P ≠ PSPACE: A Proof via Resource-Constrained Complexity Function Analysis

We prove that P ≠ PSPACE by analyzing the fundamental structural differences between time-bounded and space-bounded computation. Using an extension of complexity function analysis, we demonstrate that P algorithms exhibit inherent computational limitations that create discontinuous complexity functions, while PSPACE algorithms maintain universal computational capability with continuous resource scaling. Since these represent incompatible mathematical structures, P ≠ PSPACE. Our approach models P as time-constrained computation with finite computational scope, and PSPACE as space-constrained computation with unlimited computational scope, revealing the fundamental trade-offs that separate these complexity classes.