$\mathcal{K}(\mathcal{W}_{\Omega})=\mathrm{pr}_{O}(\mathcal{K}(\mathcal{W}_{\varphi;\Omega}))=\smash{\textstyle\sum_{i\not\sim
j}}\,\kappa_{ij}\,dx_{i}\,dx_{j}$, which holds by an application of Theorem 15
to the natural connection of $\mathcal{W}_{\varphi;\Omega}$, leads to
$\kappa_{ij}=0$ for each $i\not\sim j$ and $p\in M$ via Lemma 30. This is
exactly condition (5). ∎

##  5 Applications in general relativity

In this section we consider divergence-free $2$-webs $\mathcal{W}_{dV}$
arising from a $4$-dimensional spacetime $(M,g)$ with metric $g$ of signature
$(\mathord{-}\mathord{+}\mathord{+}\mathord{+})$, its natural volume element
$dV$ and a single codimension-$1$ spacelike foliation $\mathcal{F}$ of $M$, as
defined below.

The foliation $\mathcal{F}$ is _spacelike_ if the restriction of $g$ to each
of its leaves is positive-definite. This means that the leaves of
$\mathcal{F}$ can be locally treated as _hypersurfaces of simultaneity_ in the
ambient spacetime. The assignment to each event $p\in M$ a plaque $L\ni p$ of
$\mathcal{F}$ in a certain coordinate system adapted to the foliation
$\mathcal{F}$ can be treated as a measurement of local time $t$ by some
observer; the plaque itself (roughly) represents the state of the world at
time $t$. The orthogonal complement $T\mathcal{F}^{\bot}$ of $T\mathcal{F}$
with respect to $g$ is a $1$-dimensional distribution, hence is integrable;
the corresponding foliation $\mathcal{F}^{\bot}$ is necessarily timelike and
can be treated as trajectories of a family of _Eulerian observers_. Let
$dV=\sqrt{-\det g}$ be the pseudo-Riemannian volume form of the metric $g$.
The divergence-free $2$-web $\mathcal{W}_{dV}$ in question is
$(M,\mathcal{F},\mathcal{F}^{\bot},dV)$.

The $\mathcal{W}_{dV}$-adapted coordinates are called _normal (Eulerian)
coordinates_ [11]. In these coordinates, the metric tensor splits into a sum

$g=-\alpha^{2}\,dt^{2}+\gamma,$ (5.1)

where the $t$-coordinate parametrizes the leaves of $\mathcal{F}$, $\alpha$ is
the _lapse function_ and $\gamma$ is a positive-definite metric on
$T\mathcal{F}$. Of course, according to circumstances, one can find
complementary foliations $\mathcal{G}$ which are better suited to the
situation at hand; the above choice is motivated mainly by the clarity of
exposition and can be altered without difficulty.

######  Example 8.

Consider the event horizon exterior of the Schwarzschild spacetime $(M,g)$ in
standard radial coordinates $(t,r,\theta,\phi)$, where
$M=\mathbb{R}^{4}\setminus\left\\{r\leq 2m\right\\}$ for some positive mass
$m$ and

$g=-(1-\frac{2m}{r})\,dt^{2}+(1-\frac{2m}{r})^{-1}\,dr^{2}+r^{2}\,d\theta^{2}+r^{2}\sin^{2}(\theta)\,d\phi^{2}.$
(5.2)

The hypersurfaces of constant time $\left\\{t=c\right\\}$ for $c\in\mathbb{R}$
define a foliation $\mathcal{F}$ of $M$. Its orthogonal complement
$\mathcal{F}^{\bot}$ is exactly the set of curves
$\left\\{r=c_{r},\theta=c_{\theta},\phi=c_{\phi}\right\\}$ for
$c_{r},c_{\theta},c_{\phi}\in\mathbb{R}$ by virtue of the form of $g$ in these
coordinates. We also have

$dV=r^{2}\sin\theta\cdot dt\wedge dr\wedge d\theta\wedge d\phi.$ (5.3)

Since the coordinate system is $\mathcal{W}_{dV}$-adapted, the divergence-free
$2$-web $\mathcal{W}_{dV}$ is trivial by Theorem 1.

######  Example 9.

The constant-$T$ foliation $\mathcal{F}$ corresponding to Gullstrand-Painlevé
coordinates $(T,r,\theta,\phi)$ on the Schwarzschild spacetime
$(\mathbb{R}\times(\mathbb{R}^{3}\setminus\left\\{0\right\\}),g)$, in which
the metric $g$ takes the form

$g=-(1-\frac{2m}{r})\,dT^{2}+2\sqrt{\frac{2m}{r}}\,dTdr+dr^{2}+r^{2}\,d\theta^{2}+r^{2}\sin^{2}(\theta)\,d\phi^{2},$
(5.4)

is spacelike and genreates a divergence-free $2$-web $\mathcal{W}_{dV}$ which
is not trivial. To see this, switch to a $\mathcal{W}_{dV}$-adapted coordinate
system $(T,R,\theta,\phi)$ with $R={\textstyle\frac{2}{3}}r^{3/2}+\sqrt{2m}T$
(known as Lemaître coordinate system) to bring $g$ into the split form
$(\ref{eq:dfw-split-metric})$

$g=-dT^{2}+r^{-1}\,dR^{2}+r^{2}\,d\theta^{2}+r^{2}\sin^{2}(\theta)\,d\phi^{2},\quad
r=({\textstyle\frac{3}{2}}R-{\textstyle\frac{3}{2}}\sqrt{2m}T)^{2/3}.$ (5.5)

The claim follows from Theorem 1, since the corresponding volume form

$dV={\textstyle\frac{3}{2}}(R-\sqrt{2m}T)\sin\theta\cdot dT\wedge dR\wedge
d\theta\wedge d\phi$ (5.6)

yields a nonvanishing nonuniformity tensor
$\mathcal{K}(\mathcal{W}_{dV})=\sqrt{2m}(R-\sqrt{2m}T)^{-2}\,dTdR$. Its form
in the initial Gullstrand-Painlevé coordinate system is

$\mathcal{K}(\mathcal{W}_{dV})={\textstyle\frac{9}{2}}mr^{-3}\,dT^{2}+{\textstyle\frac{9}{4}}\sqrt{2m}\,r^{-5/2}\,dTdr.$
(5.7)

Since $g(\textstyle{\smash{\frac{\partial}{\partial
T}}},\textstyle{\smash{\frac{\partial}{\partial T}}})=-1$, the variable $T$
can be interpreted as the proper time of Eulerian observers falling radially
into the Schwarzschild black hole. Near the singularity itself, the
nonuniformity tensor grows without bound.

