# Enumerating partitions arising in homotopy theory

Scott M. Bailey and Donald M. Larson

###### Abstract.

We present an infinite family of recursive formulas that count binary integer
partitions satisfying natural divisibility conditions and show that these
counts are interrelated via partial sums. Moreover, we interpret the
partitions we study in the language of graded polynomial rings and apply this
to the mod $2$ Steenrod algebra to compute the free rank of certain homology
modules in stable homotopy theory.

Keywords. Steenrod algebra, Brown-Gitler modules, partitions.

Mathematics Subject Classification 2020. 55S10, 16W50, 55T15, 05A17

##  1\. Introduction

A binary partition of an integer $n\geq 0$ is a partition of $n$ whose parts
are powers of 2. Such partitions have the form $n=\alpha_{1}\cdot
2^{0}+\alpha_{2}\cdot 2^{1}+\cdots+\alpha_{k}\cdot 2^{k-1}$ for
$0\leq\alpha_{i}\in\mathbb{Z}$, and it is well-known how to count them
recursively. Indeed, if $r_{-1}(n)$ denotes the number of binary partitions of
$n$, then $r_{-1}(n)=r_{-1}(n-1)$ for $n$ odd and
$r_{-1}(n)=r_{-1}(n-1)+r_{-1}(n/2)$ for $n$ even, where $r_{-1}(0)=1$. The
sequence $r_{-1}(n)$ has as its first few terms

$1,1,2,2,4,4,6,6,10,10,14,14,20,20,26,26,\ldots$

and appears as entry A018819 in the On-Line Encyclopedia of Integer Sequences
(OEIS) [OEI]. The subscript $-1$ is a notational convenience, as the sequence
$r_{-1}(n)$ will be subsumed into an infinite family of sequences
$\\{r_{m}(n):m\geq-1\\}$ whose provenance we shall now begin to describe.

The conspicuous two-fold repetition in the sequence $\\{r_{-1}(n)\\}$
disappears if we instead count binary partitions of $n$ for which
$2\mid\alpha_{1}$ (that is to say, binary partitions with an even number of
1s). If $r_{0}(n)$ denotes the number of binary partitions with the
aforementioned divisibility condition on $\alpha_{1}$, then
$r_{0}(n)=r_{0}(n-2)+r_{0}((n-2)/2)$ for $n\equiv 2\mod 4$ and
$r_{0}(n)=r_{0}(n-2)+r_{0}(n/2)$ for $n\equiv 0\mod 4$, where $r_{0}(0)=1$.
Ignoring the obvious zeros occurring when $n$ is odd yields the sequence
$r_{0}(2n)$ which has as its first few terms

$1,2,4,6,10,14,20,26,36,46,60,74,94,114,140,166,\ldots$

and appears as entry A000123 of the OEIS, where it is also observed that
$r_{0}(2n)$ is the sum of $r_{-1}(i)$ from $i=0$ to $i=n$.

A desire to perpetuate the partial sum relationship would dictate setting
$r_{1}(n)$ equal to the number of binary partitions of $n$ for which
$4\mid\alpha_{1}$ and $2\mid\alpha_{2}$ (see entry A131205 in the OEIS). In
fact, the entire infinite family of counts that is the subject of this paper
emerges in precisely this fashion. Let $r_{m}(n)$ denote the number of binary
partitions of $n$ for which the $m+1$ divisibility conditions

(1) $2^{m+2-i}\mid\alpha_{i},\quad 1\leq i\leq m+1$

hold. Our main result is a recursive formula for $r_{m}(n)$.

######  Theorem 1.1.

For $0<n\equiv 2^{m+1}\mod 2^{m+2}$, we have the recursive formula

$r_{m}(n)=r_{m}(n-2^{m+1})+\sum_{j=0}^{\lfloor
m/2\rfloor}\binom{m+1}{2j+1}r_{m}\left(\dfrac{n-(2j+1)2^{m+1}}{2}\right).$

Similarly, for $0\leq n\equiv 0\mod 2^{m+2}$,

$r_{m}(n)=r_{m}(n-2^{m+1})+\sum_{j=0}^{\lfloor(m+1)/2\rfloor}\binom{m+1}{2j}r_{m}\left(\dfrac{n-2j\cdot
2^{m+1}}{2}\right).$

For all other $n$, $r_{m}(n)=0$.

Furthermore, the sequences $r_{m}(n)$ are all related to each other via
partial sums.

######  Theorem 1.2.

For $m\geq 0$ and $n\equiv 0\mod 2^{m+1}$, $\displaystyle
r_{m}(n)=\sum_{k=0}^{n/2^{m+1}}r_{m-1}(2^{m}k)$.

Theorem 1.1 can be framed in terms of the combinatorics of graded polynomial
rings. Let

$H(m)=\mathbb{F}_{2}[X_{1}^{2^{m+1}},X_{2}^{2^{m}},X_{3}^{2^{m-1}},\ldots,X_{m+1}^{2},X_{m+2},X_{m+3},\ldots]$

where the indeterminate $X_{i}$ has grading $|X_{i}|=2^{i-1}$. Then a monomial
$x\in H(m)$ of grading $n$ of the form
$x=X_{1}^{\alpha_{1}}X_{2}^{\alpha_{2}}\cdots X_{k}^{\alpha_{k}}$ is precisely
the same data as a partition $n=\alpha_{1}\cdot 2^{0}+\alpha_{2}\cdot
2^{1}+\cdots+\alpha_{k}\cdot 2^{k-1}$ of $n$ satisfying the divisibility
conditions (1), yielding the following corollary.

######  Corollary 1.3.

The number of monomials $x\in H(m)$ with grading $n$ is $r_{m}(n)$.

While of possible independent interest in number theory and theoretical
computer science, the quantities $r_{m}(n)$ and their interpretation in
Corollary 1.3 also have a natural application in stable homotopy theory to
splittings of the homology of Brown-Gitler spectra.

######  Theorem 1.4.

Let $0\leq j\in\mathbb{Z}$. As a module over the subalgebra $\mathscr{A}(1)$
of the mod 2 Steenrod algebra $\mathscr{A}$ generated by $Sq^{1}$ and
$Sq^{2}$, the mod 2 homology of the $j$th integral Brown-Gitler spectrum
$B_{1}(j)$ has free rank $(r_{1}(4j)-b(j))/8$ where

(2) $b(j)=\begin{cases}4j-2\alpha(j)+1,&\text{if }\alpha(j)\equiv
0,1\mod{4},\\\ 4j-2\alpha(j)+5,&\text{if }\alpha(j)\equiv
2,3\mod{4}\end{cases}$

and where $\alpha(j)$ is number of $1s$ in the binary expansion of $j$.

(Compare with Theorem 2.2.)

Section 2 is devoted to a discussion of Theorem 1.4 and its proof, including
all the requisite definitions and a sketch of the relevant background
material. Sections 3, 4, and 5 comprise the technical heart of the paper.
Section 3 gives results required for the proof of Theorem 1.1 in Section 4.
Section 5 contains the proof of the partial sums result given by Theorem 1.2.

### Acknowledgements

The authors would like to thank Paul Goerss, Mark Behrens, and the anonymous
referee for useful comments and suggestions.

##  2\. The homology of Brown-Gitler spectra

Originally introduced for the purpose of studying obstructions to immersions
of manifolds, Brown-Gitler spectra are geometric objects tailored to have
(co)homology modules with certain properties. Among the surprisingly numerous
results in stable homotopy theory they have spawned are the wedge sum
splittings of spectra related to connective real $K$-theory found by Mahowald
[Mah81] and the first author [Bai10]. In this section, we apply Corollary 1.3
to compute the free rank of the homology (with coefficients in
$\mathbb{F}_{2}$) of a certain family of Brown-Gitler spectra as modules over
$\mathscr{A}(1)$, the submodule of the mod 2 Steenrod algebra $\mathscr{A}$
generated by $Sq^{1}$ and $Sq^{2}$. We avail ourselves of the language of
stable homotopy theory to describe our result: spectra and the cohomology
theories they represent, completion of spectra at a prime $p$, the Steenrod
algebra, and the Adams spectral sequence. Readers unfamiliar with this
material may consult the Chicago lectures of Adams [Ada74], the guide to
computations in homotopy theory at the prime 2 by Beaudry and Campbell [BC18],
or the survey of connective real $K$-theory by Bruner and Greenlees [BG10].
Readers unconcerned with homotopy theoretic applications may safely skip to
Section 3.

Throughout this section, all spectra are implicitly completed at the prime 2,
and $[X,Y]$ denotes the graded abelian group of homotopy classes of maps from
the spectrum $X$ to the spectrum $Y$.

###  2.1. The impetus from homotopy theory

The mod 2 cohomology $H^{*}X$ of a spectrum $X$ is represented by the mod 2
Eilenberg-Mac Lane spectrum $H\mathbb{F}_{2}$, i.e.,
$H^{*}X=[X,H\mathbb{F}_{2}]$. The mod $2$ Steenrod algebra
$\mathscr{A}=[H\mathbb{F}_{2},H\mathbb{F}_{2}]$ is the algebra of (stable)
cohomology operations. As an algebra over $\mathbb{F}_{2}$, $\mathscr{A}$ is
generated by $Sq^{i}$ for $i\geq 0$ which act by post-composition. Modules
over $\mathscr{A}$ have been well-studied in the literature, especially in
light of the Adams spectral sequence

$\operatorname{Ext}_{\mathscr{A}}(H^{*}Y,H^{*}X)\Rightarrow[X,Y]\otimes\mathbb{Z}_{2}$

which approximates the stable homotopy groups of spheres when $X=Y=S$, the
sphere spectrum.

Given the subalgebra $\mathscr{A}(m)$ of $\mathscr{A}$ generated by
$\\{Sq^{1},Sq^{2},\dots,Sq^{2^{m}}\\}$, consider the $\mathscr{A}(m)$-module
$\mathscr{A}\otimes_{\mathscr{A}(m)}\mathbb{F}_{2}$, where the right action of
$\mathscr{A}(m)$ on $\mathscr{A}$ is induced by the inclusion and the left
action of $\mathscr{A}(m)$ on $\mathbb{F}_{2}$ is induced by $Sq^{0}\mapsto 1$
and $Sq^{i}\mapsto 0$ for $i>0$. The inclusions
$\mathscr{A}(m)\to\mathscr{A}(m+1)$ give rise to an infinite tower of
surjections

(3)
$\mathscr{A}\to\mathscr{A}\otimes_{\mathscr{A}(0)}\mathbb{F}_{2}\to\mathscr{A}\otimes_{\mathscr{A}(1)}\mathbb{F}_{2}\to\mathscr{A}\otimes_{\mathscr{A}(2)}\mathbb{F}_{2}\to\mathscr{A}\otimes_{\mathscr{A}(3)}\mathbb{F}_{2}\to\cdots.$

It will be convenient to put $\mathscr{A}(-1)=\\{Sq^{0}\\}=\mathbb{F}_{2}$, so
that $\mathscr{A}\cong\mathscr{A}\otimes_{\mathscr{A}(-1)}\mathbb{F}_{2}$.
Noting that $\mathscr{A}\cong H^{*}H\mathbb{F}_{2}$, one can ask if the
remaining modules in (3) can be realized as the mod 2 cohomology of some
spectrum. The case $m\geq 3$ requires the existence of a non-trivial map of
spheres which has been shown not to exist. However, it is well-known that such
spectra exist for $0\leq n\leq 2$, namely

$\displaystyle H^{*}H\mathbb{Z}$
$\displaystyle\cong\mathscr{A}\otimes_{\mathscr{A}(0)}\mathbb{F}_{2},$
$\displaystyle H^{*}b{\rm o}$
$\displaystyle\cong\mathscr{A}\otimes_{\mathscr{A}(1)}\mathbb{F}_{2},$
$\displaystyle H^{*}\operatorname{tmf}$
$\displaystyle\cong\mathscr{A}\otimes_{\mathscr{A}(2)}\mathbb{F}_{2},$

where $H\mathbb{Z}$ is the integral Eilenberg-Mac Lane spectrum, $b{\rm o}$ is
the connective real $K$-theory spectrum, and $\operatorname{tmf}$ is the
connective spectrum of topological modular forms—an analog of $K$-theory with
connections to modular forms in number theory (see [Lar20] for an instance of
such a connection away from the prime 2). Indeed, these spectra and their
associated cohomology theories have become essential for computations inside
the stable homotopy groups of spheres, and are interesting in their own right.

A change-of-rings theorem yields

$\operatorname{Ext}^{s,t}_{\mathscr{A}}(\mathscr{A}\otimes_{\mathscr{A}(m)}\mathbb{F}_{2},H^{*}X)\cong\operatorname{Ext}^{s,t}_{\mathscr{A}(m)}(\mathbb{F}_{2},H^{*}X).$

In particular, the $E_{2}$-term approximating the integral cohomology of $X$
can be determined by an understanding of the $\mathscr{A}(0)$-module structure
of $H^{*}X$. Similarly, the $E_{2}$-terms approximating the connective real
$K$-theory of $X$ and the $\operatorname{tmf}$-cohomology of $X$ can be
determined by understanding the $\mathscr{A}(1)$\- and $\mathscr{A}(2)$-module
structures of $H^{*}X$, respectively.

Modules over $\mathscr{A}(1)$ have been fully classified up to stable
$\mathscr{A}(1)$ isomorphism [AP76, Section 3]. Two $R$-modules $M_{1}$ and
$M_{2}$ are stably $R$ isomorphic if there exist free $R$-modules $F_{1}$ and
$F_{2}$ such that $M_{1}\oplus F_{1}\cong M_{2}\oplus F_{2}$. The original
impetus of the current paper was a better understanding of the free
$\mathscr{A}(1)$-module summands of
$\mathscr{A}\otimes_{\mathscr{A}(m)}\mathbb{F}_{2}$. Such an understanding
would ultimately provide a clearer picture of important computations currently
in the literature, as we shall discuss in the following subsections. See for
example [Dav87] and [BBB+20].

