Pensieve header: Fuller output of for V and Cap. Even fuller: <a href=VCapSolution-to-12.m>VCapSolution-to-12.m</a>.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\WKO4"];
<< FreeLie.m;
<< AwCalculus.m;
Rs[a_,b_]:= Es[⟨a->LS[0], b->LS[LW@a]⟩, CWS[0]];
α=LS[{x,y}, αs]; β=LS[{x,y}, βs];γ=CWS[{x,y}, γs];
V=Es[⟨x->α, y->β⟩, γ];
κ=CWS[{x}, κs]; Cap=Es[⟨x->LS[0]⟩,κ];
R4Eqn=V**(Rs[x,z] // dΔ[x,x,y]) ≡ Rs[y,z]**Rs[x,z]**V ;UnitarityEqn=(V**(V//dA[x]//dA[y]) ≡ Es[⟨x->LS[0], y->LS[0]⟩,CWS[0]]);
CapEqn=((V**(Cap//dΔ[x,x,y])//dc[x]//dc[y]) ≡ (Cap*(Cap//dσ[x,y])//dc[x]//dc[y]));
βs[x]= 1/2; βs[y]=0;
SeriesSolve[{α, β,γ, κ}, ((ℏ^(-1))R4Eqn) && UnitarityEqn && CapEqn];
{V, κ}

V@{8} // Timing

κ@{8}

The same thing, copy-paste ready and machine readable:

Sum[{(Subscript[V[[1]], x])[k], (Subscript[V[[1]], y])[k], V[[2]][k]}, {k,8}] // InputForm

Sum[κ[k], {k,8}] // InputForm

The same thing, machine readable and to degree 11, is in the file quoted below, in the same folder:

Put[Sum[{(Subscript[V[[1]], x])[k], (Subscript[V[[1]], y])[k], V[[2]][k]}, {k,11}], "VCapSolution-to-11.m"];
PutAppend[Sum[κ[k], {k,11}], "VCapSolution-to-11.m"]

{TimeUsed[], MaxMemoryUsed[]}

TwistEqn = (V≡Rs[x,y]**(V//dσ[{x,y}->{y,x}])**(El[⟨x->LS[-LW@y/2], y->LS[-LW@x/2]⟩, CWS[0]] // Γ));
TwistEqn@{6}

TrueQ[TwistEqn]@{6}

TrueQ[TwistEqn]@{11}

{TimeUsed[], MaxMemoryUsed[]}

The same thing, machine readable and to degree 12, is in the file quoted below, in the same folder:

Put[Sum[{(Subscript[V[[1]], x])[k], (Subscript[V[[1]], y])[k], V[[2]][k]}, {k,12}], "VCapSolution-to-12.m"];
PutAppend[Sum[κ[k], {k,12}], "VCapSolution-to-12.m"]

{TimeUsed[], MaxMemoryUsed[]}

TrueQ[TwistEqn]@{12}

Added July 2023: Just for fun, testing the unitarity of (Subscript[Φ, V]):

V

(Subscript[V, 12])=V//dσ[{x,y}->{1,2}];
(Subscript[Φ, V]) = ((((Subscript[V, 12])//dA))^(σ[12,3]))**((((Subscript[V, 12])//dA))^(σ[1,2]))**((Subscript[V, 12])^(σ[2,3]))**((Subscript[V, 12])^(σ[1,23]))

((Subscript[Φ, V])**dA[(Subscript[Φ, V])])@{4}

Es[⟨1->LS[0], 2->LS[0],3->LS[0]⟩,CWS[0]]

UnitarityOfPhiV=((Subscript[Φ, V])**dA[(Subscript[Φ, V])])≡Es[⟨1->LS[0], 2->LS[0],3->LS[0]⟩,CWS[0]]

UnitarityOfPhiV@{7}