From 0158f84d98cbed28e1212b830f7367501566302f Mon Sep 17 00:00:00 2001
From: Chenqi Meng <mcqgravity@gmail.com>
Date: Tue, 31 Mar 2026 19:50:49 +0800
Subject: [PATCH 01/11] Added some aliases

---
 src/aliases.jl | 36 ++++++++++++++++++++++++++++++++++++
 1 file changed, 36 insertions(+)

diff --git a/src/aliases.jl b/src/aliases.jl
index 401326b..ce1a211 100644
--- a/src/aliases.jl
+++ b/src/aliases.jl
@@ -6,6 +6,7 @@ const Z5 = PMFC{5, 1, 4, 1, 0, 0}
 const Z6 = PMFC{6, 1, 4, 1, 0, 0}
 
 const Z2xZ2 = PMFC{4, 1, 0, 1, 0, 0}
+const Z2xsVec = PMFC{4, 1, 0, 1, 0, 1}
 
 # Rep[Dₙ]
 const RepD3 = PMFC{3, 1, 0, 2, 0, 0}
@@ -36,3 +37,38 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 # Varia
 @objectnames Fib = PMFC{2, 1, 0, 2, 0, 0} I τ
 @objectnames Ising = PMFC{3, 1, 0, 1, 1, 3} I ψ σ
+@objectnames sVec = PMFC{2, 1, 0, 1, 0, 1} I ψ
+@objectnames Semion = PMFC{2, 1, 0, 1, 1, 0} I ϵ
+
+_umtcs_lowrank_centralcharge = [
+    Dict(
+        1//1 => PMFC{2, 1, 0, 1, 1, 0}, # Semion
+        -1//1 => PMFC{2, 1, 0, 1, 1, 1},
+        -14//5 => PMFC{2, 1, 0, 2, 0, 0}, # TimeReversed{Fibonacci}
+        14//5 => PMFC{2, 1, 0, 2, 0, 1}, # Fibonacci
+    ),
+    Dict(
+        2//1 => PMFC{3, 1, 2, 1, 0, 1}, # Z₃ MTC
+        -2//1 => PMFC{3, 1, 2, 1, 0, 2},
+        5//2 => PMFC{3, 1, 0, 1, 0, 0}, # The following are 8-fold ways
+        -3//2 => PMFC{3, 1, 0, 1, 0, 1},
+        -5//2 => PMFC{3, 1, 0, 1, 0, 2},
+        3//2 => PMFC{3, 1, 0, 1, 0, 3},
+        7//2 => PMFC{3, 1, 0, 1, 1, 0},
+        -1//2 => PMFC{3, 1, 0, 1, 1, 1}, # TimeReversed{Ising}
+        -7//2 => PMFC{3, 1, 0, 1, 1, 2},
+        1//2 => PMFC{3, 1, 0, 1, 1, 3}, # Ising
+        8//7 =>PMFC{3, 1, 0, 3, 0, 0}, # PSU(2)₅
+        -8//7 => PMFC{3, 1, 0, 3, 0, 1},
+    ),
+    Dict(
+        (0//1, 1) => PMFC{4, 1, 0, 1, 0, 2}, # Toric code ≅ Z(Rep[Z₂])
+        (0//1, 2) => PMFC{4, 1, 0, 1, 1, 0}, # Z(Semion) ≅ Semion ⊠ TimeReversed{Semion}
+        2//1 => PMFC{4, 1, 0, 1, 1, 1}, # Semion ⊠ Semion
+        -2//1 => PMFC{4, 1, 0, 1, 1, 2}, # TODO: TimeReversed{Semion} ⊠ TimeReversed{Semion}, but with error!
+        1//1 => PMFC{4, 1, 2, 1, 2, 0}, # Four Z₄ MTCs
+        3//1 => PMFC{4, 1, 2, 1, 2, 1},
+        -3//1 => PMFC{4, 1, 2, 1, 2, 2},
+        -1//1 => PMFC{4, 1, 2, 1, 2, 3},
+    )
+]

From ec46154e897da6ffc391445960feee7f2afaef96 Mon Sep 17 00:00:00 2001
From: Chenqi Meng <mcqgravity@gmail.com>
Date: Tue, 31 Mar 2026 22:39:52 +0800
Subject: [PATCH 02/11] Finished UMTCs lower than rank 5

---
 src/aliases.jl | 34 ++++++++++++++++++++++++++++------
 1 file changed, 28 insertions(+), 6 deletions(-)

diff --git a/src/aliases.jl b/src/aliases.jl
index ce1a211..a3deac3 100644
--- a/src/aliases.jl
+++ b/src/aliases.jl
@@ -39,17 +39,20 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 @objectnames Ising = PMFC{3, 1, 0, 1, 1, 3} I ψ σ
 @objectnames sVec = PMFC{2, 1, 0, 1, 0, 1} I ψ
 @objectnames Semion = PMFC{2, 1, 0, 1, 1, 0} I ϵ
+@objectnames ToricCode = PMFC{4, 1, 0, 1, 0, 0} I e m ψ
+@objectnames Z3MTC = PMFC{3, 1, 2, 1, 0, 1} 0 1 2
+@objectnames SU2_4 = PMFC{5, 1, 0, 3, 0, 0} 0 4 3 1 2
 
 _umtcs_lowrank_centralcharge = [
     Dict(
         1//1 => PMFC{2, 1, 0, 1, 1, 0}, # Semion
-        -1//1 => PMFC{2, 1, 0, 1, 1, 1},
-        -14//5 => PMFC{2, 1, 0, 2, 0, 0}, # TimeReversed{Fibonacci}
-        14//5 => PMFC{2, 1, 0, 2, 0, 1}, # Fibonacci
+        -1//1 => PMFC{2, 1, 0, 1, 1, 1}, # TimeReversed{Semion}
+        -14//5 => PMFC{2, 1, 0, 2, 0, 0}, # TimeReversed{Fib}
+        14//5 => PMFC{2, 1, 0, 2, 0, 1}, # Fib
     ),
     Dict(
-        2//1 => PMFC{3, 1, 2, 1, 0, 1}, # Z₃ MTC
-        -2//1 => PMFC{3, 1, 2, 1, 0, 2},
+        2//1 => PMFC{3, 1, 2, 1, 0, 1}, # Z3MTC
+        -2//1 => PMFC{3, 1, 2, 1, 0, 2}, # TimeReversed{Z3MTC}
         5//2 => PMFC{3, 1, 0, 1, 0, 0}, # The following are 8-fold ways
         -3//2 => PMFC{3, 1, 0, 1, 0, 1},
         -5//2 => PMFC{3, 1, 0, 1, 0, 2},
@@ -63,12 +66,31 @@ _umtcs_lowrank_centralcharge = [
     ),
     Dict(
         (0//1, 1) => PMFC{4, 1, 0, 1, 0, 2}, # Toric code ≅ Z(Rep[Z₂])
+        # 4//1 => PMFC{4, 1, 0, 1, 0, ?}, # ThreeFermion, but with error!
         (0//1, 2) => PMFC{4, 1, 0, 1, 1, 0}, # Z(Semion) ≅ Semion ⊠ TimeReversed{Semion}
         2//1 => PMFC{4, 1, 0, 1, 1, 1}, # Semion ⊠ Semion
-        -2//1 => PMFC{4, 1, 0, 1, 1, 2}, # TODO: TimeReversed{Semion} ⊠ TimeReversed{Semion}, but with error!
+        # -2//1 => PMFC{4, 1, 0, 1, 1, ?}, # TODO: TimeReversed{Semion} ⊠ TimeReversed{Semion}, but with error!
         1//1 => PMFC{4, 1, 2, 1, 2, 0}, # Four Z₄ MTCs
         3//1 => PMFC{4, 1, 2, 1, 2, 1},
         -3//1 => PMFC{4, 1, 2, 1, 2, 2},
         -1//1 => PMFC{4, 1, 2, 1, 2, 3},
+        -9//5 => PMFC{4, 1, 0, 2, 1, 0}, # Semion ⊠ TimeReversed{Fib}
+        19//5 => PMFC{4, 1, 0, 2, 1, 1}, # Semion ⊠ Fib
+        -19//5 => PMFC{4, 1, 0, 2, 1, 2}, # TimeReversed{Semion} ⊠ TimeReversed{Fib}
+        9//5 => PMFC{4, 1, 0, 2, 1, 3}, # TimeReversed{Semion} ⊠ Fib
+        12//5 => PMFC{4, 1, 0, 5, 0, 0}, # TimeReversed{Fib} ⊠ TimeReversed{Fib}
+        (0//1, 3) => PMFC{4, 1, 0, 5, 0, 1}, # TimeReversed{Fib} ⊠ Fib
+        -12//5 => PMFC{4, 1, 0, 5, 0, 2}, # Fib ⊠ Fib
+        10//3 => PMFC{4, 1, 0, 6, 0, 0}, # PSU2_7
+        -10//3 => PMFC{4, 1, 0, 6, 0, 1}, # TimeReversed{PSU2_7}
+    ),
+    Dict(
+        4//1 => PMFC{5, 1, 4, 1, 0, 1},
+        0//1 => PMFC{5, 1, 4, 1, 0, 2},
+        (2//1, 1) => PMFC{5, 1, 0, 3, 0, 0}, # SU2_4
+        (2//1, 2) => PMFC{5, 1, 0, 3, 1, 0}, # TimeReversed{JK4} http://dx.doi.org/10.1103/PhysRevA.92.012301
+        # Cannot find TimeReversed{SU2_4} or JK4, but they should be here somewhere with PMFC{5, 1, 0, 3, x, y}
+        # Cannot find PSU2_9 and its time-reversal, but they should be here somewhere with PMFC{5, 1, 0, x, y, z} with c = ± 16//11
+        # Cannot find c = ± 18//7 UMTCs. They are not SimpleFusion. Should be PMFC{5, 2, ?, ?, ?, ?}
     )
 ]