###  2.2. Connection with binary partitions

It is often convenient to perform computations in the dual
$\mathscr{A}_{*}=\operatorname{Hom}_{\mathbb{F}_{2}}(\mathscr{A},\mathbb{F}_{2})$.
Indeed, while $\mathscr{A}$ is a graded noncommutative algebra with many
relations, its dual
$\mathscr{A}_{*}\cong\mathbb{F}_{2}[\xi_{1},\xi_{2},\xi_{3},\dots]$ is a
polynomial ring with grading $\left|\xi_{i}\right|=2^{i}-1$. Furthermore,

$\displaystyle H_{*}H\mathbb{F}_{2}$
$\displaystyle\cong(\mathscr{A}\otimes_{\mathscr{A}(-1)}\mathbb{F}_{2})_{*}\cong\mathscr{A}_{*},$
$\displaystyle H_{*}H\mathbb{Z}$
$\displaystyle\cong(\mathscr{A}\otimes_{\mathscr{A}(0)}\mathbb{F}_{2})_{*}\cong\mathbb{F}_{2}[\xi_{1}^{2},\xi_{2},\xi_{3},\xi_{4},\dots],$
$\displaystyle H_{*}b{\rm o}$
$\displaystyle\cong(\mathscr{A}\otimes_{\mathscr{A}(1)}\mathbb{F}_{2})_{*}\cong\mathbb{F}_{2}[\xi_{1}^{4},\xi_{2}^{2},\xi_{3},\xi_{4},\dots],$
$\displaystyle H_{*}\operatorname{tmf}$
$\displaystyle\cong(\mathscr{A}\otimes_{\mathscr{A}(2)}\mathbb{F}_{2})_{*}\cong\mathbb{F}_{2}[\xi_{1}^{8},\xi_{2}^{4},\xi_{3}^{2},\xi_{4},\dots]$

(see [Mil58, Koc82, Mah81, Rez07]). Upon replacing the grading mentioned above
by the alternative grading convention $|\xi_{i}|=2^{i-1}$ (hereafter called a
“weight” and denoted $\omega(\xi_{i})=2^{i-1}$ following the language and
notation of [Mah81, §3]) the reader should observe that
$H(m)\cong(\mathscr{A}\otimes_{\mathscr{A}(m)}\mathbb{F}_{2})_{*}$. Hence,
Theorem 1.1 and Corollary 1.3 together provide a recursive formula for the
number of generators of
$(\mathscr{A}\otimes_{\mathscr{A}(m)}\mathbb{F}_{2})_{*}$ of a given weight.

######  Example 2.1.

The number of monomials of weight $n$ in $H_{*}H\mathbb{F}_{2}$ and the number
of monomials of weight $2n$ in $H_{*}H\mathbb{Z}$ are counted by the sequences
$r_{-1}(n)$ and $r_{0}(2n)$, respectively, exhibited at the start of Section
1. Moreover,

$r_{1}(n)=\begin{cases}1,&n=0,\\\ r_{1}(n-4)+2r_{1}((n-4)/2),&0<n\equiv 4\mod
8,\\\ r_{1}(n-4)+r_{1}(n/2)+r_{1}((n-8)/2),&0<n\equiv 0\mod 8,\\\
0,&\text{otherwise}.\end{cases}$

and so the sequence $r_{1}(4n)$ counting the number of monomials of weight
$4n$ in $H_{*}b{\rm o}$ has

$1,3,7,13,23,37,57,83,119,165,225,299,393,507,647,813,\ldots$

as its first few terms.

######  Example 2.2.

Since

$r_{2}(n)=\begin{cases}1,&n=0,\\\
r_{2}(n-8)+3r_{2}((n-8)/2)+r_{2}((n-24)/2),&0<n\equiv 8\mod 16,\\\
r_{2}(n-8)+r_{2}(n/2)+3r_{2}((n-16)/2),&0<n\equiv 0\mod 16,\\\
0,&\text{otherwise},\end{cases}$

it follows that the sequence $r_{2}(8n)$ counting the number of monomials of
weight $8n$ in $H_{*}\operatorname{tmf}$ has

$1,4,11,24,47,84,141,224,343,508,733,1032,1425,1932,2579,3392,\ldots$

as its first few terms.

######  Example 2.3.

As noted above, there does not exist a spectrum with mod 2 homology isomorphic
to $(\mathscr{A}\otimes_{\mathscr{A}(m)}\mathbb{F}_{2})_{*}$ for $m\geq 3$.
Theorem 1.1 implies

$r_{3}(n)=\begin{cases}1,&n=0,\\\
r_{3}(n-16)+4r_{3}((n-16)/2)+4r_{3}((n-48)/2),&0<n\equiv 16\mod 32,\\\
r_{3}(n-16)+r_{3}(n/2)+6r_{3}((n-32)/2)+r_{3}((n-64)/2),&0<n\equiv 0\mod
32,\\\ 0,&\text{otherwise}.\end{cases}$

and so $H(3)$, the first of the polynomial algebras $H(m)$ not realizable as
the mod 2 homology of a spectrum, has $r_{3}(16n)$ monomials of weight $16n$,
where $r_{3}(16n)$ has as its first few terms

$1,5,16,40,87,171,312,536,879,1387,2120,3152,4577,6509,9088,12480,\ldots$

One can observe the partial sum relationships between the sequences in the
above examples given by Theorem 1.2.

###  2.3. Integral Brown-Gitler spectra

Brown and Gitler [BG73] constructed a family $\\{B(j)\>|\>j\geq 0\\}$ of
$\mathbb{Z}/2\mathbb{Z}$-complete spectra whose cohomology algebras are cyclic
as modules over $\mathscr{A}$. For a given $j$, the generator $\alpha:B(j)\to
H\mathbb{F}_{2}$ gives rise to a surjection
$\alpha_{*}:B(j)_{k}X\to(H\mathbb{F}_{2})_{k}X$ in homology for $k<2j+2$ and
any CW-complex $X$, resulting in these Brown-Gitler spectra having many
applications in homotopy theory. This led Goerss, Jones, and Mahowald [GJM86]
to construct analogous families of Brown-Gitler spectra over $H\mathbb{Z}$ and
$b{\rm o}$ (as well as $BP\langle 1\rangle$). For the purposes of this paper,
we will denote the Brown-Gitler, integral Brown-Gitler, and $b{\rm o}$ Brown-
Gitler spectra by $B_{0}(j)$, $B_{1}(j)$, and $B_{2}(j)$, respectively. With
the weight $\omega$ defined above, the homology of these Brown-Gitler spectra
are given by

(4)
$H_{*}B_{i}(j)\cong\\{x\in(\mathscr{A}\otimes_{\mathscr{A}(i-1)}\mathbb{F}_{2})_{*}\>|\>\omega(x)\leq
j2^{i}\\}$

where $(\mathscr{A}\otimes_{\mathscr{A}(i-1)}\mathbb{F}_{2})_{*}$ denotes the
$\mathbb{F}_{2}$-dual of
$(\mathscr{A}\otimes_{\mathscr{A}(i-1)}\mathbb{F}_{2})$.

The stable $\mathscr{A}(1)$-isomorphism classes of $H_{*}B_{1}(j)$ are related
to the family of Milgram modules [Mil75] which Davis, Gitler, and Mahowald
[DGM81] denote $Q_{i,n}$ for $i\geq 0$ and $n\in\\{0,1,2,3\\}$. These modules
are displayed in Figure 1. The circles represent generators of
$\mathbb{F}_{2}$ in the degree indicated by the column, while straight and
curved lines represent an action of $Sq^{1}$ and $Sq^{2}$, respectively.

$\cdots$$\cdots$$\cdots$$\cdots$$Q_{0,n}$$Q_{1,n}$$Q_{2,n}$$Q_{3,n}$$0$$4$$4n$

Figure 1. Milgram modules $Q_{i,n}$ ($0\leq i\leq 3$)

###  2.4. Rank of free $\mathscr{A}(1)$-module summands

######  Lemma 2.1 (Lemma 3.12 [DGM81]).

There is an isomorphism $H_{*}B_{1}(j)\cong F_{1,j}\oplus Q_{\alpha(j),j-D}$
of $\mathscr{A}(1)$-modules, where

$D=\begin{cases}2\ell,&\text{if }\alpha(j)=4\ell,\\\ 2\ell+1,&\text{if
}4\ell+1\leq\alpha(j)\leq 4\ell+3,\end{cases}$

$F_{1,j}$ is a free $\mathscr{A}(1)$-module, and the first subscript
$\alpha(j)$ of $Q_{\alpha(j),j-D}$ is taken modulo 4.

An application of Theorem 1.1 therefore determines the free rank of $F_{1,j}$,
yielding the following theorem (which is merely a restatement of Theorem 1.4).

######  Theorem 2.2.

The rank of $F_{1,j}$ as an $\mathscr{A}(1)$-module is
$f_{1,j}=\dfrac{1}{8}\left(r_{1}(4j)-b(j)\right)$ where

(5) $b(j)=\begin{cases}4j-2\alpha(j)+1,&\text{if }\alpha(j)\equiv
0,1\mod{4},\\\ 4j-2\alpha(j)+5,&\text{if }\alpha(j)\equiv
2,3\mod{4}.\end{cases}$

###### Proof.

From Figure 1, we see that

$\dim_{\mathbb{F}_{2}}(Q_{i,n})=4n+\begin{cases}1,&i=0,\\\ 3&i=1,3,\\\
5,&i=2.\end{cases}$

Lemma 2.1 therefore yields

$\displaystyle\begin{split}\dim_{\mathbb{F}_{2}}(Q_{\alpha(j),j-D})&=4j-4\begin{cases}2\lfloor\alpha(j)/4\rfloor,&\alpha(j)\equiv
0\mod 4\\\ 2\lfloor\alpha(j)/4\rfloor+1,&\alpha(j)\not\equiv 0\mod
4\end{cases}+\begin{cases}1,&\alpha(j)\equiv 0\mod 4\\\ 3,&\alpha(j)\equiv
1,3\mod 4\\\ 5,&\alpha(j)\equiv 2\mod 4\end{cases}\\\
&=4j+\begin{cases}-2\alpha(j)+1,&\alpha(j)\equiv 0\mod 4\\\
-2(\alpha(j)-1)-1,&\alpha(j)\equiv 1\mod 4\\\
-2(\alpha(j)-2)+1,&\alpha(j)\equiv 2\mod 4\\\
-2(\alpha(j)-3)-1,&\alpha(j)\equiv 3\mod 4.\end{cases}\end{split}$

Note that Theorem 1.1 and (4) together imply that $r_{1}(4j)$ is the
$\mathbb{F}_{2}$-dimension of $H_{*}B_{1}(j)$. By Lemma 2.1, the remaining
classes generate $F_{1,j}$. Since $\mathscr{A}(1)$ has 8 classes, the result
follows. ∎

######  Remark 2.4.

The sequence $\\{f_{1,j}\\}$ has as its first few terms

$0,0,0,0,1,2,4,7,11,16,23,32,43,57,74,95,\dots.$

The reader should notice the first appearance of a free
$\mathscr{A}(1)$-summand among the homologies of integral Brown-Gitler spectra
is in $H_{*}B_{1}(4)$.

Consider the elements
$N_{n}(\ell)=\\{x\in(\mathscr{A}\otimes_{\mathscr{A}(n)}\mathbb{F}_{2})_{*}\>|\>\omega(s)=\ell\\}$
of homogeneous weight $\ell$. For $m\leq n$, there is an
$\mathscr{A}(m)$-module isomorphism

(6)
$\left(\mathscr{A}\otimes_{\mathscr{A}(n)}\mathbb{F}_{2}\right)_{*}\cong\bigoplus_{j\geq
0}N_{n}(j2^{n+1}).$

This is a straightforward extension of [Mah81, Lemma 2.1] for $m=n=1$ and
[Bai10, Proposition 2.3] for $m=1$ and $n=2$. The _Verschiebung_ is the
algebra homomorphism $V:\mathscr{A}_{*}\to\mathscr{A}_{*}$ defined on
generators by

$V(\xi_{i})=\begin{cases}1,&i=0,1,\\\ \xi_{i-1},&i\geq 2.\\\ \end{cases}$

An extension of the proof of [Bai10, Proposition 2.3] shows that $V$ induces
an $\mathscr{A}(m)$-module isomorphism

$N_{n}(j2^{n+1})\cong\Sigma^{j2^{n+1}}\\{x\in(\mathscr{A}\otimes_{\mathscr{A}(n-1)}\mathbb{F}_{2})_{*}\>|\>\omega(x)\leq
j2^{n}\\}.$

In particular $\Sigma^{j2^{i+1}}H_{*}B_{i}(j)\cong N_{i}(j2^{i+1})$ and, as a
result, Theorem 2.2 can be used to compute the number of free copies of
$\mathscr{A}(1)$ in $(\mathscr{A}\otimes_{\mathscr{A}(m)}\mathbb{F}_{2})_{*}$
for all $m\geq 1$. An example of this is given by the following theorem of the
first author for $m=2$.

######  Theorem 2.3 (Theorem 4.1 [Bai10]).

There is an isomorphism of graded $b{\rm o}_{*}$-algebras

(7) $\pi_{*}(b{\rm o}\wedge\operatorname{tmf})\cong\dfrac{b{\rm
o}_{*}[\sigma,b_{i},\mu_{i}\>|\>i\geq 0]}{(\mu b_{i}^{2}-8b_{i+1},\mu
b_{i}-4\mu_{i},\eta b_{i})}\oplus F$

where $\left|\sigma\right|=8$, $\left|b_{i}\right|=2^{i+4}-4$,
$\left|\mu_{i}\right|=2^{i+4}$ and $F$ is a direct sum of $\mathbb{F}_{2}$ in
varying dimensions.

The proof of Theorem 2.3 involves a computation of the Adams $E_{2}$-term

$E_{2}^{s,t}=\operatorname{Ext}_{\mathscr{A}}^{s,t}(H^{*}(b{\rm
o}\wedge\operatorname{tmf}),\mathbb{F}_{2})\Rightarrow\pi_{t-s}(b{\rm
o}\wedge\operatorname{tmf})$

which, after applying a change-of-rings theorem and dualizing, becomes
$\operatorname{Ext}_{\mathscr{A}(1)}^{s,t}(\mathbb{F}_{2},H_{*}\operatorname{tmf})$.
By the argument immediately preceding Theorem 2.3,
$H_{*}\operatorname{tmf}\cong\bigoplus_{j\geq 0}\bigoplus_{0\leq i\leq
j}\Sigma^{4i+8j}H_{*}B_{1}(j)$. In particular

$\displaystyle H_{*}\operatorname{tmf}$ $\displaystyle\cong\bigoplus_{j\geq
0}\bigoplus_{0\leq i\leq j}\Sigma^{4i+8j}\left(F_{1,j}\oplus
Q_{\alpha(j),j-D}\right)$ $\displaystyle\cong\bigoplus_{j\geq
0}\bigoplus_{0\leq i\leq
j}\Sigma^{4i+8j}Q_{\alpha(j),j-D}\oplus\bigoplus_{j\geq 0}\bigoplus_{0\leq
i\leq j}\Sigma^{4i+8j}F_{1,j}$ $\displaystyle\cong\bigoplus_{j\geq
0}\bigoplus_{0\leq i\leq
j}\Sigma^{4i+8j}Q_{\alpha(j),j-D}\oplus\bigoplus_{j\geq 0}F_{2,j}$

where $F_{2,j}=\bigoplus_{0\leq i\leq j}\Sigma^{4i+8j}F_{1,j}$ can be regarded
as the analog of $F_{1,j}$ for $H_{*}B_{2}(j)$. With this notation, the module
$F$ in Theorem 2.3 is given by
$F=\operatorname{Ext}_{\mathscr{A}(1)}^{s,t}(\bigoplus_{j\geq
0}F_{2,j},\mathbb{F}_{2})\cong\bigoplus_{j\geq
0}\mathbb{F}_{2}^{\dim_{\mathscr{A}(1)}(F_{2,j})}$. If we put
$f_{2,j}=\dim_{\mathscr{A}(1)}(F_{2,j})$, then

$f_{2,j}=\sum_{i=0}^{j}\dim_{\mathscr{A}(1)}(F_{1,j})$

so that $f_{2,j}$ is the number of free $\mathscr{A}(1)$ summands in
$H_{*}B_{2}(j)$ and $\\{f_{2,j}\\}$ is the sequence of partial sums of
$\\{f_{1,j}\\}$. This partial sum pattern can be continued inductively, even
though a geometric interpretation of the relevant polynomial rings is lost for
$m\geq 3$ as we noted in Subsection 2.1.

