Figure 17:  Ratio of spherically enclosed gas mass $M_{\mathrm{gas}}$ to total
mass $M_{\mathrm{tot}}$,
$f_{\mathrm{gas}}(r)=M_{\mathrm{gas}}(<r)/M_{\mathrm{tot}}(<r)$, as a function
of spherical radius $r$ derived from combined Chandra X-ray and Subaru weak-
lensing observations. The orange and blue shaded areas represent the
marginalized $1\sigma$ confidence regions based on triaxial NFW modeling of
the $\Sigma(\mbox{\boldmath$\theta$})$ map (Figure 9) with and without using
the LOS prior, respectively. The dashed line shows the posterior mean based on
spherical NFW modeling of the $\Sigma(\mbox{\boldmath$\theta$})$ map. The red
hatched area represents the marginalized $1\sigma$ confidence region based on
the hydrostatic total mass $M_{\mathrm{HE}}(<r)$ obtained from forward
modeling of Chandra X-ray data. The horizontal dotted line shows the cosmic
baryon fraction $f_{\mathrm{b}}=\Omega_{\mathrm{b}}/\Omega_{\mathrm{m}}$
determined by Planck Collaboration et al. (2020).

In Figure 17, we show the ratio of spherically enclosed gas mass
$M_{\mathrm{gas}}(<r)$ to total mass $M_{\mathrm{tot}}(<r)$ as a function of
spherical radius $r$:

$f_{\mathrm{gas}}(<r)=\frac{M_{\mathrm{gas}}(<r)}{M_{\mathrm{tot}}(<r)}.$ (50)

Here the total mass $M_{\mathrm{tot}}$ is taken to be either the weak-lensing
mass $M_{\mathrm{WL}}$ or the hydrostatic mass $M_{\mathrm{HE}}$ (see Section
7.1) and the gas mass $M_{\mathrm{gas}}$ is derived from the X-ray forward
model (Section 6.3). We find that the gas mass fraction $f_{\mathrm{gas}}(<r)$
increases progressively outward, indicating that the hot gas is more extended
than the underlying matter distribution.

Our lensing results, when combined with the X-ray gas mass measurements, yield
a direct estimate for $f_{\mathrm{gas}}(<r)$, free from the assumption of
hydrostatic equilibrium. Using the total mass derived from triaxial lens
modeling, the gas mass fraction enclosed within a sphere of radius
$r=0.7\,h^{-1}\,\mathrm{Mpc}\sim 0.7r_{500}$ is found to be
$f_{\mathrm{gas}}(<r)=(8.4\pm 1.0)\%$ and $(7.6\pm 1.0)\%$, with and without
using the LOS prior, respectively. Extrapolating the gas mass measurements out
to $r_{500}$, we find $f_{\mathrm{gas}}(<r_{500})=(9.6\pm 1.2)\%$ and $(9.0\pm
1.2)\%$ with and without the LOS prior, respectively.

When compared to the cosmic baryon fraction
$f_{\mathrm{b}}\equiv\Omega_{\mathrm{b}}/\Omega_{\mathrm{m}}=0.156\pm 0.002$
determined by the Planck mission (Planck Collaboration et al., 2020), our
constraint on the gas mass fraction indicates
$f_{\mathrm{gas}}(<r_{500})/f_{\mathrm{b}}=0.62\pm 0.08$ and $0.58\pm 0.08$,
with and without using the LOS prior, respectively. These are significantly
lower than the typical values of $f_{\mathrm{gas}}/f_{\mathrm{b}}\sim
0.8$–$0.9$ observed for high-mass galaxy clusters (Chiu et al., 2018a; Tian et
al., 2020; Akino et al., 2022). Such a high degree of gas depletion can be
caused by the adiabatic expansion of the post-shock gas (Ricker & Sarazin,
2001; Umetsu et al., 2010). It would take of the order of Gyrs for the gas to
fall back into the gravitational potential well of the cluster.

By contrast, the gas mass fraction based on the X-ray hydrostatic mass,
$f_{\mathrm{gas}}(<r)=M_{\mathrm{gas}}(<r)/M_{\mathrm{HE}}(<r)$, reaches the
cosmic baryon fraction $f_{\mathrm{b}}$ at $r\approx
0.8\,h^{-1}\,\mathrm{Mpc}$ and increasingly exceeds it at larger cluster
radii.

##  8 Summary and Conclusions

The Frontier Fields cluster A370 is a superlens characterized by a large
Einstein radius ($\theta_{\mathrm{Ein}}=33.9\arcsec\pm 1.1\arcsec$ for
$z_{s}=2$; Table 1) and is one of the most massive known lenses on the sky.
Recent dedicated numerical simulations of binary cluster mergers constrained
by multi-probe observations suggest that the cluster is a post-major merger of
two similar-mass clusters (Molnar et al., 2020). These results also suggest
that A370 is in a highly disturbed dynamical state and is elongated along the
current direction of the collision axis, which is closely aligned with the
line of sight in their best-matching simulation.

In this paper, we have carried out a detailed weak-lensing and X-ray study of
A370 using wide-field $BR_{\mathrm{C}}z^{\prime}$ Subaru/Sprime-Cam (Section
3) and Chandra X-ray (Section 6) observations. By combining 2D shear and
azimuthally averaged magnification constraints derived from the Subaru data,
we have performed a lensing mass reconstruction in a free-form manner (Section
4; Figures 1 and 9), which allows us to determine both radial structure and 2D
morphology of the cluster mass distribution.

In a parametric triaxial framework assuming an NFW density profile, we have
constrained the intrinsic structure, shape, and orientation of the cluster
halo by forward modeling the reconstructed $\Sigma(\mbox{\boldmath$\theta$})$
map (Section 5; Tables 6 and 7). We obtain a halo mass $M_{200}=(1.54\pm
0.29)\times 10^{15}\,h^{-1}\,M_{\odot}$ and a halo concentration
$c_{200}=5.27\pm 1.28$ with uninformative uniform priors. Using a prior on the
LOS alignment of the halo major axis derived from the binary merger
simulations of Molnar et al. (2020), we find that the data favor a more
prolate geometry with lower mass and lower concentration, $M_{200}=(1.38\pm
0.20)\times 10^{15}\,h^{-1}\,M_{\odot}$ and $c_{200}=4.45\pm 0.93$.

