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6
Marcelo Amaral
[WSS21] On charge -spin networks from multiway systems branchial graphs
Marcelo Amaral, Quantum Gravity Research
Posted
1 year ago
2118 Views
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On charge -spin networks from multiway systems branchial graphs
by
Marcelo Amaral
Quantum Gravity Research
Abstract: We find spin networks with support on multiway branchial graphs, then we solve as a constraint for SU(3) representations that have support on the same base spin network. The interpretation of SU(3) representations network respecting their respective intertwiners at nodes is that of a charge network over the spin quantum number one. We implemented new functions to compute N-valent intertwiner for both SU(3) and SU(2) or SO(3) as well as a natural way to assign labels to branchial graphs from the causal structure in line with its quantum mechanics interpretation suggesting good asymptotics in the large spin/charge quantum number limit - here the large graph updating limit.
1 From Branchial Graphs to Spin Network Structure in a Simple Way
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n
[
]
:
=
B
r
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n
c
h
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a
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[
g
b
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n
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_
,
g
c
a
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]
:
=
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[
{
v
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p
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d
j
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[
g
c
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]
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,
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}
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]
1.1 Branchial Graph Labeling for Fibonacci Golden Rule
The Fibonacci Golden Rule is given below and the causal state graph and branchial graph computed to certain level in the update multiway system:
I
n
[
]
:
=
r
u
l
e
=
{
"
A
"
"
A
B
"
,
"
B
"
"
A
"
}
;
i
n
i
t
i
a
l
C
o
n
d
i
t
i
o
n
=
"
A
"
;
l
e
v
e
l
B
r
a
n
c
h
i
a
l
=
5
;
g
c
a
u
s
a
l
F
G
R
=
R
e
s
o
u
r
c
e
F
u
n
c
t
i
o
n
[
"
M
u
l
t
i
w
a
y
S
y
s
t
e
m
"
]
[
r
u
l
e
,
i
n
i
t
i
a
l
C
o
n
d
i
t
i
o
n
,
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e
v
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l
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r
a
n
c
h
i
a
l
,
"
S
t
a
t
e
s
G
r
a
p
h
"
]
;
g
b
r
a
n
c
h
i
a
l
F
G
R
=
R
e
s
o
u
r
c
e
F
u
n
c
t
i
o
n
[
"
M
u
l
t
i
w
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y
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y
s
t
e
m
"
]
[
r
u
l
e
,
i
n
i
t
i
a
l
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n
d
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t
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o
n
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e
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l
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r
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n
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h
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a
l
,
"
B
r
a
n
c
h
i
a
l
G
r
a
p
h
"
]
O
u
t
[
]
=
Now we use BranchialEdgeLabelsFromCausalGraph function to get the labeling:
I
n
[
]
:
=
b
r
a
n
c
h
i
a
l
S
p
i
n
L
a
b
e
l
s
=
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r
a
n
c
h
i
a
l
E
d
g
e
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a
b
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l
s
F
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o
m
C
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s
a
l
G
r
a
p
h
[
g
b
r
a
n
c
h
i
a
l
F
G
R
,
g
c
a
u
s
a
l
F
G
R
]
O
u
t
[
]
=
{
{
2
,
1
,
2
,
2
,
1
}
,
{
2
,
1
,
2
,
1
,
1
,
1
}
,
{
1
,
1
,
2
,
1
,
1
,
1
}
,
{
2
,
2
,
1
,
2
,
1
}
,
{
2
,
1
,
2
,
1
,
2
,
1
,
1
}
,
{
1
,
1
,
1
,
2
,
2
,
2
,
1
}
,
{
1
,
1
,
1
,
1
,
2
,
1
}
,
{
1
,
1
,
1
,
1
}
}
branchialSpinLabels gives a list with the labels organized by intertwiner nodes.
