No-Go Theorem Requires Modification: An Additional Constant Term $C$ and Its Laurent Series Structure in f(R) gravity

No-go theorems often assert that certain solutions or configurations in gravitational and field-theoretic models cannot exist under a prescribed set of assumptions~\cite{NoGoClassic}. However, recent analyses using a \emph{modified} Robson-Villari-Biancalana (RVB) approach~\cite{RobWilcz1,BanKul1,BanKul2} indicate the presence of an additional integration constant, $C$, that may appear in topological or horizon-based integrals (notably through expansions in Laurent series). This subtle term invalidates the usual dismissal of boundary or horizon contributions and demands a revised understanding of no-go arguments. In this work, we demonstrate how $C$ can be revealed through careful Laurent expansions about the horizon and discuss why ignoring this term undermines the classical no-go statement. We illustrate these ideas with examples in $f(R)$ gravity~\cite{fR-Review1,fR-Review2,fR-Review3}, emphasizing how familiar no-go theorems must be updated to accommodate the extra freedom that $C$ provides.If the threshold for the No-Go theorem can be further lowered and this reduction can be validated through the modified uncertainty principle, it would provide experimental evidence supporting \( f(R) \) gravity theory. This implies that the corrections introduced by these modifications are not merely theoretical constructs but could be tested in high-precision experiments, offering a potential observational window into the validity of \( f(R) \) gravity.