ST/V000861/1, which also partially supports HZ. IB acknowledges support from a
“Pathways to Research” fellowship from the University of Bonn, during which
the primary statistical analysis was largely coded. BF and RI acknowledge
funding from the Agence Nationale de la Recherche (ANR projects
ANR-18-CE31-0006 and ANR-19-CE31-0017) and from the European Research Council
(ERC) under the European Union’s Horizon 2020 Framework programme (grant
agreement number 834148).

The authors thank the referee for highlighting some important deficiencies
with an earlier version of this work regarding the modelling of undetected
companions. They also thank Pavel Kroupa for comments on the article and for
lending computational resources from the Stellar Populations and Dynamics
Research Group at the University of Bonn, where all the statistical analyses
were conducted. IB is very grateful to Elena Asencio for helping to check
every line of the statistical analysis code and for lending algorithms used
for making several figures in this paper, especially the triangle plots made
using pygtc (Bocquet & Carter, 2016). He thanks Haixia Ma for providing Figure
22. The authors are grateful for a masters project by Nathan Findlay which
gave a much better understanding of how to analyse the dataset and what
quality cuts would be appropriate. IB acknowledges significant computational
resources provided by Kyu-Hyun Chae to simulate the impact of undetected
companions. Most of the figures were prepared using matlab.

## Data availability

The WB dataset used in this contribution will be made available upon
reasonable request, showing _Gaia_ catalogue information for the sample of
19786 WBs described in the text and an extra flag to indicate if each WB is
one of the 8611 systems used in our main analysis. The WB dataset used in the
nominal analysis of Chae (2023) was provided by Kyu-Hyun Chae. _Gaia_ DR3 is
publicly available.

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##  Appendix A Exactly equal mass binaries

Although we might expect the mass ratio distribution of binaries to be smooth,
observations indicate a population of exactly equal mass WBs (El-Badry et al.,
2019). We therefore include an allowance for a $\delta$-function in the
$\widetilde{q}$ distribution at $1/2$, where $\widetilde{q}$ is the fraction
of the CB mass in the less massive star. To find the likelihood $P_{\rm{eqm}}$
of a binary having exactly equal mass stars, we plot the cumulative
distribution of $0.5-\widetilde{q}$ for the WBs in our sample, which we
restrict to $\widetilde{v}<1$ to try and mitigate CB contamination. The idea
is to extrapolate the distribution down to zero and look for a positive
intercept. Since a $\delta$-function would be smeared somewhat by
observational uncertainties, we need to start our fit at some slightly
positive value of $0.5-\widetilde{q}$. We also need to avoid extending our fit
to very high $0.5-\widetilde{q}$ because our polynomial fitting function might
become inaccurate.

Figure 24: _Top_ : The cumulative distribution of $0.5-\widetilde{q}$ for WBs
with $\widetilde{v}<1$ (black line). We only show results over the range
$0.05-0.2$ because a $\delta$-function in the probability distribution at zero
would be smeared somewhat by observational uncertainties. The red line shows a
cubic fit over this range, but also extrapolates it down to zero. Notice that
the intercept is slightly positive, indicating the presence of a population of
WBs with an exactly equal mass (consistent with El-Badry et al., 2019).
_Bottom_ : The above-mentioned intercept is plotted as a function of the upper
limit to the fitting range. Different lines show results with different lower
limits, as indicated in the legend. Notice that for a wide range of fitting
ranges, the results converge around 0.04, which we assume is the likelihood
$P_{\rm{eqm}}$ of a binary having exactly equal mass components.

The results in the top panel of Figure 24 show real data and a cubic fit over
the range $0.05-0.2$. The data are shown in black, while the cubic fit is
shown in red $-$ a cubic is the lowest degree polynomial which provides an
accurate match to the data. The bottom panel of Figure 24 shows the intercept
as a function of the upper limit to $0.5-\widetilde{q}$, with each coloured
line used to show results for a different lower limit. All analysis variants
show a positive intercept, whose value converges at close to 0.04 for a wide
range of lower and upper limits to $0.5-\widetilde{q}$. We therefore adopt
$P_{\rm{eqm}}=0.04$ for the CB model in our main analysis. This assumption is
not relevant to modelling uncontaminated WBs, whose actual mass ratio
distribution is used when needed (Section 3.2.4). An equal mass fraction of a
few percent is in line with the double main sequence apparent in the colour-
magnitude diagram of our WB sample (Figure 2).

