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##  Appendix A Radial Profiles for Full Cloud Sample

In Figure 3, we present the radial density profiles for two clouds in the
sample — Perseus and Chamaeleon. In Figures 9, 10, and 11, we show the same
radial density profiles for the remaining nine clouds.

Figure 9: Same as in Figure 3 except for the Lupus molecular cloud (top), the
Taurus molecular cloud (middle), and the Cepheus molecular cloud (bottom)
Figure 10: Same as in Figure 3 except for the Ophiuchus molecular cloud (top),
the Pipe nebula (middle), and the Musca dark cloud (bottom) Figure 11: Same as
in Figure 3 except for the Orion A molecular cloud (top), the Orion B
molecular cloud (middle), and the $\lambda$ Orionis molecular cloud (bottom).
These clouds are at the edge of the Leike et al. (2020) 3D dust grid, so the
results should be treated with more caution than our “complete” cloud sample.

##  Appendix B Cloud Widths as a Function of Distance

To determine whether the distance of the cloud influences the widths we
derive, we plot the four width parameters fitted for in this study
($\sigma_{1}$ and $\sigma_{2}$ from the two-component Gaussian function,
$R_{\mathrm{flat}}$ from the Plummer function, and $\sigma$ from the single-
component Gaussian) as a function of distance. The results are shown in Figure
12. If distance was influencing the cloud widths, we would expect the cloud
widths to increase as a function of distance. As we show in Figure 12, we find
no such correlation, indicating that the cloud distance is not strongly
biasing our results.

Figure 12: The derived width parameters as a function of cloud distance, with
the “inner” width $\sigma_{1}$ of the two-component Gaussian fit, the “outer”
width $\sigma_{2}$ of the two-component Gaussian fit, the flattening radius
$R_{\mathrm{flat}}$ of the Plummer fit, and the standard deviation $\sigma$ of
the single-component Gaussian fit shown from top to bottom.

##  Appendix C Comparison between 3D Dust Correlation Kernel and Measured
Profiles

In this section, we compare the shape of the correlation kernel of the Leike
et al. (2020) 3D dust reconstruction with the shape of our cloud profiles. Our
goal is to determine the extent to which our cloud profiles are driven by data
local to the cloud, versus the prior on the dust extinction density. Leike et
al. (2020) assume the prior on the dust extinction density $s_{x}$ to be
positive and spatially correlated, enforced via a log-normal Gaussian process
prior of the form:

$\displaystyle s_{x}$ $\displaystyle=\rho_{0}\,\text{exp}(\tau_{x})$
$\displaystyle\tau$ $\displaystyle\curvearrowleft\mathcal{G}(\tau|0,T)$

where $\rho_{0}$ is the prior median extinction density and $T$ is the
correlation kernel of the Gaussian Process $\tau_{x}$. The $\rho_{0}$ term is
a hyperparameter of the model, and is set to $\rho_{0}=\frac{1}{\rm
1000\;pc^{-1}}$. The correlation kernel $T$ is inferred during their
reconstruction. To parallelize the reconstruction, Leike et al. (2020) split
the local volume into eight octants, with the correlation kernel inferred on
an octant by octant basis. We take samples of the correlation kernel inferred
for the octant in which each cloud resides, which is equivalent to obtaining
samples of the two-point correlation matrix of the logarithmic dust density.
The two-point correlation matrix directly indicates how the density at one
point is correlated with another point in 3D space.

To facilitate a comparison between the two-point correlation matrix and our
profiles, we compute the correlation of each measured profile $C_{p}(r)$.
Before doing so, we convert the radial profiles as a function of gas density
$n(r)$, utilized throughout this study, back to the native units of the Leike
et al. (2020) 3D dust map, by dividing each profile by a factor of 880 (see
§2) to obtain $s_{x}(r)$. Since the two-point correlation matrix describes the
logarthmic dust density, rather than the dust density, we further transform
$s_{x}(r)$ into logarithmic density space, hereafter $p(r)$:

$p(r)=\text{log}(s_{x}(r)\times 1000)$ (C1)

where the factor of 1000 accounts for the scaling of the hyperparameter
$\rho_{0}$. To calculate the correlation of the profile $C_{p}(r)$, we
compute:

$C_{p}(r)=p(r)\times p(0)$ (C2)

where $p(0)$ is the logarithmic density of the radial profile in Equation C1
at a radial distance of 0 pc. Given the overdensity of the dust at
$r=0\,\text{pc}$, this comparison describes how much of the dust at larger
radial distances can be explained by the correlation kernel of the dust map.
In Figure 13 we plot $T$ versus $C_{p}(r)$ for each cloud, where $T$
constitutes samples of two-point correlation matrix inferred in the cloud’s
octant, and $C_{p}(r)$ is the correlation of the profile we construct in
§3.2.1. We normalize both $T$ and $C_{p}(r)$ to a peak value of 1.0.

We observe that the shape of $C_{p}(r)$ and $T$ are qualitatively different
between radial distances of 2 and 10 pc, indicating that the “inner” widths of
our two-component Gaussian results are driven by data local to the cloud,
rather than the kernel used in the 3D dust’s reconstruction. However, for
radial distances $>$ 10 pc, we see the slopes of $C_{p}(r)$ and $T$ become
increasingly similar, particularly for the Musca, Taurus, Ophiuchus, and Orion
A clouds. The similarity between the shape of $C_{p}(r)$ and $T$ at larger
radial distances could indicate that the kernel is driving the structure of
clouds at lower densities. However, since the kernel is inferred using data
inclusive of the cloud’s structure (in addition to structure beyond each cloud
of interest) the outer envelopes could still be physically meaningful. More
validation in future work will be needed to fully verify the relationship, if
any, between the outer envelopes we observe in our radial profiles and the
atomic gas envelopes of local molecular clouds, as discussed in §6.

Figure 13: Comparison between samples of the two-point correlation function
derived from the Leike et al. (2020) 3D dust kernel (black lines) and the
derived correlation of each cloud’s radial profile obtained in this work (blue
points). Both the 3D dust map’s correlation kernel and correlation of the
cloud radial profiles have been scaled to a peak value of one. At smaller
radial distances, our measured profiles deviate significantly from the
underlying correlation kernel used to derive the map, indicating that the
inner widths we obtain for the clouds are robust. However, at larger radial
distances, the shape of the kernel and profiles become similar, indicating the
dust at larger radial distances may be prior-dominated. However, since the
prior is inferred using real cloud structure, the outer envelopes could still
be physically meaningful. See Appendix C for full details.

##  Appendix D Radial Profile Fitting Results for the Full Profile

In §4, we discuss the width results where the inner profiles (radial distances
$<$ 2 pc) are excluded from the fit, due to the Leike et al. (2020) map’s 1 pc
grid and its caveat that features less than 2 pc in size should be treated
with caution. For completeness, in Table 4, we report the results of the full
profile fits, including radial distance bins between 0 and 20 pc.

In general, we find that including the inner 2 pc of each radial profile in
the fit has a modest effect on the model parameters. When including the inner
2 pc for the two-component Gaussian-fit (in comparison to the results from
§4), the inner width $\sigma_{1}$ changes on average by 15%, while the outer
width typically differs by 17%. For the single-component Gaussian, $\sigma$
differs on average by 20%. We see the largest difference in the Plummer model,
with $R_{flat}$ changing by 28% on average. Overall, however, the quality of
all model fits decreases when including the inner 2 pc, which is why we report
the results excluding the inner 2 pc, over the full profile results, in the
main text. The decline in the quality of the fit, as determined using the same
procedure outlined in §4.2.1, is consistent with our understanding that the
Leike et al. (2020) dust map is unreliable at scales $<2$ pc, and is also less
accurate at reconstructing the highest gas densities, which are closest to the
spine. Thus, it is not surprising that the fit improves when only higher
fidelity measurements, farther from the spine, are included.

=1.5in

Table 4: Radial Volume Density Fitting (Full Profile)

