Resolution of Singularities in Reciprocal Functions via ..

This technical note demonstrates the application of the Pillar Derivative ($plf'$) framework to the fundamental singularity of the reciprocal function $f(x) = 1/x$. Unlike traditional Newtonian-Leibniz calculus, which encounters numerical divergence at $x=0$ due to the limit $\lim_{h \to 0}$, the Pillar Derivative utilizes a symmetric imaginary displacement $i\epsilon$. By treating $i\epsilon$ as a structural constant rather than a vanishing parameter, we derive a determinate value for the derivative at the origin: $plf'(0) = -1/\epsilon^2$. This result suggests that singularities in real-valued functions can be interpreted as stable projections of a complex-phase manifold, providing a concrete mathematical basis for information preservation in non-continuous systems.