##  Appendix B When orbital motion becomes non-linear

We assume uniform rectilinear motion for the WB and any undetected CB
companions, implicitly assuming that the orbital periods are very long. This
should be a very good approximation for the WB given the kAU separations of
even the tightest WBs we consider. However, CB orbital periods can be much
shorter. This can substantially reduce the mean motion of the CB over the
_Gaia_ DR3 observing baseline of $t_{G}=34$ months. In this section, we
estimate the CB orbital period below which we can no longer safely assume a
very long orbital period $P$. This is used to estimate a minimum CB semi-major
axis $a_{\rm{int}}$ when setting up its distribution in Section 3.2.3.

For simplicity, we assume that the CB is on a circular orbit and that the line
connecting its components rotates by some angle $\theta$ over the _Gaia_
observing baseline. We consider the impact of increasing $\theta$ by reducing
the CB separation while not altering the other CB orbital parameters. The
induced recoil velocity on the star detected as part of the WB depends on the
CB orbital velocity $v\propto 1/\sqrt{a_{\rm{int}}}$. Bearing in mind Kepler’s
Third Law that $P\propto a_{\rm{int}}^{3/2}$, we get that $v\propto P^{-1/3}$.
Since the CB rotation angle $\theta$ over a fixed duration of time varies as
$\theta\propto 1/P$, we get that

$\displaystyle v~{}\propto~{}\theta^{1/3}\,.$ (44)

The positive exponent captures the fact that larger $\theta$ is associated
with a tighter CB on a faster orbit.

However, the mean velocity $\overline{v}$ will in general involve some
additional shape factor $s\left(\theta\right)$ that accounts for the extent to
which the orbital arc over the _Gaia_ observing baseline deviates from a
straight line. Basic trigonometry tells us that the linear distance between
two points on a circle is smaller than the distance between them along the
circumference by a factor of $\operatorname{sinc}\left(\theta/2\right)$.
Putting this extra factor into Equation 44 tells us that

$\displaystyle\overline{v}~{}\propto~{}\theta^{-2/3}\left|\sin\left(\frac{\theta}{2}\right)\right|\,.$
(45)

Treating this as a function of $u\equiv\theta/2$, we find that the maximum
occurs when $\tan u=3u/2$. The first non-trivial solution is $u=0.97$ or
$\theta=111^{\circ}$. Rotation by this angle over $t_{G}$ implies
$P=3.25\,t_{G}$, so we expect our linear motion approximation to break down
when $P<110$ months or 9.2 years.

The corresponding orbital separation depends on the mass of the CB. For a low-
mass undetected companion, the CB total mass would only slightly exceed the
mass of the contaminated star, which is just one of the two stars detected as
a WB. The blue bars in Figure 5 show that the stars in our WB sample easily
reach down to $0.4\,M_{\odot}$. At this mass, an orbital period of 9.2 years
corresponds to $a_{\rm{int}}=3.2$ AU. While CBs with even smaller separations
would still affect the inferred WB orbital velocity, the impact rapidly
becomes smaller than implied by Equation 44.

We note that 3.2 AU is much larger than the minimum $a_{\rm{int}}$ of 0.1 AU
considered in the orbit integrations of PS23 (see their section 3.2). CBs with
a separation of 0.1 AU would complete many orbits over $t_{G}$, so the effect
of such tight CBs may have been overestimated in their analysis.