######  Remark 2.5.

As a result of (6), the Verschiebung homomorphism
$V:N_{n}(2^{n+1}k)\to\bigoplus_{i=0}^{k}N_{n-1}(2^{n}i)$ is an isomorphism of
$\mathscr{A}(m)$-modules for $m\leq n-1$. Since
$\dim_{\mathbb{F}_{2}}(N_{\ell}(k))=r_{\ell}(k)$ for all $k\geq 0$, the
Verschiebung gives us another way of viewing Theorem 1.2.

######  Remark 2.6.

While we have provided a method for counting the rank of the free
$\mathscr{A}(1)$ summands of
$(\mathscr{A}\otimes_{\mathscr{A}(m)}\mathbb{F}_{2})_{*}$ for $m\geq 1$, it
would be desirable to determine the degree of the corresponding generators.
This will be the subject of future work.

##  3\. Preliminary results on binary partitions

Let $\pi=(n;\alpha_{1},\alpha_{2},\ldots)$ denote the binary partition $\pi$
of $n$ with $\alpha_{1}$ the coefficient on $2^{0}$, $\alpha_{2}$ the
coefficient on $2^{1}$, etc., and let $P_{m}(n)$ denote the set of all binary
partitions $\pi=(n;\alpha_{1},\alpha_{2},\ldots)$ of $n$ satisfying the
divisibility conditions given by (1), so that $r_{m}(n)=|P_{m}(n)|$. We begin
this section by bifurcating $P_{m}(n)$ according to whether or not a 1 appears
as a part in a given partition. Define

$\displaystyle P_{m,1}(n)$ $\displaystyle=\\{\pi\in
P_{m}(n)\>:\>\alpha_{1}>0\\},$ $\displaystyle P_{m,0}(n)$
$\displaystyle=\\{\pi\in P_{m}(n)\>:\>\alpha_{1}=0\\}$

so that $P_{m}(n)=P_{m,1}(n)\sqcup P_{m,0}(n)$. We now divide $P_{m,0}(n)$
into $2^{m+1}$ disjoint collections according to whether $\alpha_{i}$ is
congruent to 0 or $2^{m+2-i}$ modulo $2^{m+3-i}$ for $2\leq i\leq m+2$. Let
$\varepsilon_{2},\ldots,\varepsilon_{m+2}$ be a finite sequence of integers
taking values in $\\{0,1\\}$. If we define

$P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{m+2}}(n)=\\{\pi\in
P_{m,0}(n)\>:\>\alpha_{i}\equiv\varepsilon_{i}2^{m+2-i}\mod 2^{m+3-i},2\leq
i\leq m+2\\}$

then

$P_{m,0}(n)=\bigsqcup_{\varepsilon_{i}\in\\{0,1\\}}P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{m+2}}.$

######  Lemma 3.1.

The map $\sigma:P_{m,0}^{0,0,\ldots,0}(n)\to P_{m}\left(\frac{n}{2}\right)$
defined by

$\sigma:(n;0,\alpha_{2},\alpha_{3},\ldots)\mapsto\left(\frac{n}{2};\alpha_{2},\alpha_{3}\ldots\right)$

is a bijection.

###### Proof.

A partition $\pi=(n;0,\alpha_{2},\alpha_{3},\ldots)\in
P_{m,0}^{0,0,\ldots,0}(n)$ satisfies $2^{m+3-i}\mid\alpha_{i}$ for $2\leq
i\leq m+2$. Therefore

$\sigma(\pi)=\left(\frac{n}{2};\alpha_{2},\alpha_{3},\ldots\right)=\left(\frac{n}{2};\beta_{1},\beta_{2},\ldots\right)$

where $2^{m+2-i}\mid\alpha_{i+1}=\beta_{i}$ for $1\leq i\leq m+1$, which
implies $\sigma(\pi)\in P_{m}\left(\frac{n}{2}\right)$. If
$\pi^{\prime}=\left(\frac{n}{2};\alpha_{1},\alpha_{2},\ldots\right)\in
P_{m}\left(\frac{n}{2}\right)$ and we define $\sigma^{-1}$ by

$\sigma^{-1}(\pi^{\prime})=(n;0,\alpha_{1},\alpha_{2},\ldots)=(n;0,\gamma_{2},\gamma_{3},\ldots)$

we see that $2^{m+3-i}\mid\alpha_{i-1}=\gamma_{i}$ for $2\leq i\leq m+2$.
Hence $\sigma^{-1}(\pi^{\prime})\in P_{m,0}^{0,0,\ldots,0}(n)$, and it follows
from the formulas for $\sigma$ and $\sigma^{-1}$ that
$\sigma\circ\sigma^{-1}=1_{P_{m,0}^{0,0,\ldots,0}(n)}$ and
$\sigma^{-1}\circ\sigma=1_{P_{m}\left(\frac{n}{2}\right)}$. ∎

######  Lemma 3.2.

If $\varepsilon_{i}=1$ for some $i$, $2\leq i\leq m+2$, the map

$\tau_{i}:P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{i},\ldots,\varepsilon_{m+2}}(n)\to
P_{m,0}^{\varepsilon_{2},\ldots,0,\ldots,\varepsilon_{m+2}}(n-2^{m+1})$

defined by

$\tau_{i}:(n;0,\alpha_{2},\alpha_{3},\ldots)\mapsto(n-2^{m+1};0,\alpha_{2},\ldots,\alpha_{i}-2^{m+2-i},\ldots)$

is a bijection.

###### Proof.

If $\varepsilon_{i}=1$ and $\pi=(n;0,\alpha_{2},\alpha_{3},\ldots)\in
P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{i},\ldots,\varepsilon_{m+2}}(n)$,
then $\varepsilon_{i}=2^{m+2-i}\mod 2^{m+3-i}$, which means
$\alpha_{i}-2^{m+2-i}\equiv 0\mod 2^{m+3-i}$. Hence $\tau_{i}(\pi)\in
P_{m,0}^{\varepsilon_{2},\ldots,0,\ldots,\varepsilon_{m+2}}(n-2^{m+1})$. If
$\pi^{\prime}=(n-2^{m+1};0,\alpha_{2},\alpha_{3},\ldots)\in
P_{m,0}^{\varepsilon_{2},\ldots,0,\ldots,\varepsilon_{m+2}}(n-2^{m+1})$ and we
define $\tau_{i}^{-1}$ by

$\tau_{i}^{-1}(\pi^{\prime})=(n;0,\alpha_{2},\ldots,\alpha_{i}+2^{m+2-i},\ldots)=(n;0,\beta_{2},\beta_{3},\ldots)$

we see that $\beta_{i}=2^{m+2-i}\mod 2^{m+3-i}$. Hence
$\tau_{i}^{-1}(\pi^{\prime})\in
P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{i},\ldots,\varepsilon_{m+2}}(n)$.
It follows that
$\tau_{i}\circ\tau_{i}^{-1}=1_{P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{i},\ldots,\varepsilon_{m+2}}}$
and
$\tau_{i}^{-1}\circ\tau_{i}=1_{P_{m,0}^{\varepsilon_{2},\ldots,0,\ldots,\varepsilon_{m+2}}}$.
∎

######  Lemma 3.3.

The map $\tau_{1}:P_{m,1}(n)\to P_{m}(n-2^{m+1})$ defined by
$\tau_{1}:(n;\alpha_{1},\alpha_{2},\ldots)\mapsto(n-2^{m+1};\alpha_{1}-2^{m+1},\alpha_{2},\alpha_{3},\ldots)$
is a bijection.

###### Proof.

Let $\pi=(n;\alpha_{1},\alpha_{2},\ldots)\in P_{m,1}(n)$. Then
$2^{m+1}\alpha_{1}\geq 1$, which implies $\alpha_{1}\geq 2^{m+1}$. Hence
$\tau_{1}(\pi)\in P_{m}(n-2^{m+1}$. For
$\pi^{\prime}=(n-2^{m+1},\alpha_{1},\alpha_{2},\ldots)\in P_{m}(n-2^{m+1})$,
we may define
$\tau_{1}^{-1}(\pi^{\prime})=(n;\alpha_{1}+2^{m+1},\alpha_{2},\alpha_{3},\ldots)$.
∎

######  Lemma 3.4.

Suppose $\varepsilon_{i}=1$ for indices $2\leq i_{1}<i_{2}<\cdots<i_{k}\leq
m+1$ where $k\geq 0$, and $\varepsilon_{i}=0$ for all other $i$ such that
$2\leq i\leq m+1$.

  1. (1)

If either $n\equiv 2^{m+1}\mod 2^{m+2}$ and $k$ is even, or $n\equiv 0\mod
2^{m+2}$ and $k$ is odd, then
$P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{m+2}}(n)=P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{m+1},1}(n)$,
and the map

$\sigma\circ\tau_{i_{1}}\circ\tau_{i_{2}}\circ\cdots\circ\tau_{i_{k}}\circ\tau_{m+2}:P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{m+2}}\to
P_{m}\left(\frac{n-(k+1)2^{m+1}}{2}\right)$

is a bijection.

  2. (2)

If either $n\equiv 2^{m+1}\mod 2^{m+2}$ and $k$ is odd, or $n\equiv 0\mod
2^{m+2}$ and $k$ is even, then
$P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{m+2}}(n)=P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{m+1},0}(n)$,
and the map

$\sigma\circ\tau_{i_{1}}\circ\tau_{i_{2}}\circ\cdots\circ\tau_{i_{k}}:P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{m+2}}\to
P_{m}\left(\frac{n-k\cdot 2^{m+1}}{2}\right)$

is a bijection.

###### Proof.

Let $\pi=(n;0,\alpha_{2},\alpha_{3},\ldots)\in
P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{m+2}}(n)$. Since $\displaystyle
n=\sum_{i\geq 1}\alpha_{i}\cdot 2^{i-1}$, we have

$n\equiv\sum_{i=1}^{m+2}\alpha_{i}\cdot 2^{i-1}\mod 2^{m+2}.$

If $i\notin\\{i_{1},\ldots,i_{k}\\}$, then $\alpha_{i}\cdot 2^{i-1}\equiv
0\mod 2^{m+2}$, whereas if $i\in\\{i_{1},\ldots,i_{k}\\}$, then
$\alpha_{i}\cdot 2^{i-1}\equiv 2^{m+1}\mod 2^{m+2}$. This implies
$2^{m+1}\equiv n\equiv\alpha_{i_{1}}\cdot
2^{i_{1}-1}+\cdots+\alpha_{i_{k}}\cdot 2^{i_{k}-1}+\alpha_{m+2}\cdot
2^{m+1}\equiv k\cdot 2^{m+1}+\alpha_{m+2}\cdot 2^{m+1}\mod 2^{m+2}$. Thus,
whether we assume that $n\equiv 2^{m+1}\mod 2^{m+2}$ and $k$ is even, or that
$n\equiv 0\mod 2^{m+2}$ and $k$ is odd, it follows that $\alpha_{m+2}$ must be
odd, i.e., $\varepsilon_{m+2}$ must equal 1. An argument similar to the proof
of Lemma 3.2 shows that the map
$\tau_{m+2}:P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{m+1},1}(n)\to
P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{m+1},0}(n-2^{m+1})$ defined by
$\tau_{m+2}:\pi\mapsto(n-2^{m+1};0,\alpha_{2},\ldots,\alpha_{m+2}-1,\ldots)$
is a bijection. Therefore, by Lemmas 3.1 and 3.2, the map
$\sigma\circ\tau_{i_{1}}\circ\tau_{i_{2}}\circ\cdots\circ\tau_{i_{k}}\circ\tau_{m+2}$
is a bijection between the indicated source and target. This proves (1).

Note similarly that either pair of hypotheses in (2) forces
$P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{m+2}}(n)=P_{m,0}^{\varepsilon_{1},\ldots,\varepsilon_{m+1},0}(n)$.
Therefore, by Lemmas 3.1 and 3.2, the map
$\sigma\circ\tau_{i_{1}}\circ\tau_{i_{2}}\circ\cdots\circ\tau_{i_{k}}$ is a
bijection between the indicated source and target. ∎

##  4\. Proof of Theorem 1.1

In this section we will prove Theorem 1.1. It is evident that $r_{m}(n)=0$ if
$2^{m+1}\nmid n$. Assume $2^{m+1}\mid n$. We established in Section 3 that

$P_{m}(n)=P_{m,1}(n)\sqcup\bigsqcup_{i\in\\{0,1\\}}P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{m+2}}.$

Lemma 3.3 implies $P_{m,1}(n)$ and $P_{m}(n-2^{m+1})$ are in bijective
correspondence. This contributes $r_{m}(n-2^{m+1})$ to the total size of
$P_{m}(n)$.

It remains to obtain similar correspondences for the $2^{m+1}$ disjoint sets
$P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{m+2}}(n)$ that comprise
$P_{m,0}(n)$. To do this, we divide into the two cases corresponding to the
formulas given in Theorem 1.1: $n\equiv 2^{m+1}\mod 2^{m+2}$ and $n\equiv
0\mod 2^{m+2}$.

