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\title{Vanishing of Higher Magnitude Homology\\
for Finite Prefix Models}
\author{Ryoji Hirayama\\[0.4em]
\small Graduate School of Mathematics, Nagoya University}
\date{}

\begin{document}
\maketitle

\begin{abstract}
Bradley and Vigneaux define a generalized distance on a finite prefix poset
using next-token probabilities. They conjecture that its higher magnitude
homology vanishes. Motivated by their expression of magnitude as
a weighted sum of homological Euler characteristics, we verify this
conjecture for their explicit complex, in which both endpoint faces are zero.
For each finite length, the homology groups in degrees two and above are zero.
The degree-one homology group is free abelian, with one generator for each
positive-probability edge of that length.
The proof constructs a contracting homotopy for each component whose endpoints
have an interior prefix. This homotopy inserts the least interior prefix.
Edge probabilities zero and one are both allowed.
\end{abstract}

\section{Introduction}

Bradley--Vigneaux \cite[\S\S2, 3.13]{BV25} model next-token probabilities
by an enriched category of texts and express its magnitude as a weighted
sum of Euler characteristics. They describe the degree-zero and degree-one
homology groups \cite[p.~22]{BV25}, following the low-degree calculations
of Leinster--Shulman \cite[Theorems 4.1 and 4.3]{LS21}, and conjecture that
the higher groups vanish. An identity of alternating sums alone does not
imply this vanishing. Theorem~\ref{thm:vanishing} verifies their conjecture
for the finite prefix model and the explicit complex of
\cite[\S3.13]{BV25}.

Our contraction is the elementary cone contraction on each nonempty open
prefix interval, in keeping with the interval order-complex viewpoint of
Kaneta--Yoshinaga \cite[\S4.3]{KY21}. We give it directly at the chain level
to include asymmetric distances and zero-length edges, with the endpoint
convention specified below.

\section{Prefix order and the probabilistic distance}

Let $M$ be a monoid. Define $a\preceq b$ if $at=b$ for some $t\in M$.
This relation is a preorder. For a free monoid, this is the prefix order.
Fix a finite alphabet $A$, distinct beginning and end symbols
$\bos,\eos\notin A$, and an integer $N\ge1$.
Put $\Tok=A\sqcup\{\eos\}$ and equip
\[
 L=\{\bos a\mid a\in A^*,\ |a|\le N-1\}
 \sqcup
 \{\bos a\eos\mid a\in A^*,\ |a|\le N-2\}
\]
with the prefix order. A set with a negative length bound is understood to
be empty. For $x\in L$, let $|x|$ denote its length, including the beginning
and end symbols. Let $I\subseteq L$ be the set of nonmaximal elements.
The elements covering $x\in I$ are precisely $xc$ for $c\in\Tok$.
If $x<y$, the interval $[x,y]$ is the unique chain
\[
 x=z_0<z_1<\cdots<z_m=y,\qquad
 z_i=z_{i-1}c_i,\quad m=|y|-|x|.
\]

For each $x\in I$, choose a probability distribution
$p_x\colon\Tok\to[0,1]$, so that
$\sum_{c\in\Tok}p_x(c)=1$.
Following Bradley--Vigneaux \cite[\S2]{BV25}, define
\[
 \pr{y}{x}=
 \begin{cases}
  1 & x=y,\\
  \displaystyle\prod_{i=1}^m p_{z_{i-1}}(c_i) & x<y,\\
  0 & x\not\le y.
 \end{cases}
\]
For $x\le y\le z$ we have
$\pr{z}{x}=\pr{y}{x}\pr{z}{y}$, and for arbitrary $x,y,z$,
\[
 \pr{y}{x}\pr{z}{y}\le\pr{z}{x}.
\]
Thus $\cL(x,y)=\pr{y}{x}$ defines a category enriched over $[0,1]$
with its usual order and multiplication.

With the convention $-\log0=\infty$, put
\[
 \operatorname{ob}\cM=L,\qquad d(x,y)=-\log\pr{y}{x}.
\]
Then $d(x,x)=0$ and $d(x,z)\le d(x,y)+d(y,z)$, so $\cM$ is a
generalized metric space in the sense of Lawvere \cite{Lawvere73}, allowing
asymmetric and infinite distances.
See also \cite[Section 2, Example 2.9]{LS21}.
For $x\le y\le z$ we moreover have
\begin{equation}\label{eq:additive}
 d(x,z)=d(x,y)+d(y,z).
\end{equation}
An edge $x\to xc$ has \emph{positive probability} if $p_x(c)>0$;
its length is $-\log p_x(c)$. Such an edge may have length zero.