| Two-Component Gaussian | Plummer | Single Gaussian
---|---|---|---
Cloud | $\sigma_{1}$ | $a_{1}$ | $\sigma_{2}$ | $a_{2}$ | $\sigma_{n,\rm G2}^{2}$ | $\frac{\sigma_{2}}{\sigma_{1}}$ | $lnZ_{G2}$ | $n_{0}$ | $R_{\rm flat}$ | $p$ | $\sigma_{n,\rm P}^{2}$ | $lnZ_{\rm P}$ | $\sigma$ | $a$ | $\sigma_{n,\rm G1}^{2}$ | $lnZ_{\rm G1}$
| pc | cm-3 | pc | cm-3 | cm-3 |  |  | cm-3 | pc |  | cm-3 |  | pc | cm-3 | cm-3 |
Chamaeleon | $2.6_{-0.1}^{+0.1}$ | $34.5_{-0.7}^{+0.7}$ | $9.4_{-0.4}^{+0.5}$ | $10.2_{-0.7}^{+0.7}$ | $0.4_{-0.1}^{+0.1}$ | 3.6 | -19.7 | $46.1_{-0.3}^{+0.3}$ | $3.4_{-0.1}^{+0.1}$ | $1.9_{-0.1}^{+0.1}$ | $0.3_{-0.1}^{+0.1}$ | -10.8 | $3.8_{-0.2}^{+0.2}$ | $41.1_{-1.6}^{+1.6}$ | $10.0_{-2.0}^{+2.8}$ | -73.1
Ophiuchus | $2.4_{-0.1}^{+0.1}$ | $33.1_{-1.0}^{+0.9}$ | $9.6_{-0.5}^{+0.5}$ | $13.6_{-0.9}^{+1.0}$ | $0.9_{-0.2}^{+0.3}$ | 3.9 | -33.8 | $48.1_{-0.5}^{+0.5}$ | $2.9_{-0.1}^{+0.1}$ | $1.5_{-0.0}^{+0.0}$ | $0.5_{-0.1}^{+0.1}$ | -22.6 | $4.3_{-0.3}^{+0.3}$ | $40.6_{-2.0}^{+2.2}$ | $16.9_{-3.5}^{+4.9}$ | -82.3
Lupus | $2.3_{-0.1}^{+0.1}$ | $31.4_{-1.3}^{+1.3}$ | $7.7_{-0.3}^{+0.4}$ | $17.1_{-1.4}^{+1.3}$ | $1.0_{-0.2}^{+0.3}$ | 3.3 | -34.3 | $49.0_{-0.5}^{+0.5}$ | $3.6_{-0.1}^{+0.2}$ | $1.9_{-0.1}^{+0.1}$ | $0.5_{-0.1}^{+0.2}$ | -24.6 | $4.2_{-0.2}^{+0.2}$ | $42.9_{-1.7}^{+1.8}$ | $12.1_{-2.5}^{+3.4}$ | -76.6
Taurus | $2.0_{-0.1}^{+0.1}$ | $43.1_{-1.1}^{+1.0}$ | $10.4_{-0.7}^{+0.8}$ | $11.2_{-0.9}^{+0.9}$ | $1.8_{-0.3}^{+0.5}$ | 5.2 | -45.3 | $56.8_{-1.0}^{+1.0}$ | $1.9_{-0.1}^{+0.1}$ | $1.4_{-0.1}^{+0.1}$ | $1.8_{-0.4}^{+0.5}$ | -46.0 | $3.1_{-0.2}^{+0.3}$ | $48.2_{-3.0}^{+3.2}$ | $24.4_{-5.0}^{+7.1}$ | -89.3
Perseus | $3.3_{-0.2}^{+0.2}$ | $35.0_{-1.9}^{+1.9}$ | $9.7_{-0.6}^{+0.8}$ | $16.5_{-2.0}^{+2.0}$ | $1.8_{-0.4}^{+0.5}$ | 3.0 | -45.0 | $52.5_{-0.6}^{+0.6}$ | $4.8_{-0.3}^{+0.3}$ | $2.0_{-0.1}^{+0.1}$ | $1.1_{-0.2}^{+0.3}$ | -37.3 | $5.2_{-0.2}^{+0.2}$ | $46.8_{-1.5}^{+1.5}$ | $13.6_{-2.8}^{+3.7}$ | -79.0
MuscaaaAt small radial distances, Musca is largely unresolved in the Leike et al. (2020) map, and at large radial distances, its structure is consistent with being prior-dominated, so we urge caution when interpreting the results. | $2.6_{-0.2}^{+0.3}$ | $42.8_{-3.5}^{+3.1}$ | $14.7_{-4.9}^{+6.1}$ | $6.1_{-1.7}^{+2.6}$ | $23.6_{-4.5}^{+6.8}$ | 5.7 | -89.0 | $49.9_{-3.3}^{+3.2}$ | $3.9_{-1.0}^{+1.6}$ | $2.6_{-0.7}^{+1.4}$ | $26.2_{-5.4}^{+7.5}$ | -91.4 | $3.2_{-0.3}^{+0.3}$ | $46.7_{-3.3}^{+3.3}$ | $33.6_{-6.7}^{+10.0}$ | -95.7
Pipe | $2.6_{-0.2}^{+0.3}$ | $24.4_{-1.8}^{+2.4}$ | $7.6_{-0.4}^{+0.6}$ | $18.1_{-2.8}^{+1.4}$ | $4.0_{-0.9}^{+1.1}$ | 2.9 | -58.3 | $42.0_{-1.2}^{+1.2}$ | $4.8_{-0.6}^{+0.7}$ | $2.2_{-0.3}^{+0.4}$ | $3.8_{-0.8}^{+1.1}$ | -57.2 | $4.9_{-0.2}^{+0.2}$ | $37.4_{-1.4}^{+1.4}$ | $10.4_{-2.2}^{+2.9}$ | -73.6
Cepheus | $2.0_{-0.1}^{+0.1}$ | $60.6_{-1.3}^{+1.3}$ | $9.1_{-0.6}^{+0.7}$ | $13.5_{-1.1}^{+1.2}$ | $2.1_{-0.4}^{+0.6}$ | 4.6 | -48.2 | $77.9_{-0.7}^{+0.8}$ | $2.1_{-0.1}^{+0.1}$ | $1.6_{-0.0}^{+0.0}$ | $0.8_{-0.2}^{+0.2}$ | -32.9 | $2.8_{-0.2}^{+0.2}$ | $67.9_{-3.1}^{+3.1}$ | $28.3_{-5.9}^{+8.2}$ | -92.3
Orion A* | $3.3_{-0.1}^{+0.1}$ | $48.6_{-2.0}^{+1.4}$ | $11.2_{-2.2}^{+3.9}$ | $5.0_{-1.5}^{+2.0}$ | $2.4_{-0.5}^{+0.7}$ | 3.4 | -48.6 | $54.2_{-1.0}^{+1.0}$ | $7.0_{-0.7}^{+0.7}$ | $4.8_{-0.7}^{+0.7}$ | $3.3_{-0.7}^{+0.9}$ | -54.3 | $3.7_{-0.1}^{+0.1}$ | $52.4_{-1.1}^{+1.1}$ | $5.4_{-1.1}^{+1.7}$ | -62.8
Orion B* | $4.5_{-0.2}^{+0.2}$ | $38.1_{-2.3}^{+2.1}$ | $12.0_{-1.1}^{+1.6}$ | $12.6_{-2.3}^{+2.4}$ | $1.2_{-0.2}^{+0.3}$ | 2.7 | -36.5 | $51.6_{-0.5}^{+0.5}$ | $7.5_{-0.4}^{+0.4}$ | $2.8_{-0.2}^{+0.2}$ | $1.0_{-0.2}^{+0.3}$ | -34.7 | $6.0_{-0.2}^{+0.2}$ | $47.8_{-1.1}^{+1.2}$ | $8.7_{-1.7}^{+2.4}$ | -70.5
$\lambda$ Orionis* | $3.3_{-0.1}^{+0.1}$ | $28.9_{-0.8}^{+0.8}$ | $11.6_{-0.6}^{+0.7}$ | $10.9_{-0.8}^{+0.8}$ | $0.5_{-0.1}^{+0.2}$ | 3.5 | -23.7 | $41.0_{-0.4}^{+0.4}$ | $4.1_{-0.2}^{+0.2}$ | $1.7_{-0.1}^{+0.1}$ | $0.4_{-0.1}^{+0.1}$ | -17.5 | $5.2_{-0.3}^{+0.3}$ | $35.9_{-1.5}^{+1.5}$ | $11.1_{-2.2}^{+3.2}$ | -74.6

**footnotetext: Clouds below the horizontal line – Orion A, Orion B, and
$\lambda$ Orionis – should be treated with caution, as they lie at the very
edge of the Leike et al. (2020) 3D dust grid and are subjected to additional
biases. Part of Orion A, toward the Orion Nebula Cluster, is also devoid of
dust in the Leike et al. (2020) 3D dust map despite being highly extinguished.
See §5 for full details.

Note. — Radial density profile fitting results for local clouds, computed over
radial distances between 0 and 20 pc. Unlike Table 2, we do not exclude the
inner 2 pc from the fit. (1) Name of the cloud. (2 - 6) Best-fit parameters
for the two-component Gaussian fit, including the standard deviation and
amplitude of the narrower Gaussian ($\sigma_{1}$, $a_{1}$), the standard
deviation and amplitude of the broader Gaussian ($\sigma_{2}$, $a_{2}$), and
the square of the modeled scatter in the density measurements ($\sigma_{n,\rm
G2}^{2}$). (7) Ratio of the outer to inner Gaussian standard deviations. (8)
Logarithm of the evidence of the two-component Gaussian fit. (9-12) Best-fit
parameters of the Plummer fit, including the peak amplitude $n_{0}$, the
flattening radius $R_{\rm flat}$, the power index of the density profile $p$,
and the square of the modeled scatter in the density measurements
($\sigma_{n,\rm P}^{2}$). (13) Logarithm of the evidence of the Plummer fit.
(14-16) Standard deviation $\sigma$, amplitude $a$, and the square of the
modeled scatter in the density measurements ($\sigma_{n,\rm G1}^{2}$) for the
single component Gaussian fit. (17) Logarithm of the evidence of the single-
component Gaussian fit. We report the single-component Gaussian results here
for completeness only, as it is a poor fit to the data particularly at large
radial distances. A machine readable version of this table is available at
https://doi.org/10.7910/DVN/V8NGLX.

##  Appendix E Radial Profile Fitting Results using Samples of the 3D Dust Map

In deriving their 3D dust map, Leike et al. (2020) generate twelve posterior
samples, or realizations, of the extinction density, and the 3D dust map
derived from the mean of these posterior samples is used in the fitting
results reported in §4. However, because the uncertainty in the underlying 3D
dust reconstruction is not accounted for when characterizing the radial
profiles, it is necessary to quantify how much our results change when
different realizations of the 3D dust distribution are used, rather than just
the mean. To do so, we create a new likelihood function, building on the
formalism described in §3.2.2. Recall that when fitting 3D dust determined by
the mean of the samples, we adopt a Gaussian log-likelihood that takes the
following form:

$\mathcal{L}_{\mathrm{mean}}=\ln\,p(n\big{|}\theta,\sigma_{n}^{2})=-\frac{1}{2}\sum_{i}\left[\frac{(n_{i}-n_{\theta,i})^{2}}{\sigma_{n}^{2}}+\ln(2\pi\sigma_{n}^{2})\right]$
(E1)

where $n_{i}$ is the measured profile in the $ith$ radial distance bin
characterized by the mean 3D dust map, $n_{\theta,i}$ is the model density
profile in the $ith$ distance bin (for the single Gaussian, two-component
Gaussian, or Plummer fit), $\theta$ constitutes the parameters characterizing
the model density profile for each fit (e.g. $\sigma$, $a$ for the single
Gaussian fit), and $\sigma_{n}^{2}$ is the square of the scatter in the
density (across all bins) that we infer as part of our modeling.

Our new likelihood function, taking into account all twelve realizations of
the 3D dust map, is of the form:

$\mathcal{L}_{\mathrm{samples}}=\ln[\exp(\mathcal{L}_{{\rm
sample}_{1}})+\exp(\mathcal{L}_{{\rm sample}_{2}})+...+\exp(\mathcal{L}_{{\rm
sample}_{12}})]-\ln(12)$ (E2)

where $L_{\mathrm{sample}_{1}}$ is calculated equivalently to
${L}_{\mathrm{mean}}$, except the measured radial density profile $n$ is
derived from the 1st sample of the 3D dust map. There are twelve samples, so
we compute the radial profile for each, assuming the same filament spine as
determined for the mean dust map. We exclude the inner 2 pc of the radial
profiles when fitting, as we do for the results reported in Table 2. Each of
the twelve samples is a possible reality with a probability of $\frac{1}{12}$,
and therefore the combined probability is their average. Thus, the total log-
likelihood is the average of the individual likelihoods derived from each
realization. We sample for the best-fit model parameters for all three
functions (single-component Gaussian, two-component Gaussian, Plummer) for all
clouds, using the same priors and the same dynesty nested sampling framework
we employ in §3.2.2. Our results are summarized in Table 5.

Overall, we find that that uncertainty in the profile fitting derived from the
uncertainty in the 3D dust map has only a modest effect on our results. For
the two-component Gaussian, the typical percent difference between the sample-
based inference and the mean-based inference is 2% for $\sigma_{2}$, 10% for
$\sigma_{1}$, 21% for $a_{2}$ and 11% for $a_{1}$. For the Plummer fit, $p$
changes by an average of 8%, $n_{0}$ by 17%, and $R_{\mathrm{flat}}$ by 22%.
However, we do note that for three clouds — Pipe, $\lambda$ Orionis, and
Musca, the $R_{\mathrm{flat}}$ parameter changes more drastically, increasing
by a factor of two when using the 3D dust samples rather than the mean. This
larger change could be tied to the degeneracy in the Plummer modeling
described in §4.3, which provides another reason not to overinterpret the
Plummer models, as we caution in §6.2. The two-component Gaussian fit is much
more robust, and the sample-based fitting results reported in Table 5 do not
affect any physical interpretation of the profile shapes described in §6.