1.2 Branchial Graph Labeling for wm148
Below a labeling for a Wolfram model rule, the rule wm148. We first compute the causal state graph and the branchial one:
I
n
[
]
:
=
g
c
a
u
s
a
l
W
M
1
4
8
=
R
e
s
o
u
r
c
e
F
u
n
c
t
i
o
n
[
"
M
u
l
t
i
w
a
y
S
y
s
t
e
m
"
]
[
"
W
o
l
f
r
a
m
M
o
d
e
l
"
{
{
{
x
,
y
}
}
{
{
x
,
y
}
,
{
y
,
z
}
}
}
,
{
{
{
0
,
0
}
}
}
,
4
,
"
S
t
a
t
e
s
G
r
a
p
h
"
]
;
g
b
r
a
n
c
h
i
a
l
W
M
1
4
8
=
R
e
s
o
u
r
c
e
F
u
n
c
t
i
o
n
[
"
M
u
l
t
i
w
a
y
S
y
s
t
e
m
"
]
[
"
W
o
l
f
r
a
m
M
o
d
e
l
"
{
{
{
x
,
y
}
}
{
{
x
,
y
}
,
{
y
,
z
}
}
}
,
{
{
{
0
,
0
}
}
}
,
4
,
"
B
r
a
n
c
h
i
a
l
G
r
a
p
h
"
]
O
u
t
[
]
=
And then the assignment:
I
n
[
]
:
=
b
r
a
n
c
h
i
a
l
S
p
i
n
L
a
b
e
l
s
2
=
B
r
a
n
c
h
i
a
l
E
d
g
e
L
a
b
e
l
s
F
r
o
m
C
a
u
s
a
l
G
r
a
p
h
[
g
b
r
a
n
c
h
i
a
l
W
M
1
4
8
,
g
c
a
u
s
a
l
W
M
1
4
8
]
O
u
t
[
]
=
{
{
1
}
,
{
1
,
1
,
1
,
1
}
,
{
1
,
1
,
1
,
1
,
1
}
,
{
1
,
1
,
1
}
,
{
1
,
1
,
1
,
1
,
1
,
1
}
,
{
1
,
1
}
,
{
1
,
1
,
1
,
1
,
1
}
,
{
1
,
1
,
1
}
,
{
1
,
1
,
1
}
}
1.3 SU(2) Spin Networks on Non-Regular N-Valent Branchial Graphs
The explicit computations will be done in a specific basis but general procedures for bases transformations is well known in the recoupling theory literature. The coefficients are not normalized.
To check if the labels generated above correspond to valid spin network we implement some functions.
We can check that they give valid spin network states by recursively checking that three valent nodes respect usual threeJsymbol selection rules. Let's first check the three-valent one:
I
n
[
]
:
=
s
u
2
T
h
r
e
e
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
Q
[
i
n
t
e
r
t
w
i
n
e
r
T
h
r
e
e
V
a
l
e
n
t
_
]
:
=
M
o
d
u
l
e
[
{
r
e
t
u
r
n
=
F
a
l
s
e
}
,
(
*
T
o
d
o
:
n
e
e
d
t
o
i
n
c
l
u
d
e
m
'
s
i
n
d
i
c
e
s
t
o
w
o
r
k
t
h
e
c
o
n
d
i
t
i
o
n
w
i
t
h
m
i
=
{
0
,
0
,
0
}
*
)
I
f
[
I
n
t
e
g
e
r
Q
[
T
o
t
a
l
[
i
n
t
e
r
t
w
i
n
e
r
T
h
r
e
e
V
a
l
e
n
t
]
]
&
&
i
n
t
e
r
t
w
i
n
e
r
T
h
r
e
e
V
a
l
e
n
t
[
[
3
]
]
≥
A
b
s
[
i
n
t
e
r
t
w
i
n
e
r
T
h
r
e
e
V
a
l
e
n
t
[
[
2
]
]
-
i
n
t
e
r
t
w
i
n
e
r
T
h
r
e
e
V
a
l
e
n
t
[
[
1
]
]
]
&
&
i
n
t
e
r
t
w
i
n
e
r
T
h
r
e
e
V
a
l
e
n
t
[
[
3
]
]
≤
(
i
n
t
e
r
t
w
i
n
e
r
T
h
r
e
e
V
a
l
e
n
t
[
[
2
]
]
+
i
n
t
e
r
t
w
i
n
e
r
T
h
r
e
e
V
a
l
e
n
t
[
[
1
]
]
)
,
r
e
t
u
r
n
=
T
r
u
e
;
]
;
R
e
t
u
r
n
[
r
e
t
u
r
n
]
;
]
For a general N-valence one we can decompose in a three-valent network, which correspond to choose a specific base:
I
n
[
]
:
=
s
u
2
N
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
Q
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
_
]
:
=
M
o
d
u
l
e
[
{
r
e
t
u
r
n
=
F
a
l
s
e
,
k
L
i
s
t
}
,
I
f
[
L
e
n
g
t
h
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
]
1
,
r
e
t
u
r
n
=
T
r
u
e
]
;
I
f
[
L
e
n
g
t
h
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
]
2
,
r
e
t
u
r
n
=
s
u
2
T
h
r
e
e
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
Q
[
I
n
s
e
r
t