When compared to the hydrostatic mass estimate $M_{\mathrm{HE}}$ from Chandra
observations (Section 7.1), our triaxial weak-lensing analysis yields
spherically enclosed mass ratios $M_{\mathrm{HE}}/M_{\mathrm{WL}}$ of
$1-b(r)=0.56\pm 0.09$ and $0.51\pm 0.09$ at $r=0.7\,h^{-1}\,\mathrm{Mpc}\sim
0.7r_{500}$, with and without using the LOS prior, respectively (Figure 16).
Extrapolating our X-ray forward model to $r_{500}$, we find
$1-b(r_{500})=0.54\pm 0.12$ and $0.50\pm 0.11$ with and without the LOS prior,
respectively. Since the cluster is in a highly disturbed dynamical state
(Section 5.1), this represents the likely maximum level of hydrostatic bias
expected in galaxy clusters.

Our lensing results, when combined with the X-ray gas mass measurements, yield
a direct estimate for the gas mass fraction, free from the assumption of
hydrostatic equilibrium. From triaxial lens modeling with the LOS prior, the
gas mass fraction enclosed within a sphere of radius
$r=0.7\,h^{-1}\,\mathrm{Mpc}\sim 0.7r_{500}$ is found to be
$f_{\mathrm{gas}}(<r)=(8.4\pm 1.0)\%$ (Section 7.2). When the gas mass
measurements are extrapolated to $r_{500}$,
$f_{\mathrm{gas}}(<r_{500})=(9.6\pm 1.2)\%$, or
$f_{\mathrm{gas}}(<r_{500})/f_{\mathrm{b}}=0.62\pm 0.08$ relative to the
cosmic baryon fraction,
$f_{\mathrm{b}}=\Omega_{\mathrm{b}}/\Omega_{\mathrm{m}}$ (Figure 17). These
are significantly lower than the typical values of
$f_{\mathrm{gas}}/f_{\mathrm{b}}\sim 0.8$–$0.9$ found in high-mass galaxy
clusters (Chiu et al., 2018a; Tian et al., 2020; Akino et al., 2022). The high
degree of gas depletion observed for A370 is in line with the post-major
merger scenario of Molnar et al. (2020).

We have also constructed the projected radial mass profile from an optimally
weighted projection of the $\Sigma(\mbox{\boldmath$\theta$})$ map (Table 5),
obtaining a model-independent constraint on the projected total mass of
$M_{\mathrm{2D}}(<r_{\perp})=(3.11\pm 0.47)\times 10^{15}\,h^{-1}\,M_{\odot}$
at $r_{\perp}\approx 2.3\,h^{-1}\,\mathrm{Mpc}\sim 1.2r_{\mathrm{vir}}$ for
the projected mass of the whole system, including any currently unbound
material around the cluster.

Combining the data products presented in this work with HST strong- and weak-
lensing data sets available from the Frontier Fields and BUFFALO programs will
allow us to conduct a multi-scale lensing reconstruction in the cluster of
exceptional projected mass. Such a full-lensing analysis can then be used to
detect and study mass substructures in the unique merging environment (e.g.,
Jauzac et al., 2016, 2018; Tam et al., 2020), for a detailed comparison with
the distribution of intracluster baryons. It will also allow us to perform a
detailed characterization of the mass profile shape and its deviation from the
equilibrium form over a wide radial range, for an improved determination of
the total mass bound to the cluster.

We thank the anonymous referee for providing insightful comments and
suggestions. We acknowledge fruitful discussions with Mauro Sereno, Mathilde
Jauzac, Liliya L. R. Williams, Nobuhiro Okabe, Jose M. Diego, and Renyue Cen.
This work is supported by the Ministry of Science and Technology of Taiwan
(grants MOST 106-2628-M-001-003-MY3 and MOST 109-2112-M-001-018-MY3) and by
the Academia Sinica Investigator award (grant AS-IA-107-M01). M.N.
acknowledges support from grants INAF 1.05.01.86.20 and PRIN MIUR 2017
”Zooming into Dark Matter and proto-galaxies with massive lensing clusters”.
M.O. acknowledges support from JSPS KAKENHI Grant Nos. JP18K03693, JP20H00181,
and JP20H05856. This paper is based on data collected at Subaru Telescope,
which is operated by the National Astronomical Observatory of Japan. This work
has made use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement.

##  Appendix A The Effect of the Brighter Magnitude Cut on Magnification Bias

In this study, we applied both bright and faint magnitude cuts to define a
magnitude-limited sample of background galaxies (Table 4). Applying an
additional bright magnitude cut is to reduce the contamination from unlensed
foreground galaxies (Medezinski et al., 2010, 2011, 2018). However, it will
also modify the signal of magnification bias because magnified source galaxies
near the bright cut will be removed from the observed sample. As a result, the
net effect of magnification bias includes the contribution from the bright cut
as well as from the faint cut (Chiu et al., 2020).

Following Chiu et al. (2020), we obtain the expression for the magnification
bias signal expected for a background sample defined in the magnitude range
$m_{\mathrm{bright}}\leqslant m<m_{\mathrm{faint}}$ as

$\frac{N_{\mu}(m_{\mathrm{bright}}\leqslant
m<m_{\mathrm{faint}})}{\overline{N}_{\mu}(m_{\mathrm{bright}}\leqslant
m<m_{\mathrm{faint}})}\simeq 1+\left(5s_{\mathrm{eff}}-2\right)\kappa,$ (A1)

where we have used the weak-lensing limit ($\mu\simeq 1+2\kappa$); the
quantity $s_{\mathrm{eff}}$ denotes the effective count slope for a background
sample defined in the magnitude range $m_{\mathrm{bright}}\leqslant
m<m_{\mathrm{faint}}$:

$s_{\mathrm{eff}}=\frac{s(m_{\mathrm{faint}})-f_{\mathrm{bright}}s(m_{\mathrm{bright}})}{1-f_{\mathrm{bright}}},$
(A2)

with
$f_{\mathrm{bright}}=\overline{N}_{\mu}(<m_{\mathrm{bright}})/\overline{N}_{\mu}(<m_{\mathrm{faint}})$.

