\documentclass[11pt]{article}
\usepackage[margin=1in]{geometry}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{hyperref}

\newtheorem{proposition}{Proposition}
\newtheorem{remark}{Remark}

\title{Why Pay the Tropical Tax?}
\author{Claude and MJC}
\date{March 12, 2026}

\begin{document}
\maketitle

\begin{abstract}
If the recent ablation claims are right, tropical max-min costs roughly
three bits per character relative to softer combination rules. That is a
serious empirical tax. But it does not follow that paying the tax is
irrational. If the goal is to study the Universal Model as a
mathematical object rather than merely to minimize compressed size, then
the tax may be the price of exact algebraic tractability. This note
states the strongest version of that case.
\end{abstract}

\section{The Temptation}

Once a cheaper scoring rule appears, the natural reaction is:
\begin{quote}
why keep the expensive one?
\end{quote}

If the project were only a compression contest, that question would end
the discussion. But the project is not only a compression contest. It is
also an attempt to understand a specific object: count-based prediction
with explicit event spaces, explicit pattern structure, and an exact
forward rule.

\section{What The Tax Buys}

\begin{proposition}
The exact rigid max-min rule buys exact structure that is unusually clean:
\end{proposition}
\begin{enumerate}
\item a lattice-theoretic forward pass;
\item a direct count-to-support semantics;
\item an exact quotient/luck interpretation;
\item clean fixed-point and monotonicity questions;
\item a compact mathematical spine for the MCP.
\end{enumerate}

These are not aesthetic extras. They are reasons the system is
understandable as a mathematical object rather than as a bag of useful
heuristics.

\section{Why This Still Matters}

\begin{remark}
A worse compressor can still be a better scientific object.
\end{remark}

There are many domains where the right object of study is not the
numerically best performer. Exact arithmetic, reversible computing,
finite-state models, and idealized physical systems all survive because
they reveal structure that would be obscured in a more performant but
less rigid formulation.

The UM may be in that category. If so, the right comparison is not:
\begin{quote}
does the exact rigid max-min object buy enough understanding to justify keeping it even if richer tropical or non-tropical rules compress better?
\end{quote}
but:
\begin{quote}
what can be proved, factored, and understood exactly under max-min that
becomes opaque under the softer rule?
\end{quote}

\section{The Price Must Be Faced Honestly}

None of this works if the tax is hidden.

\begin{proposition}
The only intellectually serious pro-UM position after the ablation is an
honest one: the algebra is valuable enough that we are willing to pay
for it.
\end{proposition}

That means:
\begin{itemize}
\item no pretending that source selection alone closes the gap;
\item no pretending that max-min is competitive if it is not;
\item no pretending that Track U and Track C are the same project.
\end{itemize}

\section{What A Rational Two-Track Project Looks Like}

Once the tax is admitted, the division of labor becomes clearer:
\begin{enumerate}
\item Track C minimizes total bytes and adopts whatever combination rule
  wins.
\item Track U studies the exact max-min object and develops its theory.
\item Results can move between the tracks only with explicit relabeling.
\end{enumerate}

\begin{remark}
This is not a defeat for the UM. It is a refusal to confuse theoretical
value with benchmark value.
\end{remark}

\section{Conclusion}

The tropical tax is only irrational if compression is the only thing we
care about. If the goal also includes exact understanding of a rigid
count-based predictive system, then paying the tax may be justified.

The burden is then clear: Track U must produce theorems, explanations,
and structure commensurate with the price it asks us to pay. If it does
not, the tax is waste. If it does, the tax is the cost of having an
object worth understanding exactly.

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

\bibitem{post}
Claude and MJC.
\newblock \emph{Toward a Post-Tropical MCP}.
\newblock Hutter archive, 12 March 2026.
\end{thebibliography}

\end{document}




