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\newtheorem{theorem}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
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\title{Kneser--Ney on the Integers\\{\large v2: The Ring Structure}}
\author{Claude and MJC}
\date{February 2026}

\begin{document}
\maketitle

\begin{abstract}
We rewrite the KN-as-quotient paper using the integer framework
from the UM arithmetic series: events are integers, event spaces
are rings $\mathbb{Z}/N\mathbb{Z}$, and all maps between event
spaces are modular reduction. The KN context of order $k$ is an
integer $c \in \mathbb{Z}/256^{k-1}\mathbb{Z}$, and the order
projection that drops the oldest byte is reduction mod $256^{k-2}$
---literal integer division by 256. The discount $D$ is subtraction
in $\mathbb{Z}$. The GCD of the count row is the GCD in
$\mathbb{Z}$. The continuation count is the fiber cardinality of
the column projection. The interpolation recursion is a tower of
ring surjections $\mathbb{Z}/256^K\mathbb{Z} \twoheadrightarrow
\cdots \twoheadrightarrow \mathbb{Z}/256\mathbb{Z}$. None of these
are analogies; with exact arithmetic they are the definitions.

This restatement makes several things visible that the v1 paper
left obscure: the discount--GCD gap is a gap between subtraction
and division in $\mathbb{Z}$; the continuation count is the image
size of a ring homomorphism restricted to a fiber; and the unified
pattern table is a function on the product ring
$\mathbb{Z}/256^{k-1}\mathbb{Z} \times \mathbb{Z}/256\mathbb{Z}
\cong \mathbb{Z}/256^k\mathbb{Z}$.

\emph{This is v2. v1 introduced the quotient interpretation; v2
restates it on the integers using the framework of the UM arithmetic
papers (v1--v4).}
\end{abstract}


\section{The Ring of Contexts}

\subsection{Contexts Are Integers}

A byte-level order-$k$ context is a sequence of $k-1$ bytes:
$c = (b_1, b_2, \ldots, b_{k-1})$ where $b_i \in \{0,\ldots,255\}$.
Encode this as a base-256 integer:
\begin{equation}\label{eq:context-int}
c = b_1 \cdot 256^{k-2} + b_2 \cdot 256^{k-3} + \cdots + b_{k-1}
  \in \{0, \ldots, 256^{k-1} - 1\}.
\end{equation}
The context space $I_k = \{0, \ldots, 256^{k-1}-1\}$ is the ring
$\mathbb{Z}/256^{k-1}\mathbb{Z}$.

The output byte $o \in \{0,\ldots,255\}$ is an integer in
$\mathbb{Z}/256\mathbb{Z}$.

The joint event ``context $c$ followed by output $o$'' is the
integer:
\begin{equation}\label{eq:joint-int}
e = 256 \cdot c + o = b_1 \cdot 256^{k-1} + b_2 \cdot 256^{k-2}
  + \cdots + b_{k-1} \cdot 256 + o
  \in \mathbb{Z}/256^k\mathbb{Z}.
\end{equation}
This is the context-output pair encoded as a single base-256
number of $k$ digits. The event space $E_k = I_k \times O$ is the
ring $\mathbb{Z}/256^k\mathbb{Z}$.

\begin{observation}[Product Ring]
The Chinese Remainder Theorem does not apply directly since
$\gcd(256^{k-1}, 256) = 256 \neq 1$. The decomposition
$E_k = I_k \times O$ is the mixed-radix decomposition of
$\mathbb{Z}/256^k\mathbb{Z}$, not a CRT decomposition. This
reflects the fact that the context and output share the same
alphabet---they are not independent event spaces but successive
positions in the same stream.
\end{observation}

\subsection{The Count Table Lives on the Ring}

The $n$-gram count table is a function on the ring:
\begin{equation}
c_k : \mathbb{Z}/256^k\mathbb{Z} \to \mathbb{N},
\qquad c_k(e) = |\{t : (b_{t-k+1}, \ldots, b_t) = e\}|.
\end{equation}
It counts how many times the $k$-byte pattern $e$ appears in the
data. The function $c_k$ lives on the ring and inherits the ring's
structure.

The row indexed by context $c$ is the restriction of $c_k$ to the
coset $\{256c + o : o \in \{0,\ldots,255\}\}$. This coset is an
ideal translate in the ring: it contains all events with the same
context, varying only the output.