\section{The magnitude complex}

We assume throughout that the length parameter $\ell$ satisfies
$0\le\ell<\infty$. For $k\ge0$, let
\begin{align*}
 G_{k,\ell}
 &=\left\{(x_0,\ldots,x_k)\in L^{k+1}\ \middle|\
 x_i\ne x_{i+1}\ (0\le i<k),\
 \sum_{i=0}^{k-1}d(x_i,x_{i+1})=\ell\right\},\\
 \MC_{k,\ell}(\cM)&=\Z[G_{k,\ell}],
\end{align*}
where $\Z[S]$ denotes the free abelian group on a set $S$.
For $k=0$ the sum is zero, so $\MC_{0,0}=\Z[L]$ and
$\MC_{0,\ell}=0$ for $\ell>0$.

\begin{lemma}\label{lem:chains}
For $k\ge1$, a tuple $(x_0,\ldots,x_k)$ belongs to $G_{k,\ell}$ if and only if
\[
 x_0<x_1<\cdots<x_k,\qquad
 \pr{x_k}{x_0}>0,\qquad d(x_0,x_k)=\ell.
\]
Every interior point of such a tuple satisfies
\[
 d(x_{i-1},x_i)+d(x_i,x_{i+1})=d(x_{i-1},x_{i+1}).
\]
\end{lemma}
\begin{proof}
The condition $d(u,v)<\infty$ is equivalent to
$u\le v$ and $\pr{v}{u}>0$.
Let $(x_0,\ldots,x_k)\in G_{k,\ell}$. Each adjacent distance is finite.
Since adjacent points are distinct, we have $x_0<x_1<\cdots<x_k$.
By \eqref{eq:additive}, its total length equals
the distance between its endpoints. The converse and the assertion about
interior points follow from the same additivity.
\end{proof}

By Lemma~\ref{lem:chains}, the magnitude differential of
\cite[\S3.13]{BV25}, specialized to $\cM$, takes the form
\begin{equation}\label{eq:boundary}
 \partial(x_0,\ldots,x_k)
 =\sum_{i=1}^{k-1}(-1)^i
 (x_0,\ldots,\widehat{x_i},\ldots,x_k).
\end{equation}
For $k\le1$, set $\partial=0$.
In particular, \emph{both endpoint faces are always zero}.
Deleting an interior point preserves the endpoints and total length and
leaves a strict chain. Deleting two interior points in either order gives
the same tuple. The corresponding terms have opposite signs, so they cancel.
Thus $\partial^2=0$.
We write
\[
 H_{k,\ell}(\cM)=
 \frac{\ker(\partial\colon\MC_{k,\ell}\to\MC_{k-1,\ell})}
 {\operatorname{im}(\partial\colon\MC_{k+1,\ell}\to\MC_{k,\ell})},
 \qquad \MC_{-1,\ell}=0.
\]

\section{Vanishing by contraction of intervals}

\begin{theorem}\label{thm:vanishing}
Let
\[
 E^+_\ell=
 \{\,x\to xc\mid x\in I,\ c\in\Tok,\
 p_x(c)>0,\ -\log p_x(c)=\ell\,\}
\]
be the set of positive-probability edges of length $\ell$.
For every $0\le\ell<\infty$,
\begin{align*}
 H_{k,\ell}(\cM)&=0 &&(k\ge2),\\
 H_{1,\ell}(\cM)&\cong\Z[E^+_\ell],\\
 H_{0,0}(\cM)&\cong\Z[L],\qquad
 H_{0,\ell}(\cM)=0 &&(\ell>0).
\end{align*}
The degree-one isomorphism identifies an edge $x\to xc$ with the homology
class of the cycle $(x,xc)$.
\end{theorem}

\begin{proof}
The differential preserves endpoints. Hence the complex in degrees at least
one splits as a direct sum of components $C(x,y)$. The sum runs over all
pairs $(x,y)$ satisfying $x<y$, $\pr{y}{x}>0$, and $d(x,y)=\ell$.
For $q\ge1$, a basis of $C(x,y)_q$ consists of the tuples
\[
 \sigma=(x,z_1,\ldots,z_{q-1},y),\qquad
 x<z_1<\cdots<z_{q-1}<y,
\]
and we set $C(x,y)_0=0$.
Let $V_\ell$ be the complex with $\Z[L]$ in degree zero when $\ell=0$
and zero in all other cases. Its differential is zero.
The full decomposition is
\begin{equation}\label{eq:endpoint-decomposition}
 \MC_{*,\ell}(\cM)\cong V_\ell\oplus
 \bigoplus_{\substack{x<y,\;\pr{y}{x}>0\\d(x,y)=\ell}}C(x,y).
\end{equation}

If the open interval $(x,y)=\{z\mid x<z<y\}$ is empty, then $y=xc$.
The component $C(x,y)$ consists solely of $\Z(x,y)$ in degree one,
contributing one copy of $\Z$ for the corresponding edge.