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
,
0
,
1
]
]
]
;
I
f
[
L
e
n
g
t
h
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
]
3
,
r
e
t
u
r
n
=
s
u
2
T
h
r
e
e
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
Q
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
]
]
;
I
f
[
L
e
n
g
t
h
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
]
3
,
r
e
t
u
r
n
=
s
u
2
T
h
r
e
e
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
Q
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
]
]
;
I
f
[
L
e
n
g
t
h
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
]
4
,
(
*
g
e
t
l
i
s
t
o
f
i
n
t
e
r
m
e
d
i
a
r
y
a
l
l
o
w
e
d
s
p
i
n
s
*
)
k
L
i
s
t
=
R
a
n
g
e
[
A
b
s
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
1
]
]
-
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
2
]
]
]
,
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
1
]
]
+
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
2
]
]
]
;
D
o
[
I
f
[
s
u
2
T
h
r
e
e
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
Q
[
{
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
1
]
]
,
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
2
]
]
,
k
L
i
s
t
[
[
i
]
]
}
]
&
&
r
e
t
u
r
n
F
a
l
s
e
,
r
e
t
u
r
n
=
s
u
2
T
h
r
e
e
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
Q
[
{
k
L
i
s
t
[
[
i
]
]
,
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
3
]
]
,
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
4
]
]
}
]
]
,
{
i
,
1
,
L
e
n
g
t
h
[
k
L
i
s
t
]
}
]
;
]
;
I
f
[
L
e
n
g
t
h
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
]
>
4
,
k
L
i
s
t
=
R
a
n
g
e
[
A
b
s
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
1
]
]
-
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
2
]
]
]
,
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
1
]
]
+
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
2
]
]
]
;
D
o
[
I
f
[
s
u
2
T
h
r
e
e
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
Q
[
{
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
1
]
]
,
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
2
]
]
,
k
L
i
s
t
[
[
i
]
]
}
]
&
&
r
e
t
u
r
n
F
a
l
s
e
,
r
e
t
u
r
n
=
s
u
2
N
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
Q
[
I
n
s
e
r
t
[
D
e
l
e
t
e
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
,
{
{
1
}
,
{
2
}
}
]
,
k
L
i
s
t
[
[
i
]
]
,
1
]
]
;
]
,
{
i
,
1
,
L
e
n
g
t
h
[
k
L
i
s
t
]
}
]
;
]
;
R
e
t
u
r
n
[
r
e
t
u
r
n
]
;
]
To finish we need to go over all the nodes:
I
n
[
]
:
=
s
u
2
S
p
i
n
N
e
t
w
o
r
k
Q
[
i
n
t
e
r
t
w
i
n
e
r
s
N
V
a
l
e
n
t
_
]
:
=
A
l
l
T
r
u
e
[
i
n
t
e
r
t
w
i
n
e
r
s
N
V
a
l
e
n
t
,
s
u
2
N
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
Q
]
;
Now we can check if the decorations we have chosen are good:
I
n
[
]
:
=
s
u
2
S
p
i
n
N
e
t
w
o
r
k
Q
[
b
r
a
n
c
h
i
a
l
S
p
i
n
L
a
b
e
l
s
]
s
u
2
S
p
i
n
N
e
t
w
o
r
k
Q
[
b
r
a
n
c
h
i
a
l
S
p
i
n
L
a
b
e
l
s
2
]
O
u
t
[
]
=
T
r
u
e
O
u
t
[
]
=
T
r
u
e
W
e
c
o
m
p
u
t
e
t
h
e
c
o
e
f
f
i
c
i
e
n
t
s
l
{
m
i
}
ι
k
l
1
.