##  Appendix C Comparing the best Newtonian and MOND models

In Figure 13, we presented the difference in log-likelihood between our best-
fitting Newtonian and Milgromian models. The panels in this figure showed the
results for each $r_{\rm{sky}}$ but with all $\widetilde{v}$ pixels summed
over, and vice versa. To give a better understanding of which pixels work
better in which theory, we show the full 2D distribution of $\Delta\ln P$
between these models in Figure 25. Since $\Delta\ln P$ is close to zero in
several pixels, we clarify which model does better by adding an open white
circle at the centre of a pixel if MOND fits it better. Out of the 540 pixels
used in our analysis (Table 1), only 221 pixels (41%) prefer MOND while the
remaining 59% prefer Newtonian gravity. However, we also need to consider that
the pixels which prefer MOND only do so to a rather small extent, while
sometimes pixels which prefer Newtonian gravity do so to a very large extent.

Figure 25: The relative performance of our best-fitting Newtonian and MOND
models, shown here for every pixel. The binomial log-likelihood of each model
is found by comparison with the observed number of WBs in each pixel (Figure
6). Open white circles indicate pixels for which MOND outperforms Newtonian
gravity. The combined difference in log-likelihood at each $r_{\rm{sky}}$
summed across all $\widetilde{v}$ (and vice versa) is shown in Figure 13.

##  Appendix D The best fit model with reduced CB contamination

The only alteration to our nominal analysis which gives an appreciable
preference for $\alpha_{\rm{grav}}>0$ is the one in which we fix
$f_{\rm{CB}}=0.3$ (Table 3). We explore this model in more detail to
understand if it plausibly fits the WB data with a gravity law closer to
Milgromian.

In Figure 26, we compare the observed $\widetilde{v}$ distribution to the
prediction of the best model with our nominal assumptions and with
$f_{\rm{CB}}=0.3$, considering both the final result of the gradient ascent
and the whole MCMC chain in each case. It is clear that fixing $f_{\rm{CB}}$
to such a low value causes a catastrophic disagreement with nearly all aspects
of the observations. The lack of sufficient CB contamination causes the peak
of the $\widetilde{v}$ distribution to be more pronounced than in the Gaia
number counts at very low $r_{\rm{sky}}$, though we see that both models
struggle somewhat in the peak region at intermediate $r_{\rm{sky}}$. The most
important differences concern the tail of the distribution, which cannot be
adequately fit with so few CBs. The analysis has to try and fit the extended
tail with substantially more LOS contamination, quadrupling the inferred
$f_{\rm{LOS}}$. This may help somewhat at low $r_{\rm{sky}}$, but it leads to
a rapidly rising $\widetilde{v}$ distribution at high $r_{\rm{sky}}$, in
strong disagreement with the observations (bottom right panel).

Figure 26: Comparison between the observed $\widetilde{v}$ distribution in
four different $r_{\rm{sky}}$ ranges (solid red bars in different panels) with
the prediction of our best model from our nominal analysis (black lines with
error bars) and from our revised analysis with fixed $f_{\rm{CB}}=0.3$ (blue
lines with error bars). This revised analysis is the only one to appreciably
shift the inferred gravity law towards MOND (Figure 18). The fit is much
poorer in this case. The results shown here are normalized based on the
observed WB distribution, helping to highlight that the $f_{\rm{CB}}=0.3$
model predicts too many WBs at high $r_{\rm{sky}}$.

These very serious issues cause the overall fit to be poorer than in our best-
fitting nominal analysis by $\Delta\ln P=413$, which implies that the latter
is preferred at a confidence equivalent to $\sqrt{2\Delta\ln P}=29\sigma$ for
a 1D Gaussian. Clearly, assuming a substantially lower likelihood of CB
contamination is not a viable proposition. In any case, this only shifts the
inferred gravity law towards MOND by a small amount, with Newtonian gravity
still strongly preferred despite facing just over $3\sigma$ tension (Figure
18). It is therefore extremely difficult to reconcile our WB results with MOND
as usually understood given other constraints.

