Pensieve header: Trying to understand the U(1)-indeterminacy of KV solutions.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2015-11"];
Get["../Projects/WKO4/FreeLie.m"];
Get["../Projects/WKO4/AwCalculus.m"];

x=LW["x"]; y=LW["y"]; z=LW["z"];
Rl[a_,b_]:= El[⟨a->LS[0], b->LS[LW@a]⟩, CWS[0]];
iRl[a_,b_]:= El[⟨a->LS[0], b->-LS[LW@a]⟩, CWS[0]];
Rs[a_,b_]:= Es[⟨a->LS[0], b->LS[LW@a]⟩, CWS[0]];
iRs[a_,b_]:= Es[⟨a->LS[0], b->-LS[LW@a]⟩, CWS[0]];
R4Eqn[V_] := V**(Rs[x,z] // dΔ[x,x,y]) ≡ Rs[y,z]**Rs[x,z]**V;UnitarityEqn[V_] := V**(V//dA) ≡ Es[⟨x->LS[0], y->LS[0]⟩,CWS[0]];
CapEqn[V_,Cap_] := (V**(Cap//dΔ[x,x,y])//dc[x]//dc[y]) ≡ (Cap*(Cap//dσ[x,y])//dc[x]//dc[y]);

κs[xs___] := κs[xs] = SeriesCoefficient[(1/4)Log[((ℏ/2)/(Sinh[ℏ/2]))], {ℏ,0,Length@{xs}}];
κ=CWS[{x}, κs];
(Subscript[Cap, 0])=Es[⟨x->LS[0]⟩,κ];
κ@{16}

SymmetryFactor[l_List] := Module[{ll=l, sf=1},
While[(ll=RotateLeft[ll]) =!= l, ++sf];
sf];
SymmetryFactor[{1,1,1,2,1,1,1,2}]

γs[xs___] := γs[xs]= If[{xs}===RotateLeft@{xs}, 0, SymmetryFactor[{xs}]]*κs[xs];
γ=CWS[{x,y}, γs];
γ@{6}

α=LS[{x,y}, αs]; β=LS[{x,y}, βs];
(Subscript[V, 0])=Es[⟨x->α, y->β⟩, γ];
βs["x"]= 1/4; βs["y"]=0;
SeriesSolve[{α, β}, ((ℏ^(-1))R4Eqn[(Subscript[V, 0])]) ∧ UnitarityEqn[(Subscript[V, 0])] ∧ CapEqn[(Subscript[V, 0]),(Subscript[Cap, 0])]];
(Subscript[V, 0])@{8} // Timing

Timing[(Subscript[V, 0])@{11};]