=1.5in

Table 5: Radial Volume Density Fitting (Computed using All Realizations of 3D
Dust Map)

| Two-Component Gaussian | Plummer | Single Gaussian
---|---|---|---
Cloud | $\sigma_{1}$ | $a_{1}$ | $\sigma_{2}$ | $a_{2}$ | $\sigma_{n,\rm G2}^{2}$ | $\frac{\sigma_{2}}{\sigma_{1}}$ | $lnZ_{\rm G2}$ | $n_{0}$ | $R_{\rm flat}$ | $p$ | $\sigma_{n,\rm P}^{2}$ | $lnZ_{\rm P}$ | $\sigma$ | $a$ | $\sigma_{n,\rm G1}^{2}$ | $lnZ_{\rm G1}$
| pc | cm-3 | pc | cm-3 | cm-3 |  |  | cm-3 | pc |  | cm-3 |  | pc | cm-3 | cm-3 |
Chamaeleon | $2.8_{-0.1}^{+0.1}$ | $29.2_{-0.6}^{+0.7}$ | $10.4_{-0.3}^{+0.4}$ | $8.0_{-0.4}^{+0.4}$ | $0.1_{-0.0}^{+0.0}$ | 3.7 | -4.1 | $46.9_{-1.9}^{+1.7}$ | $2.6_{-0.1}^{+0.2}$ | $1.7_{-0.0}^{+0.1}$ | $0.1_{-0.0}^{+0.0}$ | 0.3 | $4.7_{-0.4}^{+0.4}$ | $28.3_{-2.3}^{+3.7}$ | $4.8_{-1.1}^{+1.6}$ | -55.7
Ophiuchus | $2.7_{-0.1}^{+0.1}$ | $34.0_{-1.5}^{+1.2}$ | $10.9_{-0.5}^{+0.5}$ | $10.3_{-0.6}^{+0.6}$ | $0.4_{-0.1}^{+0.1}$ | 4.1 | -19.7 | $56.6_{-2.5}^{+2.7}$ | $2.3_{-0.1}^{+0.2}$ | $1.4_{-0.0}^{+0.0}$ | $0.2_{-0.0}^{+0.1}$ | -8.2 | $5.2_{-0.4}^{+0.5}$ | $28.7_{-2.3}^{+2.3}$ | $7.9_{-1.8}^{+2.4}$ | -63.0
Lupus | $3.5_{-1.4}^{+0.2}$ | $27.6_{-1.4}^{+14.6}$ | $10.0_{-1.4}^{+1.3}$ | $9.4_{-1.8}^{+4.0}$ | $0.4_{-0.1}^{+0.1}$ | 2.9 | -19.5 | $42.3_{-3.8}^{+11.7}$ | $3.5_{-0.9}^{+1.3}$ | $1.8_{-0.2}^{+0.4}$ | $0.3_{-0.1}^{+0.1}$ | -13.3 | $5.4_{-0.2}^{+0.3}$ | $29.5_{-1.5}^{+1.4}$ | $4.2_{-1.0}^{+1.2}$ | -54.0
Taurus | $2.3_{-0.1}^{+0.1}$ | $36.2_{-1.5}^{+1.2}$ | $11.5_{-0.4}^{+0.6}$ | $8.7_{-0.4}^{+0.3}$ | $0.2_{-0.0}^{+0.1}$ | 5.0 | -9.7 | $54.4_{-4.7}^{+7.2}$ | $1.6_{-0.2}^{+0.2}$ | $1.3_{-0.0}^{+0.0}$ | $0.2_{-0.0}^{+0.1}$ | -10.2 | $5.2_{-0.7}^{+0.7}$ | $24.0_{-2.8}^{+4.1}$ | $7.9_{-1.8}^{+2.4}$ | -63.0
Perseus | $3.7_{-0.1}^{+0.1}$ | $26.3_{-0.9}^{+0.9}$ | $11.4_{-0.6}^{+0.7}$ | $11.5_{-1.0}^{+1.0}$ | $0.4_{-0.1}^{+0.1}$ | 3.1 | -20.3 | $40.8_{-0.9}^{+1.0}$ | $4.7_{-0.3}^{+0.3}$ | $1.8_{-0.1}^{+0.1}$ | $0.3_{-0.1}^{+0.1}$ | -17.5 | $6.4_{-0.6}^{+0.4}$ | $31.1_{-1.7}^{+3.2}$ | $6.1_{-1.4}^{+1.9}$ | -59.3
MuscaaaAt small radial distances, Musca is largely unresolved in the Leike et al. (2020) map, and at large radial distances, its structure is consistent with being prior-dominated, so we urge caution when interpreting the results. | $2.9_{-0.1}^{+0.1}$ | $36.1_{-0.8}^{+0.8}$ | $20.9_{-2.9}^{+2.7}$ | $3.3_{-0.2}^{+0.3}$ | $0.4_{-0.1}^{+0.1}$ | 7.1 | -17.1 | $145.3_{-22.9}^{+16.9}$ | $1.0_{-0.1}^{+0.2}$ | $1.6_{-0.0}^{+0.0}$ | $0.5_{-0.1}^{+0.2}$ | -24.3 | $3.6_{-0.2}^{+0.2}$ | $34.5_{-2.5}^{+2.7}$ | $5.2_{-1.1}^{+1.6}$ | -56.9
Pipe | $2.9_{-0.2}^{+0.4}$ | $29.5_{-9.3}^{+8.4}$ | $9.9_{-0.7}^{+0.8}$ | $8.3_{-1.1}^{+1.3}$ | $0.5_{-0.1}^{+0.2}$ | 3.4 | -21.8 | $49.5_{-3.1}^{+3.4}$ | $2.6_{-0.2}^{+0.3}$ | $1.8_{-0.1}^{+0.1}$ | $0.3_{-0.1}^{+0.1}$ | -16.0 | $6.6_{-0.2}^{+0.2}$ | $21.8_{-0.8}^{+0.8}$ | $2.3_{-0.5}^{+0.7}$ | -44.9
Cepheus | $2.8_{-0.6}^{+0.1}$ | $41.6_{-1.4}^{+8.7}$ | $10.9_{-1.6}^{+1.4}$ | $6.7_{-0.9}^{+1.9}$ | $0.6_{-0.1}^{+0.2}$ | 3.9 | -25.1 | $159.1_{-12.6}^{+8.0}$ | $1.0_{-0.0}^{+0.1}$ | $1.6_{-0.0}^{+0.0}$ | $0.2_{-0.0}^{+0.1}$ | -6.5 | $3.9_{-0.2}^{+0.2}$ | $38.9_{-2.8}^{+2.7}$ | $6.5_{-1.4}^{+2.1}$ | -61.0
Orion A* | $3.1_{-0.6}^{+0.2}$ | $48.0_{-14.4}^{+19.7}$ | $8.7_{-1.0}^{+3.3}$ | $7.0_{-3.5}^{+2.3}$ | $0.4_{-0.1}^{+0.1}$ | 2.8 | -18.9 | $80.1_{-34.0}^{+3.4}$ | $4.2_{-0.3}^{+0.6}$ | $3.3_{-0.4}^{+0.3}$ | $0.4_{-0.1}^{+0.1}$ | -17.7 | $3.9_{-0.1}^{+0.1}$ | $46.9_{-1.3}^{+1.5}$ | $2.1_{-0.5}^{+0.6}$ | -43.4
Orion B* | $3.8_{-0.1}^{+0.1}$ | $35.5_{-1.2}^{+1.1}$ | $11.1_{-0.7}^{+0.8}$ | $11.7_{-1.3}^{+1.3}$ | $0.6_{-0.1}^{+0.2}$ | 2.9 | -25.3 | $50.8_{-1.1}^{+1.1}$ | $5.4_{-0.3}^{+0.3}$ | $2.3_{-0.1}^{+0.1}$ | $0.5_{-0.1}^{+0.2}$ | -24.4 | $6.0_{-0.2}^{+0.2}$ | $39.9_{-1.4}^{+1.5}$ | $5.0_{-1.2}^{+1.6}$ | -57.0
$\lambda$ Orionis* | $3.8_{-0.1}^{+0.1}$ | $30.3_{-0.7}^{+0.6}$ | $13.9_{-1.2}^{+1.6}$ | $5.9_{-0.6}^{+0.7}$ | $0.3_{-0.1}^{+0.1}$ | 3.7 | -14.3 | $85.1_{-11.7}^{+22.9}$ | $1.2_{-0.3}^{+0.2}$ | $1.3_{-0.0}^{+0.0}$ | $0.2_{-0.0}^{+0.1}$ | -8.7 | $5.7_{-0.3}^{+0.4}$ | $28.0_{-6.0}^{+2.7}$ | $3.5_{-0.8}^{+1.1}$ | -50.5

**footnotetext: Clouds below the horizontal line – Orion A, Orion B, and
$\lambda$ Orionis – should be treated with caution, as they lie at the very
edge of the Leike et al. (2020) 3D dust grid and are subjected to additional
biases. Part of Orion A, toward the Orion Nebula Cluster, is also devoid of
dust in the Leike et al. (2020) 3D dust map despite being highly extinguished.
See §5 for full details.

Note. — Radial density profile fitting results for local clouds, computed
using the likelihood function shown in Equation E2 and described in Appendix
E. The likelihood in Equation E2 utilizes all realizations of the 3D dust map
distribution from Leike et al. (2020), rather than just the mean distribution,
allowing us to quantify the effect of the underlying uncertainty in the 3D
dust reconstruction on our results. (1) Name of the cloud. (2 - 6) Best-fit
parameters for the two-component Gaussian fit, including the standard
deviation and amplitude of the narrower Gaussian ($\sigma_{1}$, $a_{1}$), the
standard deviation and amplitude of the broader Gaussian ($\sigma_{2}$,
$a_{2}$), and the square of the modeled scatter in the density measurements
($\sigma_{n,\rm G2}^{2}$). (7) Ratio of the outer to inner Gaussian standard
deviations. (8) Logarithm of the evidence of the two-component Gaussian fit.
(9-12) Best-fit parameters of the Plummer fit, including the peak amplitude
$n_{0}$, the flattening radius $R_{\rm flat}$, the power index of the density
profile $p$, and the square of the modeled scatter in the density measurements
($\sigma_{n,\rm P}^{2}$). (13) Logarithm of the evidence of the Plummer fit.
(14-16) Standard deviation $\sigma$, amplitude $a$, and the square of the
modeled scatter in the density measurements ($\sigma_{n,\rm G1}^{2}$) for the
single component Gaussian fit. (17) Logarithm of the evidence of the single-
component Gaussian fit. We report the single-component Gaussian results here
for completeness only, as it is a poor fit to the data particularly at large
radial distances. A machine readable version of this table is available at
https://doi.org/10.7910/DVN/94OKZD.

