A Structural unified budgetary proof for the P vs NP Problem

We resolve the Pvs NP problem by mapping computational complexity to the Mandatory Overlap Tax (MOT) framework established in the budgetary proof of the Collatz Conjecture(https://doi.org/10.5281/zenodo.18736801). By defining a deterministic algorithm as a "Growth Surplus" (S) system and the verification of NPspace as an "Overlap Tax" (W) system, we derive a Universal Deficit Ratio of W=2S. We prove that the structural bit-erasure required to resolve an NP singularity is exactly double the bit-generation available to a Palgorithm, necessitating a permanent complexity gap ( P≠ NP).