From 71868dacee04317dfea331b2af4cbfe9e55a3ecd Mon Sep 17 00:00:00 2001
From: Chenqi Meng <mcqgravity@gmail.com>
Date: Tue, 31 Mar 2026 23:02:07 +0800
Subject: [PATCH 03/11] Update SU2_4 and JK_4

---
 src/aliases.jl | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/src/aliases.jl b/src/aliases.jl
index a3deac3..aee8c3a 100644
--- a/src/aliases.jl
+++ b/src/aliases.jl
@@ -88,8 +88,9 @@ _umtcs_lowrank_centralcharge = [
         4//1 => PMFC{5, 1, 4, 1, 0, 1},
         0//1 => PMFC{5, 1, 4, 1, 0, 2},
         (2//1, 1) => PMFC{5, 1, 0, 3, 0, 0}, # SU2_4
+        # Cannot find TimeReversed{SU2_4}, but it should be here with PMFC{5, 1, 0, 3, 0, 1}
         (2//1, 2) => PMFC{5, 1, 0, 3, 1, 0}, # TimeReversed{JK4} http://dx.doi.org/10.1103/PhysRevA.92.012301
-        # Cannot find TimeReversed{SU2_4} or JK4, but they should be here somewhere with PMFC{5, 1, 0, 3, x, y}
+        # Cannot find JK4, but it should be here somewhere with PMFC{5, 1, 0, 3, 1, 1}
         # Cannot find PSU2_9 and its time-reversal, but they should be here somewhere with PMFC{5, 1, 0, x, y, z} with c = ± 16//11
         # Cannot find c = ± 18//7 UMTCs. They are not SimpleFusion. Should be PMFC{5, 2, ?, ?, ?, ?}
     )

From b54fbfa9eef13c8155683c7f3c63927947e5d91c Mon Sep 17 00:00:00 2001
From: Chenqi Meng <mcqgravity@gmail.com>
Date: Wed, 1 Apr 2026 13:54:35 +0800
Subject: [PATCH 04/11] Update aliases.jl

---
 src/aliases.jl | 13 +++++++++----
 1 file changed, 9 insertions(+), 4 deletions(-)

diff --git a/src/aliases.jl b/src/aliases.jl
index aee8c3a..4257c0f 100644
--- a/src/aliases.jl
+++ b/src/aliases.jl
@@ -39,7 +39,12 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 @objectnames Ising = PMFC{3, 1, 0, 1, 1, 3} I ψ σ
 @objectnames sVec = PMFC{2, 1, 0, 1, 0, 1} I ψ
 @objectnames Semion = PMFC{2, 1, 0, 1, 1, 0} I ϵ
-@objectnames ToricCode = PMFC{4, 1, 0, 1, 0, 0} I e m ψ
+@objectnames ToricCode = PMFC{4, 1, 0, 1, 0, 2} I e m ψ
+# @objectnames ThreeFermion = PMFC{4, 1, 0, 1, 0, 3} I f1 f2 f3 
+@objectnames ZSemion = PMFC{4, 1, 0, 1, 1, 0} II ϵI ϵϵ Iϵ
+@objectnames SemionSemion = PMFC{4, 1, 0, 1, 1, 1} II ϵI ϵϵ Iϵ
+@objectnames ZFib = PMFC{4, 1, 0, 5, 0, 1} II τI Iτ ττ
+# @objectnames FibFib = PMFC{4, 1, 0, 5, 0, 2} II τI Iτ ττ
 @objectnames Z3MTC = PMFC{3, 1, 2, 1, 0, 1} 0 1 2
 @objectnames SU2_4 = PMFC{5, 1, 0, 3, 0, 0} 0 4 3 1 2
 
@@ -66,10 +71,10 @@ _umtcs_lowrank_centralcharge = [
     ),
     Dict(
         (0//1, 1) => PMFC{4, 1, 0, 1, 0, 2}, # Toric code ≅ Z(Rep[Z₂])
-        # 4//1 => PMFC{4, 1, 0, 1, 0, ?}, # ThreeFermion, but with error!
+        # 4//1 => PMFC{4, 1, 0, 1, 0, 3}, # ThreeFermion
         (0//1, 2) => PMFC{4, 1, 0, 1, 1, 0}, # Z(Semion) ≅ Semion ⊠ TimeReversed{Semion}
         2//1 => PMFC{4, 1, 0, 1, 1, 1}, # Semion ⊠ Semion
-        # -2//1 => PMFC{4, 1, 0, 1, 1, ?}, # TODO: TimeReversed{Semion} ⊠ TimeReversed{Semion}, but with error!
+        # -2//1 => PMFC{4, 1, 0, 1, 1, ?}, # TimeReversed{Semion} ⊠ TimeReversed{Semion}, but with error!
         1//1 => PMFC{4, 1, 2, 1, 2, 0}, # Four Z₄ MTCs
         3//1 => PMFC{4, 1, 2, 1, 2, 1},
         -3//1 => PMFC{4, 1, 2, 1, 2, 2},
@@ -80,7 +85,7 @@ _umtcs_lowrank_centralcharge = [
         9//5 => PMFC{4, 1, 0, 2, 1, 3}, # TimeReversed{Semion} ⊠ Fib
         12//5 => PMFC{4, 1, 0, 5, 0, 0}, # TimeReversed{Fib} ⊠ TimeReversed{Fib}
         (0//1, 3) => PMFC{4, 1, 0, 5, 0, 1}, # TimeReversed{Fib} ⊠ Fib
-        -12//5 => PMFC{4, 1, 0, 5, 0, 2}, # Fib ⊠ Fib
+        # -12//5 => PMFC{4, 1, 0, 5, 0, 2}, # Fib ⊠ Fib
         10//3 => PMFC{4, 1, 0, 6, 0, 0}, # PSU2_7
         -10//3 => PMFC{4, 1, 0, 6, 0, 1}, # TimeReversed{PSU2_7}
     ),

From 24f128721587b53a1e52729b48b07451d9a98ad0 Mon Sep 17 00:00:00 2001
From: Chenqi Meng <mcqgravity@gmail.com>
Date: Wed, 1 Apr 2026 15:10:04 +0800
Subject: [PATCH 05/11] Update aliases.jl