##  Appendix F Comparison to CO Radial Profiles

We build radial profiles of the CO emission in order to test the HI-to-$\rm
H_{2}$ transition hypothesis presented in §6.1.1. Specifically, if CO emission
is detected significantly beyond the boundaries of the inner Gaussian profile,
the two component Gaussian would not represent a phase transition between
atomic and molecular gas, disproving the hypothesis discussed in §6.1.1. For
this test, we leverage data from the 1.2 m CfA CO survey (Dame et al., 2001),
selecting all clouds which have broad enough CO coverage to probe potential
emission at large radial distances (seven of eleven clouds in the sample). For
each cloud, we produce integrated intensity maps of the $\rm{}^{12}CO$
emission over the velocity range of $\rm-15$ to $15\;\rm km\;s^{-1}$. We then
smooth each map to the resolution of the Leike et al. (2020) data at the
distance of the cloud (see Table 1) and define 2D masks by thresholding the
integrated intensity maps at a level of $10\rm\;K\;km\;s^{-1}$. We then follow
the procedure for building radial profiles in 2D space, as presented in Zucker
& Chen (2018), which involves skeletonizing the mask and taking cuts across
the spine. We employ the “shift” option to account for CO emission offset from
the main spine, ensuring the emission peaks at a radial distance of zero.

The results are presented in Figure 14. In the top left hand corner of Figure
14 we show an example of what the 2D skeletonization of the CO integrated
intensity masks looks like for a single cloud in the sample — the Perseus
Molecular Cloud. In the subsequent panels of Figure 14, we show normalized CO
integrated intensity profiles (blue) alongside the normalized density profiles
(black) for the inner Gaussian component based on the 3D dust mapping. Both
profiles have been scaled to a peak value of one for comparison purposes. We
find no extended CO emission beyond the bounds of the inner Gaussian, with
most CO profiles showing smaller widths. The one exception is the Taurus
Molecular Cloud, whose CO profile is almost identical to the 3D dust profile
out to a radial distance of 5 pc, but also shows a secondary peak at larger
radial distances. We find that the secondary peak is entirely due to
projection effects, stemming from the two components of Taurus (separated by
about 10-15 pc along the line of sight) which project to similar areas on the
plane of the sky. Thus, we are unable to disprove the HI-to-$\rm H_{2}$
transition hypothesis given the results of the CO radial profile analysis.

Figure 14: Comparison to CO radial profiles for seven clouds in the sample.
The top left hand panel shows a schematic for building 2D radial profiles from
the $\rm{}^{12}CO$ integrated intensity maps for an example cloud (the Perseus
Molecular Cloud), including the spine (thick red line), the distribution of
integrated CO emission inside the filament mask (background grayscale), the
perpendicular cuts across the spine (thin red lines) and the shifted peak
integrated emission inside the filament mask (blue points). Note that the
radial profiles are measured well beyond the boundaries of the mask, and the
mask extent is only used for defining the spine and the optional shifting
procedure. The remaining panels show normalized CO integrated intensity radial
profiles (navy blues curves) alongside the inner component of the 3D dust
derived profiles (black dotted curves) for the two-component Gaussian fits
presented in the main text. Both components have been normalized to a peak
value of one. We find no significant CO emission beyond the boundary of the
3D-dust-derived inner Gaussian profile, finding no direct evidence against the
HI-to-$\rm H_{2}$ transition scenario presented in §6.1.1


We continue by substituting $\qo = 0$ in <ref> to obtain the difference in utility between voting early and late in the current case, and get:

\begin{align}
\label{eq:EarlyWaitDiffQoZero}
\Uearly - \Unotearly
\left(
\ptp - 1 +
\left(\frac{1}{2}  \left(\pta + \ptf\right) - \ptp\right) 0
\right) \cost
\nonumber \\&
- \left(
\left(3 \pta + \ptf - 2 \ptp - 1\right) \frac{0}{2}
+ \left(\ptp - \pta\right)
\right) \cost \po
\nonumber \\&
+ 1 - 0
- \left(1 - 0 - \qt\right) \pta
\nonumber \\&
\left( \ptp - 1 \right) \cost - \left(\ptp - \pta\right)\cost \po + 1 - \left(1 - \qt\right) \pta
\end{align}

Note that <ref> implies that voting early dominates waiting when $\pta=\ptp$:

* When $\pta=1=\ptp$, the difference in utility is $\qt$, implying that later uninformed arrivals contribute to the informed voters' decision to vote early.

* When $\pta=0=\ptp$, the difference in utility equals $1-\cost$, which is strictly positive as $\cost \in \left[0,1\right)$.
Furthermore, the utility is obviously monotonically increasing in $\cost$.

We now attempt to analyze cases where voters are indifferent between voting early and late.
Thus, we substitute $\qo = 0$ in <ref>, :

\begin{align}
\label{eq:QoZeroStrategiesTurnOne}
\po
\frac{
(1-\ptp)\cost + (1-\qt)\pta - 1
\end{align}

Given $\qt$ and $\cost$, we can use <ref> to derive equilibria values for $\pta$ and $\ptp$.
These, in turn, can be substituted in <ref> to obtain equilibria values for $\po$.
Having ruled out the cases where $\ptp = \pta$, we turn to substitute $\ptp = 1$ and $\pta = 0$ in <ref>, and receive $\po = \frac{1}{\cost}$.
But, <ref> shows that these values are only for $\cost \in \left(1-\qt,1-\frac{\qt}{2}\right)$, implying that $\frac{1}{\cost} > 1$, which is not feasible.
Note that substituting $\ptp=1$ and $\pta=0$ into the utility difference formula results in a utility of $1$, implying that voting dominates.
The proof of <ref> gives us two additional cases to examine.
The first is when $\cost = 1 - \qt$, $\ptp = 1$ and voters are indifferent wrt $\pta$, allowing us to get:
$\po = \frac{\pta(\qt - 1) + 1}{(\pta - 1)(\qt - 1)}$.
By requiring that $\qo \le 1$ and solving the resulting equation: $\frac{\pta(\qt - 1) + 1}{(\pta - 1)(\qt - 1)} \le 1$, one can see that the condition is satisfied only for $\qt = 0$, and even then we get that $\qo = 1$.
The second and final case is when voters are indifferent wrt $\ptp$, $\pta = 0$ and $\cost = 1 - \frac{\qt}{2}$, which imply $\po = 1 - \frac{\qt}{\ptp(\qt - 2)}$, but again we reach a degenerate case that is only feasible when $\qt = 0$, and which implies $\po = 1$.

<ref> show that equilibrium strategies may vary in different regions of our parameter space (i.e., $\cost$ and $\vec{\uninformed}$).
In particular, <ref> shows that the values assigned in equilibria to $\ptp$ depend solely on our exogenous parameters for the entire parameter space.
On the other hand, <ref> shows that this is true only when $\qo = 0$ for $\ptf$, in which case voters are indifferent to the value of $\ptf$.
This is because when $\qo = 0$, uninformed voters do not vote early, implying that the information set that $\ptf$ applies to does not exist.
This extends to $\pta$ and $\ptp$, as well; by substituting $\qo = 0$ in <ref>, we reach:

\begin{equation}
\label{eq:QoZeroStrategiesTurnTwo}
\pta
\begin{cases}
1,                   & \cost < 1 - \qt
\\
0,                   & \cost > 1 - \qt
\\
\text{indifference}, & \cost = 1 - \qt
\end{cases}
\qquad
\ptp
\begin{cases}
1,                   & \cost < 1 - \frac{\qt}{2}
\\
0,                   & \cost > 1 - \frac{\qt}{2}
\\
\text{indifference}, & \cost = 1 - \frac{\qt}{2}
\end{cases}
\end{equation}

We turn to characterize the feasible combinations for $\ptp$ and $\pta$ by using <ref>.

* If voters are indifferent wrt both $\pta$ and $\ptp$, then $\cost = 1-\qt = 1 - \frac{\qt}{2}$, but that is possible only when $\qt = 0$.
Moreover, this implies that $\cost = 1$, which is outside our feasible parameter range, thereby ruling-out this possibility.

* If voters are both indifferent wrt $\pta$ and $\ptp = 0$, then this implies $\cost = 1 - \qt > 1 - \frac{\qt}{2}$, which in turn implies $\qt < 0$, which cannot be as $\qt$ is a probability.

* If voters are indifferent wrt $\pta$ and $\ptp = 1$, then by combining the constraints for both, we get $\cost = 1-\qt < 1 -\frac{\qt}{2}$, implying that this can only hold when $0 < \qt$.

* If $\pta = 0$ and voters are indifferent wrt $\ptp$, then $1-\frac{\qt}{2} = \cost > 1 - \qt$, therefore the current condition can only be satisfied when $\qt > 0$.

* If $\pta = 0$ and $\ptp = 0$, we can deduce $\cost > \max\left(1 - \qt,1-\frac{\qt}{2}\right)$, which in our case can be written succinctly: $\cost > 1 - \frac{\qt}{2}$.

* If $\pta = 0$ and $\ptp = 1$, then we get a feasible range: $ 1-\frac{\qt}{2} > \cost > 1 - \qt$, where this range is non-empty for $\qt > 0$.

* If $\pta = 1$ and voters are indifferent wrt $\ptp$, we get a contradiction: $1-\frac{\qt}{2} = \cost < 1 - \qt$.

* If $\pta = 1$ and $\ptp = 0$, a contradiction is reached: $1-\frac{\qt}{2} < \cost < 1 - \qt$.

* If $\pta = 1$ and $\ptp = 1$, we can deduce: $\cost < \min\left(1 - \qt, 1-\frac{\qt}{2}\right)$, which in the current case can be abbreviated to: $\cost < 1 - \qt$.
For $\cost$ to be feasible, we must also require that $\qt < 1$.

We prove this result by ruling out almost all strategies for the current case in <ref>.
Then, we prove that there are feasible parameter regimes which admit such equilibria, and characterize the equilibria strategies for each region of our parameter space.

Notably, <ref> implies that in the current case, voters exhibit a “free-rider” mentality that is dependent on the cost of voting: if $\cost$ is high, then voters assign a low probability of voting, even when the interim tally shows their preference is the leading candidate.

In an equilibrium strategy profile where waiting dominates voting early, if $\qo > 0$, then $\po = 0$, $\pta = \cost$ and $\ptf = 1 - \cost$.

Given the current assumption that waiting dominates early voting, we analyze all possible strategies for the case where $\qo > 0$, as given by <ref>.
We show that the conditions implied by for all but one strategy allow us to reach contradictions.