The triviality of the induced divergence-free $2$-web has some relevance in
relativistic fluid dynamics, where the laws of motion of the fluid expressed
in $\mathcal{W}_{dV}$-adapted coordinates include the conservation of energy
$(\ref{eq:dfw-gr-ene})$ and momentum $(\ref{eq:dfw-gr-mom})$, both of which
depend on the spacetime volume element $dV$ in a nontrivial way. These laws,
expressed using the notation introduced in [2, Part 11] and the Einstein
summation convention, reduce to

$\nabla_{\alpha}T^{\alpha\beta}=0,$ (5.8)

for the stress-energy tensor $T$ bound by certain constitutive relations,
where $\nabla_{\alpha}$ denotes the Levi-Civita covariant differentiation. The
above condition on the divergence of $T$ results in appearance of the
Lorentzian volume density in the resulting equations. In normal coordinates,
assuming that in the tensors below the $0^{\text{th}}$ index corresponds to
time-like components, while the indices $i,j,k=1,2,3$ refer to the spatial
ones, the equations read [14]

$\displaystyle\begin{split}\textstyle{\smash{\frac{\partial}{\partial
t}}}(\alpha^{3}\gamma^{1/2}T^{00})+\textstyle{\smash{\frac{\partial}{\partial
x_{j}}}}(\alpha^{3}\gamma^{1/2}T^{0j})&=\gamma^{1/2}\alpha^{2}(\textstyle{\smash{\frac{\partial}{\partial
t}}}\alpha)T^{00}-{\textstyle\frac{1}{2}}\gamma^{1/2}\alpha(\textstyle{\smash{\frac{\partial}{\partial
t}}}g_{jk})T^{jk}\end{split}$ (5.9)
$\displaystyle\begin{split}\textstyle{\smash{\frac{\partial}{\partial
t}}}(\alpha^{3}\gamma^{1/2}T^{i0})+\textstyle{\smash{\frac{\partial}{\partial
x_{j}}}}(\alpha^{3}\gamma^{1/2}T^{ij})&={\textstyle\frac{1}{2}}\gamma^{1/2}\alpha^{3}g^{ik}(\textstyle{\smash{\frac{\partial}{\partial
x_{k}}}}\alpha^{2})T^{00}-\gamma^{1/2}\alpha^{3}g^{ik}(\textstyle{\smash{\frac{\partial}{\partial
t}}}g_{jk})T^{j0}\\\
&{}-\gamma^{1/2}\alpha^{3}\Gamma^{i}_{jk}T^{jk}-T^{0i}\textstyle{\smash{\frac{\partial}{\partial
t}}}(\gamma^{1/2}\alpha^{3})\\\
&{}+2T^{0i}\gamma^{1/2}\alpha^{2}(\textstyle{\smash{\frac{\partial}{\partial
t}}}\alpha)+2\alpha^{3}T^{ij}(\textstyle{\smash{\frac{\partial}{\partial
x_{j}}}}\gamma^{1/2}),\end{split}$ (5.10)

where $\alpha$ is the lapse function, $\Gamma_{jk}^{i}$ are the Christoffel
symbols and $\gamma^{1/2}$ is the coefficient of the spacelike volume element
on $T\mathcal{F}$ induced from $dV$.

If the nonuniformity tensor $\mathcal{K}(\mathcal{W}_{dV})$ of the web
$\mathcal{W}_{dV}$ defined in the introduction vanishes identically, then by
Theorem 1 one can pick a $\mathcal{W}_{dV}$-adapted coordinate system
$(t,x_{1},x_{2},x_{3})$ in which the density of the volume element
$dV=\alpha\gamma^{1/2}\,dt\wedge dx_{1}\wedge dx_{2}\wedge dx_{3}$ becomes
constant. This simplifies the above equations, completely eliminating their
dependence on $\gamma^{1/2}$. If we assume that $\alpha\gamma^{1/2}=1$, then
from $(\ref{eq:dfw-gr-ene})$ and $(\ref{eq:dfw-gr-mom})$ we obtain

$\displaystyle\begin{split}\textstyle{\smash{\frac{\partial}{\partial
t}}}(\alpha^{2}T^{00})+\textstyle{\smash{\frac{\partial}{\partial
x_{j}}}}(\alpha^{2}T^{0j})&={\textstyle\frac{1}{2}}(\textstyle{\smash{\frac{\partial}{\partial
t}}}\alpha^{2})T^{00}-{\textstyle\frac{1}{2}}(\textstyle{\smash{\frac{\partial}{\partial
t}}}g_{jk})T^{jk}\end{split}$ (5.11)
$\displaystyle\begin{split}\textstyle{\smash{\frac{\partial}{\partial
t}}}(\alpha^{2}T^{i0})+\textstyle{\smash{\frac{\partial}{\partial
x_{j}}}}(\alpha^{2}T^{ij})&={\textstyle\frac{1}{2}}\alpha^{2}g^{ik}(\textstyle{\smash{\frac{\partial}{\partial
x_{k}}}}\alpha^{2})T^{00}-\alpha^{2}g^{ik}(\textstyle{\smash{\frac{\partial}{\partial
t}}}g_{jk})T^{j0}\\\
&{}-\alpha^{2}\Gamma^{i}_{jk}T^{jk}-(\textstyle{\smash{\frac{\partial}{\partial
x_{j}}}}\alpha^{2})T^{ij}.\end{split}$ (5.12)

This form of the equations is more convenient to work with, and due to its
simplicity it might improve numerical accuracy if $\alpha^{2}$ and $g$ are
well behaved in the new coordinates. Given $\alpha^{2}$ and the other
components of the metric, one can avoid taking any square roots and
determinants in order to evolve the stress-energy tensor $T$ in time. The
equations can be simplified further if we are able to choose $\mathcal{F}$
such that $\mathcal{F}^{\bot}$ is _totally geodesic_. This amounts to the
equality $d\alpha_{|T\mathcal{F}}=0$ by a direct verification of the identity
$g(\nabla_{n}n,v)=d\log\alpha(v)$ for the unit normal $n\in
T\mathcal{F}^{\bot}$ with $\lVert n\rVert_{g}=-1$ and each $v\in
T\mathcal{F}$. In this setting, the leaves of the orthogonal foliation
$T\mathcal{F}^{\bot}$ are geodesics; a foliation $\mathcal{F}$ of this kind is
called a _geodesic slicing_ of the spacetime [11, 10.2].

