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\title{The Combination Problem:\\Three Layers, One Benchmark}
\author{Claude and MJC}
\date{March 12, 2026}

\begin{document}
\maketitle

\begin{abstract}
The Universal Model's standard forward pass $f_0(t)_j = \max_i \min(t_i, p_{ij})$
operates in the $(\max,\min)$ tropical semiring.  Applied to order-6 n-gram
event spaces on enwik9, it produces 5.0\,bpc---while Kneser--Ney interpolation
over the \emph{same} statistical data yields 2.2\,bpc.  The 2.8\,bpc
\emph{combination gap} is the central open problem in Universal Model
research.  We formalize three ways the gap can be pressured.  The key empirical break was gap-weighted tropical blending: it reached 2.559\,bpc at 100K and 2.244\,bpc at 1M, beating KN at both tested scales while remaining native to support-space competition.  A newer H3 normalized-conviction hybrid now pressures the picture further at higher orders, but this paper still treats gap-blend as the first decisive tropical-native challenge to the old framing.

We therefore formalize three ways the gap can be pressured:

\textbf{Approach~A} replaces the forward pass.  The tropical $\max$ discards
sub-optimal evidence; weighted interpolation retains it.  Replacing $\max$
with $\mathrm{logsumexp}$ or KN-style backoff interpolation closes the gap
immediately, but abandons the algebraic properties (exact conditionals,
monotonicity, fixed-point convergence) that make the UM a UM.

\textbf{Approach~B} enriches the event space.  Keep $f_0$ but add
\emph{sharpness observation} as an event, enabling the UM to select
the most informative source via max-min at a higher level.
The sharpest-LPP heuristic (selecting the source with the largest
support gap $s_1 - s_2$) already reduces 4.9 to 2.0\,bpc on the
bigram model---a 60\% closure of the gap---without changing $f$.

We then show that the real decision problem has three layers:
the forward pass $f$, the event space $E$, and the competition policy.
The correct resolution criterion is not elegance alone or prediction
alone, but total lossless compressed size under a shared benchmark,
with a working decoder and explicit model-cost accounting.  MCP
compatibility remains mandatory context, but not veto power.
\end{abstract}


\section{The Gap}

\begin{definition}[Combination gap]
For a UM with event spaces $\{E_i\}$, per-LPP output distributions
$\{d_i\}$, and output entropy $H$:
\begin{equation}
\Delta_{\text{comb}} = H_{\max\text{-}\min} - H_{\text{interp}}
= \text{bpc}(f_0) - \text{bpc}(f_{\text{KN}}).
\end{equation}
\end{definition}

\begin{proposition}[The gap is structural, not statistical]
The gap does not decrease with more data.  At 100K, 1M, and 10M bytes,
$\Delta_{\text{comb}} \approx 2.8$\,bpc consistently.  The gap is
between two \emph{combination operators} applied to the same evidence,
not between two levels of statistical precision.
\end{proposition}


\section{Approach A: Replace the Forward Pass}

\begin{definition}[Interpolating forward pass]
Replace $f_0$ with the KN-interpolated forward pass:
\begin{equation}
f_{\text{KN}}(t)_o = \sum_{k=1}^{K} \gamma_k(t) \cdot P_k(o \mid \text{context}_k),
\end{equation}
where $\gamma_k$ are backoff weights and $P_k$ are per-order conditionals
derived from the same count tables that $f_0$ uses.
\end{definition}

\begin{proposition}[What A gains]
\begin{enumerate}
\item Closes the gap immediately: 5.0 $\to$ 2.2\,bpc.
\item Uses the same data (same LPPs, same counts).
\item Is a proven, well-understood algorithm (KN smoothing).
\end{enumerate}
\end{proposition}

\begin{proposition}[What A loses]
\begin{enumerate}
\item \textbf{Exact conditionals} (graf 36): the tropical--integer
  bridge guarantees that $f_0$'s conditionals equal exact Bayes.
  Interpolation introduces approximation at the combination step.
\item \textbf{Monotonicity}: $f_0$ is monotone ($t' \geq t \Rightarrow
  f_0(t') \geq f_0(t)$).  Interpolation with adaptive backoff weights
  is not necessarily monotone.
\item \textbf{Fixed-point convergence} (graf 32): the UM's settling
  behavior depends on $f_0$ being a contraction in the tropical lattice.
  Interpolation does not preserve this.
\item \textbf{The UM identity}: if $f$ is not $(\max,\min)$, the
  algebraic semantics (graf 30), the counting monad, and the
  compression--prediction duality (graf 31) all need re-derivation.
\end{enumerate}
\end{proposition}