---
 src/aliases.jl | 39 +++++++++++++++++++++++++++------------
 1 file changed, 27 insertions(+), 12 deletions(-)

diff --git a/src/aliases.jl b/src/aliases.jl
index 4257c0f..612fc00 100644
--- a/src/aliases.jl
+++ b/src/aliases.jl
@@ -6,7 +6,6 @@ const Z5 = PMFC{5, 1, 4, 1, 0, 0}
 const Z6 = PMFC{6, 1, 4, 1, 0, 0}
 
 const Z2xZ2 = PMFC{4, 1, 0, 1, 0, 0}
-const Z2xsVec = PMFC{4, 1, 0, 1, 0, 1}
 
 # Rep[Dₙ]
 const RepD3 = PMFC{3, 1, 0, 2, 0, 0}
@@ -36,21 +35,37 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 
 # Varia
 @objectnames Fib = PMFC{2, 1, 0, 2, 0, 0} I τ
-@objectnames Ising = PMFC{3, 1, 0, 1, 1, 3} I ψ σ
 @objectnames sVec = PMFC{2, 1, 0, 1, 0, 1} I ψ
 @objectnames Semion = PMFC{2, 1, 0, 1, 1, 0} I ϵ
+@objectnames U1_1 = PMFC{2, 1, 0, 1, 1, 0} 0 1
+@objectnames SU2_1 = PMFC{2, 1, 0, 1, 1, 0} 0 1
+@objectnames PSU2_3 = PMFC{2, 1, 0, 2, 0, 0} 0 2
+
+@objectnames Z3MTC = PMFC{3, 1, 2, 1, 0, 1} 0 1 2
+@objectnames Ising = PMFC{3, 1, 0, 1, 1, 3} I ψ σ
+@objectnames SU2_2 = PMFC{3, 1, 0, 1, 0, 3} 0 2 1
+@objectnames PSU2_5 = PMFC{3, 1, 0, 3, 0, 0} 0 4 2
+
 @objectnames ToricCode = PMFC{4, 1, 0, 1, 0, 2} I e m ψ
 # @objectnames ThreeFermion = PMFC{4, 1, 0, 1, 0, 3} I f1 f2 f3 
+@objectnames Z2sVec = PMFC{4, 1, 0, 1, 0, 1}
 @objectnames ZSemion = PMFC{4, 1, 0, 1, 1, 0} II ϵI ϵϵ Iϵ
 @objectnames SemionSemion = PMFC{4, 1, 0, 1, 1, 1} II ϵI ϵϵ Iϵ
+# @objectnames Z2Semion = PMFC{4, 1, 0, 1, 1, ?}
+# @objectnames sVecSemion = = PMFC{4, 1, 0, 1, 1, ?}
+@objectnames SemionFib = PMFC{4, 1, 0, 2, 1, 1} II ϵI ϵτ Iτ
+@objectnames U1_2 = PMFC{4, 1, 2, 1, 2, 0} 0 1 2 3
 @objectnames ZFib = PMFC{4, 1, 0, 5, 0, 1} II τI Iτ ττ
 # @objectnames FibFib = PMFC{4, 1, 0, 5, 0, 2} II τI Iτ ττ
-@objectnames Z3MTC = PMFC{3, 1, 2, 1, 0, 1} 0 1 2
+@objectnames PSU2_7 = PMFC{4, 1, 0, 6, 0, 0} 0 6 2 4
+
 @objectnames SU2_4 = PMFC{5, 1, 0, 3, 0, 0} 0 4 3 1 2
+# @objectnames JK4
+# @objectnames PSU2_9
 
 _umtcs_lowrank_centralcharge = [
     Dict(
-        1//1 => PMFC{2, 1, 0, 1, 1, 0}, # Semion
+        1//1 => PMFC{2, 1, 0, 1, 1, 0}, # Semion, U1_1, SU2_1
         -1//1 => PMFC{2, 1, 0, 1, 1, 1}, # TimeReversed{Semion}
         -14//5 => PMFC{2, 1, 0, 2, 0, 0}, # TimeReversed{Fib}
         14//5 => PMFC{2, 1, 0, 2, 0, 1}, # Fib
@@ -58,27 +73,27 @@ _umtcs_lowrank_centralcharge = [
     Dict(
         2//1 => PMFC{3, 1, 2, 1, 0, 1}, # Z3MTC
         -2//1 => PMFC{3, 1, 2, 1, 0, 2}, # TimeReversed{Z3MTC}
-        5//2 => PMFC{3, 1, 0, 1, 0, 0}, # The following are 8-fold ways
-        -3//2 => PMFC{3, 1, 0, 1, 0, 1},
+        5//2 => PMFC{3, 1, 0, 1, 0, 0}, # The following are 8-fold ways of Ising-type UMTCs
+        -3//2 => PMFC{3, 1, 0, 1, 0, 1}, # TimeReversed{SU2_2}
         -5//2 => PMFC{3, 1, 0, 1, 0, 2},
-        3//2 => PMFC{3, 1, 0, 1, 0, 3},
+        3//2 => PMFC{3, 1, 0, 1, 0, 3}, # SU2_2
         7//2 => PMFC{3, 1, 0, 1, 1, 0},
         -1//2 => PMFC{3, 1, 0, 1, 1, 1}, # TimeReversed{Ising}
         -7//2 => PMFC{3, 1, 0, 1, 1, 2},
         1//2 => PMFC{3, 1, 0, 1, 1, 3}, # Ising
-        8//7 =>PMFC{3, 1, 0, 3, 0, 0}, # PSU(2)₅
-        -8//7 => PMFC{3, 1, 0, 3, 0, 1},
+        8//7 =>PMFC{3, 1, 0, 3, 0, 0}, # PSU2_5
+        -8//7 => PMFC{3, 1, 0, 3, 0, 1}, # TimeReversed{PSU2_5}
     ),
     Dict(
         (0//1, 1) => PMFC{4, 1, 0, 1, 0, 2}, # Toric code ≅ Z(Rep[Z₂])
         # 4//1 => PMFC{4, 1, 0, 1, 0, 3}, # ThreeFermion
         (0//1, 2) => PMFC{4, 1, 0, 1, 1, 0}, # Z(Semion) ≅ Semion ⊠ TimeReversed{Semion}
         2//1 => PMFC{4, 1, 0, 1, 1, 1}, # Semion ⊠ Semion
-        # -2//1 => PMFC{4, 1, 0, 1, 1, ?}, # TimeReversed{Semion} ⊠ TimeReversed{Semion}, but with error!
-        1//1 => PMFC{4, 1, 2, 1, 2, 0}, # Four Z₄ MTCs
+        # -2//1 => PMFC{4, 1, 0, 1, 1, ?}, # TimeReversed{Semion} ⊠ TimeReversed{Semion}
+        1//1 => PMFC{4, 1, 2, 1, 2, 0}, # U1_2
         3//1 => PMFC{4, 1, 2, 1, 2, 1},
         -3//1 => PMFC{4, 1, 2, 1, 2, 2},
-        -1//1 => PMFC{4, 1, 2, 1, 2, 3},
+        -1//1 => PMFC{4, 1, 2, 1, 2, 3}, # TimeReversed{U1_2}
         -9//5 => PMFC{4, 1, 0, 2, 1, 0}, # Semion ⊠ TimeReversed{Fib}
         19//5 => PMFC{4, 1, 0, 2, 1, 1}, # Semion ⊠ Fib
         -19//5 => PMFC{4, 1, 0, 2, 1, 2}, # TimeReversed{Semion} ⊠ TimeReversed{Fib}

From e5115f23f3c3fc5e0dec33407b983a86fe6ce0e4 Mon Sep 17 00:00:00 2001
From: Chenqi Meng <mcqgravity@gmail.com>
Date: Wed, 1 Apr 2026 18:46:34 +0800
Subject: [PATCH 06/11] Update aliases.jl