.
k
N
q
l
f
r
o
m
e
q
u
a
t
i
o
n
(
1
)
f
o
r
n
o
n
-
r
e
g
u
l
a
r
g
r
a
p
h
s
b
y
d
e
c
o
m
p
o
s
i
n
g
i
n
t
h
r
e
e
-
v
a
l
e
n
t
o
n
e
s
a
n
d
c
h
e
c
k
i
n
g
i
f
e
a
c
h
c
o
m
b
i
n
a
t
i
o
n
w
i
t
h
i
n
t
e
r
n
a
l
s
p
i
n
s
a
r
e
v
a
l
i
d
.
T
h
e
r
e
f
o
r
e
w
e
e
n
d
h
a
v
i
n
g
o
n
l
y
t
h
e
v
a
l
i
d
o
n
e
s
.
W
e
u
s
e
a
r
e
c
u
r
s
i
v
e
f
u
n
c
t
i
o
n
t
h
a
t
i
n
t
h
e
e
n
d
u
s
e
s
t
h
e
M
a
t
h
e
m
a
t
i
c
a
T
h
r
e
e
J
S
y
m
b
o
l
f
u
n
c
t
i
o
n
.
S
o
w
e
s
t
a
r
t
w
i
t
h
a
f
u
n
c
t
i
o
n
f
o
r
c
o
m
p
u
t
i
n
g
f
o
r
t
h
r
e
e
-
v
a
l
e
n
t
n
o
d
e
s
:
I
n
[
]
:
=
s
u
2
T
h
r
e
e
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
E
v
a
l
u
a
t
i
o
n
[
i
n
t
e
r
t
w
i
n
e
r
T
h
r
e
e
V
a
l
e
n
t
_
]
:
=
M
o
d
u
l
e
[
{
c
o
e
f
f
i
c
i
e
n
t
s
=
{
}
,
m
i
L
i
s
t
,
m
L
i
s
t
,
t
h
r
e
e
J
T
e
m
p
}
,
I
f
[
s
u
2
T
h
r
e
e
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
Q
[
i
n
t
e
r
t
w
i
n
e
r
T
h
r
e
e
V
a
l
e
n
t
]
,
m
i
L
i
s
t
=
R
a
n
g
e
[
-
#
,
#
]
&
/
@
i
n
t
e
r
t
w
i
n
e
r
T
h
r
e
e
V
a
l
e
n
t
;
m
L
i
s
t
=
S
e
l
e
c
t
[
T
u
p
l
e
s
[
{
m
i
L
i
s
t
[
[
1
]
]
,
m
i
L
i
s
t
[
[
2
]
]
,
m
i
L
i
s
t
[
[
3
]
]
}
]
,
T
o
t
a
l
[
#
]
0
&
]
;
D
o
[
I
f
[
N
o
t
[
O
d
d
Q
[
T
o
t
a
l
[
i
n
t
e
r
t
w
i
n
e
r
T
h
r
e
e
V
a
l
e
n
t
]
]
&
&
m
L
i
s
t
[
[
m
i
]
]
{
0
,
0
,
0
}
]
,
t
h
r
e
e
J
T
e
m
p
=
Q
u
i
e
t
[
T
h
r
e
e
J
S
y
m
b
o
l
[
{
i
n
t
e
r
t
w
i
n
e
r
T
h
r
e
e
V
a
l
e
n
t
[
[
1
]
]
,
m
L
i
s
t
[
[
m
i
,
1
]
]
}
,
{
i
n
t
e
r
t
w
i
n
e
r
T
h
r
e
e
V
a
l
e
n
t
[
[
2
]
]
,
m
L
i
s
t
[
[
m
i
,
2
]
]
}
,
{
i
n
t
e
r
t
w
i
n
e
r
T
h
r
e
e
V
a
l
e
n
t
[
[
3
]
]
,
m
L
i
s
t
[
[
m
i
,
3
]
]
}
]
]
;
c
o
e
f
f
i
c
i
e
n
t
s
=
A
p
p
e
n
d
T
o
[
c
o
e
f
f
i
c
i
e
n
t
s
,
{
i
n
t
e
r
t
w
i
n
e
r
T
h
r
e
e
V
a
l
e
n
t
,
m
L
i
s
t
[
[
m
i
]
]
,
t
h
r
e
e
J
T
e
m
p
}
]
;
(
*
T
o
d
o
:
t
h
r
e
e
J
T
e
m
p
W
i
t
h
C
o
n
t
r
a
c
t
e
d
F
a
c
t
o
r
=
t
h
r
e
e
J
T
e
m
p
*