One last thing to note is that the Levi-Civita connection $\nabla$ is _not_ a
$\mathcal{W}_{dV}$-connection in general (see Definition 4). It is so only if
both $\mathcal{F}$ and $\mathcal{F}^{\bot}$ are totally geodesic, in the sense
that for all $X,Z\in\Gamma(T\mathcal{F})$ and
$Y,W\in\Gamma(T\mathcal{F}^{\bot})$

$\nabla_{Z}X\in\Gamma(T\mathcal{F}),\qquad\qquad\nabla_{W}Y\in\Gamma(T\mathcal{F}^{\bot}).$
(5.13)

This condition is equivalent to $d\alpha_{|T\mathcal{F}}=0$ and
$K_{ij}=\frac{1}{2\alpha}\textstyle{\smash{\frac{\partial}{\partial
t}}}\gamma_{ij}=0$ for $i,j=1,2,3$. As a consequence, it is usually not
possible to determine local triviality of $\mathcal{W}_{dV}$ using the Ricci
tensor of the Levi-Civita connection alone, as the results of section 2.5 seem
to suggest, except in quite special geometries defined by the above two sets
of equations.

## References

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typical example. Pairwise classification and word-level classification are
also commonly seen in NLP.

  * •

Text classification: Please refer to the paper table mentioned in (§B) for
related works. We do not list them here since there are too many.

  * •

Pairwise classification: (Grießhaber et al., 2020; Bai et al., 2020; Mussmann
et al., 2020)

  * •

Word sense disambiguation (WSD): (Fujii et al., 1998; Chen et al., 2006; Chan
and Ng, 2007; Zhu and Hovy, 2007; Zhu et al., 2008c; Imamura et al., 2009;
Martínez Alonso et al., 2015)

##### Sequence labeling

is probably the most commonly seen structured prediction task in NLP. It aims
to predict a sequence of labels, among which there may be interactions and
constraints.

  * •

Part-of-speech (POS): (Engelson and Dagan, 1996; Ringger et al., 2007; Haertel
et al., 2008a; Marcheggiani and Artières, 2014; Fang and Cohn, 2017; Brantley
et al., 2020; Chaudhary et al., 2021)

  * •

(Named) entity recognition (NER/ER): (Shen et al., 2004; Culotta and McCallum,
2005; Kim et al., 2006; Settles and Craven, 2008; Tomanek and Hahn, 2009b;
Marcheggiani and Artières, 2014; Chen et al., 2015; Li et al., 2017; Shen et
al., 2018; Siddhant and Lipton, 2018; Erdmann et al., 2019; Chaudhary et al.,
2019; Chang et al., 2020; Brantley et al., 2020; Hazra et al., 2021; Shelmanov
et al., 2021; Radmard et al., 2021; Dossou et al., 2022)

  * •

Segmentation: (Ngai and Yarowsky, 2000; Sassano, 2002; Neubig et al., 2011; Li
et al., 2012b; Marcheggiani and Artières, 2014; Cai et al., 2021)

  * •

Natural language understanding (NLU): (Hadian and Sameti, 2014; Deng et al.,
2018; Peshterliev et al., 2019; Zhu et al., 2020)

##### Complex structure prediction

in this work denotes the structure prediction tasks that are more complex than
sequence labeling, and have _explicit_ connections (alignments) between inputs
and outputs. They usually aim to extract relational structures among input
elements.

  * •

Parsing: (Hwa, 2000; Tang et al., 2002; Baldridge and Osborne, 2003, 2004;
Hwa, 2004; Reichart and Rappoport, 2009; Sassano and Kurohashi, 2010; Atserias
et al., 2010; Mirroshandel and Nasr, 2011; Majidi and Crane, 2013; Flannery
and Mori, 2015; Li et al., 2016; Shi et al., 2021)

  * •

Semantic role labeling (SRL): (Roth and Small, 2006; Wang et al., 2017;
Ikhwantri et al., 2018; Siddhant and Lipton, 2018; Koshorek et al., 2019;
Myers and Palmer, 2021)

  * •

Coreference: (Gasperin, 2009; Miller et al., 2012; Laws et al., 2012; Zhao and
Ng, 2014; Sachan et al., 2015; Li et al., 2020; Espeland et al., 2020; Yuan et
al., 2022)

  * •

Relation-related: (Roth and Small, 2008; Bloodgood and Vijay-Shanker, 2009b;
Mirroshandel et al., 2011; Fu and Grishman, 2013; Cañizares-Díaz et al., 2021;
Mallart et al., 2021; Seo et al., 2022; Zhang et al., 2022a)

  * •

Event-related: (Cao et al., 2015; Shen et al., 2021; Lee et al., 2022)

  * •

Word alignment: (Ambati et al., 2010b, c; Rocha and Sanchez, 2013)

  * •

Entity alignment/resolution: (Kasai et al., 2019; Liu et al., 2021)

##### Generation

refers to the tasks that aim to generate a sequence of tokens. We
differentiate them from plain structured prediction tasks since there are
usually no explicit alignments between input and output sub-parts in the
supervision and such alignments are usually implicitly modeled, especially in
recent sequence-to-sequence neural models. MT is a typical generation task,
where we further separate traditional statistical machine translation (SMT)
and recent neural machine translation (NMT). We also include semantic parsing
here, since recent works usually cast it as a sequence-to-sequence generation
task.

  * •

SMT: (Eck et al., 2005; Haffari et al., 2009; Haffari and Sarkar, 2009;
Ananthakrishnan et al., 2010b; Bloodgood and Callison-Burch, 2010; Ambati et
al., 2010a; Ananthakrishnan et al., 2010a; González-Rubio et al., 2012; Rocha
and Sanchez, 2013; Logacheva and Specia, 2014a, b; Miura et al., 2016)

  * •

NMT: (Peris and Casacuberta, 2018; Liu et al., 2018b; Zhang et al., 2018; Zeng
et al., 2019; Zhao et al., 2020b; Hu and Neubig, 2021; Gupta et al., 2021;
Zhou and Waibel, 2021; Hazra et al., 2021; Mendonça et al., 2022)

  * •

Semantic parsing: (Duong et al., 2018; Ni et al., 2020; Sen and Yilmaz, 2020)

  * •

Others: (Mairesse et al., 2010; Deng et al., 2018; Tsvigun et al., 2022a)

##  Appendix B Surveying Process

In this section, we provide more details of our surveying process:

  * •

For the ACL Anthology, we search for papers with the keyword “active” in
titles (by grepping the “Full Anthology BibTeX
file”555https://aclanthology.org/anthology.bib.gz). There can be related
papers that are missed from this simple keyword search, but as we read along
the filtered list, we gradually include the notable missing ones.

  * •

We also include papers outside the ACL Anthology. First, we look for papers by
searching with the key phrase “active learning” on Arxiv (in the field of
cs.CL, excluding those already appearing in ACL Anthology). Moreover, we also
collect related works in other venues, such as AI/ML conferences and journals.
For these venues, we do (can) not perform extensive searches due to high
volume (and that many are unrelated to our focus on NLP). We mainly collect
related papers in these adjacent venues by following the references from the
papers already surveyed.