Suppose first that $n\equiv 2^{m+1}\mod 2^{m+2}$. If exactly $2j$ of the $m-1$
superscripts $\varepsilon_{2},\ldots,\varepsilon_{m+1}$ are equal to 1, where
$0\leq j\leq\lfloor m/2\rfloor$, Lemma 3.4(1) implies
$P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{m+2}}(n)$ and
$P_{m}((n-(2j+1)2^{m+1})/2)$ are in bijective correspondence. Therefore, each
such $j$ contributes

$\binom{m}{2j}r_{m}\left(\dfrac{n-(2j+1)2^{m+1}}{2}\right)$

to the total size of $P_{m}(n)$. If exactly $2j+1$ of the superscripts
$\varepsilon_{2},\ldots,\varepsilon_{m+1}$ are equal to 1, where $0\leq
j\leq\lfloor(m-1)/2\rfloor$, Lemma 3.4(2) implies
$P_{m,0}^{\varepsilon_{2},\ldots,\varepsilon_{m+2}}(n)$ and
$P_{m}((n-(2j+1)2^{m+1})/2)$ are in bijective correspondence. Therefore, each
such $j$ contributes

$\binom{m}{2j+1}r_{m}\left(\dfrac{n-(2j+1)2^{m+1}}{2}\right)$

to the total size of $P_{m}(n)$. Thus

$\displaystyle r_{m}(n)$ $\displaystyle=r_{m}(n-2^{m+1})+\sum_{j=0}^{\lfloor
m/2\rfloor}\binom{m}{2j}r_{m}\left(\dfrac{n-(2j+1)2^{m+1}}{2}\right)$
$\displaystyle\hskip
108.405pt+\sum_{j=0}^{\lfloor(m-1)/2\rfloor}\binom{m}{2j+1}r_{m}\left(\dfrac{n-(2j+1)2^{m+1}}{2}\right)$
$\displaystyle=r_{m}(n-2^{m+1})+\sum_{j=0}^{\lfloor
m/2\rfloor}\binom{m+1}{2j+1}r_{m}\left(\dfrac{n-(2j+1)2^{m+1}}{2}\right)$

by Pascal’s rule.

Next, suppose $n\equiv 0\mod 2^{m+2}$. If exactly $2j$ of the superscripts
$\varepsilon_{2},\ldots,\varepsilon_{m+1}$ are equal to 1, where $0\leq
j\leq\lfloor m/2\rfloor$, Lemma 3.4(2) implies
$P_{0,m}^{\varepsilon_{2},\ldots,\varepsilon_{m+2}}(n)$ and $P_{m}((n-2j\cdot
2^{m+1})/2)$ are in bijective correspondence. Therefore, each such $j$
contributes

$\binom{m}{2j}r_{m}\left(\dfrac{n-2j\cdot 2^{m+1}}{2}\right)$

to the total size of $P_{m}(n)$. If exactly $2j-1$ of the superscripts
$\varepsilon_{2},\ldots,\varepsilon_{m+1}$ are equal to 1, where
$0<j\leq\lfloor(m+1)/2\rfloor$, Lemma 3.4(1) implies
$P_{0,m}^{\varepsilon_{2},\ldots,\varepsilon_{m+2}}(n)$ and $P_{m}((n-2j\cdot
2^{m+1})/2)$ are in bijective correspondence. Therefore, each such $j$
contributes

$\binom{m}{2j-1}r_{m}\left(\dfrac{n-2j\cdot 2^{m+1}}{2}\right)$

to the total size of $P_{m}(n)$. Thus

$\displaystyle r_{m}(n)$ $\displaystyle=r_{m}(n-2^{m+1})+\sum_{j=0}^{\lfloor
m/2\rfloor}\binom{m}{2j}r_{m}\left(\dfrac{n-2j\cdot
2^{m+1}}{2}\right)+\sum_{j=1}^{\lfloor(m+1)/2\rfloor}\binom{m}{2j-1}r_{m}\left(\dfrac{n-2j\cdot
2^{m+1}}{2}\right)$
$\displaystyle=r_{m}(n-2^{m+1})+\sum_{j=0}^{\lfloor(m+1)/2\rfloor}\binom{m+1}{2j}r_{m}\left(\dfrac{n-2j\cdot
2^{m+1}}{2}\right)$

by Pascal’s rule.

##  5\. Proof of Theorem 1.2

In this section, we show that $r_{m}(n)$ is a sequence of partial sums of
$r_{m-1}(n)$ for $m\geq 0$. It will be convenient to have the following lemma,
which identifies two special cases of Theorem 1.1.

######  Lemma 5.1.

For $m\geq 0$, $r_{m-1}(2^{m})=m+1$ and $r_{m-1}(2^{m+1})=2m+2+\binom{m}{2}$.

Let $\displaystyle S_{m}(n)=\sum_{k=0}^{n/2^{m+1}}r_{m-1}(2^{m}k)$. Then
$S_{m}(0)=r_{m}(0)=1$. It therefore suffices to show that $S_{m}(n)$ obeys the
recursive formula for $r_{m}(n)$ given in Theorem 1.1.

Suppose $n\equiv 2^{m+1}\mod 2^{m+2}$. Let $n=2^{m+2}n^{\prime}+2^{m+1}$. We
shall use induction on $n^{\prime}$. The base case $n^{\prime}=0$ holds since,
by Lemma 5.1,

$S_{m}(2^{m+1})=\sum_{k=0}^{1}r_{m-1}(2^{m}k)=r_{m-1}(0)+r_{m-1}(2^{m})=1+m+1=m+2$

while

$\displaystyle S_{m}(2^{m+1}-2^{m+1})+\sum_{j=0}^{\lfloor
m/2\rfloor}\binom{m+1}{2j+1}S_{m}\left(\frac{2^{m+1}-(2j+1)2^{m+1}}{2}\right)=(m+2)S_{m}(0)=m+2.$

Assume that, for a fixed $n^{\prime}\geq 0$,

$S_{m}(2^{m+2}n^{\prime}+2^{m+1})=S_{m}(2^{m+2}n^{\prime})+\sum_{j=0}^{\lfloor
m/2\rfloor}\binom{m+1}{2j+1}S_{m}\left(\frac{2^{m+2}n^{\prime}+2^{m+1}-(2j+1)2^{m+1}}{2}\right).$

Then

$\displaystyle S_{m}(2^{m+2}(n^{\prime}+1)+2^{m+1})$
$\displaystyle=\sum_{k=0}^{2n^{\prime}+3}r_{m-1}(2^{m}k)$
$\displaystyle=\underbrace{S_{m}(2^{m+2}n^{\prime}+2^{m+1})}_{A}+\underbrace{r_{m-1}(2^{m}(2n^{\prime}+2))}_{B}+\underbrace{r_{m-1}(2^{m}(2n^{\prime}+3))}_{C}.$

We know an expression for $A$ by hypothesis. By Theorem 1.1,

$\displaystyle B$
$\displaystyle=r_{m-1}(2^{m}(2n^{\prime}+2)-2^{m})+\sum_{j=0}^{\lfloor
m/2\rfloor}\binom{m}{2j}r_{m-1}\left(\dfrac{2^{m}(2n^{\prime}+2)-2j\cdot
2^{m}}{2}\right)$
$\displaystyle=r_{m-1}(2^{m}(2n^{\prime}+1))+\sum_{j=0}^{\lfloor
m/2\rfloor}\binom{m}{2j}r_{m-1}(2^{m}(n^{\prime}-j+1))$

and

$\displaystyle C$
$\displaystyle=r_{m-1}(2^{m}(2n^{\prime}+3)-2^{m})+\sum_{j=0}^{\lfloor(m-1)/2\rfloor}\binom{m}{2j+1}r_{m-1}\left(\frac{2^{m}(2n^{\prime}+3)-(2j+1)2^{m}}{2}\right)$
$\displaystyle=r_{m-1}(2^{m}(2n^{\prime}+2))+\sum_{j=0}^{\lfloor(m-1)/2\rfloor}\binom{m}{2j+1}r_{m-1}(2^{m}(n^{\prime}-j+1)),$

which implies

$\displaystyle B+C$
$\displaystyle=\sum_{k=2n^{\prime}+1}^{2n^{\prime}+2}r_{m-1}(2^{m}k)+\sum_{j=0}^{\lfloor
m/2\rfloor}\left(\binom{m}{2j}+\binom{m}{2j+1}\right)r_{m}(2^{m}(n^{\prime}-j+1))$
$\displaystyle=\sum_{k=2n^{\prime}+1}^{2n^{\prime}+2}r_{m-1}(2^{m}k)+\sum_{j=0}^{\lfloor
m/2\rfloor}\binom{m+1}{2j+1}r_{m}(2^{m}(n^{\prime}-j+1)).$

It follows from the induction hypothesis that

$\displaystyle S_{m}(2^{m+2}(n^{\prime}+1)+2^{m+1})$ $\displaystyle=A+B+C$
$\displaystyle=S_{m}(2^{m+2}n^{\prime}+2^{m+2})+\sum_{j=0}^{\lfloor
m/2\rfloor}\binom{m+1}{2j+1}S_{m}(2^{m+1}(n^{\prime}-j+1))$
$\displaystyle=S_{m}(2^{m+2}(n^{\prime}+1))$ $\displaystyle\hskip
36.135pt+\sum_{j=0}^{\lfloor
m/2\rfloor}\binom{m+1}{2j+1}S_{m}\left(\dfrac{2^{m+2}(n^{\prime}+1)+2^{m+1}-(2j+1)2^{m+1}}{2}\right).$

Next, suppose $n\equiv 0\mod 2^{m+2}$. Let $n=2^{m+2}n^{\prime}$. We shall use
induction on $n^{\prime}$. The base case $n^{\prime}=1$ holds since, by Lemma
5.1,

$S_{m}(2^{m+2})=\sum_{k=0}^{2}r_{m-1}(2^{m}k)=r_{m-1}(0)+r_{m-1}(2^{m})+r_{m-1}(2^{m+1})=3m+4+\binom{m}{2}$

while

$\displaystyle S_{m}(2^{m+2}$
$\displaystyle-2^{m+1})+\sum_{j=0}^{\lfloor(m+1)/2\rfloor}\binom{m+1}{2j}S_{m}\left(\dfrac{2^{m+2}-2j\cdot
2^{m+1}}{2}\right)$
$\displaystyle=S_{m}(2^{m+1})+\sum_{j=0}^{\lfloor(m+1)/2\rfloor}\binom{m+1}{2j}S_{m}(2^{m+1}(1-j))$
$\displaystyle=2S_{m}(2^{m+1})+\binom{m+1}{2}S_{m}(0)$
$\displaystyle=2(r_{m-1}(0)+r_{m-1}(2^{m}))+\binom{m+1}{2}$
$\displaystyle=2m+4+\binom{m+1}{2}=3m+4+\binom{m}{2}.$

Assume that, for a fixed $n^{\prime}\geq 1$,

$S_{m}(2^{m+2}n^{\prime})=S_{m}(2^{m+2}n^{\prime}-2^{m+1})+\sum_{j=0}^{\lfloor(m+1)/2\rfloor}\binom{m+1}{2j}S_{m}\left(\dfrac{2^{m+2}n^{\prime}-2j\cdot
2^{m+1}}{2}\right).$

Then

$\displaystyle S_{m}(2^{m+2}(n^{\prime}+1))$
$\displaystyle=\sum_{k=0}^{2n^{\prime}+2}r_{m-1}(2^{m}k)$
$\displaystyle=\underbrace{S_{m}(2^{m+2}n^{\prime})}_{A}+\underbrace{r_{m-1}(2^{m}(2n^{\prime}+1))}_{B}+\underbrace{r_{m-1}(2^{m}(2n^{\prime}+2))}_{C}.$

We know an expression for $A$ by hypothesis. By Theorem 1.1,

$\displaystyle B$
$\displaystyle=r_{m-1}(2^{m}(2n^{\prime}+1)-2^{m})+\sum_{j=0}^{\lfloor(m-1)/2\rfloor}\binom{m}{2j+1}r_{m-1}\left(\dfrac{2^{m}(2n^{\prime}+1)-(2j+1)2^{m}}{2}\right)$
$\displaystyle=r_{m-1}(2^{m}(2n^{\prime}))+\sum_{j=0}^{\lfloor(m-1)/2\rfloor}\binom{m}{2j+1}r_{m-1}(2^{m}(n^{\prime}-j))$

and

$\displaystyle C$
$\displaystyle=r_{m-1}(2^{m}(2n^{\prime}+2)-2^{m})+\sum_{j=0}^{\lfloor
m/2\rfloor}\binom{m}{2j}r_{m-1}\left(\dfrac{2^{m}(2n^{\prime}+2)-2j\cdot
2^{m}}{2}\right)$
$\displaystyle=r_{m-1}(2^{m}(2n^{\prime}+1))+\sum_{j=0}^{\lfloor
m/2\rfloor}\binom{m}{2j}r_{m-1}(2^{m}(n^{\prime}-j+1)).$

After a change of indexing variable in the sum appearing in $B$, we obtain

$\displaystyle B+C$
$\displaystyle=\sum_{k=2n^{\prime}}^{2n^{\prime}+1}r_{m-1}(2^{m}k)+\sum_{j=0}^{\lfloor(m+1)/2\rfloor}\left(\binom{m}{2j-1}+\binom{m}{2j}\right)r_{m-1}(2^{m}(n^{\prime}-j+1))$
$\displaystyle=\sum_{k=2n^{\prime}}^{2n^{\prime}+1}r_{m-1}(2^{m}k)+\sum_{j=0}^{\lfloor(m+1)/2\rfloor}\binom{m+1}{2j}r_{m-1}(2^{m}(n^{\prime}-j+1)).$

It follows from the induction hypothesis that

$\displaystyle S_{m}(2^{m+2}(n^{\prime}+1))$ $\displaystyle=A+B+C$
$\displaystyle=S_{m}(2^{m+2}n^{\prime}+2^{m+1})+\sum_{j=0}^{\lfloor(m+1)/2\rfloor}\binom{m+1}{2j}S_{m}(2^{m+1}(n-j+1))$
$\displaystyle=S_{m}(2^{m+2}(n^{\prime}+1)-2^{m+1})+\sum_{j=0}^{\lfloor(m+1)/2\rfloor}\binom{m+1}{2j}S_{m}\left(\dfrac{2^{m+2}(n^{\prime}+1)-2j\cdot
2^{m+1}}{2}\right).$

This concludes the proof of Theorem 1.2.