---
 src/aliases.jl | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/src/aliases.jl b/src/aliases.jl
index 612fc00..15e3479 100644
--- a/src/aliases.jl
+++ b/src/aliases.jl
@@ -35,7 +35,7 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 
 # Varia
 @objectnames Fib = PMFC{2, 1, 0, 2, 0, 0} I τ
-@objectnames sVec = PMFC{2, 1, 0, 1, 0, 1} I ψ
+@objectnames sVec = PMFC{2, 1, 0, 1, 0, 1} I ψ # Non-modular
 @objectnames Semion = PMFC{2, 1, 0, 1, 1, 0} I ϵ
 @objectnames U1_1 = PMFC{2, 1, 0, 1, 1, 0} 0 1
 @objectnames SU2_1 = PMFC{2, 1, 0, 1, 1, 0} 0 1
@@ -48,11 +48,11 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 
 @objectnames ToricCode = PMFC{4, 1, 0, 1, 0, 2} I e m ψ
 # @objectnames ThreeFermion = PMFC{4, 1, 0, 1, 0, 3} I f1 f2 f3 
-@objectnames Z2sVec = PMFC{4, 1, 0, 1, 0, 1}
+@objectnames Z2sVec = PMFC{4, 1, 0, 1, 0, 1} 0I 1I 0ψ 1ψ # Non-modular
 @objectnames ZSemion = PMFC{4, 1, 0, 1, 1, 0} II ϵI ϵϵ Iϵ
 @objectnames SemionSemion = PMFC{4, 1, 0, 1, 1, 1} II ϵI ϵϵ Iϵ
+@objectnames sVecSemion = PMFC{4, 1, 0, 1, 1, 2} II Iϵ ψI ψϵ # Non-modular
 # @objectnames Z2Semion = PMFC{4, 1, 0, 1, 1, ?}
-# @objectnames sVecSemion = = PMFC{4, 1, 0, 1, 1, ?}
 @objectnames SemionFib = PMFC{4, 1, 0, 2, 1, 1} II ϵI ϵτ Iτ
 @objectnames U1_2 = PMFC{4, 1, 2, 1, 2, 0} 0 1 2 3
 @objectnames ZFib = PMFC{4, 1, 0, 5, 0, 1} II τI Iτ ττ
@@ -68,7 +68,7 @@ _umtcs_lowrank_centralcharge = [
         1//1 => PMFC{2, 1, 0, 1, 1, 0}, # Semion, U1_1, SU2_1
         -1//1 => PMFC{2, 1, 0, 1, 1, 1}, # TimeReversed{Semion}
         -14//5 => PMFC{2, 1, 0, 2, 0, 0}, # TimeReversed{Fib}
-        14//5 => PMFC{2, 1, 0, 2, 0, 1}, # Fib
+        14//5 => PMFC{2, 1, 0, 2, 0, 1}, # Fib, PSU2_3
     ),
     Dict(
         2//1 => PMFC{3, 1, 2, 1, 0, 1}, # Z3MTC

From f5d44e1c2ad087b392d4b7ee187e70ea29587627 Mon Sep 17 00:00:00 2001
From: Chenqi Meng <mcqgravity@gmail.com>
Date: Thu, 2 Apr 2026 15:53:26 +0800
Subject: [PATCH 07/11] Identified PSU2_9

---
 src/aliases.jl | 8 ++++++--
 1 file changed, 6 insertions(+), 2 deletions(-)

diff --git a/src/aliases.jl b/src/aliases.jl
index 15e3479..7daa965 100644
--- a/src/aliases.jl
+++ b/src/aliases.jl
@@ -14,6 +14,9 @@ const RepD5 = PMFC{4, 1, 0, 3, 0, 0}
 const RepD6 = PMFC{6, 1, 0, 2, 0, 0}
 const RepD7 = PMFC{5, 1, 0, 4, 0, 0}
 
+# Rep[Q₈]
+# const RepQ8 = PMFC{5, 1, 0, 1, ?, ?}
+
 # Vec[Dₙ]
 const VecD3 = UFC{6, 1, 2, 1, 0}
 
@@ -61,7 +64,7 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 
 @objectnames SU2_4 = PMFC{5, 1, 0, 3, 0, 0} 0 4 3 1 2
 # @objectnames JK4
-# @objectnames PSU2_9
+@objectnames PSU2_9 = PMFC{5, 1, 0, 10, 0, 1} 0 8 2 6 4
 
 _umtcs_lowrank_centralcharge = [
     Dict(
@@ -111,7 +114,8 @@ _umtcs_lowrank_centralcharge = [
         # Cannot find TimeReversed{SU2_4}, but it should be here with PMFC{5, 1, 0, 3, 0, 1}
         (2//1, 2) => PMFC{5, 1, 0, 3, 1, 0}, # TimeReversed{JK4} http://dx.doi.org/10.1103/PhysRevA.92.012301
         # Cannot find JK4, but it should be here somewhere with PMFC{5, 1, 0, 3, 1, 1}
-        # Cannot find PSU2_9 and its time-reversal, but they should be here somewhere with PMFC{5, 1, 0, x, y, z} with c = ± 16//11
+        -16//11 => PMFC{5, 1, 0, 10, 0, 0}, # TimeReversed{PSU2_9}
+        16//11 => PMFC{5, 1, 0, 10, 0, 1}, # PSU2_9
         # Cannot find c = ± 18//7 UMTCs. They are not SimpleFusion. Should be PMFC{5, 2, ?, ?, ?, ?}
     )
 ]

From aa0eff3b1ee059568efbf0083b2b75bd15cce772 Mon Sep 17 00:00:00 2001
From: Chenqi Meng <mcqgravity@gmail.com>
Date: Thu, 2 Apr 2026 18:20:57 +0800
Subject: [PATCH 08/11] Update aliases.jl

---
 src/aliases.jl | 23 +++++++++++++++--------
 1 file changed, 15 insertions(+), 8 deletions(-)

diff --git a/src/aliases.jl b/src/aliases.jl
index 7daa965..8422078 100644
--- a/src/aliases.jl
+++ b/src/aliases.jl
@@ -37,6 +37,8 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 @objectnames ZVecS3 = ZVecS3 A B C F G H D E
 
 # Varia
+
+# All level-2 PMFCs are identified
 @objectnames Fib = PMFC{2, 1, 0, 2, 0, 0} I τ
 @objectnames sVec = PMFC{2, 1, 0, 1, 0, 1} I ψ # Non-modular
 @objectnames Semion = PMFC{2, 1, 0, 1, 1, 0} I ϵ
@@ -44,26 +46,31 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 @objectnames SU2_1 = PMFC{2, 1, 0, 1, 1, 0} 0 1
 @objectnames PSU2_3 = PMFC{2, 1, 0, 2, 0, 0} 0 2
 
+# All level-3 PMFCs are identified
 @objectnames Z3MTC = PMFC{3, 1, 2, 1, 0, 1} 0 1 2
 @objectnames Ising = PMFC{3, 1, 0, 1, 1, 3} I ψ σ
 @objectnames SU2_2 = PMFC{3, 1, 0, 1, 0, 3} 0 2 1
+@objectnames PSU2_4 = PMFC{3, 1, 0, 2, 0, 1} 0 4 2 # Non-modular. {0, 4} form a Rep[Z₂] braided subcategory.
 @objectnames PSU2_5 = PMFC{3, 1, 0, 3, 0, 0} 0 4 2
 
+
 @objectnames ToricCode = PMFC{4, 1, 0, 1, 0, 2} I e m ψ
-# @objectnames ThreeFermion = PMFC{4, 1, 0, 1, 0, 3} I f1 f2 f3 
+# @objectnames ThreeFermion = PMFC{4, 1, 0, 1, 0, 3} I f1 f2 f3 # Not found. Should be here.
 @objectnames Z2sVec = PMFC{4, 1, 0, 1, 0, 1} 0I 1I 0ψ 1ψ # Non-modular
 @objectnames ZSemion = PMFC{4, 1, 0, 1, 1, 0} II ϵI ϵϵ Iϵ
 @objectnames SemionSemion = PMFC{4, 1, 0, 1, 1, 1} II ϵI ϵϵ Iϵ
 @objectnames sVecSemion = PMFC{4, 1, 0, 1, 1, 2} II Iϵ ψI ψϵ # Non-modular
-# @objectnames Z2Semion = PMFC{4, 1, 0, 1, 1, ?}
+# @objectnames Z2Semion = PMFC{4, 1, 0, 1, 1, ?} # Non-modular and not found. Should be here.
 @objectnames SemionFib = PMFC{4, 1, 0, 2, 1, 1} II ϵI ϵτ Iτ
-@objectnames U1_2 = PMFC{4, 1, 2, 1, 2, 0} 0 1 2 3
+@objectnames PSU2_6 = PMFC{4, 1, 0, 4, 0, 0} 0 6 2 4 # Non-modular. Hard to distinguish 2 and 4. They seems to be equivalent. {0, 6} form a sVec braided subcategory.
 @objectnames ZFib = PMFC{4, 1, 0, 5, 0, 1} II τI Iτ ττ
-# @objectnames FibFib = PMFC{4, 1, 0, 5, 0, 2} II τI Iτ ττ
+# @objectnames FibFib = PMFC{4, 1, 0, 5, 0, 2} II τI Iτ ττ # Not found. Should be here.
 @objectnames PSU2_7 = PMFC{4, 1, 0, 6, 0, 0} 0 6 2 4
+@objectnames U1_2 = PMFC{4, 1, 2, 1, 2, 0} 0 1 2 3
 