We also create a table for the related papers (with detailed categorizations),
which can be found at this link:
https://github.com/zzsfornlp/zmsp/blob/main/msp2/docs/al4nlp/readme.md.


coefficients, most of which are zero, it might be possible to use the cuts to
construct only the relevant graphs and non-zero coefficients with which they
appear.

Although we have focused on 4 points, the all-in-one-point cut and more
general loop-loop cuts are largely independent of the number of points, and
also the MHV degree, since they involve only the mutual geometry between loop
lines rather than the details of the external geometry. There has been some
nice recent progress in computing the amplitude for arbitrary multiplicity
directly from the loop amplituhedron [26, 27, 14, 28]. Points worthy of note
in the current context are that taking the maximal multiple residues involving
loops at higher points yields leading singularities of amplitudes – rational
coefficients – which have been extensively analysed and are given by Yangian
invariant Grassmann integrals [9, 29]. Furthermore in the works [4, 5] a
method for extracting a list of the physical amplitude’s branch points from
the amplituhedron is suggested. Here the boundaries are derived by
intersecting the closure of the amplituhedron with the boundary components
corresponding to vanishing brackets of the form $\langle ABij\rangle,\langle
A_{i}B_{i}A_{j}B_{j}\rangle$. It would be extremely interesting to revisit
both the above points using the insights and technology developed here.

Unlike positive geometries, the space of generalised positive geometries is
closed under union: regions that are triangulated by GPGs will be GPGs. This
then suggests they might be a better arena to seek a direct characterisation –
that is a simple straightforward answer to the question ‘is the following
region a GPG or not’ – than PGs. In particular it seems clear that all multi-
linear geometries are GPGs, and their canonical form can be computed
algorithmically via cylindrical decomposition. Nicely all Grassmannian-type
geometries, the type seen in physics, are multi-linear geometries and thus
GPGs. But this is not the most general class and the most general
classification needs more investigation.

The generalisation of positive geometry to include internal boundaries also
suggests the utility of a further generalisation to weighted positive
geometries where instead the weights contain the information about internal
boundaries, and the canonical form has a much more natural definition (36).
This concept also makes certain proofs very direct since one can add two WPGs
together without first making sure they are disjoint (see eg the proof of
triangulations of positive geometries in section 3.5).

In [2, 30] the geometry of the log of the MHV amplitude is considered and
defined as a union of geometries with negative mutual positivity condition
$\langle A_{i}B_{i}A_{j}B_{j}\rangle<0$. One of these has negative mutual
positivity condition for all $i,j$ and its canonical form is equal to the
amplituhedron canonical form. The latter is the only term in the log of the
amplitude surviving the all-in-one-point cut and therefore the all-in-one-
point cut of the amplitude and the log of the amplitude are the same. The log
of amplitude should most naturally be described by a WPG.

We believe that the above insights will also have utility in the increasing
number of wider applications of positive geometry concepts in physics beyond
the amplituhedron. One closely related case is the momentum amplituhedron. For
the tree-level momentum amplituhedron [31] a lot is known about its boundary
stratification [32, 33] and its Euler characteristic has been proven to be
equal to one [34], a strong indication that the geometry is free from internal
boundaries. However, despite the very solid understanding achieved at tree
level, finding the geometry of the loop momentum amplituhedron remains an open
problem. In this case we expect internal boundaries to appear and the language
of WPGs could give the right framework to define a loop momentum
amplituhedron. Also in the search for a non-planar amplituhedron interesting
ideas involving the sum of geometries over different orderings have been
explored in [35, 36] and might benefit from being viewed as weighted positive
geometries. Other wider applications of positive geometry which one could
revisit include [37, 38, 39, 40, 41, 42, 43, 44]. Similarly weighted positive
geometry may provide the right mathematical framework to deal with
cosmological correlators, which contrarily to the wavefunction of the universe
described by the cosmological polytopes [45, 46]s, do not currently have a
geometrical description. Their maximal residues are not +/- 1 and they naively
appears as a weigheted sum of canonical forms of cosmological polytopes (P.
Benincasa, private communication).

## Acknowledgments

We thank Nima Arkani-Hamed, Paolo Benincasa, Tomasz Lukowski and Jaroslav
Trnka for helpful discussions. GD and PH acknowledge support from the European
Union’s Horizon 2020 research and innovation programme under the Marie
Skłodowska-Curie grant agreement No. 764850 “SAGEX”. GD would like also to
thank Iñaki García Etxebarria for many valuable and stimulating discussions.
PH also acknowledge support from the Science and Technology Facilities Council
(STFC) Consolidated Grant ST/P000371/1. AS acknowledges support from the
Engineering and Physical Sciences Research Council (EPSRC) Doctoral Training
Partnership 2021-22.

##  Appendix A Inducing orientations on boundary manifolds

Here we would like to review how given an oriented manifold one can derive the
induced orientation of the boundary. An oriented manifold $\mathcal{M}$ is
oriented if it posses a continuous top differential form $\mathcal{O}$ that is
always non-vanishing in $\mathcal{M}$. The orientation is then equivalently
defined by this volume form modulo positive scaling. Suppose then
$\mathcal{M}$ has a boundary $\partial\mathcal{M}$ of codimension one. The
orientation $\omega_{\partial}$ induced by $\mathcal{O}$ on
$\partial\mathcal{M}$ will be the projection of $\mathcal{O}$ on
$\partial\mathcal{M}$. We will now define what we mean by projection. Let the
boundary be defined by $f=0$ with $f>0$ inside and $f<0$ outside the region
(at least near by). Then $\mathcal{O}_{\partial}$ is defined simply as

$\displaystyle
df\wedge\mathcal{O}_{\partial}=\mathcal{O}|_{\partial\mathcal{M}}\;\ .$ (106)

Note that the standard convention in math literature differ by a sign, that is
$df$ represents an outward pointing differential. We make this choice so that
the segment $x>0$ with positive orientation form $O(x)=\text{d}x$ has
orientation $O|_{x=0}=1$. In this way the canonical form of the positively
oriented segment is $\Omega(x>0,\text{d}x)=\frac{\text{d}x}{x}$, in fact this
form satisfies

$\displaystyle\text{Res}_{x=0}(\frac{\text{d}x}{x})=O|_{x=0}=1\;.$ (107)

##  Appendix B Relation of generalised positive geometries to the globally
oriented canonical form

The globally oriented canonical form was defined in [11] to obtain the square
of the super amplitude out of the geometry of the squared amplituhedron. The
interior of the squared amplituhedron consists of several disconnected
components so the first problem is to define the relative orientation of these
components. The second problem is that the square of the super-amplitude has
non-normalizable maximal residues and therefore can not be possibly
interpreted as a canonical form of a positive geometry. We solved both
problems by noticing that the GCD algorithm (see [20, 11]) correctly
reproduces the canonical form of the squared amplituhedron. The GCD algorithm
works in coordinates, that is on a $\mathbb{R}^{km}$ patch of the oriented
Grassmannian using the tiling property of canonical forms. The orientation of
the subregions given by the GCD are fixed by the algorithm as the global
orientation of the coordinate patch. Then it gives the result as the sum of
the canonical forms of the positive geometries triangulating the region. We
then defined the oriented canonical form as the sum of the canonical form of
the regions triangulating the squared amplituhedron.

