Toward P≠NP: An Observer-Theoretic Separation via SPDP Rank and a ZFC-Equivalent Foundation within the N-Frame Model

This preprint presents a comprehensive observer-theoretic framework for proving lower bounds in computational complexity, introducing the SPDP (Shifted-Partial-Derivative Projection) rank as a unifying analytic tool. Within this framework, the paper develops a ZFC-equivalent foundation for compiler-based width analysis and establishes a polynomial-time upper bound for the SPDP rank of all P-time computations. It then constructs explicit hard instances exhibiting exponential SPDP rank, thereby demonstrating a formal separation between P and NP under standard assumptions. The work integrates techniques from algebraic complexity, expander-based identity minors, and diagonal compilation to provide a structured, verifiable pathway toward resolving the P vs NP question inside ZFC.