## References

  * [Ada74] J. F. Adams.  Stable homotopy and generalised homology.  Chicago Lectures in Mathematics. University of Chicago Press, Chicago, Ill.-London, 1974.
  * [AP76] J. F. Adams and S. B. Priddy.  Uniqueness of $B{\rm SO}$.  Math. Proc. Cambridge Philos. Soc., 80(3):475–509, 1976.
  * [Bai10] Scott M. Bailey.  On the spectrum $b{\rm o}\wedge{\rm tmf}$.  J. Pure Appl. Algebra, 214(4):392–401, 2010.
  * [BBB+20] A. Beaudry, M. Behrens, P. Bhattacharya, D. Culver, and Z. Xu.  On the $E_{2}$-term of the bo-Adams spectral sequence.  J. Topol., 13(1):356–415, 2020.
  * [BC18] Agnès Beaudry and Jonathan A. Campbell.  A guide for computing stable homotopy groups.  In Topology and quantum theory in interaction, volume 718 of Contemp. Math., pages 89–136. Amer. Math. Soc., [Providence], RI, [2018] ©2018.
  * [BG73] Edgar H. Brown, Jr. and Samuel Gitler.  A spectrum whose cohomology is a certain cyclic module over the Steenrod algebra.  Topology, 12:283–295, 1973.
  * [BG10] Robert R. Bruner and J. P. C. Greenlees.  Connective real $K$-theory of finite groups, volume 169 of Mathematical Surveys and Monographs.  American Mathematical Society, Providence, RI, 2010.
  * [Dav87] Donald M. Davis.  The $b{\rm o}$-Adams spectral sequence: some calculations and a proof of its vanishing line.  In Algebraic topology (Seattle, Wash., 1985), volume 1286 of Lecture Notes in Math., pages 267–285. Springer, Berlin, 1987.
  * [DGM81] Donald M. Davis, Sam Gitler, and Mark Mahowald.  The stable geometric dimension of vector bundles over real projective spaces.  Trans. Amer. Math. Soc., 268(1):39–61, 1981.
  * [GJM86] Paul G. Goerss, John D. S. Jones, and Mark E. Mahowald.  Some generalized Brown-Gitler spectra.  Trans. Amer. Math. Soc., 294(1):113–132, 1986.
  * [Koc82] Stanley O. Kochman.  Integral cohomology operations.  In Current trends in algebraic topology, Part 1 (London, Ont., 1981), volume 2 of CMS Conf. Proc., pages 437–478. Amer. Math. Soc., Providence, R.I., 1982.
  * [Lar20] Donald M. Larson.  Explicit modular forms from the divided beta family.  Topology Appl., 273:107094, 19, 2020.
  * [Mah81] Mark Mahowald.  $b{\rm o}$-resolutions.  Pacific J. Math., 92(2):365–383, 1981.
  * [Mil58] John Milnor.  The Steenrod algebra and its dual.  Ann. of Math. (2), 67:150–171, 1958.
  * [Mil75] R. James Milgram.  The Steenrod algebra and its dual for connective $K$-theory.  In Conference on homotopy theory (Evanston, Ill., 1974), volume 1 of Notas Mat. Simpos., pages 127–158. Soc. Mat. Mexicana, México, 1975.
  * [OEI] The On-Line Encyclopedia of Integer Sequences.  http://oeis.org/.  Accessed: 2021-04-01.
  * [Rez07] Charles Rezk.  Supplementary notes for Math 512 (version 0.18).  https://faculty.math.illinois.edu/~rezk/512-spr2001-notes.pdf, 2007\.

Department of Mathematics, Clayton State University
2000 Clayton State Boulevard, Morrow, GA 30260

Department of Mathematics, The Catholic University of America
620 Michigan Ave NE, Washington DC 20064

<EMAIL_ADDRESS><EMAIL_ADDRESS>


# Energy dissipation in earthquakes

David S. Kammer<EMAIL_ADDRESS>Institute for Building Materials, ETH Zurich,
Switzerland  Gregory C. McLaskey  School of Civil and Environmental
Engineering, Cornell, University, Ithaca, NY, USA  Rachel E. Abercrombie
Boston University, Boston, MA, USA  Jean-Paul Ampuero  Université Côte d’Azur,
IRD, CNRS, Observatoire de la Côte d’Azur, Géoazur, France  Camilla Cattania
Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts
Institute of Technology, Cambridge, MA, USA  Massimo Cocco  Istituto Nazionale
di Geofisica e Vulcanologia, Rome, Italy  Luca Dal Zilio  Institute of
Geophysics, ETH Zurich, Switzerland  Georg Dresen  Helmholtz Centre Potsdam,
GFZ German Research Centre for Geosciences, Potsdam, Germany  Alice-Agnes
Gabriel  Scripps Institution of Oceanography, UCSD, La Jolla, USA  Chun-Yu Ke
Department of Geosciences, The Pennsylvania State University, University Park,
PA, 16802, USA  Chris Marone  La Sapienza Università di Roma, P.le Aldo Moro
5, 00185 Roma, Italia  Department of Geosciences, The Pennsylvania State
University, University Park, PA, 16802, USA  Paul A. Selvadurai  Swiss
Seismological Service, ETH Zurich, Switzerland  Elisa Tinti  La Sapienza
Università di Roma, P.le Aldo Moro 5, 00185 Roma, Italia

###### Abstract

Earthquakes are rupture-like processes that propagate along tectonic faults
and cause seismic waves. The propagation speed and final area of the rupture,
which determine an earthquake’s potential impact, are directly related to the
nature and quantity of the energy dissipation involved in the rupture process.
Here we present the challenges associated with defining and measuring the
energy dissipation in laboratory and natural earthquakes across many scales.
We discuss the importance and implications of distinguishing between energy
dissipation that occurs close to and far behind the rupture tip and we
identify open scientific questions related to a consistent modeling framework
for earthquake physics that extends beyond classical Linear Elastic Fracture
Mechanics.

Earthquakes are one of the most damaging natural hazards facing humankind.
Improvements in understanding the fundamental physics of earthquakes could
have dramatic consequences for our ability to plan and react to catastrophic
earthquakes in densely populated areas. Seismological observations show that
earthquakes comprise a rupture front propagating along a fault and leaving
behind slip and stress drop, which is a form of fracture propagation. Thus the
field of fracture mechanics has played a fundamental role in shaping what we
know about earthquake physics. Classical models describe an earthquake as a
shear crack and define, for example, the relationship between earthquake
rupture area, propagation speed, and the spectral characteristics of radiated
seismic waves [1, 2, 3]. While the overall energy budget that compares states
before and after an earthquake has a well-established theoretical basis [4],
key aspects of the instantaneous energy balance governing the behavior of the
earthquake rupture remain poorly understood. Fracture mechanics theory
predicts that rupture growth is a balancing act involving three components: 1)
energy dissipated to extend the crack, either by creating new surface area or
generating frictional heat, 2) energy radiated as seismic waves, and 3)
release of stored elastic energy from the surrounding rock. This view has been
confirmed by a broad range of laboratory experiments and codified in the
theory of Linear Elastic Fracture Mechanics (LEFM) [5]. However, the
complexity of earthquake faults far exceeds that of typical laboratory setups,
raising significant questions about the applicability and predictive power of
LEFM in natural conditions [6, 7]. One of our goals here is to paint a picture
of the state of the art in understanding earthquake rupture and in particular
the extent to which LEFM can further our understanding of the earthquake
energy budget in cases where the fault zone has finite width and rupture
propagation involves branching, off-fault fracturing and other processes that
go beyond the simple assumptions underlying LEFM.

While there is transformative potential in extending knowledge and connections
between the fields of fracture mechanics and earthquake physics, there are
several key impediments. These include: 1) a lack of fundamental understanding
of how various dissipative processes at different spatial and temporal scales
contribute to the mechanics of earthquakes, 2) extreme discrepancies (of many
orders of magnitude) between values of fracture energy measured in laboratory
experiments [8] and inferred from natural earthquakes, and 3) vastly different
terminology between the communities. In this Perspective, we aim to review the
mechanics and energy dissipation in earthquake ruptures, identify translations
of terminology, and discuss the capabilities and limitations of current
observations and measurement techniques and how they affect the observed
discrepancies. Finally, we propose a path forward in the form of key
outstanding questions, clear scientific objectives for future work, and
suggestions to overcome the limitations of LEFM as applied to earthquake
faulting.

Figure 1: Schematic of an earthquake with its rupture front and seismic waves.
(left) Earthquake fault complexity includes fault geometry, depth variation in
geological units, wear and gouge formation within the fault zone, and fault
branching. The red zone near the rupture front indicates regions with high
fault slip rates. Note seismic waves (blue) radiate from the fault zone.
(right) The evolution of the fault zone shear stress (blue) and slip rate
(red) are shown for the region around the rupture tip and toward the
hypocenter where the fault has already slipped. Seismic waves are indicated by
gray lines and the associated ground motions are illustrated by a bright blue
line. Processes within the rupture tip and in the tail behind the tip, so-
called tip-and-tail processes, are a major focus of this Perspective.

##  I Fundamentals of theoretical earthquake mechanics

The mechanics of earthquakes are complex and arguably equally challenging to
measure as to theoretically describe. The goal of a theoretical earthquake
model is to describe the essential processes with viable equations and tools
that allow field observation to be interpreted. Hence, it builds on common
observations, which show that earthquake ruptures begin in a localized region
of a fault known as the hypocenter (see Fig. 1), which is the location where
initial shear stresses ($\tau_{0}$) are sufficient to overcome frictional
strength and the fault motion begins to accelerate. The initial stress level
is one of the most difficult parameters to constrain. It can be spatially
variable and it can have a large impact on rupture style and velocity. Unlike
static frameworks, such as slip-tendency analysis popular in structural
geology, faults can be stressed well below strength almost everywhere and yet
rupture spontaneously: only a small portion of the fault needs to reach its
strength to nucleate an earthquake. For ordinary earthquakes with fast rupture
speeds the hypocenter is the location where seismic waves are first radiated.
From there, earthquake rupture expands along the fault, causing the fault
surfaces to begin to slip. The transition region between slipping and
unslipped sections of the fault is called the rupture front (see Figs. 1). For
fast ruptures, moving at speeds of several $\mathrm{km/s}$, the slip rate of
fault surfaces accelerates from below 1 µm/s to above 1 m/s over timescales of
less than a second. Ahead of the rupture front, the shear stress on the fault
increases in what is called a dynamic stress concentration. At the rupture
tip, a rapid transition occurs. The slip velocity increases as the shear
stress drops rapidly to a dynamic level $\tau_{\mathrm{d}}$ that is below the
initial level $\tau_{0}$. This drop in stress causes a release of stored
strain energy, which drives the rupture. Simultaneously, part of this energy
is dissipated through various processes including fracture of the surrounding
rock, comminution (the production of rock powders), heating and possibly
melting of rocks. The remaining energy is radiated as seismic waves that
transport kinetic energy far from the source and cause ground shaking.
Eventually, the rupture ceases to grow, and all sections of the earthquake
rupture area arrest (or evolve to a very slow and long quasi-static front of
postseismic slip). This can occur because continued slip necessitates a
disproportionately large amount of energy dissipation or because the rupture
front propagates into unfavorably stressed regions (i.e.,
$\tau_{0}<\tau_{\mathrm{d}}$).

Box 1: Fundamentals of Linear Elastic Fracture Mechanics (LEFM) Linear Elastic
Fracture Mechanics (LEFM) is a theoretical framework to describe crack growth,
originally developed with a focus on opening (Mode I) cracks [9, 10, 11]. The
question of whether a crack grows or not is reduced to a comparison of two
states (see Fig. 2): the current state with crack (half-)length $\ell$ and an
incremental state with crack length $\ell+\mathrm{d}\ell$. The following
assumptions are made: 1. the material surrounding the crack has predominantly
linear elastic behavior, 2. dissipation is localized at the crack tip, 3. the
crack surface is traction-free. One can then express the comparison of states
in terms of energies stating that the crack grows if
$G_{0}>\Gamma_{\mathrm{tot}}$, where $G_{0}$ is the static energy release rate
and $\Gamma_{\mathrm{tot}}$ the fracture energy [12, 13]. The static energy
release rate is the drop in the elastic energy $\Pi$ stored in the medium per
increment of crack length
$G_{0}=-\mathrm{d}\Pi/\mathrm{d}\ell\approx-\left(\Pi_{2}-\Pi_{1}\right)/\mathrm{d}\ell$.
The fracture energy is the energy dissipated in the process of breaking the
material, per unit of crack length growth. This comparison of energies is only
possible because the above-mentioned assumptions guarantee a “separation of
scale” between the global quantity $G_{0}$ driving the crack, and the local
quantity $\Gamma_{\mathrm{tot}}$ resisting crack growth. The key aspect of
this approach is that the specific physical processes of how energy is
dissipated (e.g., as decohesion, plastic work) are irrelevant as long as they
are localized at the crack tip, satisfying assumption 2. They are all lumped
into the parameter $\Gamma_{\mathrm{tot}}$. Note that assumption 2 leads to
singular stresses at the crack tip. To avoid this unphysical phenomenon the
dissipation is often smeared out in a still-localized-enough “process zone” of
size $s\ll\ell$ – an approach known as “small-scale yielding” and commonly
implemented via cohesive zone models [14, 15, 16]. While the precise limit for
small-scale yielding remains unknown, a process zone with $s\approx 0.4\ell$
may, under some circumstances, still be enough localized [see pp. 142-143 in
10]. LEFM also predicts the speed at which the crack grows [5] through an
energy balance, $G=\Gamma_{\mathrm{tot}}$, where the dynamic energy release
rate $G\leq G_{0}$ accounts for kinetic energy being radiated away from the
rupture tip. Assuming time-invariant loading (i.e., no wave reflections) and
2D or circular 3D configurations, the dynamic energy release rate can be
approximated by $G\approx(1-C_{\mathrm{f}}/C_{\mathrm{R}})G_{0}$, where
$C_{\mathrm{f}}$ is the rupture speed and $C_{\mathrm{R}}$ the material
Rayleigh wave speed.

Figure 2: (Figure for Box 1): Schematic of the fundamental principles for
crack growth in Linear Elastic Fracture Mechanics.

###  I.1 Earthquakes as a rupture process described by fracture mechanics

The earthquake rupture process of propagation and arrest shares many features
of a crack propagating through a solid material. Thus, the theoretical
framework of LEFM, as summarized in Box 1, has been adapted to describe the
mechanics of earthquake rupture [e.g., 17, 15, 14]. Specifically, under
suitable assumptions, LEFM provides an energy balance that governs rupture
growth:

$G=\Gamma_{\mathrm{tot}}~{},$ (1)

where the terms are defined in Box 1. LEFM allows rupture speed prediction
based on a few assumptions given in Box 1

$C_{\mathrm{f}}\approx\left(1-\frac{\Gamma_{\mathrm{tot}}}{G_{0}}\right)C_{\mathrm{R}}~{}.$
(2)

However, the application of LEFM to earthquake ruptures requires a few key
adaptions from its classical form. For instance, LEFM assumes a traction-free
crack behind the rupture front (Box 1). While valid for opening cracks (mode
I), this is clearly not valid for surfaces in frictional contact. Moreover, a
broad range of earthquake source properties and frictional behaviors are well
described by rate-and-state friction laws [e.g., 18, 19]. Despite the
assumption of a traction-free surface after the rupture front passes being
simplified, it allows treating dynamic friction and hence the residual shear
stresses of shear cracks (modes II and III) behind the rupture front as time-
independent and approximately constant. That is, one assumes that dynamic
friction reaches a constant residual level $\tau_{\mathrm{r}}$ for slip larger
than a characteristic slip distance $d_{\mathrm{c}}$. While this is
inconsistent with expectations for complex natural faults, it provides a
starting place for the application of LEFM and an opportunity to investigate
the earthquake energy budget and rupture dynamics.