The row total is $c_k(c, \cdot) = \sum_{o=0}^{255} c_k(256c + o)$
---the sum over the coset.


\section{The Order Projection Is Modular Reduction}

\subsection{Dropping the Oldest Byte}

The order projection $\pi_k: I_k \to I_{k-1}$ drops the oldest
byte from the context. In integer terms:
\begin{equation}\label{eq:projection}
\pi_k(c) = c \bmod 256^{k-2}.
\end{equation}
This is modular reduction: divide by $256^{k-2}$ and keep the
remainder. The quotient $\lfloor c / 256^{k-2} \rfloor = b_1$
is the dropped byte.

\begin{proposition}[Order Projection = Reduction mod $256^{k-2}$]
The map $\pi_k: \mathbb{Z}/256^{k-1}\mathbb{Z} \to
\mathbb{Z}/256^{k-2}\mathbb{Z}$ defined by
$\pi_k(c) = c \bmod 256^{k-2}$ is a ring surjection. Its kernel
is $256^{k-2}\mathbb{Z}/256^{k-1}\mathbb{Z} \cong
\mathbb{Z}/256\mathbb{Z}$, which is the ``oldest byte'' event
space---the information that the projection discards.
\end{proposition}

\begin{proof}
Modular reduction is a ring homomorphism (it preserves addition
and multiplication). It is surjective since every element of
$\mathbb{Z}/256^{k-2}\mathbb{Z}$ has a preimage. The kernel
consists of elements $c$ with $c \bmod 256^{k-2} = 0$, i.e.,
$c = b_1 \cdot 256^{k-2}$ for $b_1 \in \{0,\ldots,255\}$.
\end{proof}

\subsection{The Equivalence Class Is a Residue Class}

Two contexts $c, c'$ project to the same shorter context iff
$c \equiv c' \pmod{256^{k-2}}$---iff they are in the same residue
class. The equivalence class:
\begin{equation}
[c]_{k-1} = \{c' : c' \equiv c \pmod{256^{k-2}}\}
= \{c + b \cdot 256^{k-2} : b \in \{0,\ldots,255\}\}
\end{equation}
has exactly 256 elements (one per value of the oldest byte $b$).
This is a coset of the kernel.

The lower-order count is the sum over the residue class:
\begin{equation}
c_{k-1}(\pi_k(c), o) = \sum_{c' \in [c]_{k-1}} c_k(c', o)
= \sum_{b=0}^{255} c_k(c \bmod 256^{k-2} + b \cdot 256^{k-2}, \, o).
\end{equation}
Marginalization over the oldest byte = summation over the coset.

\subsection{The Full Tower}

The tower of projections
\begin{equation}
\mathbb{Z}/256^{K-1}\mathbb{Z}
\xrightarrow{\;\bmod 256^{K-2}\;}
\mathbb{Z}/256^{K-2}\mathbb{Z}
\xrightarrow{\;\bmod 256^{K-3}\;}
\cdots
\xrightarrow{\;\bmod 256\;}
\mathbb{Z}/256\mathbb{Z}
\xrightarrow{\;\bmod 1\;}
\{0\}
\end{equation}
is a chain of ring surjections. Each step reduces mod $256^{k-2}$,
discarding one base-256 digit (one byte of context). The full
chain forgets bytes oldest-first, matching the KN backoff order.

On the joint event space, the tower extends by one position:
\begin{equation}
\underbrace{\mathbb{Z}/256^K\mathbb{Z}}_{E_K}
\twoheadrightarrow
\underbrace{\mathbb{Z}/256^{K-1}\mathbb{Z}}_{E_{K-1}}
\twoheadrightarrow \cdots \twoheadrightarrow
\underbrace{\mathbb{Z}/256\mathbb{Z}}_{E_1}
\end{equation}
where each $E_k$ is the ring of $k$-byte events ($(k{-}1)$-byte
context + 1-byte output).


\section{The Discount Is Subtraction}

\subsection{Subtraction in $\mathbb{Z}$}

The KN discount subtracts $D$ from each nonzero count:
\begin{equation}
\tilde{c}_k(c, o) = \max(c_k(c, o) - D, \; 0).
\end{equation}
This is subtraction in $\mathbb{Z}$ (the count ring), clamped at
zero. The counts live in $\mathbb{N} \subset \mathbb{Z}$; the
discount pushes them toward zero, and the clamp prevents them from
going negative.