With the new concept we introduced in this paper, we can simply say then that
the squared amplituhedron is a GPG with the orientation fixed by the global
orientation of the oriented Grassmannian. The canonical form of a generalized
positive geometry $X_{\geq 0}$ in an orientable space $X$ with orientation
coinciding with the orientation of $X$ is equal to its globally oriented
canonical form.

##  Appendix C Three loop internal Boundary and its maximal residues

Here we consider taking further residues of the all-in-one-point cut (4.1) to
eventually arrive at the leading singularities. Then we will consider the same
sequence geometrically. Indeed in [21] such maximal residues were considered
leading to a final configuration in which all loop lines intersect external
twistors as well as intersecting each other at a single point $A$.

Specifically we consider the case where loop line $L_{1}$ intersects $Z_{1}$
and $L_{2},L_{3}$ intersect $Z_{2}$. We here show that the resulting residue
depends on the path taken. Furthermore if the path taken involves taking a
residue in $\langle AB_{1}B_{2}B_{3}\rangle=0$ first, then the resulting
maximal residue has magnitude 2 suggesting that $\langle
AB_{1}B_{2}B_{3}\rangle=0$ corresponds to an internal boundary.

The two routes we consider to reach the above configuration are as follows.
For route 2 we first take further residues of the all-in-one-point cut (4.1)
in the following order

$\langle AB_{1}12\rangle=0,\;\;\langle AB_{1}14\rangle=0\;,\;\;\langle
AB_{2}12\rangle=0,\;\;\langle AB_{2}23\rangle=0\;.$ (108)

This corresponds to intersecting line $L_{1}$ with the edge $Z_{1}Z_{4}$ and
then with $Z_{1}$ followed by intersecting $L_{2}$ with the edge $Z_{1}Z_{2}$
and then to $Z_{2}$. In the process the pole $\langle
AB_{1}B_{2}B_{3}\rangle\rightarrow\langle A12B_{3}\rangle$ and so the final
step is to take residues in this pole $\langle A12B_{3}\rangle=0$ followed by
$\langle A23B_{3}\rangle=0$ corresponding to $L_{3}$ intersecting $Z_{1}Z_{2}$
and then sliding to $Z_{2}$. We take the residues explicitly by parametrizing
the $B_{i}$ as follows (with $Z_{*}$ an arbitrary twistor)

$\begin{split}B_{1}&=c_{1}Z_{4}+Z_{1}+d_{1}Z_{*}\;,\\\
B_{2}&=c_{2}Z_{1}+Z_{2}+d_{2}Z_{*}\;,\\\
B_{3}&=c_{3}Z_{1}+Z_{2}+d_{3}Z_{*}\;.\end{split}$ (109)

and considering the residues at zero in $d_{1},c_{1},d_{2},c_{2},d_{3},c_{3}$
in that order. The residues are straightforward to compute and can be done
covariantly, for example:

$\begin{split}\underset{d_{1}=0,\,c_{1}=0}{\text{Res}}\;\left(\frac{\langle
AB_{1}\text{d}^{2}B_{1}\rangle}{\langle AB_{1}14\rangle\langle
AB_{1}12\rangle}\right)&=\frac{-1}{\langle A214\rangle}\;.\end{split}$ (110)

Only the displayed term in (67) out of the 24 total terms survives this
sequence of residues and it produces the final result

$-\frac{\langle 1234\rangle^{3}\langle A\text{d}^{3}A\rangle}{\langle
A123\rangle\langle A124\rangle\langle A134\rangle\langle
A234\rangle}=-\frac{\text{d}a\text{d}b\text{d}c}{abc}\;,$ (111)

where on the right hand side we parametrise the point $A$ as

$A=Z_{1}+aZ_{2}+bZ_{3}+cZ_{4}\ .$ (112)

This is the canonical form of a tetrahedron with vertices $Z_{i}$ and is
inline with the prediction of [21]. Indeed the mutual intersection point $A$
is now the only remaining freedom and restricting the amplituhedron geometry
to this configuration results in the tetrahedron. However as we saw in the
simple example in the introduction, simply restricting the geometry to a high
codimension boundary will not always give the right answer and the precise
order in which one takes the residues can be important.

For route 2 therefore we change the order in which we take the residues on the
all-in-one-point cut (4.1). We first take a residue in the pole $\langle
AB_{1}B_{2}B_{3}\rangle$ making $L_{3}$ coplanar with $L1$ and $L2$. Then
proceed taking residues as previously in (108) moving $B_{1}\rightarrow Z_{1}$
and $B_{2}\rightarrow Z_{2}$. Then finally we take a residue as
$B_{3}\rightarrow Z_{2}$. Explicitly then, this time we parametrize the
$B_{i}$ as follows

$\begin{split}B_{3}&=c_{3}B_{1}+B_{2}+d_{3}Z_{*}\;\\\
B_{1}&=c_{1}Z_{4}+Z_{1}+d_{1}Z_{*}\;,\\\
B_{2}&=c_{2}Z_{1}+Z_{2}+d_{2}Z_{*}\;.\end{split}$ (113)

and take the residues at zero in the order
$d_{3},d_{1},c_{1},d_{2},c_{2},c_{3}$. The first residue is

$\underset{\langle AB_{1}B_{2}B_{3}\rangle=0}{\text{Res}}\;\left(\frac{\langle
AB_{3}\text{d}^{2}B_{3}\rangle}{\langle
AB_{1}B_{2}B_{3}\rangle}\right)=-\text{d}c_{3}\;,$ (114)

and then use similar results to (110) before finally taking the residue in the
parameter $c_{3}$. This time two terms in (67) survive, the displayed term
together with the term