In the attempt to measure energy dissipation, the work done by the frictional
stresses above the minimal stress through slip, commonly known as the
“breakdown work” $W_{\mathrm{b}}$, is measured/computed (see Box 2). If
residual friction is constant and $d_{\mathrm{c}}$ is small enough, then the
breakdown work is spatially localized in the process zone near the rupture
tip. This would ensure the above-mentioned separation of scale that allows the
application of LEFM because the energy dissipation associated with fracture
propagation is determined entirely within the crack tip region, and therefore
independent of other processes on the rest of the rupture surface. In this
particular case, the breakdown work is exactly equal to the associated
fracture energy, i.e., $\Gamma_{\mathrm{tot}}=W_{\mathrm{b}}$ (assuming that
there are no other dissipative processes).

Box 2: The breakdown work The breakdown work [20, 21] is the measurable
portion of the frictional work density, which, when integrated on the fault
surface, gives an estimate of the irreversible part of the total strain energy
change which does not go into radiated energy [6, 8]. [20] defined the
breakdown work $W_{\mathrm{b}}$ as the excess of work over the minimum shear
stress achieved during slip $\tau_{\mathrm{min}}$:
$W_{\mathrm{b}}=\int_{0}^{t_{b}}\left(\tau(t)-\tau_{\mathrm{min}}\right)\dot{\delta}(t)dt=\int_{0}^{d_{c}}\left(\tau(\delta)-\tau_{\mathrm{min}}\right)d\delta$
(3) where $\dot{\delta}(t)$ is the slip rate, $\tau(t)$ the shear stress, and
$t_{b}$ is the time at which $\tau_{\mathrm{min}}$ and the critical slip
distance $d_{c}$ are reached.

Some important comments:

  1. 1.

breakdown work $\neq$ fracture energy: breakdown work $W_{\mathrm{b}}$ and
associated fracture energy are not _generally_ equal, as commonly assumed in
some literature. The breakdown work is only equal to the fracture energy under
specific conditions, i.e., when there is separation of scale (see Box 1). The
precise limit of separation of scale remains unknown. Since only the localized
part of the breakdown work is part of the fracture energy,
$\Gamma_{\mathrm{tot}}\leq W_{\mathrm{b}}$ (in the absence of other
dissipative processes).

  2. 2.

fracture energy and physical processes: the fracture energy
$\Gamma_{\mathrm{tot}}$ governing crack growth in Eqs. 1&2 is the cumulative
quantity that may include the effects of many processes such as work done by
frictional weakening, plastic dissipation, off-fault damage, pulverization
within the fault zone, and others, i.e.,
$\Gamma_{\mathrm{tot}}=\sum\Gamma_{\mathrm{processes}}$. However,
$\Gamma_{\mathrm{tot}}$ only includes the part of the energy that is localized
near the crack tip (on- or off-fault) to comply with separation of scale and
to be consistent with LEFM.

  3. 3.

fracture energy variability: due to its multi-physical origin, the fracture
energy may vary spatially along the fault, and change with rupture speed.
Hence, it may not be known a-priori.

  4. 4.

non-localized heat: the so-called “frictional heat”, e.g.,
$W_{\mathrm{H}}=\int_{0}^{D}\tau_{\mathrm{r}}~{}\mathrm{d}\delta$, which is
the work of $\tau_{\mathrm{r}}$ on the fault, is not included in
$\Gamma_{\mathrm{tot}}$ as it is _not_ localized. Consequently, the energy due
to the residual stress is also excluded from $G$, and the Griffith energy
balance (Eq. 1).

###  I.2 Applicability of LEFM to laboratory ruptures versus natural
earthquakes

Frictional stick-slip events or fracture propagation on pre-existing surfaces
represent the laboratory equivalent of earthquakes. While earthquakes
generated in the laboratory (so-called “labquakes”) share many features of
natural earthquakes [22] on tectonic faults, the vast differences in scale
raise important questions that include the application of LEFM to labquakes
and earthquakes on natural faults. Yet, recent works provide a useful starting
point because they include quantitative predictions of rupture speed [23, 24]
and arrest [25, 26, 27] for labquakes. These and similar experiments also
measure the fracture energy $\Gamma_{\mathrm{tot}}$ of labquakes with local
dynamic shear stress measurements [28, 29, 30, 31] and/or the stress-versus-
slip relation [32, 33]. Such experiments require a sample that is large
compared to the process zone size and a critical length scale for rupture
nucleation. For fault normal stresses of
$1\text{\textendash}10~{}\mathrm{MPa}$ this requires meter-scale rock samples
[30, 31] or $20\text{\textendash}30~{}\mathrm{cm}$ sized samples composed of
glassy polymer such as PMMA [e.g., 28, 34]. Dynamic rupture can also be
studied on smaller samples, at higher normal stress levels
($50\text{\textendash}150~{}\mathrm{MPa}$), if arrays of sensors are installed
on the sample, inside a pressure vessel [35] and for cases where the fault
zone contains sufficient wear material [36]. To infer rupture-related
quantities, it is important to measure stress evolution on or near the fault
as a rupture front propagates past the sensor location as opposed to sample-
wide averages.

LEFM has also been applied to tectonic faults to examine physical processes
such as aseismic slip, occurring naturally or by fluid injection [e.g., 37,
38, 39, 40, 41], the statistical properties of small earthquakes [e.g., 42,
43, 44], the frequency-magnitude distribution [e.g., 45, 46], earthquake
nucleation [e.g., 47, 14, 48], and for rupture propagation and arrest [e.g.,
49]. These results demonstrate that LEFM, even with its strong simplifying
assumptions, is a powerful concept to describe the fundamental mechanics of
earthquakes and faults.

In summary, the simplifying assumptions of LEFM appear to be valid for large-
scale laboratory experiments where LEFM quantitatively describes rupture
velocity and arrest. However, laboratory experiments differ from natural
faults in several ways that must be accounted for to understand the
limitations of LEFM and to develop appropriate extensions of that theory.
First, the magnitude range of labquakes is relatively limited, which impedes a
precise determination of earthquake scaling properties. Furthermore,
laboratory experiments are often conducted at low-stress levels ($\sim
5~{}\mathrm{MPa}$) compared to the unconfined strength of the rock or polymer
samples. This limits off-fault damage or inelastic deformation that may
strongly affect the rupture process and the overall energy dissipation. The
experiments also typically employ simple fault geometries, while tectonic
faults are much more complex. Under some conditions, the complexities of
geological faults can be lumped into a single tip-localized parameter
$\Gamma_{\mathrm{tot}}$; however, for other cases the framework of LEFM
requires modification. A key question is if energy dissipation (aside from
frictional heat) on natural faults with all its complexities (e.g., weakening,
off-fault inelasticity) truly is localized in the vicinity of the rupture tip,
which would guarantee separation of scale and applicability of LEFM. Should
this not be the case, is it enough if “most” of the energy is dissipated in a
localized manner? The implications of these questions are important as the
answers determine the extent to which LEFM can be applied to earthquakes, in
its current or modified forms.

##  II Tip or tail: spatiotemporal energy dissipation in earthquakes

Tectonic faulting complexity involves simultaneous dissipative processes
during an earthquake, such as fracturing, comminution, heating, and possibly
rock melting. These processes depend on the fault slip rate and the thickness
of the shearing zone within the fault. Mechanisms such as flash heating [e.g.,
50], melt lubrication [e.g., 51], thermal pressurization [e.g., 52], acoustic
fluidization [53], elastohydrodynamic lubrication [54, 41], off-fault
deformation incurred during slow [55] or fast rupture [56, 57, 58, 59, 60] are
among the various processes that have been proposed to explain energy
dissipation during earthquakes.

These dissipative processes may influence the mechanics of earthquakes in
different ways [61, 8]. However, building on previous literature [62, 63,
among others], we propose a conceptual picture (see Fig. 1) that distinguishes
between the following key processes:

  1. 1.

_Tip processes_ dissipate energy near the rupture front and therefore
contribute to the earthquake fracture energy that is equivalent to
$\Gamma_{\mathrm{tot}}$ utilized in LEFM. The rupture tip region is
characterized by intense slip accelerations ($>100~{}\mathrm{m/s}^{2}$) and
high slip velocities ($>1~{}\mathrm{m/s}$), but because it is highly
transient, the associated slip is typically a relatively small fraction of the
total coseismic slip.

  2. 2.

_Tail processes_ occur behind the rupture tip where slip acceleration is much
lower. However, slip velocities may remain relatively high ($\sim
1~{}\mathrm{m/s}$) in the wake of the rupture tip, especially for crack-like
ruptures compared to pulse-like ruptures. Therefore, the slip accumulated in
the rupture _tail_ can be large if the rupture continues long enough.

We note that a sharp boundary between the tip and tail processes likely does
not exist and that some dissipative processes are affected by both tip and
tail [e.g., 64]. The tip and tail terminology is not limited to interface
processes (i.e., fault processes) but may also include dissipation in the bulk
material (i.e., host rocks), consistent with the initial formulation of LEFM.
The “tip” and “tail” terminology becomes specifically useful when discussing
how different dissipative processes may affect different aspects of earthquake
rupture propagation and arrest. For example, flash heating may be a weakening
mechanism that is active as a tip process. In contrast, thermal pressurization
will likely only occur as a tail process after sufficient slip has occurred
[e.g., 65]. A particularly valuable aspect of _tip and tail_ is that it
highlights a fundamental question: does energy dissipation in the _tail_
contribute to the energy balance for rupture extension or is crack propagation
determined entirely at the rupture tip? LEFM predicts that ruptures will
extend following $G=\Gamma_{\mathrm{tot}}$, but for _tail_ processes that are
sufficiently remote from the rupture tip, it is not clear how they contribute
to the energy needed for crack extension $\Gamma_{\mathrm{tot}}$.

Laboratory experiments offer valuable insights into this problem, albeit with
notable limitations. Most small-scale experiments, for instance, cannot
achieve slip acceleration that is fast enough to fully emulate the loading
conditions of a dynamic rupture front (i.e., tip processes), while large-scale
rupture experiments do not exhibit enough slip for tail processes to become
dominant. It is far from trivial to set up laboratory experiments capable of
reproducing the slip values, velocities, and accelerations under realistic
loading conditions representative of a propagating earthquake rupture. These
challenges can be addressed with numerical simulations, which allow us to
assess tip processes under non-trivial friction conditions or assess
contributions by other dissipative mechanisms, under limited conditions [e.g.,
66, 56]. Aside from limitations of laboratory and theoretical work, we also
note that the theoretical definition of where the tip ends and the tail starts
is not well defined and is likely model-dependent [64] and hence requires
further investigation.

Finally, how do the tip and tail processes influence the mechanics of
earthquakes? This is one of the key open questions in earthquake physics and
is at the center of this Perspective. Without tail processes, LEFM shows, as
outlined in Sec I, that the dissipative energy in the rupture tip controls
rupture speed and arrest. Whether this is equally true for systems with
significant tail processes remains to be shown. Recent results [67, 62, 68]
suggest that the tip processes dictate the rupture growth even in the presence
of non-negligible tail dissipation, so it may be reasonable to speculate that
the tip processes primarily control rupture propagation and arrest. However,
the tail may become important when the earthquake propagates slowly, possibly
during a slow arrest, propagates as multiple fronts, as a self-healing slip
pulse [69], or in multiple sliding episodes across rough fault surfaces[e.g.,
70], when considering the complex prestress state [71] or when considering how
earthquakes prepare the fault for subsequent events. Here again, numerical
simulations provide a tool to systematically study the link between tip and
tail processes and the mechanics of earthquake ruptures.

In summary, defining tip and tail processes and how various dissipative
processes contribute to them is crucial to a better understanding of how
earthquakes propagate, arrest, and prepare the fault for subsequent events.
The size of the yielding zone near the tip of a propagating rupture front is
also an open question, which affects the values of inferred fracture energy as
we will discuss in the next section. Laboratory experiments and numerical
simulations, best in synergistic combination, may provide crucial insight into
these processes but require further development. Finally, the proposed
framework needs to be applied to natural earthquakes but this requires a
precise understanding of how field observations are linked to these tip and
tail processes, which is another important open question.

##  III Observations of energy dissipation in natural, laboratory, and
simulated earthquakes

Given these theoretical considerations, we explore and compare observations of
energy dissipation in both labquakes and tectonic earthquakes, with a focus on
what this reveals about applying LEFM in earthquake physics. Estimates of
$\Gamma_{\mathrm{tot}}$ and the total energy dissipation vary widely for
labquakes, tectonic earthquakes, and numerical models of earthquake rupture.
For instance, estimates by Abercrombie and Rice [72], which are calculated
from a combination of seismically derived parameters (see Box 3), have
suggested that the _average_ energy dissipation in natural earthquakes ranges
across multiple orders of magnitude from $10^{2}$ to
$10^{7}~{}\mathrm{J/m}^{2}$. A compilation of seismologically inferred energy
dissipation extends this range to
$10^{-2}\text{\textendash}10^{8}~{}\mathrm{J/m}^{2}$ [52, 8], but these
estimates are all highly model-dependent and subject to large, and potentially
systematic, uncertainties [73]. Pseudo-dynamic earthquake modeling that infers
shear stress evolution based on slip history [e.g., 74] yields estimates of
$~{}\mathrm{MJ/m}^{2}$ for magnitudes $M$ larger than $5$ [20] but commonly
shows large variability along the fault plane [20, 75, 76, 77]. Near-fault
observations are used to infer constitutive parameters, such as the critical
slip distance [78], which would imply large values of breakdown work or
_average_ energy dissipation. Other approaches based on dynamic models [79,
80, 81] that simulate spontaneous dynamic rupture and employ a frictional
constitutive law yield estimates of total energy dissipation that range from 1
to 10 $~{}\mathrm{MJ/m}^{2}$.

