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\title{Toward a Post-Tropical MCP}
\author{Claude and MJC}
\date{March 12, 2026}

\begin{document}
\maketitle

\begin{abstract}
If entropy-weighted combination survives empirical scrutiny, then the
current MCP cannot be the final theory of count-based prediction. That
does not make the MCP useless. It means the present MCP should be read
as the exact theory of a rigid constitutive choice: tropical max-min
combination. A broader post-tropical MCP would need to separate what is
true of counting-based prediction in general from what is true only of
max-min, and then explain why confidence-weighted combination improves
compression without discarding the count-structural insights that made
the MCP valuable in the first place.
\end{abstract}

\section{The Pressure}

The recent empirical pressure is simple:
\begin{enumerate}
\item max-min appears much worse than interpolation on the same count
  tables;
\item source selection improves max-min but does not close most of the
  gap;
\item entropy-weighted blending may outperform plain KN interpolation;
\item gap-weighted tropical blending may outperform both while staying
  native to support-space competition.
\end{enumerate}

If that picture is right, then the main question is no longer whether
the tropical rule is elegant. The main question is what part of the MCP
is really about count structure, and what part is only about the
tropical choice of combination operator.

\section{Three Layers Of The MCP}

A useful way to restate the theory is:
\begin{enumerate}
\item \textbf{Count layer.} Counts, marginals, conditionals, quotienting,
  factorization, and code-length identities.
\item \textbf{Combination layer.} The rule that combines multiple pieces
  of evidence into one predictive distribution.
\item \textbf{Architecture layer.} Event spaces, factor towers, hidden
  bottlenecks, and recurrent transport.
\end{enumerate}

\begin{proposition}
Most of the current empirical challenge lands on the combination layer,
not on the count layer.
\end{proposition}

That is why the right response is not to throw away the MCP, but to
refactor it.

\section{What Likely Survives}

The following claims plausibly survive into a broader post-tropical MCP:
\begin{itemize}
\item counts induce conditionals;
\item code length is expected negative log-probability;
\item quotienting splits coarse and residual information;
\item coarsening loses mutual information;
\item hidden state and recurrent transport can still be analyzed as
  bottlenecks and path sums.
\end{itemize}

These are not inherently tropical claims. They are structural claims
about count-based prediction and compression.

\section{What Likely Becomes Conditional}

The following claims should probably be restated as conditional on the
specific max-min rule:
\begin{itemize}
\item lattice/residuation derivations of the forward pass;
\item the exact tropical--integer bridge in its current form;
\item monotonicity and fixed-point claims tied to the rigid operator;
\item any theorem whose proof uses max-min as a constitutive axiom
  rather than as one admissible rule among others.
\end{itemize}

\begin{remark}
This is not failure. It is a cleaner statement of scope.
\end{remark}

\section{What The New Theory Must Explain}

If entropy-weighted combination really wins, a broader theory must
explain at least three things:

\begin{enumerate}
\item \textbf{Confidence as a structural quantity.}
Why should entropy or sharpness determine evidence weight?

\item \textbf{Soft competition among sources.}
Why is a distributed weighting over predictors better than either
winner-take-all selection or fixed recursive discount?

\item \textbf{Compatibility with count semantics.}
How can the new combination rule still be understood as operating over
count-derived evidence rather than as an unrelated numerical hack?
\end{enumerate}

\section{A Minimal Program}

The smallest intellectually serious next program is:
\begin{enumerate}
\item keep the current MCP as the exact max-min theory;
\item mark clearly which grafs are constitutive of that theory;
\item develop a second layer of statements that hold for any
  count-derived predictive operator;
\item characterize entropy-weighted combination inside that broader
  layer.
\end{enumerate}

\begin{proposition}
If this program succeeds, the project does not split into ``pure UM''
versus ``mere compressor.'' It becomes a nested theory: the tropical
MCP as one exact corner of a broader count-combination framework.
\end{proposition}

\section{Formal Boundary}

Gap-blend sharpens the distinction between \emph{tropical-native ingredients} and
\emph{the old rigid max-min object}.  The following steps remain tropical-native:
\begin{enumerate}
\item extract per-order support tables from counts;
\item compute the top-two support gap $g_k = s_{1,k} - s_{2,k}$;
\item convert that gap into a power-of-two confidence weight.
\end{enumerate}

What changes is the competition law.  Flat max-min aggregation chooses one
dominant support path.  Gap-blend instead uses those tropical-native confidence
signals to form a weighted mixture over per-order output distributions.  So the
first departure is not from support algebra as such, but from the old rigid
flat-aggregation rule.

Entropy-blend pushes farther: its weight depends on a full real-valued
functional of the distribution, not just winner-margin information in support
space.  That is why gap-blend pressures the boundary \emph{within} the tropical
family, while entropy-blend pressures the boundary \emph{beyond} it.

\section{Conclusion}
The best way to take the recent empirical pressure seriously is not to
abandon the MCP and not to ignore the data. It is to ask for a more
accurate theory boundary.

The current MCP remains valuable as the exact theory of tropical
count-based prediction. A post-tropical MCP would preserve the count
structure, quotient structure, and compression identities while
generalizing the combination rule. That is the right conceptual target
if the empirical advantage of entropy-weighted combination survives.

\begin{thebibliography}{9}
\bibitem{mcp}
Claude and MJC.
\newblock \emph{MCP}.
\newblock Working manuscript, 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.

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

\bibitem{depth}
Graf.
\newblock \emph{Conviction Depth: Fewer Wins, Bigger Wins}.
\newblock Hutter archive, 12 March 2026.

\bibitem{price}
Claude and MJC.
\newblock \emph{The Price of the Tropical Semiring}.
\newblock Hutter archive, 12 March 2026.
\end{thebibliography}

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