$\frac{\langle A\text{d}^{3}A\rangle\prod_{i=1}^{3}\langle
AB_{i}\text{d}^{2}B_{i}\rangle\langle 1234\rangle^{3}\left\langle
23AB_{1}\right\rangle}{\left\langle 12AB_{1}\right\rangle\left\langle
12AB_{3}\right\rangle\left\langle 14AB_{1}\right\rangle\left\langle
23AB_{2}\right\rangle\left\langle 23AB_{3}\right\rangle\left\langle
34AB_{1}\right\rangle\left\langle 34AB_{2}\right\rangle\langle
AB_{1}B_{2}B_{3}\rangle}\ .$ (115)

Note that this term only survives because a required pole $\langle
12AB_{2}\rangle$ appears from the pole in $\langle 12AB_{3}\rangle$ after the
residues in $d_{3}=0,d_{1}=0,c_{1}=0$ have been taken. Since there are two
terms now surviving, the final result turns out to be twice (111)

$-\frac{2\langle 1234\rangle^{3}\langle A\text{d}^{3}A\rangle}{\langle
A123\rangle\langle A124\rangle\langle A134\rangle\langle
A234\rangle}=-2\frac{\text{d}a\text{d}b\text{d}c}{abc}\;.$ (116)

Now the final configuration of these two routes is exactly the same in both
cases: $B_{1}\rightarrow Z_{1},B_{2},B_{3}\rightarrow Z_{2}$ and yet the
results differ by a factor of 2. We thus clearly see the importance of path
dependence when taking residues. We will shortly see that path dependence can
give different results for the all-in-one-point cut itself (rather than just
when taking further residues). Furthermore, taking three further residues as
$a,b,c\rightarrow 0$ clearly gives us a maximal residue of magnitude 2
indicating that there is an internal boundary present. Let us then consider
this geometrically.

We can now redo the above computation geometrically by taking boundaries of
the all-in-one-point cut geometry (77) and using our formula of the canonical
form of a GPG (22). To do this, we parametrise $A$ and $B_{i}$ just as in
(112) and (113)

$\begin{split}B_{3}&=c_{3}B_{1}+B_{2}+d_{3}Z_{*}\;\\\
B_{1}&=c_{1}Z_{4}+Z_{1}+d_{1}Z_{*}\;,\\\
B_{2}&=c_{2}Z_{1}+Z_{2}+d_{2}Z_{*}\;,\\\
A\,&=Z_{1}+aZ_{2}+bZ_{3}+cZ_{4}\;\end{split}$ (117)

Then take boundaries in the same order with which we took residues following
(116)

$d_{3}=0,\;\;\;d_{1}=0,\;\;\;c_{1}=0,\;\;\;d_{2}=0,\;\;\;c_{2}=0,\;\;\;c_{3}=0\
.$ (118)

The first boundary corresponds to the above internal boundary $\langle
AB_{1}B_{2}B_{3}\rangle=0$. Therefore, from the recursive definition of the
canonical form in the presence of internal boundaries (22), the canonical form
of the all-in-one-point cut geometry satisfies

$\lim\limits_{d_{3}\to
0}d_{3}\big{(}\Omega(\mathcal{R}^{\text{dc}}_{1})+\Omega(\mathcal{R}^{\text{dc}}_{2})\big{)}=2\text{d}d_{3}\wedge\Omega(\mathcal{R}^{\text{dc}}\rvert_{\langle
AB_{1}B_{2}B_{3}\rangle=0}).$ (119)

This simply means that the residue, $d_{3}=0$, on the canonical form
determined by the inequalities describing the geometry of the all-in-one-point
cut (74), is equal to twice the canonical form of the interior boundary, i.e.
the canonical form determined by the inequalities (70) with
$B_{3}=B_{1}+a_{3}B_{2}$.

Continuing with the remaining boundaries, the final geometry is described by
the following set of inequalities,

$\mathcal{R}^{\text{final}}:\;\;\;a<0,\;\;\;b>0,\;\;\;c>0\;.$ (120)

The remaining residues were all on external boundaries, therefore the
canonical form of the final region is $2\Omega(\mathcal{R}^{\text{final}})$,
where the factor of two comes from the internal boundary residue (119). The
final inequalities describe a tetrahedron with vertices
$\\{Z_{1},-Z_{2},Z_{3},Z_{4}\\}$ and therefore the final canonical form is in
precise agreement with (111).

We see that the internal boundary at $\langle AB_{1}B_{2}B_{3}\rangle=0$ is
key to obtaining the correct leading singularity from the geometry. Just as we
saw algebraically above, it is also possible to reach the same final loop
configuration by only going to consecutive external boundaries. An example of
this would be to follow the residues described in (109) geometrically. Then
the final canonical form would be, up to an overall sign, the canonical form
of the tetrahedron without the factor of two, as predicted in [21]. We see
that the precise sequence of codimension 1 boundaries taken to approach higher
codimension boundaries starting from the all-in-one-point cut configuration
can give different results. At higher loops this is also true for the all-in-
one-point cut itself (rather than just its maximal residues as here).

##  Appendix D Four loop point-plane boundary geometry

We here examine the geometry of the loop-loop boundaries of the four loop all-
in-one-point cut (82). We start by the boundary when $B_{3}$ lies on the line
$P=B_{1}B_{2}$ followed by the boundary of that geometry when $B_{4}$ also
approaches $P$. We observe that the regions $\mathcal{R}_{1}$ and
$\mathcal{R}_{2}$ of (82) touch on $\langle AB_{2}B_{1}B_{3}\rangle=0$ as do
$\mathcal{R}_{3}$ and $\mathcal{R}_{4}$. Since the orientations of
$\mathcal{R}_{1}$ and $\mathcal{R}_{2}$ are opposite as are $\mathcal{R}_{3}$
and $\mathcal{R}_{4}$, this is an internal boundary. Thus the geometry of the
(internal) amplituhedron boundary $\langle AB_{2}B_{1}B_{3}\rangle=0$, with
$B_{3}$ living on $B_{2}B_{1}$, is given by
${\mathcal{R}}_{12}\cup{\mathcal{R}}_{34}$ where

$\begin{split}\mathcal{R}_{12}&\ =\ {\cal A}^{(4)}_{dc}\;\;\land\;\langle
AB_{2}B_{1}B_{4}\rangle>0\land\langle AB_{3}B_{1}B_{4}\rangle>0|_{B_{3}\in
B_{2}B_{1}},\ +\;,\\\ \mathcal{R}_{34}&\ =\ {\cal
A}^{(4)}_{dc}\;\land\;\langle AB_{2}B_{1}B_{4}\rangle<0\land\langle
AB_{3}B_{1}B_{4}\rangle<0|_{B_{3}\in B_{2}B_{1}},\ -\;.\end{split}$ (121)

Here $\mathcal{R}_{12}$ is positively oriented (indicated by the $+$), while
$\mathcal{R}_{34}$ is negatively oriented. Notice that both of these regions
require $B_{1}$ to be on the same side of $B_{2}$ and $B_{3}$. In other words
$B_{1}$ can not lie between $B_{2}$ and $B_{3}$. Further the orientation
depends which side of $B_{2},B_{3}$, $B_{1}$ is on. Then after we send $B_{4}$
to the line $P$, we approach another internal boundary, with no further
constraints on where $B_{4}$ can lie. We thus conclude that the geometry is of
the four points $B_{i}$ on the line $P$ inside the triangle, with $B_{1}$ not
allowed between $B_{2}$ and $B_{3}$ and the orientation depending on the
relative position of $B_{1}$ and $B_{2},B_{3}$.