In typical observations of magnification bias based on deep multi-band imaging
(Umetsu et al., 2014; Chiu et al., 2016), we require $m_{\mathrm{bright}}$ to
be $2$–$3$ magnitudes brighter than $m_{\mathrm{faint}}$ (Table 4). In this
work, we have $f_{\mathrm{bright}}\approx 1.0\%$ for the lensing-cut sample
and $f_{\mathrm{bright}}\approx 3.3\%$ for the null-test sample, so that we
can safely ignore the correction terms proportional to $f_{\mathrm{mask}}$. In
the limit $f_{\mathrm{bright}}\to 0$, we have

$\frac{N_{\mu}(m_{\mathrm{bright}}\leqslant
m<m_{\mathrm{faint}})}{\overline{N}_{\mu}(m_{\mathrm{bright}}\leqslant
m<m_{\mathrm{faint}})}\to\frac{N_{\mu}(<m_{\mathrm{faint}})}{\overline{N}_{\mu}(<m_{\mathrm{faint}})}.$
(A3)

In this study, we interpret the magnification bias signal using the
approximation $s_{\mathrm{eff}}\simeq s(m_{\mathrm{faint}})$.

##  Appendix B Photometric Zero Point Calibration

Figure 18: Magnitude difference between the Suprime-Cam $z^{\prime}$ and the
reference $z^{\prime}$ derived from Pan-STARRS $grizy$ photometry, shown for a
calibration sample of point sources with reference $z^{\prime}$ magnitudes in
the range $\in[18.5,20.0]$. Figure 19: Binned distribution of stars in
color–color space for the COSMOS field (left) and A370 (right), after the
color matching. The locations of the peak density (redder colors) in the two
fields are different because of the field-to-field variations of stellar
populations as well as of different magnitude cuts applied to both samples. By
contrast, the shapes of the stellar locus in the two fields are similar.
Therefore, the cross-correlation technique can be used to match the color
distribution of stars in the cluster field to the reference COSMOS photometry.

The zero point for the Suprime-Cam $z^{\prime}$ filter was calibrated by
matching sextractor’s AUTO fluxes for point sources to their PSF fluxes from
the Pan-STARRS DR1 catalog (Flewelling et al., 2020). Since the transmission
curves of the Pan-STARRS $z$ filter and the Suprime-Cam $z^{\prime}$ filter
are different, we followed the procedure of Umetsu et al. (2010) to infer
$z^{\prime}$-band fluxes from the Pan-STARRS photometry. We use the hyperz
code (New-Hyperz ver. 11; Bolzonella et al., 2000) to perform a spectral
energy distribution (SED) fitting to Pan-STARRS $grizy$ photometry, with
stellar templates from the Pickles library (Pickles, 1998). Pan-STARRS’s
$z^{\prime}$ fluxes were obtained using the transmission curve of the Suprime-
Cam $z^{\prime}$ filter. Point sources with Pan-STARRS’s $z^{\prime}$
magnitudes in the range $[18.5,20.0]$ are used for calibration, because stars
with $z^{\prime}$ brighter than $18.5$ mag are saturated. Figure 18 compares
the calibrated Suprime-Cam $z^{\prime}$ magnitudes and the reference
$z^{\prime}$ magnitudes derived from Pan-STARRS $grizy$ photometry. The
residual rms scatter is 0.034 mag.

The zero points for the Suprime-Cam $B$ and $R_{\mathrm{C}}$ filters were
first derived by matching the stellar locus in the $B-R_{\mathrm{C}}$ vs.
$R_{\mathrm{C}}-z^{\prime}$ diagram to the COSMOS2020 photometry (Weaver et
al., 2022). The uniformity of the colors of Galactic stars permits a reliable
color calibration between fields with $|b|>30^{\circ}$ (for detail, see
Gilbank et al., 2011). To this end, we use isophotal fluxes for better color
measurements. Since the COSMOS photometry does not cover the Suprime-Cam
$R_{\mathrm{C}}$ band, we needed to estimate $R_{\mathrm{C}}$ magnitudes for
COSMOS field objects. We use again the hyperz code with SEDs from the Pickles
library (Pickles, 1998) to obtain the best-fit model for each star using
COSMOS2020 isophotal photometry in 4 Suprime-Cam intermediate bands
$(IB574,IA624,IA679,IB709)$. The $R_{\mathrm{C}}$ photometry was derived using
the transmission curve of the Suprime-Cam $R_{\mathrm{C}}$ filter. Since the
wavelength coverage of the Suprime-Cam $R_{\mathrm{C}}$ band is well sampled
by the 4 Suprime-Cam intermediate filters, this $R_{\mathrm{C}}$ estimation is
regarded as an interpolation of data. The $R_{\mathrm{C}}$ magnitudes obtained
with this method are thus model independent and sufficiently accurate for our
purpose (see Umetsu et al., 2010). The matching was performed using the cross
correlation in CC space between our Suprime-Cam data in A370 and the
COSMOS2020 data. Figure 19 shows the result of color matching. Once the color
offsets are determined, the zero points for the Suprime-Cam $B$ and
$R_{\mathrm{C}}$ filters were derived from the color offsets.

We repeat the same procedure to obtain Suprime-Cam $R_{\mathrm{C}}$ magnitudes
for galaxies in the COSMOS field. The SED fitting for each COSMOS galaxy was
performed with the hyperz code by using spectral templates from the GALAXEV
library (Bruzual & Charlot, 2003) and by fixing the redshift to each
photometric redshift (computed with the lephare code; Ilbert et al. 2006) from
the COSMOS2020 farmer catalog. A Galactic extinction correction was applied to
galaxies in both data sets according to Schlegel et al. (1998). By matching
the distributions of galaxies in the $B-R_{\mathrm{C}}$ vs.
$R_{\mathrm{C}}-z^{\prime}$ diagram for both data sets, we find that
additional offsets of $-0.02$ mag and $+0.10$ mag need to be added to the
Suprime-Cam $B$ and $R_{\mathrm{C}}$ magnitudes, respectively. We have
corrected for the residual offsets by adding $-0.02$ mag and $+0.10$ mag to
the respective magnitudes in our data set. These additional offsets may be due
to the bias in our Suprime-Cam $R_{\mathrm{C}}$ estimation based on the SED
fitting and the uncertainty of the cross correlation matching.

