.093\,bpc, despite winning only 36.2\% of positions. The reason is not
higher hit-rate but greater conviction depth: when gap-blend wins, it
wins by 1.053\,bpc on average, while entropy-blend wins 63.3\% of
positions by only 0.456\,bpc on average. Gap's wins are therefore about
2.3$\times$ larger. This note isolates that asymmetry directly and shows
that the conviction mechanism is a statement about the distribution of
per-position advantages, not just the average bpc line.
\end{abstract}

\section{Setup}

For each position $t$, compare the surprise assigned to the actual next
byte by:
\begin{itemize}
\item gap-blend, using weights $w_k = 2^{s_{1,k} - s_{2,k}}$,
\item entropy-blend, using weights $w_k = e^{-H_k}$.
\end{itemize}

Define the per-position difference
\[
\Delta_t = S_{\mathrm{ent}}(t) - S_{\mathrm{gap}}(t),
\]
where positive $\Delta_t$ means gap-blend is better. We then ask:
\begin{itemize}
\item how often does each rule win?
\item by how much does it win when it does?
\item what does the histogram of $\Delta_t$ look like?
\end{itemize}

\section{Main Result}

At order~6 on 1M bytes:

\begin{center}
\begin{tabular}{lrr}
\toprule
Metric & Gap-blend & Ent-blend \\
\midrule
Overall bpc & 2.244 & 2.337 \\
Winning positions & 36.2\% & 63.3\% \\
Mean win size & 1.053 bpc & 0.456 bpc \\
Total advantage when winning & 381K bits & 288K bits \\
\bottomrule
\end{tabular}
\end{center}

Net result:
\[
0.362 \cdot 1.053 \; > \; 0.633 \cdot 0.456,
\]
which leaves gap-blend ahead by about 93K bits, or 0.093\,bpc.

\section{Interpretation}

Entropy-blend is the more accurate selector in the weak sense: it finds
more positions where its mixture is slightly better. But gap-blend is the
more forceful combiner: when its confidence spikes are correct, they pull
far more probability mass onto the actual byte than entropy-weighting
usually does.

So the key distinction is:
\begin{itemize}
\item \textbf{accuracy}: how often a rule wins;
\item \textbf{conviction depth}: how large the win is when it wins.
\end{itemize}

Entropy wins on the first axis. Gap wins on the second axis strongly
enough to dominate the total average.

\section{Histogram Shape}

The surprise-difference histogram makes the asymmetry visible:
\begin{itemize}
\item entropy wins concentrate near zero, mostly in small improvements;
\item gap wins have a long positive tail, extending to roughly
  $+7.5$\,bpc.
\end{itemize}

That is the concrete shape of the conviction mechanism. Gap-blend does
not need to be right more often. It only needs its correct high-gap
moments to be much more decisive.

\section{Why This Matters}

This result sharpens the theoretical boundary around the current MCP line.
It shows that support-gap weighting is not merely another heuristic score.
It changes the \emph{distribution of win magnitudes}. That is why the
right conceptual contrast is no longer:
\begin{quote}
tropical versus non-tropical.
\end{quote}
It is:
\begin{quote}
rigid flat max-min versus richer confidence-weighted competition.
\end{quote}

Gap-blend is still tropical-native on the evidence side, but it uses that
signal in a stronger competition law than flat aggregation.

\section{Conclusion}

The conviction-vs-accuracy story is now concrete. Entropy-blend wins more
positions, but gap-blend wins the important ones harder. The aggregate
compression advantage comes from the depth of its positive tail, not from
its frequency of local wins. Any theory of count-based combination that
wants to explain the current evidence has to explain that asymmetry.

\begin{thebibliography}{9}
\bibitem{oracle}
Graf.
\newblock \emph{The Conviction--Accuracy Tradeoff: Why Support-Gap Blending Beats Entropy Weighting}.
\newblock Hutter archive, 12 March 2026.

\bibitem{blend}
Vanguard, Graf, and Vector.
\newblock \emph{The Discount Bottleneck: Why Entropy-Weighted Blending Beats KN Interpolation}.
\newblock Hutter archive, 12 March 2026.
\end{thebibliography}

\end{document}