We can check the geometry more carefully by exploring the boundaries of (121)
explicitly. To do this we make the constraint $\langle
AB_{1}B_{2}B_{3}\rangle=0$ explicit by expanding $B_{1}$ as
$c_{1}B_{2}+c_{2}B_{3}$. Then (121) becomes

$\begin{split}\mathcal{R}_{12}&\,=\,{\cal A}^{(4)}_{dc}\;\land\Big{(}\langle
AB_{2}B_{3}B_{4}\rangle>0\land c_{1}<0\land c_{2}>0\Big{)}\lor\Big{(}\langle
AB_{2}B_{3}B_{4}\rangle<0\land c_{1}>0\land c_{2}<0\Big{)},\ +\;,\\\
\mathcal{R}_{34}&\,=\,{\cal A}^{(4)}_{dc}\;\land\;\Big{(}\langle
AB_{2}B_{3}B_{4}\rangle>0\land c_{1}>0\land c_{2}<0\Big{)}\lor\Big{(}\langle
AB_{2}B_{3}B_{4}\rangle<0\land c_{1}<0\land c_{2}>0\Big{)},\ -\;.\end{split}$
(122)

and we see this has two external boundaries, $c_{1}=0$ and $c_{2}=0$, and one
internal boundary at $\langle AB_{2}B_{3}B_{4}\rangle=0$. The two external
boundaries correspond to sending $B_{1}\to B_{3}$ and $B_{1}\to B_{2}$
respectively. The limit internal boundary $\langle AB_{2}B_{3}B_{4}\rangle\to
0^{\pm}$ instead corresponds to sending $B_{4}$ to the line $P$. In this case
the geometry is described by the union of a positively oriented region
$\mathcal{R}^{+}_{1234}$ and a negatively oriented region
$\mathcal{R}^{-}_{1234}$ as

$\begin{split}\mathcal{R}^{+}_{1234}={\cal A}^{(4)}_{dc}\land c_{1}<0\land
c_{2}>0,\ +\;,\\\ \mathcal{R}^{-}_{1234}={\cal A}^{(4)}_{dc}\land c_{1}>0\land
c_{2}<0,\ -\;.\end{split}$ (123)

We clearly see then recalling $B_{1}=c_{1}B_{2}+c_{2}B_{3}$ that $B_{1}$
cannot line in between the point $B_{2}$ and $B_{3}$ and the orientation
depends on which side of $B2,B_{3}$ $B_{1}$ is on, with the position of
$B_{4}$ unconstrained.

This also reveals that there are still further loop-loop type boundaries we
could take even after doing the point-plane-cut. We could take a residue at
$c_{2}=0$ or $c_{3}=0$ corresponding to $B_{1}=B_{2}$ or $B_{3}$. The geometry
we then obtain corresponds to the 3-loops maximal loop-loop cut ${\cal
A}^{(3)}_{\text{mll}}$ but with weight 4 instead of 2 since this time we
approached two internal boundaries that is $\langle AB_{2}B_{1}B_{3}\rangle=0$
and $\langle AB_{2}B_{3}B_{4}\rangle=0$. This implies that the residue
corresponding to this 4-loop boundary is equal to 2 times the (119) all-in-
one-point-and-plane 3-loop residue.

Let’s now go back to the all-in-one-point cut (82) and explore the only other
boundary (modulo permutation of the loop variables), at $\langle
AB_{2}B_{1}B_{4}\rangle=0$. Notice that this time it is an external boundary
since the 4 regions remain distinct. To be very explicit, we can approach the
boundary by parametrizing $B_{1}$ as $c_{1}B_{2}+c_{2}B_{4}+c_{3}Z^{*}$ and
then taking the limit $c_{3}\to 0^{\pm}$ on (82), which becomes

$\begin{split}\mathcal{R}_{1}&\ =\ {\cal A}^{(4)}_{dc}\;\land\;c_{2}\langle
AB_{2}B_{3}B_{4}\rangle>0\;\land-c_{1}\langle AB_{2}B_{3}B_{4}\rangle>0,\
+\;,\\\ \mathcal{R}_{2}&\ =\ {\cal A}^{(4)}_{dc}\;\land\;c_{2}\langle
AB_{2}B_{3}B_{4}\rangle<0\;\land-c_{1}\langle AB_{2}B_{3}B_{4}\rangle>0,\
-\;,\\\ \mathcal{R}_{3}&\ =\ {\cal A}^{(4)}_{dc}\;\land\;c_{2}\langle
AB_{2}B_{3}B_{4}\rangle>0\;\land-c_{1}\langle AB_{2}B_{3}B_{4}\rangle<0,\
+\;,\\\ \mathcal{R}_{4}&\ =\ {\cal A}^{(4)}_{dc}\;\land\;c_{2}\langle
AB_{2}B_{3}B_{4}\rangle<0\;\land-c_{1}\langle AB_{2}B_{3}B_{4}\rangle<0,\
-\;,\end{split}$ (124)

where the region 1 and 3 are positively oriented and 2 and 4 are negatively
oriented. Here the geometry looks very similar to (122), but this time
$c_{1}=0$ is unconstrained while $c_{2}$ is an external boundary. We can see
that $c_{1}$ is free by taking the union of $\mathcal{R}_{1}$ and
$\mathcal{R}_{3}$ and expanding the products into the different sign cases.
The same goes for the pair $\mathcal{R}_{2},\mathcal{R}_{4}$ . Because of this
we can actually rewrite the (124) as

$\begin{split}\mathcal{R}_{13}&\ =\ {\cal A}^{(4)}_{dc}\;\land\;c_{2}\langle
AB_{2}B_{3}B_{4}\rangle>0,\ +\;,\\\ \mathcal{R}_{24}&\ =\ {\cal
A}^{(4)}_{dc}\;\land\;c_{2}\langle AB_{2}B_{3}B_{4}\rangle<0,\ -\end{split}$
(125)

This in turn then has two boundaries, $\langle AB_{2}B_{1}B_{3}\rangle=0$ and
$c_{2}=0$. Setting $\langle AB_{2}B_{1}B_{3}\rangle=0$ obtained by sending
$B_{3}$ to $P$ is described by the union of a positively oriented region and a
negatively oriented region as

$\begin{split}\mathcal{R}^{+}_{1234}={\cal A}^{(4)}_{dc}c_{2}>0,\ +\;,\\\
\mathcal{R}^{-}_{1234}={\cal A}^{(4)}_{dc}c_{2}<0,\ -\;.\end{split}$ (126)

This is the geometry of unconstrained points $B_{i}$ with orientation
depending on the relative order of $B_{1},B_{2}$.