(
-
1
)
^
(
j
k
-
m
L
i
s
t
[
[
m
i
,
3
]
]
)
;
*
)
]
,
{
m
i
,
1
,
L
e
n
g
t
h
[
m
L
i
s
t
]
}
]
;
]
;
R
e
t
u
r
n
[
c
o
e
f
f
i
c
i
e
n
t
s
]
;
]
For the evaluation will be useful to have a four-valent intertwiner, which evaluate independently the 2 three-valent ones with the intermediary spins (this function is not really necessary):
I
n
[
]
:
=
s
u
2
F
o
u
r
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
E
v
a
l
u
a
t
i
o
n
[
i
n
t
e
r
t
w
i
n
e
r
F
o
u
r
V
a
l
e
n
t
_
]
:
=
M
o
d
u
l
e
[
{
c
o
e
f
f
i
c
i
e
n
t
s
=
{
}
,
k
L
i
s
t
,
c
o
e
f
f
i
c
i
e
n
t
s
T
e
m
p
1
,
c
o
e
f
f
i
c
i
e
n
t
s
T
e
m
p
2
}
,
I
f
[
L
e
n
g
t
h
[
i
n
t
e
r
t
w
i
n
e
r
F
o
u
r
V
a
l
e
n
t
]
4
,
(
*
H
e
r
e
w
e
g
e
t
a
l
i
s
t
o
f
i
n
t
e
r
m
e
d
i
a
r
y
a
l
l
o
w
e
d
s
p
i
n
s
*
)
k
L
i
s
t
=
R
a
n
g
e
[
A
b
s
[
i
n
t
e
r
t
w
i
n
e
r
F
o
u
r
V
a
l
e
n
t
[
[
1
]
]
-
i
n
t
e
r
t
w
i
n
e
r
F
o
u
r
V
a
l
e
n
t
[
[
2
]
]
]
,
i
n
t
e
r
t
w
i
n
e
r
F
o
u
r
V
a
l
e
n
t
[
[
1
]
]
+
i
n
t
e
r
t
w
i
n
e
r
F
o
u
r
V
a
l
e
n
t
[
[
2
]
]
]
;
D
o
[
I
f
[
s
u
2
T
h
r
e
e
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
Q
[
{
i
n
t
e
r
t
w
i
n
e
r
F
o
u
r
V
a
l
e
n
t
[
[
1
]
]
,
i
n
t
e
r
t
w
i
n
e
r
F
o
u
r
V
a
l
e
n
t
[
[
2
]
]
,
k
L
i
s
t
[
[
i
]
]
}
]
,
c
o
e
f
f
i
c
i
e
n
t
s
T
e
m
p
1
=
s
u
2
T
h
r
e
e
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
E
v
a
l
u
a
t
i
o
n
[
{
i
n
t
e
r
t
w
i
n
e
r
F
o
u
r
V
a
l
e
n
t
[
[
1
]
]
,
i
n
t
e
r
t
w
i
n
e
r
F
o
u
r
V
a
l
e
n
t
[
[
2
]
]
,
k
L
i
s
t
[
[
i
]
]
}
]
;
I
f
[
s
u
2
T
h
r
e
e
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
Q
[
{
k
L
i
s
t
[
[
i
]
]
,
i
n
t
e
r
t
w
i
n
e
r
F
o
u
r
V
a
l
e
n
t
[
[
3
]
]
,
i
n
t
e
r
t
w
i
n
e
r
F
o
u
r
V
a
l
e
n
t
[
[
4
]
]
}
]
,
c
o
e
f
f
i
c
i
e
n
t
s
T
e
m
p
2
=
s
u
2
T
h
r
e
e
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
E
v
a
l
u
a
t
i
o
n
[
{
k
L
i
s
t
[
[
i
]
]
,
i
n
t
e
r
t
w
i
n
e
r
F
o
u
r
V
a
l