##  Appendix C Weak-lensing Magnification Analysis

Here we detail our magnification analysis. Following the procedure outlined in
Umetsu et al. (2014, 2016), we account for the Poisson, intrinsic clustering,
and additional systematic contributions to the total uncertainty
$\sigma_{\mu}$ (Section 2.5). First, we estimate
$\sigma_{\mu,i}^{\mathrm{int}}$ dominated by intrinsic clustering from the
azimuthal variation of the counts in cell. A positive tail of $>\nu\sigma$
cells is then removed in each bin using iterative $\sigma$ clipping with
$\nu=2$. This is to alleviate the bias due to angular clustering of red
galaxies. The Poisson noise term $\sigma_{\mu,i}^{\mathrm{stat}}$ is estimated
from the clipped mean counts in each annular bin. The difference between the
mean counts estimated with and without $\sigma$ clipping is taken as a
systematic error,
$\sigma_{\mu,i}^{\mathrm{sys}}=|n_{\mu,i}^{(\nu)}-n_{\mu,i}^{(\infty)}|/\nu$,
with $n_{\mu,i}^{(\nu)}$ and $n_{\mu,i}^{(\infty)}$ the clipped and unclipped
mean counts in the $i$th annulus, respectively. Finally, these errors are
combined in quadrature as

$\sigma_{\mu,i}^{2}=(\sigma_{\mu,i}^{\mathrm{int}})^{2}+(\sigma_{\mu,i}^{\mathrm{stat}})^{2}+(\sigma_{\mu,i}^{\mathrm{sys}})^{2}.$
(C1)

Our magnification analysis is insensitive to the particular choice of $\nu$
because of the inclusion of the $\sigma_{\mu}^{\mathrm{sys}}$ term (Umetsu et
al., 2014, 2016). Note that by including the $\sigma_{\mu}^{\mathrm{stat}}$
term in Equation (C1), we are in effect double counting the contribution of
Poisson fluctuations in estimating the errors. We find that including the
$\sigma_{\mu}^{\mathrm{stat}}$ term increases the estimated total uncertainty
by $10\%$–$20\%$. This slight overestimate of the uncertainty is not expected
to significantly affect our joint mass reconstruction, because our lensing
constraints are dominated by the shear measurements.

Masking of observed sky is corrected for using the method of Umetsu et al.
(2011, Method B of Appendix A), which is fully automated once the
configuration parameters of sextractor (Bertin & Arnouts, 1996) are optimally
tuned (Umetsu et al., 2014, 2016; Chiu et al., 2016). We find that the masked
area fraction $f_{\mathrm{mask}}$ is $\sim 4\%$ of the sky at
$\theta>10\arcmin$, increasing toward the cluster center up to $\sim 10\%$ at
$\theta\lower 2.15277pt\hbox{$\;\buildrel<\over{\sim}\;$}2\arcmin$. The masked
area fraction averaged over the radial range
$\theta\in[\theta_{\mathrm{min}},\theta_{\mathrm{max}}]$ is $\approx 4.6\%$.
This is similar to the results of Umetsu et al. (2016) for the CLASH sample at
the median redshift $\overline{z_{l}}\approx 0.35$.

The mask-corrected magnification bias profile
$b_{\mu,i}=n_{\mu,i}/\overline{n}_{\mu}$ is proportional to
$(1-f_{\mathrm{mask,back}})/(1-f_{\mathrm{mask},i})\equiv 1+\Delta
f_{\mathrm{mask},i}$ with $f_{\mathrm{mask,back}}$ estimated in the reference
background region at $\theta\in[12\arcmin,16\arcmin]$. Thus, the mask
correction essentially depends on the difference of the $f_{\mathrm{mask}}$
values, $\Delta f_{\mathrm{mask},i}\simeq
f_{\mathrm{mask},i}-f_{\mathrm{mask,back}}$, which is insensitive to the
particular choice of the configuration parameters for source extraction.
Accordingly, the systematic uncertainty on the mask correction is not expected
to significantly bias our magnification measurements.

##  Appendix D Two-dimensional to One-dimensional Projection

To enable a direct comparison between the results from 1D and 2D mass
reconstructions, we construct a surface mass density profile $\Sigma(\theta)$
from an optimally weighted projection of the
$\Sigma(\mbox{\boldmath$\theta$})$ field as (Umetsu et al., 2015)

$\mbox{\boldmath$\Sigma$}_{(1)}=\left[A^{t}C_{(2)}^{-1}A\right]^{-1}A^{t}C_{(2)}^{-1}\mbox{\boldmath$\Sigma$}_{(2)},$
(D1)

where
$\mbox{\boldmath$\Sigma$}_{(2)}=\\{\Sigma(\mbox{\boldmath$\theta$}_{m})\\}_{m=1}^{N_{\mathrm{pix}}}$
is a pixelized mass map, $C_{(2)}$ is the pixel-to-pixel covariance matrix of
$\mbox{\boldmath$\Sigma$}_{(2)}$, $\mbox{\boldmath$\Sigma$}_{(1)}$ is a data
vector containing radially binned $\Sigma$ values, and $A$ is a mapping matrix
whose elements $A_{mi}$ represent the area fraction of the $m$th pixel lying
within the $i$th clustercentric radial bin (Section 2.5). The bin-to-bin
covariance matrix for $\mbox{\boldmath$\Sigma$}_{(1)}$ is given by

$C_{(1)}=\left[A^{t}C_{(2)}^{-1}A\right]^{-1}.$ (D2)

##  Appendix E Chandra X-ray Brightness Profiles

Figure 20 shows the radial X-ray surface brightness profiles of A370 measured
in 10 energy bands ($0.5$ to $7$ keV) from Chandra observations. The binned
total and background X-ray brightness profiles in each energy band are plotted
in each panel, along with the best-fit model derived from simultaneous forward
modeling of the 10 energy bands.