 @objectnames SU2_4 = PMFC{5, 1, 0, 3, 0, 0} 0 4 3 1 2
-# @objectnames JK4
+# @objectnames JK4 = PMFC{5, 1, 0, 3, 1, 1} 0 4 3 1 2 # Not found. Should be here.
+@objectnames PSU2_8 = PMFC{5, 1, 0, 7, 0, 0} 0 8 2 6 4 # Non-modular. Hard to distinguish 2 and 6. They seems to be equivalent. {0, 8} form a Rep[Z₂] braided subcategory.
 @objectnames PSU2_9 = PMFC{5, 1, 0, 10, 0, 1} 0 8 2 6 4
 
 _umtcs_lowrank_centralcharge = [
@@ -89,10 +96,10 @@ _umtcs_lowrank_centralcharge = [
     ),
     Dict(
         (0//1, 1) => PMFC{4, 1, 0, 1, 0, 2}, # Toric code ≅ Z(Rep[Z₂])
-        # 4//1 => PMFC{4, 1, 0, 1, 0, 3}, # ThreeFermion
+        # 4//1 => PMFC{4, 1, 0, 1, 0, 3}, # ThreeFermion Not found. Should be here.
         (0//1, 2) => PMFC{4, 1, 0, 1, 1, 0}, # Z(Semion) ≅ Semion ⊠ TimeReversed{Semion}
         2//1 => PMFC{4, 1, 0, 1, 1, 1}, # Semion ⊠ Semion
-        # -2//1 => PMFC{4, 1, 0, 1, 1, ?}, # TimeReversed{Semion} ⊠ TimeReversed{Semion}
+        # -2//1 => PMFC{4, 1, 0, 1, 1, ?}, # TimeReversed{Semion} ⊠ TimeReversed{Semion} Not found. Should be here.
         1//1 => PMFC{4, 1, 2, 1, 2, 0}, # U1_2
         3//1 => PMFC{4, 1, 2, 1, 2, 1},
         -3//1 => PMFC{4, 1, 2, 1, 2, 2},
@@ -103,7 +110,7 @@ _umtcs_lowrank_centralcharge = [
         9//5 => PMFC{4, 1, 0, 2, 1, 3}, # TimeReversed{Semion} ⊠ Fib
         12//5 => PMFC{4, 1, 0, 5, 0, 0}, # TimeReversed{Fib} ⊠ TimeReversed{Fib}
         (0//1, 3) => PMFC{4, 1, 0, 5, 0, 1}, # TimeReversed{Fib} ⊠ Fib
-        # -12//5 => PMFC{4, 1, 0, 5, 0, 2}, # Fib ⊠ Fib
+        # -12//5 => PMFC{4, 1, 0, 5, 0, 2}, # Fib ⊠ Fib Not found. Should be here.
         10//3 => PMFC{4, 1, 0, 6, 0, 0}, # PSU2_7
         -10//3 => PMFC{4, 1, 0, 6, 0, 1}, # TimeReversed{PSU2_7}
     ),

From c3d59006e669b404586277e4e7699f9b5fd7d00f Mon Sep 17 00:00:00 2001
From: Chenqi Meng <mcqgravity@gmail.com>
Date: Thu, 2 Apr 2026 19:58:41 +0800
Subject: [PATCH 09/11] Update aliases.jl

---
 src/aliases.jl | 18 ++++++++++--------
 1 file changed, 10 insertions(+), 8 deletions(-)

diff --git a/src/aliases.jl b/src/aliases.jl
index 8422078..5fe4776 100644
--- a/src/aliases.jl
+++ b/src/aliases.jl
@@ -38,7 +38,7 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 
 # Varia
 
-# All level-2 PMFCs are identified
+# All rank-2 PMFCs are identified
 @objectnames Fib = PMFC{2, 1, 0, 2, 0, 0} I τ
 @objectnames sVec = PMFC{2, 1, 0, 1, 0, 1} I ψ # Non-modular
 @objectnames Semion = PMFC{2, 1, 0, 1, 1, 0} I ϵ
@@ -46,27 +46,29 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 @objectnames SU2_1 = PMFC{2, 1, 0, 1, 1, 0} 0 1
 @objectnames PSU2_3 = PMFC{2, 1, 0, 2, 0, 0} 0 2
 
-# All level-3 PMFCs are identified
-@objectnames Z3MTC = PMFC{3, 1, 2, 1, 0, 1} 0 1 2
-@objectnames Ising = PMFC{3, 1, 0, 1, 1, 3} I ψ σ
+# All rank-3 PMFCs are identified
 @objectnames SU2_2 = PMFC{3, 1, 0, 1, 0, 3} 0 2 1
+@objectnames Ising = PMFC{3, 1, 0, 1, 1, 3} I ψ σ
 @objectnames PSU2_4 = PMFC{3, 1, 0, 2, 0, 1} 0 4 2 # Non-modular. {0, 4} form a Rep[Z₂] braided subcategory.
 @objectnames PSU2_5 = PMFC{3, 1, 0, 3, 0, 0} 0 4 2
+@objectnames Z3MTC = PMFC{3, 1, 2, 1, 0, 1} 0 1 2
 
-
+@objectnames Z2sVec = PMFC{4, 1, 0, 1, 0, 1} 0I 1I 0ψ 1ψ # Non-modular
 @objectnames ToricCode = PMFC{4, 1, 0, 1, 0, 2} I e m ψ
 # @objectnames ThreeFermion = PMFC{4, 1, 0, 1, 0, 3} I f1 f2 f3 # Not found. Should be here.
-@objectnames Z2sVec = PMFC{4, 1, 0, 1, 0, 1} 0I 1I 0ψ 1ψ # Non-modular
 @objectnames ZSemion = PMFC{4, 1, 0, 1, 1, 0} II ϵI ϵϵ Iϵ
 @objectnames SemionSemion = PMFC{4, 1, 0, 1, 1, 1} II ϵI ϵϵ Iϵ
 @objectnames sVecSemion = PMFC{4, 1, 0, 1, 1, 2} II Iϵ ψI ψϵ # Non-modular
 # @objectnames Z2Semion = PMFC{4, 1, 0, 1, 1, ?} # Non-modular and not found. Should be here.
-@objectnames SemionFib = PMFC{4, 1, 0, 2, 1, 1} II ϵI ϵτ Iτ
+@objectnames Z2Fib = PMFC{4, 1, 0, 2, 0, 2} 0I 1I 1τ 0τ # Non-modular.
+@objectnames sVecFib = PMFC{4, 1, 0, 2, 0, 3} II ψI Iτ ψτ # Non-modular.
+@objectnames SemionFib = PMFC{4, 1, 0, 2, 1, 1} II ϵI ϵτ Iτ # Others can be obtained by taking braided reverse to each layers respectively.
 @objectnames PSU2_6 = PMFC{4, 1, 0, 4, 0, 0} 0 6 2 4 # Non-modular. Hard to distinguish 2 and 4. They seems to be equivalent. {0, 6} form a sVec braided subcategory.
 @objectnames ZFib = PMFC{4, 1, 0, 5, 0, 1} II τI Iτ ττ
 # @objectnames FibFib = PMFC{4, 1, 0, 5, 0, 2} II τI Iτ ττ # Not found. Should be here.
 @objectnames PSU2_7 = PMFC{4, 1, 0, 6, 0, 0} 0 6 2 4
-@objectnames U1_2 = PMFC{4, 1, 2, 1, 2, 0} 0 1 2 3
+@objectnames sRepZ4 = PMFC{4, 1, 2, 1, 0, 2} 0 2 1 3
+@objectnames U1_2 = PMFC{4, 1, 2, 1, 2, 0} 0 2 1 3
 
 @objectnames SU2_4 = PMFC{5, 1, 0, 3, 0, 0} 0 4 3 1 2
 # @objectnames JK4 = PMFC{5, 1, 0, 3, 1, 1} 0 4 3 1 2 # Not found. Should be here.