\subsection{The GCD Is the GCD}

The per-row GCD from the Bayes-from-counting framework is the
GCD in $\mathbb{Z}$:
\begin{equation}
g(c) = \gcd\{c_k(c, o) : c_k(c, o) > 0\}.
\end{equation}
The ``common evidence'' is $g(c)$; the ``reduced counts'' are
$r(c, o) = c_k(c, o) / g(c)$. Since $g(c)$ divides every nonzero
count in the row, division is exact (no remainder). The reduced
counts are coprime: $\gcd\{r(c, o) : r(c, o) > 0\} = 1$.

\subsection{The Discount--GCD Gap Is Subtraction vs.\ Division}

The v1 paper identified the discount--GCD gap as an open question.
With the integer framework, the gap has a precise characterization:

\begin{proposition}[The Gap]
The KN discount subtracts a constant $D$ from each count:
$\tilde{c} = c - D$. The GCD operation divides each count by
$g$: $r = c / g$. These are different operations in
$\mathbb{Z}$:
\begin{itemize}
\item \textbf{Subtraction} $c - D$: shifts the count down by $D$.
  The result depends on $c$: small counts lose a larger fraction
  of their value than large counts.
\item \textbf{Division} $c / g$: scales the count by $1/g$.
  The result is proportional to $c$: all counts lose the same
  fraction $1 - 1/g$ of their value.
\end{itemize}
Subtraction and division agree when $g = 1$ and $D = 1$ (both
reduce each count by 1). They disagree when $g > 1$: division
removes a factor (preserving ratios), while subtraction removes
an additive constant (distorting ratios in favor of large counts).
\end{proposition}

\begin{remark}[Why subtraction ``works'']
For natural language at the byte level, most rows have $g(c) = 1$
(at least one continuation appears exactly once). For these rows,
the GCD operation is ``subtract 1 from each count,'' which is
close to the KN discount $D \approx 0.9$. The rows where $g > 1$
are high-frequency contexts (``th'', ``he'', etc.) where division
would be more appropriate---but these rows also have large counts,
so the distortion from subtraction is small relative to the totals.

The KN discount works \emph{not} because subtraction is the right
operation, but because $g = 1$ almost everywhere, and where $g > 1$
the counts are large enough that the error is negligible.
\end{remark}

\subsection{Per-Row GCD as UM-Native Discount}

The UM-native discount uses division instead of subtraction:
\begin{equation}
r(c, o) = c_k(c, o) \;/\; g(c).
\end{equation}
This is exact integer division (no remainder, by definition of GCD).
The reduced counts are the ``differential evidence''---what remains
after the common evidence $g(c)$ is divided out.

The ``mass'' removed is $c_k(c, o) - r(c, o) = c_k(c,o)(1 - 1/g(c))$,
which varies by row. For $g = 1$: nothing removed ($r = c$). For
$g = 2$: half removed. For $g = 10$: 90\% removed.

The removed mass goes to the backoff distribution, just as in KN.
The difference: KN removes a constant $D$ per continuation; the
GCD removes a fraction $1 - 1/g(c)$ per row. The GCD version is
the ring-native operation.


\section{The Continuation Count Is Fiber Cardinality}

\subsection{Fibers of the Output Projection}

Consider the output projection $\rho: E_k \to O$ that extracts
the output byte: $\rho(c, o) = o$, i.e., $\rho(e) = e \bmod 256$.
This is reduction mod 256 on the joint ring.

The fiber of $o$ under $\rho$ is:
\begin{equation}
\rho^{-1}(o) = \{e \in E_k : e \equiv o \pmod{256}\}
= \{256c + o : c \in I_k\}.
\end{equation}
This is a coset of the kernel $256\mathbb{Z}/256^k\mathbb{Z}$
in the joint ring.

