Pensieve header: Solving R4, Cap, and Unitarity without assuming $W=1$. With Dancso, October 2023.

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];
ω=CWS[{x}, ωs]; Wen=Es[⟨x->LS[0]⟩,ω];

R4Eqn=V**(Rs[x,z] // dΔ[x,x,y]) ≡ Rs[y,z]**Rs[x,z]**V ;
CapWen=Cap≡((Wen**Cap)//dA[x]//dS[x]);
CapEqn=((V**(Cap//dΔ[x,x,y])//dc[x]//dc[y]) ≡ (Cap*(Cap//dσ[x,y])//dc[x]//dc[y]));
UnitarityEqn=((V//dA[x]//dA[y])**(Wen*(Wen//dσ[x,y]))**V) ≡ (Wen//dΔ[x,x,y]);

βs[x]= 1/2; βs[y]=0;
ωs[x,x,x]=1;
SeriesSolve[{α, β,γ, κ,ω}, ((ℏ^(-1))R4Eqn)∧CapWen∧CapEqn∧UnitarityEqn];
Column@{V@{5}, κ@{5},ω@{5}}