Figure 20: Radial X-ray surface brightness profiles of A370 measured in 10
energy bands from Chandra observations. Red (gray) vertical bars in each panel
show the $1\sigma$ confidence range of the total (background) counts in radial
bins. The red (gray) shaded area in each panel shows the marginalized
$2\sigma$ confidence region of the model for the total (background) counts
obtained from simultaneous forward fitting of the 10-band brightness profiles.
The red dashed line in each panel shows the posterior mean profile of the
cluster contribution.

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# Glamour muscles: why having a body is not what it means to be embodied

Shawn L. Beaulieu1 and Sam Kriegman2

1University of Vermont, Burlington, VT 05446
2Northwestern University, Evanston, IL 60208
<EMAIL_ADDRESS>
###### Abstract

Embodiment has recently enjoyed renewed consideration as a means to amplify
the faculties of smart machines. Proponents of embodiment seem to imply that
optimizing for movement in physical space promotes something more than the
acquisition of niche capabilities for solving problems in physical space.
However, there is nothing in principle which should so distinguish the problem
of action selection in physical space from the problem of action selection in
more abstract spaces, like that of language. Rather, what makes embodiment
persuasive as a means toward higher intelligence is that it promises to
capture, but does not actually realize, contingent facts about certain bodies
(living intelligence) and the patterns of activity associated with them. These
include an active resistance to annihilation and revisable constraints on the
processes that make the world intelligible. To be theoretically or practically
useful beyond the creation of niche tools, we argue that “embodiment” cannot
be the trivial fact of a body, nor its movement through space, but the
perpetual negotiation of the function, design, and integrity of that body—that
is, to participate in what it means to constitute a given body. It follows
that computer programs which are strictly incapable of traversing physical
space might, under the right conditions, be more embodied than a walking,
talking robot.

## Meet the new paradigm. Same as the old paradigm.

The accomplishments of artificial intelligence are legion. But so too are its
deficiencies. To say this is not to impugn the hard work of those who advanced
the field to its present state, nor to doubt the material gains this work has
secured. It is instead to acknowledge that every technology has costs and
benefits that are intrinsic to both its design and the context in which it
functions [70]. While we have witnessed tremendous progress in the generation
of text and images [16, 22], event classification and forecasting [54, 41,
86], planning [92, 42, 79, 8], product design [67, 58], and scientific
discovery [43, 21, 20, 5, 32] we have also seen that AI programs behave in
ways that cast doubt on their reliability [6, 52, 63, 26, 30], and show a
persistent brittleness that curbs their widespread utility [33, 48, 64, 100].
Researchers hoping to address these issues often look to living intelligence
for inspiration, and among the most distinguishing features of living
intelligence is the collection of traits referred to as embodiment.

Embodied machines are meant to be systems whose expanded component function
exhibits or obtains special qualities that are inaccessible to more
constrained systems [29, 28, 24, 2, 99, 81, 49, 62, 38, 3, 27]. It is thought
that embodied machines will better mirror the traits of their living
counterparts insofar as classically cognitive phenomena like memory,
perception, and representation are causally wedded to bodily action.
Autonomous movement among and between otherwise immobile parts is what enables
these systems to acquire novel information about the world they inhabit, and
to change its state by way of interaction. Observations are thus not merely
subject to accurate or useful processing; rather, the details of observation
are actively generated and transformed by bodily action. Embodiment is then
the total imbrication of action and perception, and is thought to bestow upon
machines the ability to grasp the implications of what they observe and the
consequences of what they do.

To speak in these terms, however, is to frame the prospective benefits of
embodiment as relating either to the type of action an embodied machine
performs (e.g. mechanical rotation versus image classification), or to how
this action affects the distribution of inputs the machine receives. That
different actions have different effects, and therefore different uses, is not
a special contribution of embodiment per se, but a generic fact about
technology in general. Thus, to argue that embodied machines afford us the
capacity to automate physical processes [29, 28] is to say nothing more than
that there are circumstances in which we would do well to use an image
classifier, a language model, or a robot. Meanwhile, to argue that what
embodied machines do, and what disembodied machines cannot do, is control the
distribution of inputs they receive [29, 24, 44, 2] is only to make an
argument about the possible efficiency of the embodied approach. If learning
what is entailed by certain events is still a function of inputs which are
specified in advance, even if these are conditioned on the actions a machine
performs, then we have not defined a qualitatively different methodology. All
that disembodied programs need to acquire such mechanistic knowledge is a
sufficiently large dataset of relevant observables from which to infer cause
and effect. Great effort might be necessary to produce such a dataset, and
embodied machines might thus more rapidly infer causal relations from the
actions they elect to perform [12, 99], but the measure of their utility still
consists in how well they map a given set of inputs to a given set of outputs.