Our assumption that waiting dominates early voting implies $\po = 0$, and that $\Uearly - \Unotearly < 0$.
By substituting both into <ref>, we can constrain the feasible voting strategies:

\begin{equation}
\label{eq:WaitConstraint}
(\pta + \ptf - 2 \cdot \ptp) \frac{\cost}{2}
+ \pta - 1
) \qo
+ (\ptp-1)\cost + (\qt - 1) \pta + 1
< 0
\end{equation}

Furthermore, substituting $\po = 0$ into <ref> allows to simplify $\pta$:

\begin{equation}
\label{eq:P2aWait}
\pta
\begin{cases}
1,                   & \cost < 1 - \ptf
\\
0,                   & \cost > 1 - \ptf
\\
\text{indifference}, & \cost = 1 - \ptf
\end{cases}
\end{equation}

Armed with <ref>, we tackle each possible strategy.

* If $\pta = 1$ and $\ptf = 1$ dominate, then by <ref> we get that $\cost < \pta = 1$, and similarly from <ref> we obtain that $\cost < 1 - \ptf = 0$.
Both imply that setting $\pta = 1$ and $\ptf = 1$ is only rational when $\cost < 0$, but this is outside our feasible parameter range.

* If $\pta = 1$ and $\ptf = 0$ dominate, then <ref> correspondingly tell us that $c > \pta = 1$ and $1 = 1 - \ptf > c$, implying a contradiction: $1 > c > 1$.

* If $\pta = 1$ dominates and voters are indifferent wrt $\ptf$, then as before,  we use <ref> to reach a contradiction: $1 = \pta = \cost < 1 - \ptf \le 1$.

* If $\pta = 0$ and $\ptf = 1$ dominate, then <ref> allow us to arrive at a contradiction: $0 = 1 - \ptf < c < \pta = 0$.

* If $\pta = 0$ and $\ptf = 0$ dominate, then from <ref>, we deduce that $\cost > 1 - \ptf = 1$, which is outside our feasible region for costs.

* If $\pta = 0$ dominates and voters are indifferent wrt $\ptf$, then <ref> imply: $0 = \pta = \cost < 1 - \ptf$.
Substituting this in <ref> gives us:

\begin{align*}
(\pta + \ptf - 2 \cdot \ptp) \frac{\cost}{2}
+ \pta - 1
) \qo
+ (\ptp-1)\cost + (\qt - 1) \pta + 1
\nonumber \\&
(\pta + \ptf - 2 \cdot \ptp)\cdot0
+ 0 - 1
) \qo
+ (\ptp-1)\cdot0 + (\qt - 1)\cdot0 + 1
\nonumber \\&
1 - \qo
\end{align*}

But, because $\qo \in \left[0,1\right]$, we reach a contradiction:
$0 > 1 - \qo \geq 0$.

* If voters are indifferent wrt $\pta$ and $\ptf = 1$ dominates, then <ref> and the fact that $\pta \in \left[0,1\right]$ produce a contradiction: $\pta < \cost = 1 - \ptf = 0$.

* If voters are indifferent wrt $\pta$ and $\ptf = 0$ dominates, then <ref> implies $\cost = 1 - \ptf = 1$, which is outside the feasible regime for $\cost$.

* If voters are indifferent wrt to both $\pta$ and $\ptf$, then we get from <ref> that $\pta = \cost = 1 - \ptf$.
Thus, we deduce that $\pta = \cost$ and $\ptf = 1 - \cost$.

In total, the only feasible strategy is the one in which $\pta = \cost$ and $\ptf = 1 - \cost$.

Now, we substitute the strategy given by <ref> in <ref>:

\begin{equation}
\label{eq:EarlyDominatesFinalIndifference}
\left(\ptp\left(1 - \qo\right) - 2 + 1.5\qo + \qt\right)\cost + 1 - \qo < 0
\end{equation}

Given this new constraint, we analyze each option given by <ref> for $\ptp$, and characterize the feasible parameter regimes that admit each via a series of results.
The proofs for the following results are given in <ref>.

According to <ref>, our choice of $\ptp$ imposes two conditions: $\qo < 1$ and $\cost > 1 + \frac{\qt}{2 \left(\qo - 1\right)} = \frac{2\qo + \qt - 2}{2 \left(\qo - 1\right)}$.
We substitute $\ptp = 0$ in <ref>, and get:
$\cost\left(1.5\qo + \qt - 2\right) < \qo - 1$.
This implies that: $\cost > \frac{\qo - 1}{1.5\qo + \qt - 2}$, where the direction of the inequality was flipped because both $\qo + \qt \le 1$ and $\qo < 1$, allowing us to deduce: $1.5\qo + \qt - 2 \le -\frac{1}{2}$.

Together with the condition imposed by <ref>, we get:
$\cost \in \left(
\max\left(
\frac{\qo - 1}{1.5\qo + \qt - 2}
\frac{2\qo + \qt - 2}{2 \left(\qo - 1\right)}
\right)
\right)$.

This range is non-empty when either $\frac{\qo - 1}{1.5\qo + \qt - 2} \le \frac{2\qo + \qt - 2}{2 \left(\qo - 1\right)} < 1$, or $\frac{2\qo + \qt - 2}{2 \left(\qo - 1\right)} \le \frac{\qo - 1}{1.5\qo + \qt - 2} < 1$.

First, examine the case where: $\frac{\qo - 1}{1.5\qo + \qt - 2} < 1$.
This requirement is equivalent to $\qo -1 > 1.5\qo + \qt - 2$, where the direction of the inequality is flipped as $1.5\qo + \qt - 2 \le -\frac{1}{2} < 0$.
This requirement implies that $1 > 0.5\qo + \qt$, which is indeed true: by definition $\qo + \qt \le 1 $, and in the current case we furthermore have $\qo > 0$.

In the other case, we have the following condition: $\frac{2\qo + \qt - 2}{2 \left(\qo - 1\right)} < 1$.
When multiplying both sides of by $\qo -1$, we flip the direction of the inequality due to our assumption that $\qo \in \left( 0, 1 \right)$, giving us: $2\qo + \qt - 2 > 2 \left(\qo - 1\right)$.
This implies that: $\qt > 0$.

Due to the asymmetry between the feasible $\qt$ values for the two cases, we characterize in <ref> the regimes for which each one is satisfied.
Note that these results are slightly more general as they are reused for additional proofs.
In particular, <ref> does not impose the restriction $\qo > 0$, which is instead accounted for only afterward.

We have that $\frac{\qo - 1}{1.5\qo + \qt - 2} < \frac{2\qo + \qt - 2}{2 \left(\qo - 1\right)}$ is satisfied when $\qt \in \left[0, 2 - \sqrt{2} \right)$, and $\qo \in \left[0, \frac{3-3.5\qt - \sqrt{8.25\qt^2 -5\qt + 1}}{2}\right) \cup \left(\frac{3-3.5\qt + \sqrt{8.25\qt^2 -5\qt + 1}}{2}, 1\right)$.

We have that $\frac{\qo - 1}{1.5\qo + \qt - 2} \geq \frac{2\qo + \qt - 2}{2 \left(\qo - 1\right)}$ is satisfied when $\qt \in \left[0,1\right]$, and
$\qo \in \left[\frac{3-3.5\qt - \sqrt{8.25\qt^2 -5\qt + 1}}{2}, \frac{3-3.5\qt + \sqrt{8.25\qt^2 -5\qt + 1}}{2} \right]$.

Proofs for both results are given in <ref>.
Intuitively, we use the various requirements to constrain our parameter space, resulting in systems of several equations which we solve.
By using them, we fully characterize when $\frac{\qo - 1}{1.5\qo + \qt - 2} < \frac{2\qo + \qt - 2}{2 \left(\qo - 1\right)}$ holds (via <ref>), and similarly, where we have $\frac{\qo - 1}{1.5\qo + \qt - 2} \geq \frac{2\qo + \qt - 2}{2 \left(\qo - 1\right)}$ (via <ref>).

Next, we account for the constraints which were left out of the aforementioned two results, to ensure their generality.
In particular, we have to consider that $\qo + \qt \in \left[0,1\right]$, and also consider the requirement of the current case that $\qo \in \left( 0, 1 \right)$.
For the former constraint, one can verify using the extreme values of our range for $\qt$ and the resulting ranges for $\qo$ that the range of values $\qo \in \left(\frac{3-3.5\qt + \sqrt{8.25\qt^2 -5\qt + 1}}{2}, 1\right)$ is infeasible.
Furthermore, for the latter constraint, one can account for it by intersecting it with our resulting parameter regions.
Finally, the characterization stated in <ref> is obtained.

For the given range of $\qo$ values, we have that $\qo \in \left[0,1\right)$.
Thus, we have $1.5\qo + \qt - 2 < 0$ and $\qo - 1 < 0$, implying that $\frac{\qo - 1}{1.5\qo + \qt - 2} < \frac{2\qo + \qt - 2}{2 \left(\qo - 1\right)}$ is equivalent to:

\begin{align*}
\left(2\qo + \qt - 2\right)\left(1.5\qo + \qt - 2\right)
2 \left(\qo - 1\right)^2
\nonumber \\&
\qo^2 + \left(3.5\qt - 3\right)\qo + \qt^2 - 4\qt + 2
\end{align*}

The roots of this polynomial are obtained at:

\begin{align*}
\qo
\frac{
3-3.5\qt \pm \sqrt{
(3.5\qt - 3)^2 - 4(\qt^2 - 4\qt + 2)
\nonumber \\&
\frac{
3-3.5\qt \pm \sqrt{
12.25\qt^2 - 21\qt + 9 - 4\qt^2 + 16\qt - 8
\nonumber \\&
\frac{1}{2}\left(3-3.5\qt \pm \sqrt{8.25\qt^2 - 5\qt + 1}\right)
\end{align*}

We always have two real roots for $\qo$, as:

\begin{align}
\label{eq:EarlyDominatesFinalIndifferenceConditionSqrtIsPositive}
8.25\qt^2 -5\qt + 1
\left(\sqrt{8.25}\qt - 1\right)^2 + \left(\sqrt{33}-5\right)\qt
\left(\sqrt{25}-5\right)\qt
\geq
\end{align}

Thus far, our feasible range for $\qo$ is:

\begin{equation}
\label{eq:EarlyDominatesQo}
\qo \in
\left(
\frac{1}{2}\left(
\sqrt{8.25\qt^2 -5\qt + 1}
\right)
\right)
\bigcup
\left(
\frac{1}{2}\left(
\sqrt{8.25\qt^2 -5\qt + 1}
\right)
\right)
\end{equation}

The left segment of <ref> is non-empty when
$0 < \frac{1}{2}\left(
\sqrt{8.25\qt^2 -5\qt + 1}
\right)$, or equivalently:
\frac{1}{2}\sqrt{8.25\qt^2 -5\qt + 1}
\frac{1}{2}\left( 3-3.5\qt \right)
Per <ref>, the term $8.25\qt^2 -5\qt + 1$ is strictly positive, while the term $3-3.5\qt$ is non-negative for $\qt \le \frac{3}{3.5}$, thus the inequality does not hold for $\qt > \frac{3}{3.5}$.
Given that $\qt \leq \frac{3}{3.5}$, we can raise the two sides by a power of two:

\begin{align*}
\frac{1}{4} \left(
\left( 3-3.5\qt \right)^2
\left( 8.25\qt^2 -5\qt + 1 \right)
\right)
\nonumber \\&
\frac{12.25\qt^2 - 21\qt + 9 - 8.25\qt^2 +5\qt - 1}{4}
\nonumber \\&
\qt^2 - 4\qt + 2
\end{align*}

This equals zero when:
\qt
% &
\frac{1}{2}\cdot\left(4 \pm \sqrt{16-4\cdot2}\right)
% \nonumber\\&
\frac{1}{2}\cdot\left(4 \pm 2\sqrt{2}\right)
% \nonumber\\&
2 \pm \sqrt{2}
$, which implies that either $\qt < 2-\sqrt{2}$, or $\qt > 2+\sqrt{2}$.
But, $\qt \le 1 < 2+\sqrt{2}$, so we remain with $\qt < 2-\sqrt{2}$.