e
n
t
[
[
3
]
]
,
i
n
t
e
r
t
w
i
n
e
r
F
o
u
r
V
a
l
e
n
t
[
[
4
]
]
}
]
;
c
o
e
f
f
i
c
i
e
n
t
s
=
A
p
p
e
n
d
T
o
[
c
o
e
f
f
i
c
i
e
n
t
s
,
J
o
i
n
[
c
o
e
f
f
i
c
i
e
n
t
s
T
e
m
p
1
,
c
o
e
f
f
i
c
i
e
n
t
s
T
e
m
p
2
]
]
(
*
H
e
r
e
w
e
w
i
l
l
n
e
e
d
t
o
i
m
p
l
e
m
e
n
t
t
h
e
m
i
s
s
i
n
g
(
-
1
)
^
(
k
L
i
s
t
[
[
i
]
]
-
m
k
)
t
h
a
t
s
u
m
o
v
e
r
c
o
m
m
o
m
m
'
s
-
p
r
o
b
a
b
l
y
i
t
w
i
l
l
b
e
b
e
t
t
e
r
t
o
r
e
w
r
i
t
e
t
h
i
s
f
u
n
c
t
i
o
n
w
i
t
h
t
h
e
e
x
p
l
i
t
d
e
p
e
n
d
e
n
c
e
o
n
i
n
t
e
r
n
a
l
k
*
)
]
]
,
{
i
,
1
,
L
e
n
g
t
h
[
k
L
i
s
t
]
}
]
;
]
;
R
e
t
u
r
n
[
c
o
e
f
f
i
c
i
e
n
t
s
]
;
]
Now we can use the two functions above to implement a recursive function to compute the coefficients for nodes of general valence:
I
n
[
]
:
=
s
u
2
N
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
E
v
a
l
u
a
t
i
o
n
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
_
]
:
=
M
o
d
u
l
e
[
{
c
o
e
f
f
i
c
i
e
n
t
s
=
{
}
,
c
o
e
f
f
i
c
i
e
n
t
s
T
e
m
p
1
,
k
L
i
s
t
,
m
l
i
s
t
}
,
I
f
[
L
e
n
g
t
h
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
]
1
,
m
l
i
s
t
=
R
a
n
g
e
[
-
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
1
]
]
,
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
1
]
]
]
;
c
o
e
f
f
i
c
i
e
n
t
s
=
T
a
b
l
e
[
{
I
n
s
e
r
t
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
,
0
,
{
{
1
}
,
{
1
}
}
]
,
{
0
,
0
,
m
l
i
s
t
[
[
i
]
]
}
,
(
-
1
)
^
(
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
1
]
]
+
m
l
i
s
t
[
[
i
]
]
)
}
,
{
i
,
1
,
L
e
n
g
t
h
[
m
l
i
s
t
]
}
]
;
]
;
I
f
[
L
e
n
g
t
h
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
]
2
,
c
o
e
f
f
i
c
i
e
n
t
s
=
s
u
2
T
h
r
e
e
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
E
v
a
l
u
a
t
i
o
n
[
I
n
s
e
r
t
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
,
0
,
1
]
]
]
;
I
f
[
L
e
n
g
t
h
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