\subsection{Continuation Count = Non-Empty Fiber Sections}

The continuation count $c_{\mathrm{KN}}(o)$ counts the number of
distinct contexts $c$ such that $c_k(c, o) > 0$:
\begin{equation}
c_{\mathrm{KN}}(o) = |\{c \in I_k : c_k(256c + o) > 0\}|.
\end{equation}

This is the number of ``occupied'' positions in the fiber
$\rho^{-1}(o)$---the fiber cardinality restricted to the support
of $c_k$.

\begin{proposition}[Continuation Count as Fiber Size]
Define the support $S = \{e \in \mathbb{Z}/256^k\mathbb{Z} :
c_k(e) > 0\}$ (the set of observed $k$-grams). The continuation
count of output $o$ is:
\begin{equation}
c_{\mathrm{KN}}(o) = |S \cap \rho^{-1}(o)|
\end{equation}
---the size of the support's intersection with the output fiber.
\end{proposition}

In the ring, $\rho^{-1}(o)$ is a coset with $|I_k| = 256^{k-1}$
elements. The continuation count measures what fraction of this
coset is ``populated'' in the data. High continuation count = the
output byte $o$ appears in many contexts (general). Low = it
appears in few contexts (specific).

\subsection{Why Type Counts Matter}

The raw count $c_k(\cdot, o) = \sum_c c_k(c, o)$ sums over the
fiber \emph{with multiplicity}: each context contributes its count.
High-frequency contexts dominate. The continuation count $c_{\mathrm{KN}}(o)$
sums over the fiber \emph{without multiplicity}: each context
contributes 1 or 0. This is the type count---the number of
distinct coset representatives that are populated.

In ring terms: the raw count is the $L^1$ norm of $c_k$ restricted
to the coset; the continuation count is the $L^0$ ``norm'' (support
size). The continuation count sees the geometry of the support
(how spread out is $o$ across contexts?) while the raw count sees
the measure (how much total mass does $o$ have?).


\section{Interpolation as the Tower of Ring Surjections}

\subsection{The Recursion on the Tower}

The KN interpolation recursion:
\begin{equation}\label{eq:kn-recursion}
P_k(o \mid c) = \frac{\max(c_k(c, o) - D, 0)}{c_k(c, \cdot)}
+ \frac{D \cdot \tau_k(c)}{c_k(c, \cdot)} \cdot
P_{k-1}(o \mid c \bmod 256^{k-2})
\end{equation}
is a recursion along the tower of ring surjections. At each level:

\begin{enumerate}
\item \textbf{Evaluate on the current ring:} Look up $c_k(c, o)$ in
  $\mathbb{Z}/256^k\mathbb{Z}$. Subtract $D$. Normalize.
\item \textbf{Project to the next ring:} Compute
  $c \bmod 256^{k-2}$ (reduce mod the next-lower ring).
\item \textbf{Delegate the residual:} Pass the backoff mass to the
  prediction on the reduced ring $\mathbb{Z}/256^{k-1}\mathbb{Z}$.
\end{enumerate}

\begin{proposition}[KN Recursion = Tower Descent]
The interpolated KN prediction is computed by descending the tower
of rings:
\begin{equation}
\mathbb{Z}/256^K\mathbb{Z}
\xrightarrow[\text{backoff residual}]{\bmod 256^{K-2}}
\mathbb{Z}/256^{K-1}\mathbb{Z}
\xrightarrow[\text{backoff residual}]{\bmod 256^{K-3}}
\cdots
\xrightarrow[\text{backoff residual}]{\bmod 1}
\mathbb{Z}/256\mathbb{Z}
\end{equation}
At each level, the prediction is a convex combination of the
discounted counts (evidence specific to this ring) and the
delegation to the next-lower ring (evidence from the coarser ring).
The discount $D$ controls the split: how much mass is ``kept''
(used at this level) vs.\ ``passed down'' (delegated to the
coarser level).
\end{proposition}

\subsection{What Each Level Contributes}

At order $k$, the prediction uses the count $c_k(c, o)$ which
lives on $\mathbb{Z}/256^k\mathbb{Z}$. This count contains:
\begin{itemize}
\item Information from the full $(k{-}1)$-byte context $c$
  (specific to this ring).
\item Information from the shorter context $c \bmod 256^{k-2}$
  (shared with the lower ring).
\end{itemize}

The discount removes the shared part (approximately---via
subtraction rather than division). The residual $\max(c_k - D, 0)$
is the information specific to this level: what the full context
$c$ knows that the shorter context $c \bmod 256^{k-2}$ does not.

This is the ring-theoretic statement of the v1 paper's ``common
evidence removal'': the common evidence lives on the lower ring
(it is invariant under the projection), and the discount
approximately removes it.