Suppose instead that $(x,y)$ is nonempty, and let $v$ be its least element.
For each $q\ge1$, we define a homomorphism
$h:C(x,y)_q\to C(x,y)_{q+1}$. This map raises the homological degree by one.
On basis elements, we set
\[
 h(\sigma)=
 \begin{cases}
  -(x,v,z_1,\ldots,z_{q-1},y)
       & v\notin\{z_1,\ldots,z_{q-1}\},\\
  0    & v\in\{z_1,\ldots,z_{q-1}\}.
 \end{cases}
\]
Inserting $v$ leaves a strict chain because $v$ is the least interior
prefix. It also preserves the total length, which depends only on the
endpoints.

If $v$ does not occur in $\sigma$, the term of $\partial h(\sigma)$
obtained by deleting $v$ is $+\sigma$: the minus sign in $h$ cancels the
boundary sign $(-1)^1$. The term obtained by deleting $z_j$ has sign
$(-1)^j$, whereas its counterpart in $h\partial(\sigma)$ has sign
$(-1)^{j+1}$. These terms cancel, giving
$\partial h(\sigma)+h\partial(\sigma)=\sigma$.
This includes $q=1$, since $\partial(-(x,v,y))=(x,y)$.

If $v$ occurs in $\sigma$, then $z_1=v$ and $h(\sigma)=0$.
Deleting $v$ from $\sigma$ gives the term $-(x,z_2,\ldots,z_{q-1},y)$ in
$\partial(\sigma)$, and $h$ sends this term to $\sigma$.
Every other term retains $v$ and is sent to zero by $h$.
Thus in both cases
\[
 \partial h+h\partial=\operatorname{id}_{C(x,y)}.
\]
Every cycle $z$ in this component therefore satisfies $z=\partial h(z)$,
so the component has zero homology in every degree.

Taking the direct sum, only the positive-probability edges contribute in
degree one, and every summand has zero homology in degrees at least two.
The assertion in degree zero follows from the remaining summand.
\end{proof}

\begin{remark}[Weights and zero distances]
The proof does not use the normalization of the probabilities, so the
same theorem holds for arbitrary edge weights in $[0,1]$.
An edge of probability one remains a generator of $H_{1,0}$.
In the presence of such edges, the complex used here must be distinguished
from the magnitude complex for general enriched categories of
Leinster--Shulman \cite[Definition 5.7]{LS21}.
In the latter complex, an endpoint face can survive when the corresponding
end edge has length zero. For example, on the two-point prefix space
$x<y$ with $d(x,y)=0$, one has
$\partial^{\mathrm{LS}}(x,y)=(y)-(x)$.
If every edge probability is strictly less than one, all distances between
distinct points are positive or infinite, and the two complexes coincide.
\end{remark}

\begin{thebibliography}{1}

\bibitem{BV25}
T.-D. Bradley and J.~P. Vigneaux.
\newblock The magnitude of categories of texts enriched by language models.
\newblock arXiv:2501.06662v2, 2025.
\newblock \url{https://arxiv.org/abs/2501.06662v2}.

\bibitem{LS21}
T.~Leinster and M.~Shulman.
\newblock Magnitude homology of enriched categories and metric spaces.
\newblock {\em Algebraic \& Geometric Topology}, 21:2175--2221, 2021.
\newblock \url{https://arxiv.org/abs/1711.00802}.

\bibitem{KY21}
R.~Kaneta and M.~Yoshinaga.
\newblock Magnitude homology of metric spaces and order complexes.
\newblock {\em Bulletin of the London Mathematical Society}, 53:893--905, 2021.
\newblock \url{https://arxiv.org/abs/1803.04247}.

\bibitem{Lawvere73}
F.~W. Lawvere.
\newblock Metric spaces, generalized logic and closed categories.
\newblock {\em Rendiconti del Seminario Matematico e Fisico di Milano},
  43:135--166, 1973.
\newblock Reprinted in \emph{Reprints in Theory and Applications of Categories}
  \textbf{1} (2002), 1--37.
  \url{https://tac.mta.ca/tac/reprints/articles/1/tr1abs.html}.

\end{thebibliography}
\end{document}