]
3
,
c
o
e
f
f
i
c
i
e
n
t
s
=
s
u
2
T
h
r
e
e
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
E
v
a
l
u
a
t
i
o
n
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
]
]
;
I
f
[
L
e
n
g
t
h
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
]
4
,
c
o
e
f
f
i
c
i
e
n
t
s
=
s
u
2
F
o
u
r
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
E
v
a
l
u
a
t
i
o
n
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
]
]
;
I
f
[
L
e
n
g
t
h
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
]
>
4
,
k
L
i
s
t
=
R
a
n
g
e
[
A
b
s
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
1
]
]
-
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
2
]
]
]
,
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
1
]
]
+
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
2
]
]
]
;
D
o
[
c
o
e
f
f
i
c
i
e
n
t
s
T
e
m
p
1
=
s
u
2
T
h
r
e
e
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
E
v
a
l
u
a
t
i
o
n
[
{
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
1
]
]
,
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
2
]
]
,
k
L
i
s
t
[
[
i
]
]
}
]
;
I
f
[
s
u
2
T
h
r
e
e
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
Q
[
{
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
1
]
]
,
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
[
[
2
]
]
,
k
L
i
s
t
[
[
i
]
]
}
]
&
&
c
o
e
f
f
i
c
i
e
n
t
s
T
e
m
p
1
!
=
{
}
,
c
o
e
f
f
i
c
i
e
n
t
s
=
A
p
p
e
n
d
T
o
[
c
o
e
f
f
i
c
i
e
n
t
s
,
J
o
i
n
[
c
o
e
f
f
i
c
i
e
n
t
s
T
e
m
p
1
,
s
u
2
N
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
E
v
a
l
u
a
t
i
o
n
[
I
n
s
e
r
t
[
D
e
l
e
t
e
[
i
n
t
e
r
t
w
i
n
e
r
N
V
a
l
e
n
t
,
{
{
1
}
,
{
2
}
}
]
,
k
L
i
s
t
[
[
i
]
]
,
1
]
]
]
]
;
]
,
{
i
,
1
,
L
e
n
g
t
h
[
k
L
i
s
t
]
}
]
;
]
;
R
e
t
u
r
n
[
c
o
e
f
f
i
c
i
e
n
t
s
]
;
]
And then to evaluate a set of intertwiners:
I
n
[
]
:
=
s
u
2
S
p
i
n
N
e
t
w
o
r
k
E
v
a
l
u
a
t
i
o
n
[
i
n
t
e
r
t
w
i
n
e
r
s
N
V
a