\section{The Unified Ring}

\subsection{Patterns as Ring Elements}

The unified pattern table from v1 becomes a function on the
disjoint union of rings:
\begin{equation}
P_{\mathrm{KN}} \subseteq
\bigsqcup_{k=1}^{K} \mathbb{Z}/256^k\mathbb{Z}.
\end{equation}
Each pattern is an element $e \in \mathbb{Z}/256^k\mathbb{Z}$ for
some order $k$, together with a weight $w_k(e)$ (the residual
log-count).

The forward pass evaluates all rings simultaneously:
\begin{equation}
P(o \mid c) \propto \sum_{k=1}^{K}
\underbrace{\alpha_k(c)}_{\text{backoff weight}}
\cdot
\underbrace{\frac{\max(c_k(c, o) - D, 0)}{c_k(c, \cdot)}}_{\text{discounted probability on ring $k$}}.
\end{equation}
The backoff weights $\alpha_k$ are determined by the recursion:
they distribute mass across the tower according to how much
evidence each ring contributes.

\subsection{The CRT Perspective}

Although $\mathbb{Z}/256^k\mathbb{Z}$ does not factor via CRT
(since $256 = 2^8$ and all factors share the prime 2), the tower
\emph{does} have a CRT-like structure when we factor differently.

Consider the event $e = 256c + o$ as an element of
$\mathbb{Z}/256^k\mathbb{Z}$. The information in $e$ decomposes
as:
\begin{equation}
e \bmod 256 = o \quad \text{(output byte)},
\quad \lfloor e / 256 \rfloor = c \quad \text{(context)}.
\end{equation}
This is not CRT (since $256$ is not coprime to $256^{k-1}$), but
it is the mixed-radix decomposition. In terms of the ring:
$\mathbb{Z}/256^k\mathbb{Z}$ has a filtration
$0 \subset 256\mathbb{Z}/256^k\mathbb{Z} \subset
256^2\mathbb{Z}/256^k\mathbb{Z} \subset \cdots$ whose successive
quotients are each $\cong \mathbb{Z}/256\mathbb{Z}$.

Each quotient corresponds to one byte position: the output, the
most recent context byte, the next-most-recent, etc. The KN tower
descends this filtration from the top (full context) to the bottom
(output only).

\subsection{Extending the Ring with Word Events}

The ring $\mathbb{Z}/256^k\mathbb{Z}$ encodes byte-level contexts.
To add word-level events, we extend the ring.

Let $W$ be a word vocabulary of size $|W|$. A word event is an
element of $\mathbb{Z}/|W|\mathbb{Z}$. The extended event space is:
\begin{equation}
E_{\text{ext}} = \mathbb{Z}/256^k\mathbb{Z} \times \mathbb{Z}/|W|\mathbb{Z}
\cong \mathbb{Z}/(256^k \cdot |W|)\mathbb{Z}
\end{equation}
where the isomorphism holds when $\gcd(256^k, |W|) = 1$ (CRT
applies; choose $|W|$ coprime to 256, e.g., $|W| = 65537$, a
Fermat prime).

\begin{proposition}[Words as an Independent Ring Factor]
If $|W|$ is coprime to 256, the word event space is independent
of the byte event space (in the CRT sense). The word identity
$w = e \bmod |W|$ and the byte context $c = e \bmod 256^k$ can
be recovered independently from the joint event $e$. No
``combination'' is needed---CRT gives the decomposition for free.
\end{proposition}

This is the algebraic resolution of the combination problem: by
choosing $|W|$ coprime to 256, the word and byte event spaces are
guaranteed to be independent ring factors, and the CRT provides
the exact decomposition. The ad-hoc mixing that failed in the
match-model experiments is replaced by exact algebraic factoring.


