# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.
 
# <math>\underline\ell = (\ell_e)_{e \in E^{\text{Morse}}}</math> is a tuple of edge lengths with <math>\ell_e \in [0,1]</math>, where the '''Morse edges''' <math>E^{\text{Morse}}</math> are those satisfying <math>P(e) \subset A_j</math> for some <math>j</math>. Moreover, for a Morse edge <math>e \in E^{\text{Morse}}</math> incident to a critical vertex, we require <math>\ell_e = 1</math>.
 
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order.  Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.
 
# For every main vertex <math>v \in V^m</math> and each edge <math>e \in E(v)</math>, <math>x_{v,e}</math> is a marked point in <math>\partial D</math> such that, for any <math>v \in V^m</math>, the sequence <math>(x_{v,e^0(v)},\ldots,x_{v,e^{|v|-1}(v)})</math> is in (strict) counterclockwise order.  Similarly, <math>\underline O</math> is a collection of unordered interior marked points -- i.e. for <math>v \in V^m</math>, <math>\underline O_v</math> is an unordered subset of <math>D^0</math>.
# <math>\underline\beta = (\beta_i)_{1\leq i\leq s}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-s+\sum_{1\leq i\leq s} p_i = m</math> holds.
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# <math>\underline\beta = (\beta_i)_{1\leq i\leq t}, \beta_i \in DM(p_i)^{\text{sph,ord}}</math> is a collection of sphere trees such that the equation <math>m'-t+\sum_{1\leq i\leq t} p_i = m</math> holds.
# <math>\underline z = (z_1,\ldots,z_s), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.
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# <math>\underline z = (z_1,\ldots,z_t), z_i \in \underline O</math> is a distinguished tuple of interior marked points, which corresponds to the places where the sphere trees are attached to the disk tree.
 
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.
 
# The disk tree satisfies the '''stability condition''', i.e. for every <math>v \in V^m</math> with <math>\underline O_v = \emptyset</math>, we have <math>\#E(v) \geq 3</math>.
    
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