From c33bb31aadbdb75ebb39da5f757f5344918742ca Mon Sep 17 00:00:00 2001
From: Chenqi Meng <mcqgravity@gmail.com>
Date: Thu, 2 Apr 2026 20:42:03 +0800
Subject: [PATCH 10/11] format

---
 src/aliases.jl | 92 +++++++++++++++++++++++++++-----------------------
 1 file changed, 49 insertions(+), 43 deletions(-)

diff --git a/src/aliases.jl b/src/aliases.jl
index 5fe4776..af24c32 100644
--- a/src/aliases.jl
+++ b/src/aliases.jl
@@ -38,7 +38,7 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 
 # Varia
 
-# All rank-2 PMFCs are identified
+# All rank-2 PMFCs are identified!
 @objectnames Fib = PMFC{2, 1, 0, 2, 0, 0} I τ
 @objectnames sVec = PMFC{2, 1, 0, 1, 0, 1} I ψ # Non-modular
 @objectnames Semion = PMFC{2, 1, 0, 1, 1, 0} I ϵ
@@ -46,10 +46,14 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 @objectnames SU2_1 = PMFC{2, 1, 0, 1, 1, 0} 0 1
 @objectnames PSU2_3 = PMFC{2, 1, 0, 2, 0, 0} 0 2
 
-# All rank-3 PMFCs are identified
+# All rank-3 PMFCs are identified!
 @objectnames SU2_2 = PMFC{3, 1, 0, 1, 0, 3} 0 2 1
 @objectnames Ising = PMFC{3, 1, 0, 1, 1, 3} I ψ σ
-@objectnames PSU2_4 = PMFC{3, 1, 0, 2, 0, 1} 0 4 2 # Non-modular. {0, 4} form a Rep[Z₂] braided subcategory.
+@objectnames Ising1 = PMFC{3, 1, 0, 1, 1, 3} I ψ σ # c = 1//2 Ising type UMTC
+@objectnames Ising3 = PMFC{3, 1, 0, 1, 0, 3} I ψ σ # c = 3//2 Ising type UMTC
+@objectnames Ising5 = PMFC{3, 1, 0, 1, 0, 0} I ψ σ # c = 5//2 Ising type UMTC
+@objectnames Ising7 = PMFC{3, 1, 0, 1, 1, 0} I ψ σ # c = 7//2 Ising type UMTC
+@objectnames PSU2_4 = PMFC{3, 1, 0, 2, 0, 1} 0 4 2 # Non-modular. {0, 4} form a Rep[Z₂] braided subcategory. Have the same fusion rule as Rep[D₃], but different braiding.
 @objectnames PSU2_5 = PMFC{3, 1, 0, 3, 0, 0} 0 4 2
 @objectnames Z3MTC = PMFC{3, 1, 2, 1, 0, 1} 0 1 2
 
@@ -62,14 +66,16 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 # @objectnames Z2Semion = PMFC{4, 1, 0, 1, 1, ?} # Non-modular and not found. Should be here.
 @objectnames Z2Fib = PMFC{4, 1, 0, 2, 0, 2} 0I 1I 1τ 0τ # Non-modular.
 @objectnames sVecFib = PMFC{4, 1, 0, 2, 0, 3} II ψI Iτ ψτ # Non-modular.
-@objectnames SemionFib = PMFC{4, 1, 0, 2, 1, 1} II ϵI ϵτ Iτ # Others can be obtained by taking braided reverse to each layers respectively.
+@objectnames SemionFib = PMFC{4, 1, 0, 2, 1, 1} II ϵI ϵτ Iτ # Others can be obtained by taking braided-reverse to each layers respectively.
+# How to understand PMFC{4, 1, 0, 3, 0, 1} and PMFC{4, 1, 0, 3, 0, 2}? They are non-modular, with the same fusion rule as Rep[D₅], with the Muger center Rep[Z₂]. What is its minimal modular extension?
 @objectnames PSU2_6 = PMFC{4, 1, 0, 4, 0, 0} 0 6 2 4 # Non-modular. Hard to distinguish 2 and 4. They seems to be equivalent. {0, 6} form a sVec braided subcategory.
 @objectnames ZFib = PMFC{4, 1, 0, 5, 0, 1} II τI Iτ ττ
 # @objectnames FibFib = PMFC{4, 1, 0, 5, 0, 2} II τI Iτ ττ # Not found. Should be here.
 @objectnames PSU2_7 = PMFC{4, 1, 0, 6, 0, 0} 0 6 2 4
-@objectnames sRepZ4 = PMFC{4, 1, 2, 1, 0, 2} 0 2 1 3
+@objectnames sRepZ4 = PMFC{4, 1, 2, 1, 0, 2} 0 2 1 3 # Non-modular. {0, 2} form a Rep[Z₂] braided subcategory.
 @objectnames U1_2 = PMFC{4, 1, 2, 1, 2, 0} 0 2 1 3
 