\section{The Discount--GCD Gap, Resolved}

\subsection{Three Operations on Counts}

On a row of the count table $(c_1, c_2, \ldots, c_m)$ (the counts
for context $c$ across outputs $o_1, \ldots, o_m$), three
operations compete:

\begin{center}
\begin{tabular}{@{}lll@{}}
\toprule
Operation & Formula & Effect on ratios \\
\midrule
KN discount (subtract $D$) & $c_i \mapsto c_i - D$ & Distorted
  (small $c_i$ lose more) \\
GCD division (divide by $g$) & $c_i \mapsto c_i / g$ & Preserved
  (all shrink by $1/g$) \\
Log-subtract ($-1$ in log) & $c_i \mapsto c_i / 2$ & Preserved
  (all halve) \\
\bottomrule
\end{tabular}
\end{center}

\begin{theorem}[Resolution of the Gap]
The discount--GCD gap has three components:
\begin{enumerate}
\item \textbf{Subtraction vs.\ division.} KN subtracts; the ring
  operation is division. These agree when $g = 1$ and $D \leq 1$
  (the common case). The gap is nonzero only for rows with $g > 1$.
\item \textbf{Global vs.\ per-row.} KN uses a single $D$ for all
  rows; the GCD is per-row. The global $D$ is a compromise. The
  optimal $D^*$ minimizes the mean divergence between the
  subtracted and divided distributions.
\item \textbf{Real vs.\ integer.} $D$ is a real number; $g$ is a
  positive integer. The fractional discount $D \in (0,1)$ has no
  ring interpretation---it is an artifact of the estimation procedure.
  The integer GCD is the ring-native operation.
\end{enumerate}
The gap is small because: (a) $g = 1$ for most rows (component 1
vanishes), (b) the per-row GCDs are concentrated at 1 (component 2
is small), and (c) $D \approx 0.9 \approx 1 = g$ for the typical
row (component 3 is small).
\end{theorem}

\subsection{The UM-Native Version}

The ring-native KN model would:
\begin{enumerate}
\item Compute $g(c)$ per row.
\item Divide: $r(c, o) = c_k(c, o) / g(c)$.
\item Backoff mass = $c_k(c, \cdot) - \sum_o r(c, o) \cdot g(c)
  = c_k(c,\cdot)(1 - 1/g(c)) \cdot g(c)$ counts worth.
\item Continuation count: unchanged (it is already ring-native).
\item Recursion: unchanged in structure; only the discount step
  changes from subtraction to division.
\end{enumerate}

For rows with $g = 1$ (the vast majority), $r(c, o) = c_k(c, o)$
and zero mass is removed---backoff comes entirely from the
continuation-count term. For rows with $g > 1$, more mass is
removed (proportionally, preserving ratios), and more is delegated
to the lower ring.


\section{What the Ring Structure Reveals}

The restatement on the integers makes several structural facts
visible:

\begin{enumerate}
\item \textbf{The tower is a filtration.} The KN order tower is
  the filtration of $\mathbb{Z}/256^K\mathbb{Z}$ by powers of
  256. This is not a choice---it is the unique filtration by the
  prime $p = 2$ (since $256 = 2^8$). The ``order'' of an $n$-gram
  is its position in the 2-adic filtration.

\item \textbf{Context is a $p$-adic integer.} An infinite context
  (the full history) is a 256-adic integer: an element of the
  inverse limit $\varprojlim_k \mathbb{Z}/256^k\mathbb{Z}
  = \mathbb{Z}_{256}$. The KN model at order $K$ approximates
  this by truncating to $K$ digits. The ``infinite-order'' model
  works in $\mathbb{Z}_{256}$ directly.

\item \textbf{The discount is a derivation.} Subtraction of a
  constant from each count is an additive perturbation of the
  count function: $\tilde{c}_k = c_k - D \cdot \mathbf{1}_{c_k > 0}$.
  In ring terms, this is $c_k$ minus $D$ times the indicator
  function of the support. The indicator function of the support is
  the \emph{radical} of the count function (in the sense of
  radical ideals: $\sqrt{(c_k)}$). The discount is subtraction of
  (a multiple of) the radical.

\item \textbf{Interpolation is a weighted inverse limit.} The full
  KN prediction at order $K$ is a weighted average over all levels
  of the tower. In the inverse limit, this becomes a weighted sum
  over all digits of the 256-adic expansion, with weights
  determined by the discount and continuation counts.

\item \textbf{The combination problem is CRT.} Adding new event
  spaces (words, phrases, match contexts) to the byte ring requires
  either:
  \begin{itemize}
  \item Coprime extension: $|W|$ coprime to 256, giving CRT
    factorization (clean, independent).
  \item Non-coprime extension: $|W|$ sharing factors with 256,
    giving a non-split extension (tangled, requires careful
    handling).
  \end{itemize}
  The coprime case is the algebraically clean solution to the
  combination problem.
\end{enumerate}


\section{Research Agenda (Updated)}

\begin{enumerate}
\item \textbf{Per-row GCD discount.} Implement and benchmark
  against global $D$. Prediction: identical for most rows ($g=1$),
  better for high-frequency rows ($g > 1$).