The right segment of <ref> is non-empty when:
\sqrt{8.25\qt^2 -5\qt + 1}
\right) < 1$, which can be simplified to:
$\sqrt{8.25\qt^2 -5\qt + 1} < 3.5\qt - 1$.
From <ref>, the left-hand side is strictly positive, implying that the inequality does not hold for $\qt \le \frac{1}{3.5} \approx 0.28$.
When $\qt > \frac{1}{3.5} \approx 0.28$, then both sides are non-negative and we can raise them to the power of 2:
$8.25\qt^2 -5\qt + 1 < \left(3.5\qt - 1\right)^2$, which can be simplified to:

\begin{align*}
12.25\qt^2 - 7\qt + 1 - \left( 8.25\qt^2 -5\qt + 1 \right)
\nonumber \\&
\nonumber \\&
4\qt\left(\qt - \frac{1}{2}\right)
\end{align*}

This implies $\qt >\frac{1}{2}$, meaning that for this case we have $\qt \in \left(\frac{1}{2}, 1\right)$.

We need at least one segment of <ref> to be non-empty, so we suffice with $\qt < 2-\sqrt{2} \approx 0.58$.
In total, our feasible range of parameters is:

\begin{align*}
\qo
\in
\left[
\frac{1}{2}\left(
\sqrt{8.25\qt^2 -5\qt + 1}
\right)
\right)
\bigcup
\left(
\frac{1}{2}\left(
\sqrt{8.25\qt^2 -5\qt + 1}
\right)
\right)
\\
\qt
\in \left[0, 2 - \sqrt{2} \right)
\end{align*}

Similar reasoning to the proof of <ref> allows us to deduce that $\qo$ values must be in the following range:

\begin{equation}
\label{eq:WaitingPtpZeroCostConstraintSecondRangeQo}
\qo
\in
\left[
\frac{1}{2}\left(
\sqrt{8.25\qt^2 -5\qt + 1}
\right)
\frac{1}{2}\left(
\sqrt{8.25\qt^2 -5\qt + 1}
\right)
\right]
\end{equation}

In particular, we get from <ref> that this range is non-empty for $\qt \in \left[0,1\right]$, i.e.

\begin{equation*}
\frac{1}{2}\left(
\sqrt{8.25\qt^2 -5\qt + 1}
\right)
\frac{1}{2}\left(
\sqrt{8.25\qt^2 -5\qt + 1}
\right).
\end{equation*}

But, for $\qo$ to be feasible, we also have to require that either one of <ref> must hold.

\begin{equation}
\label{eq:WaitingPtpZeroCostConstraintSecondLe1}
0 \le \frac{1}{2}\left(3-3.5\qt - \sqrt{8.25\qt^2 -5\qt + 1}\right) \le 1
\end{equation}

\begin{equation}
\label{eq:WaitingPtpZeroCostConstraintSecondGe0}
0 \le \frac{1}{2}\left(3-3.5\qt + \sqrt{8.25\qt^2 -5\qt + 1}\right) \le 1
\end{equation}

\begin{equation}
\label{eq:WaitingPtpZeroCostConstraintSecondSurround}
\frac{1}{2}\left(3-3.5\qt - \sqrt{8.25\qt^2 -5\qt + 1}\right)
\le 0 < 1 \le
\frac{1}{2}\left(3-3.5\qt + \sqrt{8.25\qt^2 -5\qt + 1}\right)
\end{equation}

Satisfying <ref>
For <ref> to hold, we can equivalently require that both <ref> hold:

\begin{equation}
\label{eq:WaitingPtpZeroCostConstraintSecondGe0Ge0}
\sqrt{8.25\qt^2 -5\qt + 1} \geq 3.5\qt - 3
\end{equation}

\begin{equation}
\label{eq:WaitingPtpZeroCostConstraintSecondGe0Le1}
\sqrt{8.25\qt^2 -5\qt + 1} \le 3.5\qt - 1
\end{equation}

For <ref>, the proof of <ref> shows it is satisfied when $\qt \in \left[\frac{1}{2},1\right]$.

The condition given in <ref> is satisfied when $\qt \in \left[0,\frac{3}{3.5}\right]$ due to the strict positivity of the left-hand side, as shown by <ref>.
For $\qt > \frac{3}{3.5}$, this is satisfied when:

\begin{align}
\label{eq:WaitingPtpZeroCostConstraintSecondGe0GtFrac}
\geq
\left(3.5\qt - 3\right)^2
\left(8.25\qt^2 -5\qt + 1\right)
\nonumber \\&
\left(12.25\qt^2 - 21\qt + 9\right)
\left(8.25\qt^2 -5\qt + 1\right)
\nonumber \\&
4\qt^2 - 16\qt + 8
\nonumber \\&
4\cdot\left(\qt^2 - 4\qt + 2\right)
\end{align}

From the proof of <ref>, we get that <ref> is satisfied when $\qt \in \left[2-\sqrt{2}, 2+\sqrt{2}\right]$.
But, $0.58 \approx 2-\sqrt{2} < \frac{3}{3.5} \approx 0.85$, and furthermore we must have $\qt \le 1 < 2+\sqrt{2} \approx 3.41$.
Therefore, we find that <ref> holds when:
$\qt \in \left[0,\frac{3}{3.5}\right]\bigcup\left(\frac{3}{3.5}, 1\right] = \left[0,1\right]$.

Recall that the satisfiability of <ref> hinges on both <ref> holding.
Thus, we intersect their feasible parameter ranges.
In total, we get that <ref> holds when $\qt \in \left[\frac{1}{2},1\right]$.

Satisfying <ref>
For <ref> to hold, we can require that both of the following are satisfied:

\begin{equation}
\label{eq:WaitingPtpZeroCostConstraintSecondLe1Ge0}
\sqrt{8.25\qt^2 -5\qt + 1} \le 3 - 3.5\qt
\end{equation}

\begin{equation}
\label{eq:WaitingPtpZeroCostConstraintSecondLe1Le1}
\sqrt{8.25\qt^2 -5\qt + 1} \geq 1 - 3.5\qt
\end{equation}

For <ref>, from the proof of <ref> we get that the condition holds for $\qt \le 2-\sqrt{2}$.

The constraint given in <ref> is always satisfied when $\qt \geq \frac{1}{3.5}$ due to <ref>.
With respect to $\qt < \frac{1}{3.5}$, both sides are positive, allowing us to raise them to the power of two, and solve the resulting inequality:

\begin{align*}
\le
8.25\qt^2 -5\qt + 1 - \left(1 - 3.5\qt\right)^2
\\&
8.25\qt^2 -5\qt + 1 - \left(1 - 7\qt + 12.25\qt^2\right)
\\&
-4\qt^2 + 2\qt
\\&
-4\qt \left(\qt - \frac{1}{2}\right)
\end{align*}

This holds for $\qt \in \left[0,\frac{1}{2}\right]$.
As <ref> holds for $\qt \geq \frac{1}{3.5}$ and $\frac{1}{3.5} < \frac{1}{2}$, thus we have $\qt \in \left[0,1\right]$.

In total, note that <ref> is satisfied only when both <ref> are satisfied.
By intersecting their feasible value ranges, we find that <ref> holds for $\qt \in \left[0,2-\sqrt{2}\right]$.

Satisfying <ref>
The condition given in <ref> is equivalent to requiring that both:

\begin{equation}
\label{eq:WaitingPtpZeroCostConstraintSecondSurroundLe}
\sqrt{8.25\qt^2 -5\qt + 1} \geq 3-3.5\qt
\end{equation}

\begin{equation}
\label{eq:WaitingPtpZeroCostConstraintSecondSurroundGe}
\sqrt{8.25\qt^2 -5\qt + 1} \geq 3.5\qt - 1
\end{equation}

The condition of <ref> holds for $\qt \geq \frac{3}{3.5}$, due to <ref>.
For $\qt < \frac{3}{3.5}$, reasoning similar to that given in the proof of <ref> shows that it is satisfied for $\qt \in \left[2-\sqrt{2}, 2+\sqrt{2}\right]$.
When combined with the previous results, we deduce that the condition holds for $\qt \in \left[2-\sqrt{2}, 1\right]$.

With respect to <ref>, again one can deduce from <ref> that it is satisfied for $\qt \le \frac{1}{3.5}$, while the proof of <ref> shows that the condition holds for $\qt \in \left[0, \frac{1}{2}\right]$.

In total, observe that the intersection of the feasible parameters for <ref> is empty.