+
 @objectnames SU2_4 = PMFC{5, 1, 0, 3, 0, 0} 0 4 3 1 2
 # @objectnames JK4 = PMFC{5, 1, 0, 3, 1, 1} 0 4 3 1 2 # Not found. Should be here.
 @objectnames PSU2_8 = PMFC{5, 1, 0, 7, 0, 0} 0 8 2 6 4 # Non-modular. Hard to distinguish 2 and 6. They seems to be equivalent. {0, 8} form a Rep[Z₂] braided subcategory.
@@ -77,54 +83,54 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 
 _umtcs_lowrank_centralcharge = [
     Dict(
-        1//1 => PMFC{2, 1, 0, 1, 1, 0}, # Semion, U1_1, SU2_1
-        -1//1 => PMFC{2, 1, 0, 1, 1, 1}, # TimeReversed{Semion}
-        -14//5 => PMFC{2, 1, 0, 2, 0, 0}, # TimeReversed{Fib}
-        14//5 => PMFC{2, 1, 0, 2, 0, 1}, # Fib, PSU2_3
+        1 // 1 => PMFC{2, 1, 0, 1, 1, 0}, # Semion, U1_1, SU2_1
+        -1 // 1 => PMFC{2, 1, 0, 1, 1, 1}, # TimeReversed{Semion}
+        -14 // 5 => PMFC{2, 1, 0, 2, 0, 0}, # TimeReversed{Fib}
+        14 // 5 => PMFC{2, 1, 0, 2, 0, 1}, # Fib, PSU2_3
     ),
     Dict(
-        2//1 => PMFC{3, 1, 2, 1, 0, 1}, # Z3MTC
-        -2//1 => PMFC{3, 1, 2, 1, 0, 2}, # TimeReversed{Z3MTC}
-        5//2 => PMFC{3, 1, 0, 1, 0, 0}, # The following are 8-fold ways of Ising-type UMTCs
-        -3//2 => PMFC{3, 1, 0, 1, 0, 1}, # TimeReversed{SU2_2}
-        -5//2 => PMFC{3, 1, 0, 1, 0, 2},
-        3//2 => PMFC{3, 1, 0, 1, 0, 3}, # SU2_2
-        7//2 => PMFC{3, 1, 0, 1, 1, 0},
-        -1//2 => PMFC{3, 1, 0, 1, 1, 1}, # TimeReversed{Ising}
-        -7//2 => PMFC{3, 1, 0, 1, 1, 2},
-        1//2 => PMFC{3, 1, 0, 1, 1, 3}, # Ising
-        8//7 =>PMFC{3, 1, 0, 3, 0, 0}, # PSU2_5
-        -8//7 => PMFC{3, 1, 0, 3, 0, 1}, # TimeReversed{PSU2_5}
+        2 // 1 => PMFC{3, 1, 2, 1, 0, 1}, # Z3MTC
+        -2 // 1 => PMFC{3, 1, 2, 1, 0, 2}, # TimeReversed{Z3MTC}
+        5 // 2 => PMFC{3, 1, 0, 1, 0, 0}, # The following are 8-fold ways of Ising-type UMTCs
+        -3 // 2 => PMFC{3, 1, 0, 1, 0, 1}, # TimeReversed{SU2_2}
+        -5 // 2 => PMFC{3, 1, 0, 1, 0, 2},
+        3 // 2 => PMFC{3, 1, 0, 1, 0, 3}, # SU2_2
+        7 // 2 => PMFC{3, 1, 0, 1, 1, 0},
+        -1 // 2 => PMFC{3, 1, 0, 1, 1, 1}, # TimeReversed{Ising}
+        -7 // 2 => PMFC{3, 1, 0, 1, 1, 2},
+        1 // 2 => PMFC{3, 1, 0, 1, 1, 3}, # Ising
+        8 // 7 => PMFC{3, 1, 0, 3, 0, 0}, # PSU2_5
+        -8 // 7 => PMFC{3, 1, 0, 3, 0, 1}, # TimeReversed{PSU2_5}
     ),
     Dict(
-        (0//1, 1) => PMFC{4, 1, 0, 1, 0, 2}, # Toric code ≅ Z(Rep[Z₂])
+        (0 // 1, 1) => PMFC{4, 1, 0, 1, 0, 2}, # Toric code ≅ Z(Rep[Z₂])
         # 4//1 => PMFC{4, 1, 0, 1, 0, 3}, # ThreeFermion Not found. Should be here.
-        (0//1, 2) => PMFC{4, 1, 0, 1, 1, 0}, # Z(Semion) ≅ Semion ⊠ TimeReversed{Semion}
-        2//1 => PMFC{4, 1, 0, 1, 1, 1}, # Semion ⊠ Semion
+        (0 // 1, 2) => PMFC{4, 1, 0, 1, 1, 0}, # Z(Semion) ≅ Semion ⊠ TimeReversed{Semion}
+        2 // 1 => PMFC{4, 1, 0, 1, 1, 1}, # Semion ⊠ Semion
         # -2//1 => PMFC{4, 1, 0, 1, 1, ?}, # TimeReversed{Semion} ⊠ TimeReversed{Semion} Not found. Should be here.
-        1//1 => PMFC{4, 1, 2, 1, 2, 0}, # U1_2
-        3//1 => PMFC{4, 1, 2, 1, 2, 1},
-        -3//1 => PMFC{4, 1, 2, 1, 2, 2},
-        -1//1 => PMFC{4, 1, 2, 1, 2, 3}, # TimeReversed{U1_2}
-        -9//5 => PMFC{4, 1, 0, 2, 1, 0}, # Semion ⊠ TimeReversed{Fib}
-        19//5 => PMFC{4, 1, 0, 2, 1, 1}, # Semion ⊠ Fib
-        -19//5 => PMFC{4, 1, 0, 2, 1, 2}, # TimeReversed{Semion} ⊠ TimeReversed{Fib}
-        9//5 => PMFC{4, 1, 0, 2, 1, 3}, # TimeReversed{Semion} ⊠ Fib
-        12//5 => PMFC{4, 1, 0, 5, 0, 0}, # TimeReversed{Fib} ⊠ TimeReversed{Fib}
-        (0//1, 3) => PMFC{4, 1, 0, 5, 0, 1}, # TimeReversed{Fib} ⊠ Fib
+        1 // 1 => PMFC{4, 1, 2, 1, 2, 0}, # U1_2
+        3 // 1 => PMFC{4, 1, 2, 1, 2, 1},
+        -3 // 1 => PMFC{4, 1, 2, 1, 2, 2},
+        -1 // 1 => PMFC{4, 1, 2, 1, 2, 3}, # TimeReversed{U1_2}
+        -9 // 5 => PMFC{4, 1, 0, 2, 1, 0}, # Semion ⊠ TimeReversed{Fib}
+        19 // 5 => PMFC{4, 1, 0, 2, 1, 1}, # Semion ⊠ Fib
+        -19 // 5 => PMFC{4, 1, 0, 2, 1, 2}, # TimeReversed{Semion} ⊠ TimeReversed{Fib}
+        9 // 5 => PMFC{4, 1, 0, 2, 1, 3}, # TimeReversed{Semion} ⊠ Fib
+        12 // 5 => PMFC{4, 1, 0, 5, 0, 0}, # TimeReversed{Fib} ⊠ TimeReversed{Fib}
+        (0 // 1, 3) => PMFC{4, 1, 0, 5, 0, 1}, # TimeReversed{Fib} ⊠ Fib
         # -12//5 => PMFC{4, 1, 0, 5, 0, 2}, # Fib ⊠ Fib Not found. Should be here.
-        10//3 => PMFC{4, 1, 0, 6, 0, 0}, # PSU2_7
-        -10//3 => PMFC{4, 1, 0, 6, 0, 1}, # TimeReversed{PSU2_7}
+        10 // 3 => PMFC{4, 1, 0, 6, 0, 0}, # PSU2_7
+        -10 // 3 => PMFC{4, 1, 0, 6, 0, 1}, # TimeReversed{PSU2_7}
     ),
     Dict(
-        4//1 => PMFC{5, 1, 4, 1, 0, 1},
-        0//1 => PMFC{5, 1, 4, 1, 0, 2},
-        (2//1, 1) => PMFC{5, 1, 0, 3, 0, 0}, # SU2_4
+        4 // 1 => PMFC{5, 1, 4, 1, 0, 1},
+        0 // 1 => PMFC{5, 1, 4, 1, 0, 2},
+        (2 // 1, 1) => PMFC{5, 1, 0, 3, 0, 0}, # SU2_4
         # Cannot find TimeReversed{SU2_4}, but it should be here with PMFC{5, 1, 0, 3, 0, 1}
-        (2//1, 2) => PMFC{5, 1, 0, 3, 1, 0}, # TimeReversed{JK4} http://dx.doi.org/10.1103/PhysRevA.92.012301
+        (2 // 1, 2) => PMFC{5, 1, 0, 3, 1, 0}, # TimeReversed{JK4} http://dx.doi.org/10.1103/PhysRevA.92.012301
         # Cannot find JK4, but it should be here somewhere with PMFC{5, 1, 0, 3, 1, 1}
-        -16//11 => PMFC{5, 1, 0, 10, 0, 0}, # TimeReversed{PSU2_9}
-        16//11 => PMFC{5, 1, 0, 10, 0, 1}, # PSU2_9
+        -16 // 11 => PMFC{5, 1, 0, 10, 0, 0}, # TimeReversed{PSU2_9}
+        16 // 11 => PMFC{5, 1, 0, 10, 0, 1}, # PSU2_9
         # Cannot find c = ± 18//7 UMTCs. They are not SimpleFusion. Should be PMFC{5, 2, ?, ?, ?, ?}
-    )
+    ),
 ]

From eb3802bd7a41789b6b84089247f260e7c8cf8d75 Mon Sep 17 00:00:00 2001
From: Chenqi Meng <mcqgravity@gmail.com>
Date: Mon, 20 Apr 2026 11:19:50 +0800
Subject: [PATCH 11/11] More systematic alias

---
 src/aliases.jl  | 59 +++++++++++++++++++++++++++++++++++++++----------
 test/aliases.jl |  5 +++++
 2 files changed, 52 insertions(+), 12 deletions(-)
 create mode 100644 test/aliases.jl

diff --git a/src/aliases.jl b/src/aliases.jl
index af24c32..fc7c711 100644
--- a/src/aliases.jl
+++ b/src/aliases.jl
@@ -38,27 +38,63 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 
 # Varia
 