Since the data on which embodied machines are trained need not be supplied
prior to their deployment, they appear to us as open-ended, or indeterminate,
in a way that disembodied machines do not. What this obscures is that the
specification of relevant inputs, and the definition of permissible actions,
is encoded in the very design of such a machine, which is not significantly
altered by the activity the machine itself realizes. One might object that
some machines are programmed with the ability to add, remove, and otherwise
modulate system components [91, 39], and so appear to escape this
characterization, but decisions regarding which phenomena to use or model, and
how to do so, are clearly delineated by the engineering which defines them as
the machines they are111It is worth quoting the late cyberneticist, Gordon
Pask, at length on this issue: “A valve, for example, accepts only an
electrical input and provides an amplified electrical output. If it also
responds to temperature or vibration, it is to this extent a bad valve. The
logical simplicity of the computer model is a consequence of being able to put
one’s finger upon a component which performs a known function and to reject
the imperfections as irrelevant…When trying to construct a physical model of a
[self-organizing system] we are beset with a peculiar difficulty…The logical
requirements force us to use media such that, when a physical model is
constructed, we cannot specify components which have a well defined function,
and we cannot separate inputs and outputs into a set which are relevant and a
set which may be discounted. It is inherent in the logical character of the
[self-organizing system] that all available methods of organization are used,
and that it cannot be realized in a single reference frame. Thus, any of the
tricks which the physical model can perform, such as learning and remembering,
may be performed by one or all of a variety of mechanisms, chemical or
electrical or mechanical. Thus, however much we try, we cannot achieve an
electrical model or a mechanical model or a chemical model of a self-
organizing system. Any physical model necessarily includes them all in varying
degrees.” [74] They therefore depend on some external process by which
relevant factors are specified [74, 18, 19]. Although the shape of the input
distribution is not given to embodied machines in advance, but results from
their interaction with the environment, what phenomena populate the
distribution is rigidly constrained by what the machine is made to perceive
and how it is made to act. This may or may not result in efficiency gains
relative to a disembodied machine with access to more or better data, but it
will yield real benefits insofar as we need a machine that moves. But nor does
defining embodiment as the push and pull of action and perception necessarily
imply a relation to physical movement. Provided that language models directly
participate in the social process of determining how language is used, rather
than merely how to use language given its social determination, they achieve a
linguistic analog to physical embodiment [89]. Physically embodied machines
are thus distinct only in what aspects of the world they are licensed to
observe and alter. And just as the manipulation of physical objects might
ground [9] or supplement [85] certain cognitive faculties, so too might the
use and manipulation of language augment visual comprehension in image models.
There is no a priori reason to believe that physical movement as such is
uniquely informative, or causally efficacious, relative to other kinds of
action, except insofar as it’s necessary to solve certain problems.

Far from transcending orthodoxy, such a shallow notion of embodiment merely
repackages the idea of intelligence as reinforced selection [73]. But this is
not altogether undesirable. Skeptics of embodiment as an alternative paradigm
can rest easy knowing that nothing new or supernatural is being proposed,
while proponents of embodiment can celebrate the fact that, under certain
conditions, their methods might secure valuable gains. To leave it at that,
however, would be to forfeit the promise of a genuine alternative to existing
theories and methods, and would reduce physically embodied machines to niche
tools for solving or reasoning about problems in physical space. Yet we know
that the capacities of living intelligence aren’t exhausted by mere mobility.
What other principles can we draw on to make machines that are meaningfully
different from those presently in existence? And what reasons do we have to
pursue such a project?

## Taking embodiment seriously.

Figure 1: Both classic AI (image classifiers, language models, etc.) and
“embodied” AI (robots) map a given set of inputs to a given set of outputs
according to a specific set of rules. Neither the rules, nor what counts as an
input or output, are modified by either type of machine unless an additional
set of rules permits relevant modifications. But for embodied AI, what counts
as an input or output, and what operations it is capable of realizing, would
not be conclusively specified prior to its deployment. These would instead
emerge through its interaction with arbitrary features of its environment,
from which the system itself cannot be cleanly separated.

A constitutive property of life is that in forming and maintaining itself it
can, within the limits of its powers, construct or exploit whatever affords it
the means to do so [71, 35, 47, 73, 74, 76, 95, 18, 69, 31, 4, 53, 11]. If it
is adaptive for an organism to develop an affinity for sound, or attend to
fluctuations in ambient temperature, then all that stands in its way are the
viability of the mechanism involved and the likelihood of its discovery. What
passes for embodiment in machine learning ignores the details of how bodies
are formed, adapted, and maintained [29, 28, 24]. This labor is instead
performed by humans, who establish the conditions for its comparatively simple
methods to work. We propose to call these machines glamour muscles, because in
their effort to emulate life they merely copy or refine its superficial form
and function; they don’t meaningfully participate in their own design,
function, and maintenance. Herein lies the conceptual slippage between
“embodiment” as a description of living intelligence, and “embodiment” as the
realization of physical movement in otherwise immobile and inert machines.
Natural systems we would regard as paradigmatically embodied don’t just have
bodies inside of which cognition occurs, nor is their physiology conscripted
to perform an immutable, or deliberately bounded, set of operations for
satisfying pre-given ends [95, 82, 60]. Rather, the definition of a body—and
all that it can sense, compute, and perform—arises from a perpetual, and
historically specific, negotiation with the environment; which itself
undergoes continual change and redefinition [19, 17, 71, 56]. This is not to
say that organisms are infinitely plastic, or that we should aspire to build
machines that are free from constraint. To the contrary, some manner of
constraint above and beyond what is physically permitted provides the
necessary conditions for life’s possibility [40, 19, 17, 78]. Rather, the
essential question is whether the space of possible actions and percepts for a
given system can be plausibly inferred from how it presently works [76, 87,
19, 88]. For existing AI the answer is yes, because the logic by which it
operates is fully determined by constraints it cannot challenge or transform
[74, 18, 87]. No existing language model will ever, without expert
modification, commit itself to anything other than language, even if it would
be adaptive to do so. Precisely why a language model produces a given string
may be opaque to us, but that it produces strings (to any degree of
coherence), and not salt water taffy, is a consequence of it being designed
for language use.