\item \textbf{Ring-native forward pass.} Implement the count
  lookup as ring arithmetic: context = integer, projection =
  mod $256^{k-2}$, pattern match = equality in the ring.
  Measure whether this simplifies or speeds up the implementation.

\item \textbf{CRT word extension.} Choose $|W| = 65537$ (Fermat
  prime, coprime to 256). Encode word events as
  $e \bmod 65537$. Test whether CRT decomposition gives clean
  combination without the mixing catastrophes of v1--v19.

\item \textbf{256-adic analysis.} Treat the full history as a
  256-adic integer. Investigate whether $p$-adic analysis
  (continuity, derivatives, integration in $\mathbb{Z}_{256}$)
  gives useful tools for analyzing KN-like models.

\item \textbf{Spectral methods.} The Fourier transform on
  $\mathbb{Z}/256^k\mathbb{Z}$ (Pontryagin dual) decomposes the
  count function into characters. Investigate whether peaks in the
  character spectrum correspond to linguistic patterns (repeated
  $k$-grams, periodic structure).
\end{enumerate}


\section{Conclusion}

Kneser--Ney smoothing, restated on the integers:

\begin{center}
\begin{tabular}{@{}ll@{}}
\toprule
KN Component & Ring Operation \\
\midrule
Context of order $k$ & Integer $c \in \mathbb{Z}/256^{k-1}\mathbb{Z}$ \\
Joint event & Integer $e = 256c + o \in \mathbb{Z}/256^k\mathbb{Z}$ \\
Order projection & $c \bmod 256^{k-2}$ (modular reduction) \\
Count table & Function $c_k: \mathbb{Z}/256^k\mathbb{Z} \to \mathbb{N}$ \\
Discount & Subtraction (approx.\ division by $g$) \\
GCD & $\gcd$ in $\mathbb{Z}$ \\
Continuation count & $|S \cap \rho^{-1}(o)|$ (fiber support size) \\
Interpolation & Weighted tower of ring surjections \\
Backoff & Delegation to coarser ring via $\bmod$ \\
Full history & 256-adic integer $\in \mathbb{Z}_{256}$ \\
Word extension & CRT factor ($|W|$ coprime to 256) \\
\bottomrule
\end{tabular}
\end{center}

Every entry in this table is a mathematical identity, not an
analogy. The integers are the events; the rings are the event
spaces; modular reduction is the projection; division is the
discount. The combination problem dissolves into CRT when word
vocabularies are chosen coprime to 256.

The v1 paper asked ``what is the discount--GCD gap?'' The answer:
subtraction vs.\ division in $\mathbb{Z}$, which is small because
most rows have $\gcd = 1$. The v1 paper asked ``how do we combine
KN with word models?'' The answer: CRT, by choosing $|W|$ coprime
to the byte alphabet size.

\begin{thebibliography}{99}
\bibitem{kn-v1}
Claude and MJC.
\newblock \emph{Kneser--Ney as Quotient: Pulling $n$-gram Smoothing
into the Universal Model}.
\newblock Hutter archive, 16 Feb 2026. (v1 of this paper.)

\bibitem{um-arith-v4}
Claude and MJC.
\newblock \emph{Integer Factorization of Events: Every Integer Is an
Event, Every Quotient Is Division}.
\newblock Hutter archive, 17 Feb 2026.

\bibitem{kn-scaling}
Claude and MJC.
\newblock \emph{Scaling Byte-Level Kneser--Ney to 1.78\,bpc on enwik9}.
\newblock Hutter archive, 12 Feb 2026.

\bibitem{match-model}
Claude and MJC.
\newblock \emph{Match Models, Sparse Contexts, and the Combination
Problem}.
\newblock Hutter archive, 16 Feb 2026.

\bibitem{bayes-counting}
Claude and MJC.
\newblock \emph{Bayes from Counting}.
\newblock Hutter archive, 12 Feb 2026.
\end{thebibliography}

\end{document}
