UnitaryMasterEquation := 
      Dt[S * C] == 
       kappaDN * Integrate[Exp[phi] * G4 \[Wedge] HodgeDual[G4], X] + 
       (3 * kappaDN^6 * Lambda^2) / 
        ((2*Pi)^7 * PlanckLength^2 * (r^2 + a^2)^4) + 
       MRBConstant * TopologicalF[Sqrt[25/Pi^2] + MRBConstant/7] + 
       (8 * Pi * GravitationalConstant * VacuumRho) / 
        (SpeedOfLight^4 * 3.3125` * 10^-122) + 
       (1729 + 4096)^(1/18)

FluxIntegral := Integrate[Exp[phi] * G4 \[Wedge] HodgeStar[G4], X]

KerrGeometry := (r^2 + a^2)

Needs["PhysicalConstants`"];
{kappaDN, PlanckLength, GravitationalConstant, SpeedOfLight} = 
  {1, NISTValue["Planck length"], NISTValue["Gravitational constant"], 
   NISTValue["Speed of light"]};

(* When flux term -> 0 *)
N[MRBConstant * (5/Pi + MRBConstant/7) + (1.04/3.3125) + (1729+4096)^(1/18), 30]
(* Output: 2.23675417393035178711474380400 *)

"integral e^phi G4 wedge *G4 over X + (3 kappa^6 Lambda^2)/((2pi)^7 l_p^2 (r^2+a^2)^4) + MRBConstant (sqrt(25/pi^2)+MRB/7) + (1.04e-122)/(3.3125e-122) + (1729+4096)^(1/18)"


\begin{align*}
\frac{d}{dt}(S \cdot C) &= \kappa_{DN} \int_X e^\phi G_4 \wedge \ast G_4 + \frac{3\kappa_{DN}^6 \Lambda^2}{(2\pi)^7 \ell_p^2 (r^2 + a^2)^4} \\
&\quad + C_{\text{MRB}} \cdot \mathcal{F}_{\text{top}}\!\left(\sqrt{\frac{25}{\pi^2}} + \frac{C_{\text{MRB}}}{7}\right) + \frac{8\pi G \rho_{\text{vac}}}{c^4 \cdot 3.3125 \times 10^{-122}} + (1729 + 4096)^{1/18}
\end{align*}

(* Load constants *)  
Needs["PhysicalConstants`"];  
kappaDN = 1;  
planckL = NISTValue["Planck length"];  
vacRho = 1.04 * 10^-122;  

(* Define terms *)  
topoTerm = MRBConstant * (Sqrt[25/Pi^2] + MRBConstant/7);  
cosmoTerm = vacRho / (3.3125 * 10^-122);  
goldenTerm = (1729 + 4096)^(1/18);  

(* Numerical output *)  
result = topoTerm + cosmoTerm + goldenTerm;  
Print["Master Equation Output: ", N[result, 30]];  
Print["Comparison to Sqrt[5]: ", N[Sqrt[5], 30]];  

Master Equation Output: 2.23675417393035178711474380400  
Comparison to Sqrt[5]: 2.23606797749978969640917366873

(((((3*((2sqrt2)/Pi)^6*(1.040*10^-122)^2))/((((2Pi)^7*(1.616255*10^-35)^2))))) + MRB const * (sqrt(25/(Pi^2))+1/7*MRB const) + (1.040*10^-122)/(3.3125*10^-122) + (1729+4096)^(1/18)) 
Input interpretation
(3 ((2 sqrt(2))/π)^6 (1.04/10^122)^2)/((2 π)^7 (1.616255×10^(-35))^2) + C_MRB (sqrt(25/π^2) + 1/7 C_MRB) + (1.04/10^122)/(3.3125/10^122) + (1729 + 4096)^(1/18)
Result
2.23675... about  √5

Further:

Input interpretation

((3 ((2 sqrt(2))/π)^6 (1.04/10^122)^2)/((2 π)^7 (1.616255×10^(-35))^2) + C_MRB (sqrt(25/π^2) + 1/7 C_MRB) + (1.04/10^122)/(3.3125/10^122) + (1729 + 4096)^(1/18)) - 1/ϕ

Result
1.61872018518045693891015696963435387311452380230307633987598097343880035...