l
e
n
t
_
]
:
=
M
o
d
u
l
e
[
{
c
o
e
f
f
i
c
i
e
n
t
s
=
{
}
}
,
I
f
[
s
u
2
S
p
i
n
N
e
t
w
o
r
k
Q
[
i
n
t
e
r
t
w
i
n
e
r
s
N
V
a
l
e
n
t
]
,
c
o
e
f
f
i
c
i
e
n
t
s
=
s
u
2
N
V
a
l
e
n
t
I
n
t
e
r
t
w
i
n
e
r
E
v
a
l
u
a
t
i
o
n
/
@
i
n
t
e
r
t
w
i
n
e
r
s
N
V
a
l
e
n
t
;
]
;
R
e
t
u
r
n
[
c
o
e
f
f
i
c
i
e
n
t
s
]
;
]
Let us apply the functions above to calculate the coefficients for the Fibonacci golden rule:
I
n
[
]
:
=
s
u
2
S
p
i
n
N
e
t
w
o
r
k
S
t
a
t
e
C
o
e
f
f
i
c
i
e
n
t
s
=
s
u
2
S
p
i
n
N
e
t
w
o
r
k
E
v
a
l
u
a
t
i
o
n
[
b
r
a
n
c
h
i
a
l
S
p
i
n
L
a
b
e
l
s
]
;
(
*
{
{
j
s
}
,
{
m
s
}
,
t
h
r
e
e
J
s
}
*
)
s
u
2
S
p
i
n
N
e
t
w
o
r
k
S
t
a
t
e
C
o
e
f
f
i
c
i
e
n
t
s
/
/
D
i
m
e
n
s
i
o
n
s
T
a
b
l
e
[
s
u
2
S
p
i
n
N
e
t
w
o
r
k
S
t
a
t
e
C
o
e
f
f
i
c
i
e
n
t
s
[
[
i
]
]
/
/
D
i
m
e
n
s
i
o
n
s
,
{
i
,
1
,
L
e
n
g
t
h
[
s
u
2
S
p
i
n
N
e
t
w
o
r
k
S
t
a
t
e
C
o
e
f
f
i
c
i
e
n
t
s
]
}
]
O
u
t
[
]
=
{
8
}
O
u
t
[
]
=
{
{
3
}
,
{
3
}
,
{
3
}
,
{
5
}
,
{
3
}
,
{
3
}
,
{
3
}
,
{
3
}
}
I
n
[
]
:
=
R
a
n
d
o
m
C
h
o
i
c
e
[
s
u
2
S
p
i
n
N
e
t
w
o
r
k
S
t
a
t
e
C
o
e
f
f
i
c
i
e
n
t
s
[
[
1
,
1
]
]
]
O
u
t
[
]
=
{
1
,
2
,
3
}
,
{
-
1
,
-
2
,
3
}
,
1
7
,
{
1
,
2
,
3
}
,
{
-
1
,
-
1
,
2
}
,
-
2
2
1
,
{
1
,
2
,
3
}
,
{
-
1
,
0
,
1
}
,
2
3
5
,
{
1
,
2
,
3
}
,
{
-
1
,
1
,
0
}
,
-
1
3
5
,
{
1
,
2
,
3
}
,
{
-
1
,
2
,
-
1
}
,
1
1
0
5
,
{
1
,
2
,
3
}
,
{
0
,
-
2
,
2
}
,
-
1
2
1
,
{
1
,
2
,
3
}
,
{
0
,
-
1
,
1
}
,
2
2
1
0
5
,
{
1
,
2
,
3
}
,
{
0
,
0
,
0
}
,
-
3
3
5
,
{
1
,
2
,
3
}
,
{
0
,
1
,
-
1
}
,
2
2
1
0
5
,
{
1
,
2
,
3
}
,
{
0
,
2
,
-
2
}
,
-
1
2
1
,
{
1
,
2
,
3
}
,
{
1
,
-
2
,
1
}
,
1
1
0
5
,
{
1
,
2
,
3
}
,
{
1
,
-
1
,
0
}
,
-
1
3
5
,
{
1
,
2
,
3
}
,
{
1
,
0
,
-
1
}
,
2
3
5
,
{
1
,
2
,
3
}
,
{
1
,
1
,
-
2
}
,
-
2
2
1
,
{
1
,
2
,
3
}
,
{
1
,
2
,
-
3
}
,
1
7
,
{
3
,
2
,
1
}
,
{
-
3
,
2
,
1
}
,
1
7
,
{
3
,
2
,
1
}
,
{
-
2
,
1
,
1
}
,
-
2
2
1
,
{
3
,
2
,
1
}
,
{
-
2
,
2
,
0
}
,
-
1