+# --- SU2 ---
+@objectnames SU2_1 = PMFC{2, 1, 0, 1, 1, 0} _0 _1
+@objectnames SU2_2 = PMFC{3, 1, 0, 1, 0, 3} _0 _2 _1
+@objectnames SU2_3 = PMFC{4, 1, 0, 2, 1, 3} _0 _3 _1 _2
+@objectnames SU2_4 = PMFC{5, 1, 0, 3, 0, 0} _0 _4 _3 _1 _2
+@objectnames SU2_5 = PMFC{6, 1, 0, 6, 0, 1} _0 _5 _1 _4 _2 _3
+
+# --- PSU2 ---
+@objectnames PSU2_3 = PMFC{2, 1, 0, 2, 0, 0} _0 _2
+@objectnames PSU2_4 = PMFC{3, 1, 0, 2, 0, 1} _0 _4 _2 # Non-modular. {0, 4} form a Rep[Z₂] braided subcategory. Isomorphic to Rep[D₃] as UFC, but has different braiding.
+@objectnames PSU2_5 = PMFC{3, 1, 0, 3, 0, 0} _0 _4 _2
+@objectnames PSU2_6 = PMFC{4, 1, 0, 4, 0, 0} _0 _6 _4 _2 # Non-modular. Hard to distinguish 2 and 4. They seems to be equivalent. {0, 6} form a sVec braided subcategory.
+@objectnames PSU2_7 = PMFC{4, 1, 0, 6, 0, 0} _0 _6 _2 _4
+@objectnames PSU2_8 = PMFC{5, 1, 0, 7, 0, 0} _0 _8 _2 _6 _4 # Non-modular. Hard to distinguish 2 and 6. They seems to be equivalent. {0, 8} form a Rep[Z₂] braided subcategory.
+@objectnames PSU2_9 = PMFC{5, 1, 0, 10, 0, 1} _0 _8 _2 _6 _4
+@objectnames PSU2_10 = PMFC{6, 1, 0, 16, 0, 0} _0 _10 _8 _2 _6 _4 # Non-modular
+@objectnames PSU2_11 = PMFC{6, 1, 0, 18, 0, 1} _0 _10 _2 _8 _4 _6
+
+# --- Kitaev 16-fold way ---
+@objectnames Kitaev16_0 = PMFC{4, 1, 0, 1, 0, 2} I e m ψ
+@objectnames Kitaev16_1 = PMFC{3, 1, 0, 1, 1, 3} I ψ σ
+@objectnames Kitaev16_2 = PMFC{4, 1, 2, 1, 2, 0} _0 _2 _1 _3
+@objectnames Kitaev16_3 = PMFC{3, 1, 0, 1, 0, 3} I ψ σ
+@objectnames Kitaev16_4 = PMFC{4, 1, 0, 1, 1, 1} II ϵI ϵϵ Iϵ
+@objectnames Kitaev16_5 = PMFC{3, 1, 0, 1, 0, 0} I ψ σ
+@objectnames Kitaev16_6 = PMFC{4, 1, 2, 1, 2, 1} _0 _2 _1 _3
+@objectnames Kitaev16_7 = PMFC{3, 1, 0, 1, 1, 0} I ψ σ
+@objectnames Kitaev16_8 = PMFC{4, 1, 0, 1, 0, :?} I e m ψ
+@objectnames Kitaev16_9 = PMFC{3, 1, 0, 1, 1, 2} I ψ σ
+@objectnames Kitaev16_10 = PMFC{4, 1, 2, 1, 2, 2} _0 _2 _1 _3
+@objectnames Kitaev16_11 = PMFC{3, 1, 0, 1, 0, 2} I ψ σ
+@objectnames Kitaev16_12 = PMFC{4, 1, 0, 1, 1, :?} II ϵI ϵϵ Iϵ
+@objectnames Kitaev16_13 = PMFC{3, 1, 0, 1, 0, 1} I ψ σ
+@objectnames Kitaev16_14 = PMFC{4, 1, 2, 1, 2, 3} _0 _2 _1 _3
+@objectnames Kitaev16_15 = PMFC{3, 1, 0, 1, 1, 1} I ψ σ
+
 # All rank-2 PMFCs are identified!
 @objectnames Fib = PMFC{2, 1, 0, 2, 0, 0} I τ
 @objectnames sVec = PMFC{2, 1, 0, 1, 0, 1} I ψ # Non-modular
 @objectnames Semion = PMFC{2, 1, 0, 1, 1, 0} I ϵ
 @objectnames U1_1 = PMFC{2, 1, 0, 1, 1, 0} 0 1
-@objectnames SU2_1 = PMFC{2, 1, 0, 1, 1, 0} 0 1
-@objectnames PSU2_3 = PMFC{2, 1, 0, 2, 0, 0} 0 2
+
+
 
 # All rank-3 PMFCs are identified!
-@objectnames SU2_2 = PMFC{3, 1, 0, 1, 0, 3} 0 2 1
+
 @objectnames Ising = PMFC{3, 1, 0, 1, 1, 3} I ψ σ
 @objectnames Ising1 = PMFC{3, 1, 0, 1, 1, 3} I ψ σ # c = 1//2 Ising type UMTC
 @objectnames Ising3 = PMFC{3, 1, 0, 1, 0, 3} I ψ σ # c = 3//2 Ising type UMTC
 @objectnames Ising5 = PMFC{3, 1, 0, 1, 0, 0} I ψ σ # c = 5//2 Ising type UMTC
 @objectnames Ising7 = PMFC{3, 1, 0, 1, 1, 0} I ψ σ # c = 7//2 Ising type UMTC
-@objectnames PSU2_4 = PMFC{3, 1, 0, 2, 0, 1} 0 4 2 # Non-modular. {0, 4} form a Rep[Z₂] braided subcategory. Have the same fusion rule as Rep[D₃], but different braiding.
-@objectnames PSU2_5 = PMFC{3, 1, 0, 3, 0, 0} 0 4 2
+ 
+
 @objectnames Z3MTC = PMFC{3, 1, 2, 1, 0, 1} 0 1 2
 
 @objectnames Z2sVec = PMFC{4, 1, 0, 1, 0, 1} 0I 1I 0ψ 1ψ # Non-modular
-@objectnames ToricCode = PMFC{4, 1, 0, 1, 0, 2} I e m ψ
+
 # @objectnames ThreeFermion = PMFC{4, 1, 0, 1, 0, 3} I f1 f2 f3 # Not found. Should be here.
 @objectnames ZSemion = PMFC{4, 1, 0, 1, 1, 0} II ϵI ϵϵ Iϵ
 @objectnames SemionSemion = PMFC{4, 1, 0, 1, 1, 1} II ϵI ϵϵ Iϵ
@@ -67,19 +103,18 @@ Base.getindex(::CenterTable, ::Type{VecS3}) = ZVecS3
 @objectnames Z2Fib = PMFC{4, 1, 0, 2, 0, 2} 0I 1I 1τ 0τ # Non-modular.
 @objectnames sVecFib = PMFC{4, 1, 0, 2, 0, 3} II ψI Iτ ψτ # Non-modular.
 @objectnames SemionFib = PMFC{4, 1, 0, 2, 1, 1} II ϵI ϵτ Iτ # Others can be obtained by taking braided-reverse to each layers respectively.
-# How to understand PMFC{4, 1, 0, 3, 0, 1} and PMFC{4, 1, 0, 3, 0, 2}? They are non-modular, with the same fusion rule as Rep[D₅], with the Muger center Rep[Z₂]. What is its minimal modular extension?
-@objectnames PSU2_6 = PMFC{4, 1, 0, 4, 0, 0} 0 6 2 4 # Non-modular. Hard to distinguish 2 and 4. They seems to be equivalent. {0, 6} form a sVec braided subcategory.
+# How to understand PMFC{4, 1, 0, 3, 0, 1} and PMFC{4, 1, 0, 3, 0, 2}? They are non-modular, isomorphic to Rep[D₅] as UFC, with the Muger center Rep[Z₂]. What is its minimal modular extension?
+
 @objectnames ZFib = PMFC{4, 1, 0, 5, 0, 1} II τI Iτ ττ
 # @objectnames FibFib = PMFC{4, 1, 0, 5, 0, 2} II τI Iτ ττ # Not found. Should be here.
-@objectnames PSU2_7 = PMFC{4, 1, 0, 6, 0, 0} 0 6 2 4
+
 @objectnames sRepZ4 = PMFC{4, 1, 2, 1, 0, 2} 0 2 1 3 # Non-modular. {0, 2} form a Rep[Z₂] braided subcategory.
+# Some other Z4PMFCs are not identified: do not know how to understand. Some are not found.
 @objectnames U1_2 = PMFC{4, 1, 2, 1, 2, 0} 0 2 1 3
 
 
-@objectnames SU2_4 = PMFC{5, 1, 0, 3, 0, 0} 0 4 3 1 2
+
 # @objectnames JK4 = PMFC{5, 1, 0, 3, 1, 1} 0 4 3 1 2 # Not found. Should be here.
-@objectnames PSU2_8 = PMFC{5, 1, 0, 7, 0, 0} 0 8 2 6 4 # Non-modular. Hard to distinguish 2 and 6. They seems to be equivalent. {0, 8} form a Rep[Z₂] braided subcategory.
-@objectnames PSU2_9 = PMFC{5, 1, 0, 10, 0, 1} 0 8 2 6 4
 
 _umtcs_lowrank_centralcharge = [
     Dict(
diff --git a/test/aliases.jl b/test/aliases.jl
new file mode 100644
index 0000000..5484b09
--- /dev/null
+++ b/test/aliases.jl
@@ -0,0 +1,5 @@
+@testset "Test all aliases" begin
+    for (alias, name) in aliases
+        @test alias == name
+    end
+end
\ No newline at end of file