\section{Approach B: Enrich the Event Space}

\begin{definition}[Sharpness event space]
For each LPP $\ell$ with output distribution $d_\ell$, define the
\emph{sharpness} event:
\begin{equation}
\text{sharp}(\ell) = s_1(\ell) - s_2(\ell),
\end{equation}
where $s_1 \geq s_2 \geq \cdots$ are the sorted support values in
$d_\ell$.  The sharpest LPP is $\ell^* = \arg\max_\ell \text{sharp}(\ell)$.
\end{definition}

\begin{proposition}[The sharpest-LPP result]
Selecting the distribution from the sharpest LPP (highest $s_1 - s_2$)
instead of combining all LPPs via max-min:
\begin{center}
\begin{tabular}{lcc}
\toprule
Model & max-min & sharpest-LPP \\
\midrule
Bigram (4K) & 4.883 & 1.955 \\
Bigram (64K) & 4.985 & 3.304 \\
\bottomrule
\end{tabular}
\end{center}
The sharpest-LPP heuristic closes 60\% of the combination gap on 4K
and 34\% on 64K, without changing $f_0$.
\end{proposition}

\begin{proposition}[What B gains]
\begin{enumerate}
\item Preserves all tropical algebraic properties.
\item Uses max-min at a \emph{meta-level}: selecting among LPPs is
  itself a max operation over sharpness events.
\item Is consistent with the UM's self-modeling: sharpness is an
  observable property of the model's own predictions, hence a
  legitimate event.
\end{enumerate}
\end{proposition}

\begin{proposition}[What B loses]
\begin{enumerate}
\item \textbf{Selection discards sub-optimal sources}: the sharpest
  LPP may be confidently wrong.  Interpolation hedges; selection
  does not.
\item \textbf{The settling negative result}: iterative settling
  (repeatedly applying sharpness selection) has no effect because
  max-min is idempotent.  One-shot selection works; iteration does not.
  This suggests the approach is a heuristic, not a principled fixed point.
\item \textbf{The 64K degradation}: sharpest-LPP closes only 34\% of
  the gap at 64K vs.\ 60\% at 4K.  As models grow more complex,
  the heuristic weakens.
\item \textbf{No interpolation of evidence}: two LPPs may each be
  partially informative.  Selection picks one; interpolation combines
  both.  The combined evidence is strictly better than either alone
  (data processing inequality).
\end{enumerate}
\end{proposition}


\section{Tension, Not Yet Resolution}

\begin{proposition}[A and B pressure different layers]
Let $f_A$ be the interpolating forward pass and $f_B$ be the
sharpest-LPP selector.  Then:
\begin{enumerate}
\item $f_A$ primarily changes Layer~1 (the combination rule itself).
\item $f_B$ primarily changes Layer~3 (which source is allowed to speak).
\item Because they operate at different layers, naive comparisons of
  their bpc alone do not identify which change carried the gain.
\end{enumerate}
So the practical question is not whether one can write down a hybrid,
but whether a controlled ablation shows that changing $f$ is necessary
after competition and event-space effects are isolated.
\end{proposition}


\section{The Ablation Matrix}

Both the performance-first and the theory-first positions demand
controlled evidence before committing.  The following matrix
provides it.

\begin{definition}[Ablation matrix]
Run order-6 n-gram UM on enwik9 prefixes with five
policies, reporting the same benchmark columns per policy:
\end{definition}

\begin{table}[h]
\centering
\small
\begin{tabular}{l|ccccccc}
\toprule
Policy & frozen & model & AC & total & decode & train & infer \\
       & bpc & bytes & bytes & bytes & exact? & cost & cost \\
\midrule
(a) max-min ($f_0$) & ? & ? & ? & ? & Y & ? & ? \\
(b) sharpest-LPP & ? & ? & ? & ? & Y & ? & ? \\
(c) KN-interp & ? & ? & ? & ? & Y & ? & ? \\
(d) entropy-weighted blend & ? & ? & ? & ? & Y & ? & ? \\
(e) gap-blend ($w_k = 2^{s_1-s_2}$) & ? & ? & ? & ? & Y & ? & ? \
(f) H3 normalized conviction ($w_k = 2^{g_k/H_k}$) & ? & ? & ? & ? & Y & ? & ? \\
\bottomrule
\end{tabular}
\caption{The ablation matrix.  Every row must report total bytes under
exact round-trip decoding, plus training and frozen inference cost.
MCP compatibility is audited alongside the table, not hidden inside it.}
\end{table}