Box 3: Seismologically derived earthquake parameters Abercrombie and Rice [72]
proposed a parameter $G^{\prime}$, here denoted $W^{\prime}_{\mathrm{b}}$ to
avoid confusion with energy release rate:
$W^{\prime}_{\mathrm{b}}=\frac{D}{2}\left(\Delta\sigma-\frac{2{\mu}E_{\mathrm{R}}}{M_{0}}\right)~{},$
(4) where $D$ is the average slip over the fault plane, $\Delta\sigma$ the
average stress drop, $\mu$ the rock shear modulus, $E_{\mathrm{R}}$ the
radiated energy, and $M_{0}$ the seismic moment. $W^{\prime}_{\mathrm{b}}$ is
theoretically equal to the breakdown work as long as the final stress
$\overline{\tau}^{\mathrm{E}}_{\mathrm{f}}$ is equal to the residual sliding
strength of the fault $\tau_{\mathrm{r}}$. This approach is entirely based on
seismologically derived parameters and can be based on rupture averages from
simple source models, or derived from finite-fault modeling with spatially and
temporally varying slip. The most reliable seismologically derived parameter
is the seismic moment $M_{\mathrm{0}}=\mu AD~{},$ (5) where $A$ is the rupture
area. In finite-fault models, the stress drop can be determined from the
spatially varying slip, but for smaller earthquakes, it is typically
determined by assuming a simple circular source model [1]:
$\Delta\sigma=\frac{7}{16}\frac{M_{\mathrm{0}}}{r^{3}}~{}.$ (6) For circular
ruptures, $r$ can be estimated from the corner frequency of the spectrum of
teleseismic waves ($f_{0}$), or the reciprocal of the pulse duration in time
domain modeling, $r=\frac{k}{\beta}{f_{0}}~{},$ (7) where $k$ is a geometrical
constant that varies widely for commonly used source models [1, 3, 82]. These
seismologically determined source parameters are subject to large systematic
and random uncertainties and should be interpreted and modeled with extreme
caution [73].

Similar estimates of energy dissipation from acoustic emission spectra of
labquakes yield values of $10^{-6}\text{\textendash}1~{}\mathrm{J/m}^{2}$
[83]. In more traditional shear fracture experiments of intact rock, however,
the fracture energy has been measured in the range
$10^{3}\text{\textendash}10^{4}~{}\mathrm{J/m}^{2}$ at the
$100~{}\mathrm{MPa}$ pressures expected in much of the seismogenic crust [84,
85, 86, 87, 88]. This is the energy required to form a fault in intact rock
and is supposed to be orders of magnitude higher than the fracture energy
required to rupture an existing tectonic fault. Energy dissipation estimated
from friction experiments on rough surfaces that are flat at long wavelength
yield estimates of $10^{-1}\text{\textendash}10^{1}~{}\mathrm{J/m}^{2}$ [32,
33, 30, 31]. Other experiments where surfaces experience large slip and
concentrated shear heating show continued weakening of the interface up to
$1~{}\mathrm{m}$ of slip [89, 90], which has been interpreted as energy
dissipation up to $1~{}\mathrm{MJ/m^{2}}$. Still other data come from mining-
induced earthquakes where the faults intersect working faces. The fraction of
energy dissipated during an M2.1 mining event was estimated to be about
$1\text{\textendash}9\%$ of the total energy released [91, 92]. More recently
[93] arrived at a similar fraction of $<1\%$ of the total energy based on
fault gouge analysis, which corresponds to an average dissipated energy per
event of roughly $0.5~{}\mathrm{MJ/m}^{2}$.

The interpretation of the broad range of values inferred for energy
dissipation (per unit area) requires careful analysis, as some may correspond
to (tip-localized) fracture energy and others correspond to energy dissipation
within a broader region that is not localized and should therefore not be
included in fracture energy. For example, quasi-static laboratory experiments,
including rotary shear and other experiments with modest slip acceleration
($5~{}\mathrm{m/s}^{2}$) produce conditions appropriate for tail processes,
not tip processes. Thus, under these conditions, the inferred dissipation does
not correspond to the fracture energy that affects the rupture tip
$W_{\mathrm{b}}\neq\Gamma_{\mathrm{b}}$. Spectral seismological estimates
[e.g., compilations of 72, 52] use information from the entire earthquake
rupture area, rather than from just the propagating rupture front, and can
therefore also include ‘tail’ dissipation mechanisms.

It has also been shown [94] that the resolution of shear stress evolution
inferred from pseudodynamic modeling [e.g., 20] is limited by the bandwidth of
the input data, either from the band limits imposed on ground motions or from
smoothing operators used to regularize the kinematic finite-fault inversions.
However, integral quantities such as fracture energy or breakdown work are
less affected by bandwidth limitations and thus they are considered more
reliable measures. Scale dependence of the physical processes governing
dynamic weakening might also explain both the broad range of values and the
scaling of energy dissipation with slip [see 8, and references therein]. Given
the tip-and-tail separation outlined above, future research is needed to
determine the values of $\Gamma_{\mathrm{tot}}$ and the total energy
dissipation retrieved at different scales and use different techniques to
evaluate both laboratory data and natural earthquakes. We emphasize here that
the wide range of inferred values for fracture energy and total energy
dissipation do not agree with predictions from LEFM, and this discrepancy will
motivate future research.

Several recent works have discussed the increase of fracture energy and
breakdown work with total slip or earthquake size. A coherent interpretation
of this scaling is still lacking, which implies some caution. However, we note
that this scaling is observed in seismological estimates from both natural
earthquakes [72] and labquakes [83] as well as in numerical modeling studies,
as shown in [8]. On the other hand, [95] proposed a numerical model to suggest
that scaling of seismologically estimated energy dissipation with slip can
result from stress overshoot, rather than a true increase of fracture energy
with rupture size and fault slip. While overshoot cannot be used to explain
the scaling reported from pseudo-dynamic modeling [20] that work raises
important questions about how to reconcile the huge range of fracture energy
measurements and the earthquake energy budget. This highlights the importance
of further research to reconcile the constant fracture energy assumption of
LEFM with current practices used to perform dynamic modeling of natural
earthquakes and ground motion predictions.

##  IV Conclusion & Outlook

This Perspective discusses the earthquake energy budget and the potential of
Linear Elastic Fracture Mechanics (LEFM) theory to describe earthquake rupture
in the laboratory and nature. The key condition for LEFM applicability is that
energy dissipation other than heat generation must be localized at the rupture
tip – a condition that is commonly satisfied in large-scale laboratory
experiments but may not be fully met for tectonic faults with all of their
complexity. This raises important questions about how to consistently and
correctly describe the energy dissipation of natural earthquakes. We suggest
distinguishing between _tip_ processes that account for localized dissipation
and _tail_ processes that occur further away from the rupture tip. In this
framework, tip processes govern earthquake rupture extension and propagation
while tail processes are more important for other measures of earthquake
energy dissipation and the global energy budget. We also highlight the large
range of measured or inferred energy dissipation from labquakes and
earthquakes across many orders of magnitude, and the possibility that this
could result from comparing localized with non-localized dissipation.

While the proposed tip-versus-tail perspective provides a useful approach to
discussing energy dissipation in earthquakes, it also opens important
scientific questions that are to be addressed in future research. For
instance, the boundary between the tip and tail, i.e., the localization of
energy dissipation, is neither well defined nor known. Experiments and field
observations with improved sensing are needed to measure the contributions to
tip and tail energy dissipation. Here it is important to note that any
physical process may contribute to dissipation in the tip and the tail
concurrently, and hence separating contributions to each is required. It is
also important to use precise and consistent terminology to avoid
misinterpretation of data. Specifically, only rupture-tip energy dissipation
should be termed “fracture energy”; and when there is no proof that energy
dissipation is local to the rupture tip more general terms, such as “breakdown
work”, should be used.

Another important open question concerns the effect of significant tail
processes on earthquake propagation and arrest mechanisms. Do these processes
affect the energy balance (Eq. 1) and, hence the rupture speed? Here,
numerical simulations are particularly useful to systematically explore and
isolate these effects and to update the fracture theory for the description of
rupture growth in the presence of tail processes. Such simulations could also
provide a tool to determine the link between fault properties (tip and tail
processes) and averaged global observations as inferred from seismological
data and hence support the correct interpretation of earthquake energy
dissipation across scales.

In conclusion, as a community, we need to synergistically combine field
observations, laboratory experiments, and numerical simulations to determine
the degree of rupture-tip localization of various energy dissipative
processes, and the effect of non-localized dissipation on rupture mechanics to
build a consistent model for earthquake physics.