Yet satisfying this condition of unpredictability in practice does not require
unpredictability in principle, or the possession of an immaterial will. It
depends on a fundamental indeterminacy222We might here distinguish between
epistemic uncertainty, which reflects the inadequacy of whatever description
we use to characterize a given system, and practical indeterminacy, which is a
source of epistemic uncertainty. Epistemic uncertainty may never be completely
resolved even for practically determinate systems, owing to the provisional
nature of all scientific hypotheses [84]. But this problem is only compounded
by practically indeterminate systems, because it cannot be entirely known what
features of the world, including what aspects of their own dynamics, these
systems are, or will be, relevantly coupled to [76, 53, 11, 55]. Predicting or
explaining the dynamics of structurally determinate systems can be done more
or less suitably with comparably determinate models, but indeterminate systems
require descriptions that change with those structural alterations that
redefine the system under investigation [77, 74, 88]. of form and function,
which resolves itself through material interaction—both with the surrounding
environment, to the extent we wish to make such a distinction, and among its
ever shifting and mutable parts [98, 71, 56, 19, 34, 73]. Even benign
mechanical operations for living things involve the risk of annihilation that
must be mitigated by the very mechanisms that compose these operations, and
can only be made sense of under conditions these operations help to construct
and maintain. Whereas the most elementary operations for machines are only
made possible or intelligible by the prior commitment of human labor. Given
this labor, the social and material conditions that enable smart machines are
causally isolated from the activity these machines themselves realize. When we
say, for instance, that a neural network takes pictures as input and produces
class labels as output, we are enforcing a special set of constraints on a
system that define what it is (a classifier) and what it does (classify
pictures). The process of classifying pictures, which is the only activity the
system is allowed to perform, has no power to directly alter these
constraints. Although we, observing its performance and finding it wanting,
may choose to do so. What then defines the system, and what enables the
activity it realizes, is independent of activity internal to the system (the
act of classifying pictures). This is true even in those cases where an
algorithm has been programmed to alter certain of its constraints, because
what the algorithm is permitted to alter is not itself open to modification
[90, 53, 68, 1, 12, 59, 66, 67, 58, 39]. And since the knowledge and resources
required to make such specifications are not uniformly distributed in the
population, it is not just our technology but we who are captive to the design
choices of those who build it. Breaking the distinction between the
specification of constraints and the activity these constraints enable will
confer upon machines the capacity to develop fundamentally new traits, and
upon everyday users the capacity to shape the development of these traits by
interacting with them. This may help us solve problems for which plausible
hypotheses, relevant observables, or appropriate actions are unknown, or for
which the problems themselves elude definition [19], while at the same time
linking the design and purpose of such machines to the immediate interests of
users [72].

For this to be possible, the constraints that specify what a given machine is
or does must be amenable to alteration by arbitrary forces acting on arbitrary
spatial and temporal scales, which carries the unending risk that such
alterations might render the machine inoperable. But the risk of annihilation
is both the cost paid by relaxing those constraints on material interaction
that usefully limit behavioral variety in machines, and the condition of
possibility for the acquisition of novel traits. Without the means to expand
the variety of possible states, the variety of possible behaviors is entailed
only by the present design, but increased variety means increased risk of
error [76, 75]. Mere susceptibility to dissolution, however, is insufficient.
Whirlpools, soufflés, and vulcanized rubber are all things that deteriorate,
but are not thereby instances of living intelligence. For the possibility of
annihilation to mean anything, for it to entail certain behaviors, there must
be reasons in light of which one acts to forestall the disintegration of that
which one values. It is not a feature of how shaving cream or chatbots are
organized that their organization is adapted or maintained to ensure their
survival. If we were to withhold the energy needed to power a computer except
on occasion it did something we desire, and reward it with energy for every
good deed it performed, we would not have trained it to do anything, because
acting to obtain energy is not built into the organization of computers. Your
phone does not “care” if you fail to charge its battery. In a shallow sense,
you can program it to care, and make it emit increasingly urgent signals
reminding you to plug it in as its battery depletes. You can even make the
content of these signals adaptive in some way, in order to increase the
probability of charging. But there are no practical consequences for the
device itself if it does not receive a charge. Even if it “dies” you can plug
it in tomorrow and it will work just fine. This is because it is designed not
to have any such practical consequences resulting from access to energy, just
like it is designed not to have any practical consequences resulting from the
texture or tensility of the surface on which it rests. This makes it highly
reliable and efficient for certain applications. But because there are no
practical consequences resulting from these aspects of the world, they cannot
be incorporated into the activity of the device unless doing so is already a
feature of its design. Except for the phenomena which are specified as
relevant to their function, such devices are characterized by a rigid
separation between the world they inhabit and the activity they realize.

This should not be confused with the idea that machines are state-determined,
while living intelligence is not [34], but that changes to the state of living
intelligence by phenomena they lack the ability to detect or act upon can
result in the means to do so if such a change in state affects viability and
can be relevantly incorporated into future activity [19, 71]. To assume that
prior knowledge of the anatomical structure which detects light, and how to
build it, is necessary for light-sensitivity to emerge is to attribute the
logistics of machine design to that of living intelligence. But the capacity
to measure and control specific phenomena is a capacity with a specific
history—one that cannot be explained by assuming the capacity already exists,
and simply must be expressed [98]. Prior work [73, 74] has shown that if an
existing component is not deliberately made to be insensitive to an
environmental factor (e.g. light, temperature, vibration, etc.) and if the
environmental factor can thus affect its physical state, this component can
develop sensitivity if the energy supplied for its continued growth and
maintenance is conditional on its change of state being useful for detecting
or acting upon the environmental factor.333Working with metal circuits grown
by electrochemical deposition—a process controlled at a given point in
solution by one among many automata competing to maximize the means of circuit
growth and persistence—Gordon Pask used this idea to generate structures that
acquired, but were not designed to have, an affinity for sonic vibration [73,
74]. By supplying energy only to those competing automata associated with
circuits whose output evidenced the desired sensitivity, automata were
compelled by selection to produce circuits that improved on, or amplified, the
sensitivity of their peers. Owing to the physical dependence of circuits on
their detection of relevant stimuli, and the pressure among automata to secure
limited resources, specialized sensors were grown and maintained using
components with no pre-given function. On the other hand, sensitivity to
temperature, unless it is specified in advance, will not emerge in an image
classifier, or the hardware on which it runs, even if it could improve its
ability to classify images. In short, the developmental history of living
intelligence is not devised and nurtured from without, but the historical
achievement of its antecedent forms acting in a world alive to their own
making.