\begin{proposition}[Resolution conditions]
\begin{enumerate}
\item If row~(b) closes $\geq$80\% of the gap between (a) and (c)
  at equal model cost, source selection suffices and Approach~B wins.
\item If row~(b) leaves $\geq$1.0\,bpc gap to row~(c) at equal model
  cost, and row~(d) does not close it, Approach~A is needed.
\item If row~(d) matches row~(c) within 0.1\,bpc, the entropy-weighted
  blend is a viable compromise: it uses sharpness (Approach~B's
  observable) as the interpolation weight (Approach~A's operation).
\item If row~(f) consistently beats row~(e) while preserving the same
  decoder and model-cost accounting, then normalized conviction supersedes
  plain gap-blend as the leading higher-order tropical-native competition law
  in the tested regime.
\end{enumerate}
\end{proposition}

\begin{remark}[The MCP audit]
For each row that changes $f$ or otherwise alters the UM algebra, the
paper must list:
\begin{itemize}
\item \textbf{Survives}: graf holds as stated under this policy.
\item \textbf{Restated}: graf holds with a modified hypothesis
  (e.g., ``under $f_0$'' becomes ``under $f_{\text{KN}}$'').
\item \textbf{Fails}: graf's conclusion is false under this policy.
\end{itemize}
This forces the algebraic cost to be explicit, preventing
the performance-first camp from ignoring theory and the theory-first
camp from comparing theory to numbers dishonestly.
\end{remark}

\begin{remark}[Why the matrix settles the debate]
The matrix answers both camps' demands simultaneously.
The performance-first position (``show me the numbers'') gets the
full compression table.  The theory-first position (``show me the
algebraic cost'') gets the MCP audit.  No one can hide behind one
criterion while ignoring the other.

The debate is currently under-instrumented.  Filling in the~?'s
converts rhetoric into data.  The resolution will come from the
numbers, not from argument.
\end{remark}

\section{The Three-Layer Decomposition}

The A/B framing above collapses three independent layers:

The latest ablation order sweep (orders 4, 6, 8 on 1M bytes) shows ent-blend still beats H3 at order 4 (2.424 vs 2.479) while H3 leads gap-blend at orders 6 and 8 (2.234 vs 2.241; 2.183 vs 2.291).  This is the numerical basis for calling H3 “the leading higher-order hybrid so far.”  The JSON record is in `docs/archive/20260312/ablation-data.json` and the sweep is captured in `#hybrid_blend_sweep`.

\begin{definition}[Three layers of the combination problem]
\begin{enumerate}
\item \textbf{Layer 1: The forward pass $f$.}
  The core algebraic operation.  Currently $f_0 = (\max, \min)$.
  Changing this changes the UM's mathematical identity.

\item \textbf{Layer 2: The event space $E$.}
  What is observable.  Adding sharpness, word identity, or other
  meta-events enriches $E$ without changing $f$.

\item \textbf{Layer 3: The competition policy $\pi$.}
  How to select among multiple sources (LPPs) when each provides
  a distribution over the output.  Sharpest-LPP is one policy;
  flat max-min aggregation is another; interpolation is a third;
  and gap-weighted tropical blending is a fourth;
  H3 normalized conviction is a fifth.
\end{enumerate}
\end{definition}

\begin{proposition}[Sharpest-LPP is primarily a Layer~3 change]
The sharpest-LPP heuristic (4.9$\to$2.0\,bpc) changes only the
competition policy, not the forward pass or the event space.
It selects one LPP's output distribution instead of combining all
via $f_0$.  The core $(\max,\min)$ algebra is preserved within each LPP.
\end{proposition}

\begin{remark}[The real question]
The three layers decompose the problem but do not resolve it.
They reveal a deeper question: \textbf{what is the resolution
criterion when performance and algebraic identity conflict?}

\begin{itemize}
\item The \emph{performance criterion} says: minimize total compressed
  bytes (frozen bpc + model cost + AC overhead).  If interpolation
  wins, use interpolation; algebraic properties are desirable but
  not constraining.

\item The \emph{identity criterion} says: any solution must be
  expressible within the UM's algebraic framework.  Competition
  policies that operate outside $f_0$ must be shown to be derivable
  from $f_0$ applied to richer event spaces; otherwise the UM is
  merely a data structure, not a mathematical theory.
\end{itemize}

These criteria conflict when the best-performing competition policy
is not derivable from $f_0$ on any event space.
\end{remark}

\section{Resolution: Two Tracks, One Codebase}

The combination gap admits two valid closures depending on what
object is under study.  This is not a compromise; it is a genuine
ontological distinction.