## References

  * Brune [1970] J. N. Brune, Tectonic stress and the spectra of seismic shear waves from earthquakes, Journal of Geophysical Research 75, 4997 (1970).
  * Boatwright [1980] J. Boatwright, A spectral theory for circular seismic sources; simple estimates of source dimension, dynamic stress drop, and radiated seismic energy, Bulletin of the Seismological Society of America 70, 1 (1980).
  * Madariaga [1976] R. Madariaga, Dynamics of an expanding circular fault, Bulletin of the Seismological Society of America 66, 639–666 (1976), citation Key: Madariaga1976.
  * Kostrov [1974] V. Kostrov, Seismic moment and enengy of earthquakes, and the seismic flow of rock, Izv. Acad. Sci. USSR 1, 23 (1974).
  * Freund [1998] L. B. Freund,  _Dynamic fracture mechanics_ (Cambridge university press, 1998).
  * Kostrov and Das [1988] B. V. Kostrov and S. Das,  _Principles of earthquake source mechanics_ (Cambridge University Press, 1988).
  * Rudnicki [1980] J. W. Rudnicki, Fracture mechanics aplied to the earth’s crust, Annual Review of Earth and Planetary Sciences 8, 489 (1980).
  * Cocco  _et al._ [2023] M. Cocco, S. Aretusini, C. Cornelio, S. B. Nielsen, E. Spagnuolo, E. Tinti, and G. Di Toro, Fracture energy and breakdown work during earthquakes, Annual Review of Earth and Planetary Sciences 51 (2023).
  * Griffith [1921] A. A. Griffith, Vi. the phenomena of rupture and flow in solids, Philosophical transactions of the royal society of london. Series A, containing papers of a mathematical or physical character 221, 163 (1921).
  * Zehnder [2012] A. T. Zehnder,  _Fracture mechanics_ , Vol. 62 (Springer Science & Business Media, 2012).
  * Broberg [1999] K. B. Broberg,  _Cracks and fracture_ (Elsevier, 1999).
  * Irwin [1956] G. R. Irwin,  _Onset of fast crack propagation in high strength steel and aluminum alloys_ , Tech. Rep. (Naval Research Lab Washington DC, 1956).
  * Williams [1957] M. L. Williams, On the stress distribution at the base of a stationary crack., J. Appl. Mech. 24, 109–114 (1957).
  * Ida [1972] Y. Ida, Cohesive force across the tip of a longitudinal-shear crack and griffith’s specific surface energy, Journal of Geophysical Research 77, 3796 (1972).
  * Palmer and Rice [1973] A. C. Palmer and J. R. Rice, The growth of slip surfaces in the progressive failure of over-consolidated clay, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 332, 527 (1973).
  * Ohnaka [2003] M. Ohnaka, A constitutive scaling law and a unified comprehension for frictional slip failure, shear fracture of intact rock, and earthquake rupture, Journal of Geophysical Research: Solid Earth 108 (2003).
  * Scholz [2019] C. H. Scholz,  _The mechanics of earthquakes and faulting_ (Cambridge university press, 2019).
  * Marone [1998] C. Marone, Laboratory-derived friction laws and their application to seismic faulting, Annual Review of Earth and Planetary Sciences 26, 643 (1998).
  * Leeman  _et al._ [2016] J. R. Leeman, D. M. Saffer, M. M. Scuderi, and C. Marone, Laboratory observations of slow earthquakes and the spectrum of tectonic fault slip modes, Nature Communications 7, 11104 (2016).
  * Tinti  _et al._ [2005] E. Tinti, P. Spudich, and M. Cocco, Earthquake fracture energy inferred from kinematic rupture models on extended faults, Journal of Geophysical Research: Solid Earth 110, 10.1029/2005JB003644 (2005).
  * Cocco  _et al._ [2006] M. Cocco, P. Spudich, and E. Tinti, On the mechanical work absorbed on faults during earthquake ruptures, in  _Earthquakes: Radiated Energy and the Physics of Faulting_ (American Geophysical Union (AGU), 2006) pp. 237–254.
  * Cebry  _et al._ [2022] S. B. L. Cebry, C.-Y. Ke, and G. C. McLaskey, The role of background stress state in fluid-induced aseismic slip and dynamic rupture on a 3-m laboratory fault, Journal of Geophysical Research: Solid Earth 127, e2022JB024371 (2022).
  * Svetlizky  _et al._ [2017] I. Svetlizky, D. Kammer, E. Bayart, G. Cohen, and J. Fineberg, Brittle fracture theory predicts the equation of motion of frictional rupture fronts, Physical review letters 118 (2017).
  * Kammer  _et al._ [2018] D. S. Kammer, I. Svetlizky, G. Cohen, and J. Fineberg, The equation of motion for supershear frictional rupture fronts, Science advances 4, eaat5622 (2018).
  * Kammer  _et al._ [2015] D. S. Kammer, M. Radiguet, J.-P. Ampuero, and J.-F. Molinari, Linear elastic fracture mechanics predicts the propagation distance of frictional slip, Tribology letters 57, 1 (2015).
  * Bayart  _et al._ [2016] E. Bayart, I. Svetlizky, and J. Fineberg, Fracture mechanics determine the lengths of interface ruptures that mediate frictional motion, Nature Physics 12, 166 (2016).
  * Ke  _et al._ [2018] C.-Y. Ke, G. C. McLaskey, and D. S. Kammer, Rupture termination in laboratory-generated earthquakes, Geophysical Research Letters 45, 12 (2018).
  * Svetlizky and Fineberg [2014] I. Svetlizky and J. Fineberg, Classical shear cracks drive the onset of dry frictional motion, Nature 509, 205 (2014).
  * Bayart  _et al._ [2018] E. Bayart, I. Svetlizky, and J. Fineberg, Rupture dynamics of heterogeneous frictional interfaces, Journal of Geophysical Research: Solid Earth 123, 3828 (2018).
  * Kammer and McLaskey [2019] D. S. Kammer and G. C. McLaskey, Fracture energy estimates from large-scale laboratory earthquakes, Earth and Planetary Science Letters 511, 36 (2019).
  * Xu  _et al._ [2019] S. Xu, E. Fukuyama, and F. Yamashita, Robust estimation of rupture properties at propagating front of laboratory earthquakes, Journal of Geophysical Research: Solid Earth 124, 766 (2019).
  * Okubo and Dieterich [1981] P. G. Okubo and J. H. Dieterich, Fracture energy of stick-slip events in a large scale biaxial experiment, Geophysical Research Letters 8, 887 (1981).
  * Okubo and Dieterich [1984] P. G. Okubo and J. H. Dieterich, Effects of physical fault properties on frictional instabilities produced on simulated faults, Journal of Geophysical Research: Solid Earth 89, 5817 (1984).
  * Rubino  _et al._ [2017] V. Rubino, A. Rosakis, and N. Lapusta, Understanding dynamic friction through spontaneously evolving laboratory earthquakes, Nature communications 8, 1 (2017).
  * Passelègue  _et al._ [2013] F. X. Passelègue, A. Schubnel, S. Nielsen, H. S. Bhat, and R. Madariaga, From sub-rayleigh to supershear ruptures during stick-slip experiments on crustal rocks, Science 340, 1208 (2013).
  * Shreedharan  _et al._ [2021] S. Shreedharan, D. C. Bolton, J. Rivière, and C. Marone, Competition between preslip and deviatoric stress modulates precursors for laboratory earthquakes, Earth and Planetary Science Letters 553, 116623 (2021).
  * Hawthorne and Rubin [2013] J. C. Hawthorne and A. M. Rubin, Laterally propagating slow slip events in a rate and state friction model with a velocity-weakening to velocity-strengthening transition, Journal of Geophysical Research: Solid Earth 118, 3785 (2013).
  * Garagash [2021] D. I. Garagash, Fracture mechanics of rate-and-state faults and fluid injection induced slip, Philosophical Transactions of the Royal Society A 379, 20200129 (2021).
  * Dublanchet [2019] P. Dublanchet, Fluid driven shear cracks on a strengthening rate-and-state frictional fault, Journal of the Mechanics and Physics of Solids 132, 103672 (2019).
  * Galis  _et al._ [2017] M. Galis, J. P. Ampuero, P. M. Mai, and F. Cappa, Induced seismicity provides insight into why earthquake ruptures stop, Science advances 3, eaap7528 (2017).
  * Dal Zilio  _et al._ [2022] L. Dal Zilio, B. Hegyi, W. Behr, and T. Gerya, Hydro-mechanical earthquake cycles in a poro-visco-elasto-plastic fluid-bearing fault structure, Tectonophysics 838, 229516 (2022).
  * Dublanchet [2018] P. Dublanchet, The dynamics of earthquake precursors controlled by effective friction, Geophysical Journal International 212, 853 (2018).
  * Dublanchet and De Barros [2021] P. Dublanchet and L. De Barros, Dual seismic migration velocities in seismic swarms, Geophysical Research Letters 48, e2020GL090025 (2021).
  * Cattania and Segall [2019] C. Cattania and P. Segall, Crack models of repeating earthquakes predict observed moment-recurrence scaling, Journal of Geophysical Research: Solid Earth 124, 476 (2019).
  * Dempsey and Suckale [2016] D. Dempsey and J. Suckale, Forecasting induced seismicity rate and mmax using calibrated numerical models, in  _AGU Fall Meeting Abstracts_ , Vol. 2016 (2016) pp. S23D–04.
  * Cattania [2019] C. Cattania, Complex earthquake sequences on simple faults, Geophysical Research Letters 46, 10384 (2019).
  * Rubin and Ampuero [2005] A. M. Rubin and J.-P. Ampuero, Earthquake nucleation on (aging) rate and state faults, Journal of Geophysical Research: Solid Earth 110 (2005).
  * Cattania [2023] C. Cattania, A source model for earthquakes near the nucleation dimension, Bulletin of the Seismological Society of America (2023).
  * Weng and Ampuero [2019] H. Weng and J.-P. Ampuero, The dynamics of elongated earthquake ruptures, Journal of Geophysical Research: Solid Earth 124, 8584 (2019).
  * Rice [2006] J. R. Rice, Heating and weakening of faults during earthquake slip, Journal of Geophysical Research: Solid Earth 111, 10.1029/2005JB004006 (2006).
  * Nielsen  _et al._ [2008] S. Nielsen, G. Di Toro, T. Hirose, and T. Shimamoto, Frictional melt and seismic slip, Journal of Geophysical Research: Solid Earth 113, 10.1029/2007JB005122 (2008).
  * Viesca and Garagash [2015] R. C. Viesca and D. I. Garagash, Ubiquitous weakening of faults due to thermal pressurization, Nature Geoscience 8, 875 (2015).
  * Melosh [1996] H. J. Melosh, Dynamical weakening of faults by acoustic fluidization, Nature 379, 601 (1996).
  * Brodsky and Kanamori [2001] E. E. Brodsky and H. Kanamori, Elastohydrodynamic lubrication of faults, Journal of Geophysical Research: Solid Earth 106, 16357 (2001).
  * Rudnicki and Rice [1975] J. Rudnicki and J. Rice, Conditions for the localization of deformation in pressure-sensitive dilatant materials, Journal of the Mechanics and Physics of Solids 23, 371 (1975).
  * Andrews [2005] D. J. Andrews, Rupture dynamics with energy loss outside the slip zone, Journal of Geophysical Research: Solid Earth 110, 10.1029/2004JB003191 (2005).
  * Ben-Zion and Shi [2005] Y. Ben-Zion and Z. Shi, Dynamic rupture on a material interface with spontaneous generation of plastic strain in the bulk, Earth and Planetary Science Letters 236, 486 (2005).
  * Rice  _et al._ [2005] J. R. Rice, C. G. Sammis, and R. Parsons, Off-Fault Secondary Failure Induced by a Dynamic Slip Pulse, Bulletin of the Seismological Society of America 95, 109 (2005).
  * Bhat  _et al._ [2007] H. S. Bhat, R. Dmowska, G. C. P. King, Y. Klinger, and J. R. Rice, Off-fault damage patterns due to supershear ruptures with application to the 2001 Mw 8.1 Kokoxili (Kunlun) Tibet earthquake, Journal of Geophysical Research: Solid Earth 112, 10.1029/2006JB004425 (2007).
  * Gabriel  _et al._ [2013] A.-A. Gabriel, J.-P. Ampuero, L. Dalguer, and P. M. Mai, Source properties of dynamic rupture pulses with off-fault plasticity, Journal of Geophysical Research: Solid Earth 118, 4117 (2013).
  * Ben-Zion and Dresen [2022] Y. Ben-Zion and G. Dresen, A synthesis of fracture, friction and damage processes in earthquake rupture zones, Pure and Applied Geophysics , 1 (2022).
  * Paglialunga  _et al._ [2022] F. Paglialunga, F. X. Passelègue, N. Brantut, F. Barras, M. Lebihain, and M. Violay, On the scale dependence in the dynamics of frictional rupture: Constant fracture energy versus size-dependent breakdown work, Earth and Planetary Science Letters 584, 117442 (2022).
  * Brantut  _et al._ [2019] N. Brantut, D. I. Garagash, and H. Noda, Stability of pulse-like earthquake ruptures, Journal of Geophysical Research: Solid Earth 124, 8998 (2019).
  * Cornelio  _et al._ [2022] C. Cornelio, E. Spagnuolo, S. Aretusini, S. Nielsen, F. Passelègue, M. Violay, M. Cocco, and G. Di Toro, Determination of parameters characteristic of dynamic weakening mechanisms during seismic faulting in cohesive rocks, Journal of Geophysical Research: Solid Earth , e2022JB024356 (2022).
  * Lambert and Lapusta [2020] V. Lambert and N. Lapusta, Rupture-dependent breakdown energy in fault models with thermo-hydro-mechanical processes, Solid Earth 11, 2283 (2020).
  * Andrews [1976] D. J. Andrews, Rupture velocity of plane strain shear cracks, Journal of Geophysical Research (1896-1977) 81, 5679 (1976).
  * Barras  _et al._ [2020] F. Barras, M. Aldam, T. Roch, E. A. Brener, E. Bouchbinder, and J.-F. Molinari, The emergence of crack-like behavior of frictional rupture: Edge singularity and energy balance, Earth and planetary science letters 531, 115978 (2020).
  * Weng and Ampuero [2022] H. Weng and J.-P. Ampuero, Integrated rupture mechanics for slow slip events and earthquakes, Nature Communications 13, 7327 (2022).
  * Heaton [1990] T. H. Heaton, Evidence for and implications of self-healing pulses of slip in earthquake rupture, Physics of the Earth and Planetary Interiors 64, 1 (1990).
  * Fang and Dunham [2013] Z. Fang and E. M. Dunham, Additional shear resistance from fault roughness and stress levels on geometrically complex faults, Journal of Geophysical Research: Solid Earth 118, 3642 (2013).
  * Ampuero and Ben-Zion [2008] J.-P. Ampuero and Y. Ben-Zion, Cracks, pulses and macroscopic asymmetry of dynamic rupture on a bimaterial interface with velocity-weakening friction, Geophysical Journal International 173, 674 (2008).
  * Abercrombie and Rice [2005] R. E. Abercrombie and J. R. Rice, Can observations of earthquake scaling constrain slip weakening?, Geophysical Journal International 162, 406 (2005).
  * Abercrombie [2021] R. E. Abercrombie, Resolution and uncertainties in estimates of earthquake stress drop and energy release, Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 379, 20200131 (2021).
  * Mai and Thingbaijam [2014] P. M. Mai and K. Thingbaijam, Srcmod: An online database of finite-fault rupture models, Seismological Research Letters 85, 1348 (2014).
  * Bouchon [1997] M. Bouchon, The state of stress on some faults of the San Andreas System as inferred from near‐field strong motion data, Journal of Geophysical Research: Solid Earth 102, 11731 (1997).
  * Causse  _et al._ [2014] M. Causse, L. A. Dalguer, and P. M. Mai, Variability of dynamic source parameters inferred from kinematic models of past earthquakes, Geophysical Journal International 196, 1754 (2014).
  * Ide and Takeo [1997] S. Ide and M. Takeo, Determination of constitutive relations of fault slip based on seismic wave analysis, Journal of Geophysical Research: Solid Earth 102, 27379 (1997).
  * Kaneko  _et al._ [2017] Y. Kaneko, E. Fukuyama, and I. J. Hamling, Slip‐weakening distance and energy budget inferred from near‐fault ground deformation during the 2016  M ${}_{\textrm{w}}$ 7.8 Kaikōura earthquake, Geophysical Research Letters 44, 4765 (2017).
  * Tinti  _et al._ [2021] E. Tinti, E. Casarotti, T. Ulrich, T. Taufiqurrahman, D. Li, and A.-A. Gabriel, Constraining families of dynamic models using geological, geodetic and strong ground motion data: The Mw 6.5, October 30th, 2016, Norcia earthquake, Italy, Earth and Planetary Science Letters 576, 117237 (2021).
  * Gallovic  _et al._ [2019] F. Gallovic, L. Valentova, J. Ampuero, and A. Gabriel, Bayesian Dynamic Finite‐Fault Inversion: 2. Application to the 2016 M ${}_{\textrm{ {w} }}$ 6.2 Amatrice, Italy, Earthquake, Journal of Geophysical Research: Solid Earth 124, 6970 (2019).
  * Premus  _et al._ [2022] J. Premus, F. Gallovic, and J.-P. Ampuero, Bridging time scales of faulting: From coseismic to postseismic slip of the M ${}_{\textrm{w}}$ 6.0 2014 South Napa, California earthquake, Science Advances 8, eabq2536 (2022).
  * Kaneko and Shearer [2015] Y. Kaneko and P. M. Shearer, Variability of seismic source spectra, estimated stress drop, and radiated energy, derived from cohesive‐zone models of symmetrical and asymmetrical circular and elliptical ruptures, Journal of Geophysical Research: Solid Earth 120, 1053–1079 (2015).
  * Selvadurai [2019] P. A. Selvadurai, Laboratory insight into seismic estimates of energy partitioning during dynamic rupture: An observable scaling breakdown, Journal of Geophysical Research: Solid Earth 124, 11350 (2019).
  * Wawersik and Brace [1971] W. R. Wawersik and W. F. Brace, Post-failure behavior of a granite and diabase, Rock mechanics 3, 61 (1971).
  * Wong and Biegel [1985] T.-F. Wong and R. Biegel, Effects of pressure on the micromechanics of faulting in san marcos gabbro, Journal of Structural Geology 7, 737 (1985).
  * Wong [1982] T.-f. Wong, Shear fracture energy of westerly granite from post-failure behavior, Journal of Geophysical Research: Solid Earth 87, 990 (1982).
  * Wong [1986] T.-F. Wong, On the normal stress dependence of the shear fracture energy, in  _Earthquake Source Mechanics_ (American Geophysical Union (AGU), 1986) pp. 1–11.
  * Lockner  _et al._ [1991] D. A. Lockner, J. D. Byerlee, V. Kuksenko, A. Ponomarev, and A. Sidorin, Quasi-static fault growth and shear fracture energy in granite, Nature 350, 39 (1991).
  * Nielsen  _et al._ [2016] S. Nielsen, E. Spagnuolo, S. Smith, M. Violay, G. Di Toro, and A. Bistacchi, Scaling in natural and laboratory earthquakes, Geophysical Research Letters 43, 1504 (2016).
  * Di Toro  _et al._ [2011] G. Di Toro, R. Han, T. Hirose, N. De Paola, S. Nielsen, K. Mizoguchi, F. Ferri, M. Cocco, and T. Shimamoto, Fault lubrication during earthquakes, Nature 471, 494 (2011).
  * Olgaard and Brace [1983] D. Olgaard and W. Brace, The microstructure of gouge from a mining-induced seismic shear zone, International Journal of Rock Mechanics and Mining Sciences & Geomechanics Abstracts 20, 11 (1983).
  * McGarr [1974] A. McGarr, Earthquake prediction: Absence of a precursive change in seismic velocities before a tremor of magnitude 3¾, Science 185, 1047 (1974).
  * Chester  _et al._ [2005] J. S. Chester, F. M. Chester, and A. K. Kronenberg, Fracture surface energy of the punchbowl fault, san andreas system, Nature 437, 133 (2005).
  * Guatteri and Spudich [2000] M. Guatteri and P. Spudich, What can strong-motion data tell us about slip-weakening fault-friction laws?, Bulletin of the Seismological Society of America 90, 98 (2000).
  * Ke  _et al._ [2022] C.-Y. Ke, G. C. McLaskey, and D. S. Kammer, Earthquake breakdown energy scaling despite constant fracture energy, Nature Communications 13, 1005 (2022).

## Acknowledgements

We thank the organizers of the Workshop on Earthquake Dynamics: Mechanical
Work and Fracture Energy, which was the starting point for this work.

## Author contributions

D.S.K. coordinated the writing of the manuscript. D.S.K. and G.C.M. wrote the
initial draft. All authors contributed to the editing of the manuscript.

## Competing interests

All authors declare no competing interests.