Vulnerability in robots [62, 46], disintegrating architectures [37], dynamic
configuration of parts [94, 57, 39], and biobots composed entirely of living
tissues [50] have all been reported; but, in general, they only passively
contend with the forces of their environment. They don’t actively resist the
tendency to disorder, or secure conditions for their continued operation.
Organisms resist the tides of entropy by growing and maintaining structures
appropriate to their circumstances, or else by evoking circumstances
appropriate to their capacity for growth and maintenance. Robots cannot yet
grow new structures, but they have been made to deform [62] or re-adhere [7]
their remnant structure to rescue behavior lost to injury. Recovering function
erased by mechanical injury through exaptation of viable parts has a rich
history in robotics [59, 14, 51]. But however much exaptation might mirror the
compensatory strategies seen in mature organisms, the development of new and
contextually determined structures, as seen in ontogeny, has no real analog in
machines. To the extent that artificial constructs exhibit this capacity it is
often by a strict division of labor, with chemical and biological processes
inserted into traditional machine pipelines—as in the case where novel machine
capacities are built by human labor, or when events are detected and redressed
by simple lifeforms [25, 93, 65, 44, 101, 45]. This later category of methods
recognizes that survival-seeking systems are inveterate problem-solvers [76,
83, 13, 55, 23], and that profitable collaboration involves translating
problems into a common language, along with the enforcement of conditions that
transform activities normally undertaken for the sake of the survival-seeking
system into those that satisfy externally imposed ends.444If administered to
entities that warrant ethical consideration, we might think this method
unpalatable, but it isn’t necessary that access to the means of persistence is
limited to those occasions in which an exogenous objective is fulfilled. An
objective may be endogenously generated, and its fulfillment may be predicated
on the grounds of preserving an identity or commitment, rather than
physiological integrity. Or an objective may be exogenous only in the sense of
not being reducible to egoistic satisfaction, but instead aimed at securing
the conditions for all to flourish [40]. Labor performed for the satisfaction
of the exogeneous objective—which answers to, but is not exhausted by,
individual aspirations—would then be equivalent to the labor necessary for the
realization of one’s own freedom. What is critical to this method, however, is
the possibility of, and resistance to, dissolution—a dynamic which, in
machines, is scrupulously avoided, or framed as an obstacle to be overcome
[97, 48]. This is done by sharply constraining their material properties and
the scope of their permitted activity. Rather than plan and manage every
detail of optimization, which requires extensive knowledge and foresight, the
inborn competencies of survival-seeking systems can be exploited for highly
specific purposes if it is a condition of their existence that these purposes
are realized [73, 80, 10]. By analogy, it is precisely that to provide value
for an employer is what it means to provide subsistence for yourself that
control over the product of your labor can be so precisely wielded [98]. Once
having constructed machines which can appropriately be called “embodied” in
the sense we’ve articulated, this mode of interaction may help structure their
behavior since, by virtue of being so embodied, they would no longer be
subject to detailed programming. Likewise, employers need not have intimate
knowledge about how you come to perform the labor you do, or for what reasons,
in order to compel specific acts of labor. Rather, your labor is entailed by
the need to secure what is required to live and pursue your goals, owing to
the way society has been organized. Different forms of behavior would be
entailed by different forms of social organization, and this is no less true
for actually embodied machines.

Thus, if resistance to annihilation is the engine of living intelligence, it
might also be used to give material force to social desire. Democratic use and
design of technology often consists in collating various user interests into
one among many constraints on the decisions of those who build and own
technology [15, 96, 61]. This tradition is admirable for its commitment to
inclusivity at the point of production, but nevertheless falls short of
enacting the principle articulated by Gordon Pask that would explode their
function as instruments of alienation [72].

In delegating decision-making to the labor of another process (machines,
committees, political representatives, etc.) it should be the case that the
power and scope of the delegate is commensurate with the practical
consequences resulting from the decisions they make. If the practical
consequences of delegate decisions are (or believed to be) undesirable, then
the function of the delegate should be progressively dissolved, until which
time it can prove its decisions will be desirable. Representative action,
insofar as it’s necessary, then becomes materially possible only if the
represented parties stand to be satisfied. Otherwise, both the claim to
represent these parties, and the capacity to act on their behalf, are
immediately rescinded without the need for deliberation. To represent another
party is thus precisely what it means to carry out their interests, rather
than to transform those interests into actions consistent with more rational
ends, or more congenial to the interests of the representative.

This has implications for even local problem areas, like AI safety. Concerns
around AI alignment [36] often assume that the production and deployment of
future technologies will be mediated by existing forms of social organization,
with products developed in isolation from the immediate interests of users,
only later to be foisted upon them. Without making light of the task before
us, it is nonetheless worth considering whether alignment might be better
achieved by popular involvement at the point of production, and subsequently
maintained by the obligate satisfaction of user desire. Social needs would
thereby become arbitrary material forces that act directly on machines, much
like light or vibration, rather than something which affects machines only
through a given technical interface, political procedure, or indirect market
exchange. Modeled on the approach taken by Pask and others, we might then
collectively build and liquefy various delegations, with powers of action and
discernment that far exceed those of any individual, on the insuperable
condition they yield material benefits we all participate in defining. This
will involve the construction of actually embodied machines: ones that do not
simply traverse physical space, but that shape, and are shaped by, arbitrary
material events, including those social factors without which they wouldn’t
even be intelligible as the machines we take them to be.

## Conclusion

Embodied machines strike us as being indeterminate in a way that supposedly
disembodied machines do not, because the distribution of inputs they receive
is not specified prior to their deployment, but results from their interaction
with the environment. However, what counts as an input or output for embodied
machines, as well as the rules for processing them, is conclusively specified
by their design. For example, a machine equipped with light sensors will not
transform them into smoke detectors or pH meters, nor will a language model
elect not to minimize prediction error (unless these are already features of
their design). On the other hand, living intelligence has the capacity to
determine, by virtue of its own activity rather than by external design, what
features of the world—including what features of its own activity—are relevant
to the form of life it sustains. It does this by growing and maintaining new
structures, and by being open to modification by the world it inhabits, which
both increases its risk of error and establishes the conditions for generating
novelty. Building machines that approach living intelligence in these respects
will result in more flexible devices, whose form and function may be
determined by their interaction with users.

## Acknowledgements and Author Statements

SLB would like to thank Kate Nolfi for discussion and editing, and Kyrill
Potapov for reading an early draft.

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