Feasible parameters for $\qo$
Finally, recall that the range <ref> includes feasible $\qo$ values when either one of <ref> is satisfied.
Note that $0.58 \approx 2-\sqrt{2} > 0.5$, thus we get that <ref> is non-empty for $\qt \in \left[0,1\right]$.
This implies that $\frac{\qo - 1}{1.5\qo + \qt - 2} \geq \frac{2\qo + \qt - 2}{2 \left(\qo - 1\right)}$ is satisfied when:

\begin{align*}
\qo
\in
\left[
\frac{1}{2}\left(
\sqrt{8.25\qt^2 -5\qt + 1}
\right)
\frac{1}{2}\left(
\sqrt{8.25\qt^2 -5\qt + 1}
\right)
\right]
\\
\qt
\in
\left[0,1\right]
\end{align*}

From <ref> and our assumption that $\ptp = 1$, we obtain two constraints: $\cost < 1+\frac{\qt}{2\left(\qo-1\right)}$ and $\qo < 1$.
By substituting $\ptp = 1$ in <ref>, we get $\left(\frac{\qo}{2} + \qt - 1\right)\cost + 1 - \qo < 0$, which we rearrange to: $1 - \qo < \left(1 - \frac{\qo}{2} - \qt\right)\cost$.
Dividing the expression by $1 - \frac{\qo}{2} - \qt$ does not change the inequality's direction, as $\qo + \qt \in \left[0,1\right]$ and as in the current case we have $\qo > 0$.
Thus, we deduce that $1 - \frac{\qo}{2} - \qt > 0$, and get a constraint on $\cost$: $\frac{1 - \qo}{1 - \frac{\qo}{2} - \qt} < \cost$.
To satisfy both the new and the previous constraints, we require: $\frac{1 - \qo}{1 - \frac{\qo}{2} - \qt} < \cost < 1+\frac{\qt}{2\left(\qo-1\right)}$.
Feasible $\cost$ values exist when:

\begin{align*}
\left(
\frac{1 - \qo}{1 - \frac{\qo}{2} - \qt}
\right)
\nonumber \\&
\frac{2\qo + \qt - 2}{2\left(\qo-1\right)}
\left(
\frac{1 - \qo}{1 - \frac{\qo}{2} - \qt}
\right)
\nonumber \\&
\frac{
\left( 2\qo + \qt - 2 \right) \left( 1 - \frac{\qo}{2} - \qt \right)
2 \left( 1 - \qo \right)^2
}{2 \left(\qo-1\right) \left( 1 - \frac{\qo}{2} - \qt \right)}
\nonumber \\&
\frac{
2 \left( 1 - \qo \right)^2
\left( 2\qo + \qt - 2 \right) \left( 1 - \frac{\qo}{2} - \qt \right)
}{2 \left(1-\qo\right) \left( 1 - \frac{\qo}{2} - \qt \right)}
\nonumber \\&
\frac{
-\qo^2 + \left(2.5\qt + 1\right)\qo + \qt^2 - 3\qt
}{2 \left(1-\qo\right) \left( 1 - \frac{\qo}{2} - \qt \right)}
\end{align*}

The denominator is positive, as $\qo \in \left(0,1\right)$, and as we have already shown that $1 - \frac{\qo}{2} - \qt > 0$.
So, we only require:
$0 < -\qo^2 + \left(2.5\qt + 1\right)\qo + \qt^2 - 3\qt$.
Solving for $\qo$, the roots of the polynomial are:

\begin{align*}
\qo
\frac{
-\left(2.5\qt + 1\right)
\pm
\sqrt{
\left(2.5\qt + 1\right)^2
4\cdot\left(-1\right)\cdot\left(\qt^2 - 3\qt\right)
\nonumber \\&
\frac{
-\left(2.5\qt + 1\right)
\pm
\sqrt{6.25\qt^2 + 5\qt + 1 + 4\qt^2 - 12\qt}
\nonumber \\&
\frac{5\qt + 2 \pm \sqrt{41\qt^2 - 28\qt + 4}}{4}
\end{align*}

The roots are in $\mathbb{R}$ when $0 \le 41\qt^2 - 28\qt + 4$, and the zeroes of this constraint are obtained at:

\begin{equation}
\label{eq:WaitingPtpOneSqrtZeroes}
\qt
\frac{
\pm
\sqrt{28^2 - 4\cdot41\cdot4}
\frac{14 \pm 4\sqrt{2}}{41}
\end{equation}

For clarity, note that $\frac{14 - 4\sqrt{2}}{41} \approx 0.2$, $\frac{14 + 4\sqrt{2}}{41} \approx 0.48$.
Thus, our parameter range is currently:
$\qo \in \left(\frac{5\qt + 2 - \sqrt{41\qt^2 - 28\qt + 4}}{4}, \frac{5\qt + 2 + \sqrt{41\qt^2 - 28\qt + 4}}{4}\right)$, $\qt \in \left[0, \frac{14 - 4\sqrt{2}}{41}\right] \cup \left[\frac{14 + 4\sqrt{2}}{41},1\right]$, $\cost \in \left(\frac{1 - \qo}{1 - \frac{\qo}{2} - \qt}, 1+\frac{\qt}{2\left(\qo-1\right)}\right)$.

For $\qo$ to be feasible, we require that the current feasible range has a non-empty intersection with $\left[0,1\right]$.
This is analyzed in <ref>, with the corresponding proof given in <ref>.

$\forall \qt \in \left[0, \frac{14 - 4\sqrt{2}}{41}\right)\cup\left(\frac{1}{2}, 1\right]: \left(\frac{5\qt + 2 - \sqrt{41\qt^2 - 28\qt + 4}}{4}, \frac{5\qt + 2 + \sqrt{41\qt^2 - 28\qt + 4}}{4}\right) \cap \left[0,1\right] \ne \emptyset$

To ensure that throughout the parameter space we furthermore require $\qo + \qt \in \left[0,1\right]$.
An immediate calculation of the various possible extreme values of $\qt$ and the corresponding extrema of $\qo$ show that our feasible range for $\qt$ is: $\qt \in \left[0, \frac{14 - 4\sqrt{2}}{41}\right)$.
This allows us to refine our feasible parameter range and conclude the proof:

\begin{align*}
\qo   & \in \left(\frac{5\qt + 2 - \sqrt{41\qt^2 - 28\qt + 4}}{4}, \frac{5\qt + 2 + \sqrt{41\qt^2 - 28\qt + 4}}{4}\right)
\\
\qt   & \in \left[0, \frac{14 - 4\sqrt{2}}{41}\right)
\\
\cost & \in \left(\frac{1 - \qo}{1 - \frac{\qo}{2} - \qt}, \frac{2\qo + \qt - 2}{2\left(\qo-1\right)}\right)
\end{align*}

For $\left(\frac{5\qt + 2 - \sqrt{41\qt^2 - 28\qt + 4}}{4}, \frac{5\qt + 2 + \sqrt{41\qt^2 - 28\qt + 4}}{4}\right) \cap \left[0,1\right] \ne \emptyset$ to hold, we require that both $\frac{5\qt + 2 - \sqrt{41\qt^2 - 28\qt + 4}}{4} < \frac{5\qt + 2 + \sqrt{41\qt^2 - 28\qt + 4}}{4}$, which is true for $\qt \in \left[0, \frac{14 - 4\sqrt{2}}{41}\right) \cup \left(\frac{14 + 4\sqrt{2}}{41}, 1 \right]$ due to <ref>, and furthermore that at least one of <ref> must hold.

\begin{equation}
\label{eq:WaitingPtpOneFirst}
\frac{5\qt + 2 - \sqrt{41\qt^2 - 28\qt + 4}}{4} \in \left[0,1\right)
\end{equation}

\begin{equation}
\label{eq:WaitingPtpOneSecond}
\frac{5\qt + 2 + \sqrt{41\qt^2 - 28\qt + 4}}{4} \in \left(0,1\right]
\end{equation}

\begin{equation}
\label{eq:WaitingPtpOneThird}
\frac{5\qt + 2 - \sqrt{41\qt^2 - 28\qt + 4}}{4}
\le 0 < 1 \le
\frac{5\qt + 2 + \sqrt{41\qt^2 - 28\qt + 4}}{4}
\end{equation}

Satisfying <ref>
Note that this condition holds when both <ref> hold.

\begin{equation}
\label{eq:WaitingPtpOneFirstZero}
0 \le \frac{5\qt + 2 - \sqrt{41\qt^2 - 28\qt + 4}}{4}
\end{equation}

\begin{equation}
\label{eq:WaitingPtpOneFirstOne}
1 > \frac{5\qt + 2 - \sqrt{41\qt^2 - 28\qt + 4}}{4}
\end{equation}

The former is equivalent to $\sqrt{41\qt^2 - 28\qt + 4} \le 5\qt + 2$, and thus is satisfied for $\qt \in \left[0,3\right]$:

\begin{align*}
\geq
41\qt^2 - 28\qt + 4 - \left(5\qt + 2\right)^2
\nonumber \\&
41\qt^2 - 28\qt + 4 - \left(25\qt^2 + 20\qt + 4\right)
\nonumber \\&
16\qt\left(\qt - 3\right)
\end{align*}

For <ref>, it is equivalent to $\sqrt{41\qt^2 - 28\qt + 4} > 5\qt - 2$, which is trivially satisfied when both $\qt < \frac{2}{5}$ and when the square root has a real solution, thus from <ref> we get that $\qt \in \left[0, \frac{14 - 4\sqrt{2}}{41}\right)$, as $0.2 \approx \frac{14 - 4\sqrt{2}}{41} < \frac{2}{5}$.
In case $\qt \geq \frac{2}{5}$, both sides of the constraint are positive, thus we can raise them to the power of two and solve to show that it is satisfied for $\qt > \frac{1}{2}$:

\begin{align*}
41\qt^2 - 28\qt + 4 - \left(5\qt - 2\right)^2
\nonumber \\&
41\qt^2 - 28\qt + 4 - \left(25\qt^2 - 20\qt + 4\right)
\nonumber \\&
16\qt^2 - 8\qt
\nonumber \\&
8\qt\left(2\qt - 1\right)
\end{align*}

In total, we get that the constraint holds when $\qt \in \left[0, \frac{14 - 4\sqrt{2}}{41}\right)\cup\left(\frac{1}{2}, 1\right]$.

Satisfying <ref>
This requirement is equivalent to satisfying both <ref>:

\begin{equation}
\label{eq:WaitingPtpOneSecondZero}
0 < \frac{5\qt + 2 + \sqrt{41\qt^2 - 28\qt + 4}}{4}
\end{equation}

\begin{equation}
\label{eq:WaitingPtpOneSecondOne}
1 \geq \frac{5\qt + 2 + \sqrt{41\qt^2 - 28\qt + 4}}{4}
\end{equation}

<ref> is satisfied when $\qt \in \left[0, \frac{14 - 4\sqrt{2}}{41}\right] \cup \left[\frac{14 + 4\sqrt{2}}{41},1\right]$, due to <ref> and as $\qt$ is a probability and thus must be in $\left[0,1\right]$.
On the other hand, <ref> is equivalent to $\sqrt{41\qt^2 - 28\qt + 4} \le 2 - 5\qt$, which does not hold for $\qt > 0.2$.
For $\qt \le 0.2$, we raise the inequality to the power of two and deduce that the condition is satisfied when $\qt \in \left[0, 0.2\right]$:

\begin{align*}
\geq
41\qt^2 - 28\qt + 4 - \left(2 - 5\qt\right)^2
\nonumber \\&
41\qt^2 - 28\qt + 4 - \left(4 - 20\qt + 25\qt^2\right)
\nonumber \\&
16\qt\left(\qt - \frac{1}{2}\right)
\end{align*}

In total, <ref> holds when $\qt \in \left[0, 0.2\right]$.

