Variational Diagnostics and Flow-Entropy Synthesis: A Dual-Facet Approach to Essential Singularities and Event Horizon Dynamics

This research introduces a novel framework for establishing the non-existence of limits for multivariable functions characterized by high-order singularities at the origin. We formulate a "Dynamic Flow Model" restricted within a Diagonal Scanning Cube, effectively superseding the traditional epsilon-delta limit formulation with a rigorous topological definition based on "Complex-Dual Potential. " Our findings demonstrate that executing a structural substitution over the region y = i induces a fundamental "Twist" in the manifold of the underlying vector field. Within the localized geometry of the cube, the Curl operator generates rotational torques that cannot be counterbalanced by external Divergence. This mathematical asymmetry, mimicking the intrinsic behavior of "Mathematical Black Holes, " drives the system into a state of high path-entropy. Consequently, we conclude that the limit is not merely conventionally undefined, but rather manifests as a state of permanent dynamic instability within the Complex-Dual space, completely precluding resolution via classical analytical methods.