Pensieve Header: Solving the two F equations with perturbative hair, where F is written as an exponential. This time with a corrected F21.
It seems that F is not unique.

SetDirectory["C:\\drorbn\\AcademicPensieve\\Projects\\ScatterAndGlow"];
<< ScatterAndGlow.m

$CutoffDegree=6;
H[h_] := PH[(h /. x[i_] :> z*x[i])+O[z]^$CutoffDegree];

f1coeffs=DeclareSeries[f1[x[1], x[2]], $CutoffDegree];
f2coeffs=DeclareSeries[f2[x[1], x[2]], $CutoffDegree];
PH[f1]

F=S[Exp[a1 Ar[2,1]+a2 Ar[1,2]+Y[1,2,1,PH[f1]]+Y[1,2,2,PH[f2]]]];
F21 = S[Exp[(
a1 Ar[1,2]+a2 Ar[2,1]+Y[2,1,2,PH[f1]]+Y[2,1,1,PH[f2]]
) /. {x[1] -> x[2], x[2] -> x[1]}
]];

lhs1 = Ar[3,0] // S[Exp[Ar[1,3]+Ar[2,3]]] // F

rhs1 = Ar[3,0] // F // S[sigma[1,3],sigma[2,3]]

lhs2=Ar[1,0] // F21 // S[sigma[1,2]]// S[Exp[1/2 Ar[1,1]]] // S[Exp[1/2 Ar[2,2]]]

rhs2= Ar[1,0] // S[Exp[Expand[1/2(Ar[1,1]+Ar[1,2]+Ar[2,1]+Ar[2,2])]]]// F

eqs = {
Coefficient[lhs1, Y[1,2,0]] == Coefficient[rhs1, Y[1,2,0]],
Coefficient[lhs2, Y[1,2,0]] == Coefficient[rhs2, Y[1,2,0]]
}

sol=PHSolve[
eqs,
Join[{a1,a2}, f1coeffs, f2coeffs]
]

CanonicalForm[
S[Exp[Ar[1,3]+Ar[2,3]]] ** F == F ** S[sigma[1,3],sigma[2,3]]/. sol
]

CanonicalForm[
F21**S[sigma[1,2]]**S[Exp[1/2 Ar[1,1]]]**S[Exp[1/2 Ar[2,2]]] == S[Exp[Expand[1/2(Ar[1,1]+Ar[1,2]+Ar[2,1]+Ar[2,2])]]]**F /. sol
]

Union[sol]

Union[sol][[1]] /. a2 -> 0