Satisfying <ref>
The current condition is equivalent to both of the following.

\begin{equation}
\label{eq:WaitingPtpOneThirdLe}
\frac{5\qt + 2 - \sqrt{41\qt^2 - 28\qt + 4}}{4} \le 0
\end{equation}

\begin{equation}
\label{eq:WaitingPtpOneThirdGe}
\frac{5\qt + 2 + \sqrt{41\qt^2 - 28\qt + 4}}{4} \ge 1
\end{equation}

<ref> complements <ref>, implying that the requirement does not hold for $\qt \in \left[0,1\right]$, and that furthermore <ref> has no feasible satisfying parameters, as its satisfiability hinges on both <ref> holding at the same time.

Feasible parameters for $\qt$
As we require that at least one out of <ref> holds, we take the union of the feasible parameter range of each of these, which equals:
\qt \in \left[0, \frac{14 - 4\sqrt{2}}{41}\right)\cup\left(\frac{1}{2}, 1\right]

§ ADDITIONAL RELATED WORK

In this section, we go over additional related papers.
The reliance of on-chain governance protocols on tokens to represent voting power bears a similarity to corporate governance mechanisms which give voting power per share.
Thus, we also go over works on corporate governance which could interest the blockchain community.

§.§ Blockchain Governance

Theoretical Studies
Kiayias and Lazos <cit.> present a SoK of blockchain governance mechanisms.
In the paper, governance is defined in the broad sense, taken to mean both the mechanism by which improvements are suggested and adopted by cryptocurrencies like Bitcoin, and the on-chain protocols employed by DeFi projects.
In the paper, several properties of interest are defined, such as who is granted suffrage by each mechanism, and the confidentiality of the voting process, and quantitatively evaluate the various mechanisms.

Wright <cit.> suggest using Posner and Weyl's <cit.> quadratic voting in blockchains.
In such voting mechanisms, the voting power of an individual is equal to the square root of the number of votes it casts.
Dimitri <cit.> analyze a PoS consensus mechanism which relies on quadratic voting, and note it is not Sybil resistant: voters can gain a disproportionate advantage by splitting their votes across multiple identities.
Indeed, similar mechanisms are in use in practice <cit.>.
Some practitioners <cit.> recommend improving Sybil-resistance by randomly choosing the subset of eligible voters for each proposal.
We note that adversaries can circumvent this measure by splitting their funds into multiple accounts in advance.

Empirical Studies
Fritsch et al. <cit.> study the distribution of voting power and the degree of decentralization of the governance mechanisms used by Compound, Uniswap and ENS.
This was done by evaluating various measures, such as the Gini coefficient <cit.> of token holders, and the frequency with which single voters have influenced decisions.
The authors found that the surveyed protocols became more decentralized with time, and that a small number of voters hold a considerable amount of voting power, but that they do not exercise it often.

Feichtinger et al. <cit.> later extended the study of <cit.> using the same metrics, but with more recent data spanning a larger amount of governance protocols.
They show that governance mechanisms are in fact still relatively centralized, with majority power concentrated in the hands of few users in $17$ mechanisms out of the total of $21$ which were examined.
These results are supplemented by an analysis of the costs associated with governance protocols, which are estimated at over $\$10$ Million.
They consider both costs directly related to participation in decision-making (e.g., paying fees for voting transactions), and indirect ones (costs paid due to the overhead of updating governance-related data structures when performing unrelated actions, like token transfers).

Fan et al. <cit.> use the case of Uniswap and SushiSwap as a natural experiment to measure how governance token issuance incentivizes the migration of users between two competing platforms.
In particular, SushiSwap was created as a fork of Uniswap, and thus by design both are identical, with the exception that SushiSwap rewarded liquidity providers with governance tokens, while Uniswap did not.
The authors note that although SushiSwap's policy quickly attracted liquidity providers, some of these providers tended to remove their liquidity a few days afterwards, especially ones who provide large amounts of liquidity.

Barbereau et al. <cit.> quantify the centralization of the governance mechanisms of Uniswap, Maker, SushiSwap, Yearn Finance and UMA by giving a high-level comparison of the token issuance strategy of each, and then performing an empirical evaluation of the distribution of tokens among users, concluding that voting rights are centralized in all examined projects.
With respect to issuance strategies, the authors found that only SushiSwap and Yearn Finance adopted “fair” strategies that do not involve some preliminary issuance of tokens by the projects' creators and then selling it to users.
They note that a fair strategy could be to the detriment of a project, and give the case of Yearn Finance as an example: the project's lack of centrally owned capital made it difficult to fund a third-party security audit.
The paper proceeds with an empirical evaluation of the distribution of tokens among users through the lens of various metrics, such as the Gini coefficient, and the distributions' similarity to a “good” distribution that represents an ideal level of decentralization <cit.>.

Barbereau et al. <cit.> expand the study of <cit.> by analyzing user participation in the surveyed mechanisms, and including additional mechanisms in their analysis, specifically 0x, Aave, Compound and Uniswap.
They found that typically, the rate of participation is low, only rarely rising above a turnout of $1\%$.
The authors ascribe the low rate to the ability of users to trade them, and justify this conclusion by showing that until January 2022, the trading volume of governance tokens is much higher when compared to the amount of tokens which are delegated.

Sun et al. <cit.> attempt to quantify the level of centralization in Maker's governance protocol, using data spanning August 2019 and October 2021, and by applying metrics like voter participation and Twitter sentiment.
The authors find that the average and median share of votes cast by the largest voter per proposal were $52.66\%$ and $48.35\%$, respectively.
Furthermore, they examine the financial impact of centralization.
They conduct regression analyses and show that Maker's trading volume and the number of transactions interacting with the platform are positively correlated with the share of users in possession of $10-100$K MKR tokens, and negatively correlated with the share of those who hold more than $100$K tokens.

Sharma et al. <cit.> conducted a mixed-methods study to evaluate the effectiveness of governance mechanisms employed by $10$ DAO: Compound, BitDAO, AssangeDAO, BanklessDAO, KrauseHouse, LivePeer, MetaGammaDelta, MolochDAO, dxDAO.
For their work, they interviewed $10$ participants of such mechanisms, and used responses to identify decentralization metrics which are considered as relevant by the community, primarily, the distribution of token ownership, token holders' participation rate in the governance process, and the geographic distribution of holders.
By these metrics, the authors find that Compound, AssangeDAO, Bankless and Krausehouse fare poorly with respect to decentralization, and show that across all examined platforms, proposal success is positively correlated with the token holdings of the proposer, and that repeated participation in votes is positively correlated with a voter's holdings.
Morevoer, the work examines the technical aspects of governance mechanisms, and shows that some are limited in that they do not allow successful proposals to execute arbitrary transactions, while others rely on third-party services to fulfil proposals.

Dotan et al. <cit.> perform a series of case-studies on vulnerabitilies of governance mechanisms and how these were exploited.
For example, the authors show that Compound's mechanism awarded users with governance tokens for taking loans, where the reward mechanism resulted in the interest-rate of the corresponding loans to be negative.
The case studies are augmented by a data-driven analysis showing that voter turnout is negatively correlated with transaction fees, and that on some occasions, voters possessed governance tokens for the span of a single voting window.

Kitzler et al. <cit.> study so-called DAO “contributors”, such as DAO owners and developers.
They find that contributors posses the majority of voting power in $7.5\%$ of the examined governance mechanisms, and have decided at least a single proposal in $20.4\%$ of them.
The authors uncover systemic evidence in support of Dotan et al. <cit.>, showing that in $14.8\%$ of the examined governance proposals, the volume of trades of governance tokens made in the $100$ days preceding each proposal exceeded the amount of votes required to obtain a majority in the corresponding proposal.

§.§ Corporate Governance

Theoretical Studies
Cvijanovic et al. <cit.> propose a theoretical model that captures the “free-rider effect”, wherein voters who have popular preferences are less inclined to vote.
The work shows that there is an equilibrium where all “underdog” voters vote and other voters only participate partially.

Dekel and Wolinsky <cit.> study the efficiency of the outcome of corporate control contests when allowing voting rights to be traded separately of shares, where an outcome was considered efficient if the actor that wins control of the corporation is the one that maximizes shareholder value.
The authors show that allowing vote trading always results in inefficient outcomes, and note that this contrasts with the intuition that in the political setting vote buying should be considered deterimental, while in the corporate setting it should enhance efficiency.

A method for assessing the value of shareholder voting rights was given by Kalay et al. <cit.>.
Their method compares the price of a share to a “synthesized” non-voting share, with the latter being constructed using put-call parity.
The method is then employed on real-world data, showing that the value of voting rights increases around special shareholder meetings and when voting on contentious mergers and acquisitions.

Empirical Studies
Brav et al. <cit.> conduct an study on retail shareholder voting, finding that turnout is higher for voters with larger holdings or when the expected benefits from voting are high, while turnout decreases when the cost of participation is high.
An analysis of shareholder voting in $17$ European countries between the years $2005$ and $2010$ was carried out by Renneboog and Szilagyi <cit.>.
The authors found that turnout by owners of free-float shares (that is, shares that can be publicly traded) was relatively low: $17\%$ in France, $10\%$ in Germany, and $4\%$ in Italy.
Furthermore, the authors note that shareholder proposals (proposals written by shareholders rather than by the management of a corporation) were mostly raised by shareholders of large and poorly performing corporations.
Additionally, dissenting votes were more frequent when the proposals advanced the adoption of anti-takeover devices and executive compensation.
A higher rate of dissent in proposals on remuneration was also found by Hewitt <cit.>, who studied OECD countries.

Li and Schwartz-Ziv <cit.> show that the daily volume, number of trades and the volatility of shares are increased around shareholder voting dates, particularly when the votes are considered important and when voting culminated in a dissenting outcome.
A similar finding that share prices decline after voting is given by Fos and Holderness <cit.>.
Additionally, the work presents data showing that corporations sometimes reveal voting dates after the fact, thereby preventing stockholders from exercising their right to vote.
We note that commonly used governance mechanisms do not allow such paractices, as proposals are announced publicly on the blockchain.

Shareholder Value Maximization (SVM)
SVM is a business goal which can be broadly defined as striving to increase the wealth of the business' shareholders.
Sundaram and Inkpen <cit.> propose SVM as the “preferred” goal for corporate finance, arguing it is ethical and easily measurable.
Freeman et al. <cit.> criticize SVM as it only focuses on shareholders and ignores corporate stakeholders, with the latter taken to mean any actor who can affect or be affected by the corporation.
Thus, shareholders are stakeholders, but not necessarily the other way around.
For example, employees of the corporation are stakeholders, even if they do not possess any shares.
Inkpen and Sundaram <cit.> discuss the relevance of SVM in the face of academic criticism and later developments in the corporate finance world, such as the supposed greater divergence in shareholder interests in recent years <cit.>.

§ GLOSSARY

A summary of all symbols and acronyms used in the paper.
alttree AAAA