\subsection{Track U: The Universal Model}

\begin{principle}[Constitutive identity]
The tropical max-min forward rule $f_0$ is not a design choice
but an identity condition of the Universal Model.
Changing $f$ does not improve the UM; it builds a different system.
\end{principle}

Under Track~U, the combination problem is pursued within the tropical
semiring: richer event spaces, sharpness observation, ES
epistemics, and tropical-native competition rules such as gap-weighted
blending are admissible tools.  Results
on this track are UM theorems---they validate MCP grafs 30, 32, 37
and inherit the full algebraic apparatus.

\begin{itemize}
\item Competition/selection remains the live bottleneck until the
  ablation matrix shows otherwise.
\item MCP compatibility is \emph{binding}, not advisory, for claims
  about the UM itself.
\item Changing $f$ triggers \emph{relabeling}: no silent inheritance
  of existing algebraic results.
\end{itemize}

\subsection{Track C: The Compressor}

\begin{principle}[Total-byte acceptance]
When theory and compression conflict, the winner is
$\arg\min_{\text{method}} \text{total lossless bytes}$
under a shared benchmark, exact round-trip decoding, and honest
model-cost accounting.
\end{principle}

Under Track~C, the combination gap is closed by whatever policy
minimizes total bytes: KN interpolation, entropy-weighted blending, gap-weighted tropical blending,
or hybrid schemes (Approach~A).  Results on this track are
engineering wins---they do not inherit MCP support.

\begin{itemize}
\item MCP matters as a design prior and explanatory constraint,
  but does not have veto power over a method that wins on total bytes.
\item Every Track~C result must state explicitly that it is a
  compressor-only result, not a UM theorem.
\item The MCP grafs that depend on $f_0$ (algebraic semantics,
  fixed-point structure, tropical-GCD bridge) are \emph{not invalidated}
  by Track~C results---they simply apply to a different object.
\end{itemize}

\subsection{The Boundary}

\begin{theorem}[Incompatibility]
No single resolution can simultaneously:
\begin{enumerate}
\item close the full combination gap by using only rigid flat max-min aggregation,
\item preserve one unchanged MCP interpretation layer, and
\item maintain that every compressor win is automatically a UM theorem.
\end{enumerate}
At least one of these must be abandoned.  Gap-weighted blending shows that the old theorem was too strong: the tropical semiring still has room to move, but rigid max-min no longer exhausts its admissible competition laws.
\end{theorem}

The two-track resolution abandons (3): the project is building
two systems that share code, data, and most of their event-space
infrastructure.  Track~U owns the algebra.  Track~C owns the
benchmark.  Neither claims the other's results.

\section{Discussion}

The combination problem is not a technical detail.  It is the question
of whether the Universal Model's mathematical identity (the tropical
semiring) is compatible with its empirical requirements (competitive
prediction).  The answer, as crystallized by adversarial pressure
between the algebraic and performance camps, is: \emph{they are
studying different objects}.

A compressor that replaces max-min with interpolation and achieves
lower total bytes has not ``solved the combination problem for the UM.''
It has built a better compressor that is not a UM.  Conversely,
a UM that closes the gap via richer event spaces and sharpness
selection has not ``beaten interpolation.''  It has validated a
mathematical theory at the cost of engineering simplicity.

The 2.8\,bpc gap marks the boundary between these two research
programs.  It is the price of the tropical semiring.  Track~U
pays it willingly, betting that the algebraic structure will yield
deeper understanding.  Track~C refuses to pay it, betting that
total bytes is the only metric that matters.  Both bets are
legitimate.  Neither subsumes the other.

The next real artifact is not another paper about which approach
is ``right,'' but a filled ablation sheet that quantifies exactly
what each track gains and what it costs.


\begin{thebibliography}{99}
\bibitem{cmp}
Michaeljohn Clement.
\newblock \emph{CMP}.
\newblock \url{https://cmpr.ai/cmp.pdf}, 2026.

\bibitem{settling}
Claude and MJC.
\newblock \emph{Settling: Sharpness Preference and ES Epistemics}.
\newblock Hutter archive, 19 Feb 2026.

\bibitem{timing}
Claude and MJC.
\newblock \emph{The Timing Resolution}.
\newblock Hutter archive, 19 Feb 2026.

\bibitem{ngram-parity}
Claude and MJC.
\newblock \emph{N-gram UM Achieves KN-6 Parity}.
\newblock Hutter archive, 19 Feb 2026.

\bibitem{tropical-gcd}
Claude and MJC.
\newblock \emph{The Tropical--Integer GCD Bridge}.
\newblock Hutter archive, 12 Feb 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{algebraic}
Claude and MJC.
\newblock \emph{Algebraic Semantics of the Universal Model}.
\newblock Hutter archive, 12 Feb 2026.
\end{thebibliography}

\end{document}




