Pensieve header: A first attempt to get $V$ from $\Phi$.

SetDirectory["C:\\drorbn\\AcademicPensieve\\2012-05\\beta5.1"];
<< betaCalculus.m
Clear[ℏ]; Unprotect[C];
$PerturbativeDegree=8;
βSimplify[expr_]:= Replace[
Series[Normal[expr], {ℏ, 0, $PerturbativeDegree}],sd_SeriesData :> MapAt[Expand, sd, 3]
];
βCollect[B[ω_, μ_]] := B[βSimplify[ω],βSimplify[μ]];
{V, C, sol} = Get[Switch[$PerturbativeDegree,
4, "SolutionToDegree4-120523.m",
6, "SolutionToDegree6-120523.m",
8, "SolutionToDegree8-120524.m"
]];
C=C/.(Subscript[κ, 1])->0;
Φ=(Inverse[V] // dP[12,3])**Inverse[V]**(V // dP[2,3])**(V // dP[1,23]);
ν = B[Series[((Sinh[(Subscript[c, 1])ℏ/2])/((Subscript[c, 1])ℏ/2)), {ℏ,0,$PerturbativeDegree}],0];

buckle=(Inverse[Φ]//dP[13,2,4])**(Φ//dP[1,3,2])**Θ[3,2]**Inverse[Φ]**(Φ//dP[12,3,4])

LuckyV=buckle // tη[1] // hη[2] // dm[1,2,1] // tη[3] // hη[4] // dm[3,4,2]

V

Simplify[LuckyV(C//dP[12])Inverse[C](Inverse[C]//dP[2])==V]