##  Appendix E The artificial velocity limit in the Chae (2023) analysis

In Section 5.2, we discussed several problems with the WB analysis of Chae
(2023). One additional problem which may be noteworthy is that the sample is
based on the WB catalogue of El-Badry et al. (2021). Each WB is assigned a
likelihood of being genuine based on several factors, one of the most
important being the magnitude of $\bm{v}_{\rm{sky}}$ (see their equation
7).202020This is estimated using the difference in proper motions without
allowing for the systemic RV, which should be a reasonable approximation given
the small angular sizes of WBs. Unlike our limit that $\widetilde{v}<5$, their
limit to $v_{\rm{sky}}$ does not easily translate into a limit on
$\widetilde{v}$ because there is no allowance for more massive WBs having a
faster Newtonian $v_{c}$. Those authors require that

$\displaystyle
v_{\rm{sky}}~{}<~{}2.1\sqrt{\frac{\rm{kAU}}{r_{\rm{sky}}}}\,\text{km/s}\,,$
(46)

which in terms of $\widetilde{v}$ corresponds to

$\displaystyle\widetilde{v}~{}<~{}\sqrt{\frac{5\,M_{\odot}}{M}}\,.$ (47)

Figure 27: The red dots show the distribution of WB mass and $\widetilde{v}$
for the nominal sample of Chae (2023). Notice how the vast majority of WBs lie
below the solid blue line (Equation 47) due to selection effects (equation 7
of El-Badry et al., 2021).

We illustrate this cutoff in Figure 27, where we plot $\widetilde{v}$ as a
function of $M$ for the 26615 WBs in the nominal sample of Chae (2023). Nearly
all of these WBs lie below the blue line showing the limit given by Equation
47. This is not a completely strict limit because each WB is required to be
consistent with this bound within the $2\sigma$ observational uncertainty on
$v_{\rm{sky}}$ (see section 2 of El-Badry et al., 2021). Even so, it should be
clear that a WB sample selected in this way is not ideal for the WBT.212121K.
El-Badry, private communication.

Figure 28: Similar to Figure 21, but restricted further to only WBs with
$M<1.25\,M_{\odot}$. This ensures that the artificial limit to the
$\widetilde{v}$ distribution in the Chae (2023) sample (blue line in Figure
27) does not influence the results for $\widetilde{v}<2$. The mass limit has
little impact on the results, which continue to show a flat trend once we
impose that
$\sigma\left(\widetilde{v}\right)<0.1\max\left(1,\widetilde{v}/2\right)$, as
done here.

To check if our results in Section 5.2 remain reliable despite this
complication, we need to impose an upper limit on the mass so that results
remain reliable up to some maximal $\widetilde{v}$. This is equivalent to
drawing a rectangular cut towards the bottom left of Figure 27. To retain most
of the WB sample while having reliable results beyond the main peak of the
$\widetilde{v}$ distribution, we impose a limit of $M<1.25\,M_{\odot}$. With
this mass limit, regardless of the mass of a WB, there is no truncation to the
$\widetilde{v}$ distribution for $\widetilde{v}<2$. We then use Figure 28 to
show the median $\widetilde{v}$ as a function of $r_{\rm{sky}}/r_{{}_{M}}$ for
the case where the sample is further limited to only those WBs where the
estimated
$\sigma\left(\widetilde{v}\right)<0.1\max\left(1,\widetilde{v}/2\right)$, as
done in Figure 21. Both figures show very similar results, indicating that the
$\widetilde{v}$ limit imposed by Equation 47 has little impact on the flat
trend of the median $\widetilde{v}$ with respect to our proxy for the internal
WB acceleration. This is presumably because the main peak of the
$\widetilde{v}$ distribution lies at $\widetilde{v}<1$, which is not much
affected by this cut (Figure 27). Even so, having a mass-dependent limit to
$\widetilde{v}$ could in principle bias the WBT because a WB with a lower mass
also has a lower acceleration at the same $r_{\rm{sky}}$. Since the main idea
of the WBT is that some typical measure of
$\widetilde{v}\propto\sqrt{g/g_{{}_{N}}}$, one should be cautious about a
sample selection that allows lower mass systems to reach a higher
$\widetilde{v}$ than higher mass systems.