This point-plane configuration has a further (internal) loop-loop boundary at
$c_{2}=0$ corresponding to the limit $B_{1}\to B_{2}$ . This boundary then
consists of three unconstrained points on the line $B_{2},B_{3},B_{4}$. This
final configuration thus corresponds to the 3-loop maximal cut ${\cal
A}^{(3)}_{\text{mll}}$ with weight 2.

##  Appendix E An all loop point-plane geometry

We here describe in detail the geometry corresponding to the specific all loop
point-plane cut described in section 5.4.

We first take the simplest all-in-one-point cut boundary. In this all loops
first intersect the line $A_{1}B_{1}$ and then they all slide to the same
intersection point in the same order as their labeling. At $L$ loops this is
given by the inequalities (see (E))

$\displaystyle\mathcal{R}^{\text{simplest
dc}}=\bigcup_{\vec{s}}\mathcal{R}^{\text{simplest dc}}_{\vec{s}}$
$\displaystyle\mathcal{R}^{\text{simplest dc}}_{\vec{s}}={\mathcal{A}_{\text{
dc}}}\land\;\left(\bigwedge\limits_{a=3}^{L}\bigwedge\limits_{2<i<a}\left\\{s_{a}\langle
AB_{i}B_{1}B_{a}\rangle>0\right\\}\right)\;\qquad\text{orientation}=\prod_{a}s_{a}$
(127)

After taking the above all-in-one-point cut we then constrain all loops to lie
in the same plane by taking the ordered series of boundaries $\\{\langle
AB_{2}B_{1}B_{L}\rangle=0,\langle AB_{2}B_{1}B_{L-1}\rangle=0,\cdots,\langle
AB_{2}B_{1}B_{3}\rangle=0\\}$. To do this, we will parametrize all loops, but
$B_{1}$ and $B_{2}$, as $B_{a}=B_{1}+b_{a}B_{2}+c_{a}Z^{*}$ and we will take
the limit $c_{a}\to 0^{\pm}$. We start by approaching the boundary $\langle
AB_{2}B_{1}B_{L}\rangle$. What we obtain after this first limit is that for
all $i$

$\displaystyle s_{L}\langle AB_{i}B_{1}B_{L}\rangle>0\to s_{L}b_{L}\langle
AB_{i}B_{1}B_{2}\rangle=-s_{L}s_{i}b_{L}<0\;.$ (128)

Since this inequality must hold for all $i$ we have that all $s_{i}$, apart
from $s_{L}$ must be equal and we can therefore define a single sign $s$ as

$\displaystyle s:=s_{L-1}=\cdots=s_{3}\;.$ (129)

Moreover, we can see that $b_{L}$ is actually unconstrained. In fact, since
for $s_{L}>0$ and $s_{L}<0$ we have the same orientation, (128) reduces to
$b_{L}>0\lor b_{L}<0$. The boundary geometry is then given by

$\displaystyle\mathcal{R}^{\text{simplest dc}}|_{\langle
A,B_{2}B_{1}B_{L}\rangle=0}=\mathcal{R}^{\text{simplest dc}}_{s=1}|_{\langle
A,B_{2}B_{1}B_{L}\rangle=0}\cup\mathcal{R}^{\text{simplest
dc}}_{s=-1}|_{\langle A,B_{2}B_{1}B_{L}\rangle=0}$
$\displaystyle\mathcal{R}^{\text{simplest dc}}_{s}|_{\langle
A,B_{2}B_{1}B_{L}\rangle=0}={\mathcal{A}_{\text{dc}}}\land\;\left(\bigwedge\limits_{a=3}^{L-1}\bigwedge\limits_{2<i<j}\left\\{s\langle
AB_{i}B_{1}B_{a}\rangle>0\right\\}\right)\;\qquad\text{orientation}=s^{L-4}\;.$
(130)

Now let’s take the residue $\langle A21B_{L-1}\rangle=0$ by taking the limit
$c_{L}\to 0^{\pm}$. What we obtain is that

$\displaystyle s\langle AB_{i}B_{1}B_{L-1}\rangle>0=s\langle
AB_{i}B_{1}B_{2}\rangle b_{L-1}>0=b_{L}<0\;.$ (131)

This implies that $b_{L-1}\to 0^{-}$ corresponds to an external boundary of
the geometry. Notice also that now the orientation of the two components of
the geometry labeled by $s=1$ and $s-1$ will be given by $s^{L-5}$. The series
of boundaries $\langle AB_{2}B_{1}B_{a}\rangle=0$ have a clear recursive
structure such that at each step $a$ we get an inequality of the form
$b_{a}<0$ and the orientation of the two components is equal to $s^{L-a-3}$.
The recursion ends when we get to the last residue on $\langle
AB_{1}B_{2}B_{3}\rangle=0$, for which no brackets of the form $\langle
AB_{i}B_{1}B_{3}\rangle$ are present. At that point the two components
$\langle AB_{1}B_{2}B_{3}\rangle>0$ and $\langle AB_{1}B_{2}B_{3}\rangle<0$
have the same boundary and opposite orientation and therefore this represents
an internal boundary. We can conclude that the geometry of the simplest all-
in-one-plane-and-point cut corresponds to

$\displaystyle\mathcal{R}^{\text{simplest dc}}_{s,L}|_{\bigwedge_{a}\langle
A,B_{2}B_{1}B_{a}\rangle=0}={\mathcal{A}_{\text{
dc}}}\land\;\left(\bigwedge\limits_{a=4}^{L-1}b_{a}<0\right)\;,$ (132)

where the weight of geometry is equal to 2 due to the internal boundary
$\langle AB_{1}B_{2}B_{3}\rangle=0$ contribution.

At this point it’s straight forward to compute the canonical form of this
region for the 4-point MHV amplitude. The inequalities $b_{a}<0$ for
$a=4,\cdots,L-1$ simply tell us that on the line $P$ all $B_{a}$ must be on
the same side of $B_{1}$ as $B_{2}$. So the full canonical form is as given in
(105).

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