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

TL;DR

This paper introduces a novel ZFC-based framework that uses SPDP rank and a new complexity measure called CEW to theoretically separate P from NP, providing a rigorous foundation for the P vs NP problem.

Contribution

It develops a self-contained, formal separation framework within ZFC that connects SPDP rank, CEW, and computational complexity, offering a new approach to P vs NP.

Findings

Bounded CEW observers correspond to P, unbounded CEW to NP.

Exponential SPDP rank for #3SAT implies P != NP.

Constructive lower bounds on SPDP rank using Ramanujan-Tseitin expanders.

Abstract

We present a self-contained separation framework for P vs NP developed entirely within ZFC. The approach consists of: (i) a deterministic, radius-1 compilation from uniform polynomial-time Turing computation to local sum-of-squares (SoS) polynomials with polylogarithmic contextual entanglement width (CEW); (ii) a formal Width-to-Rank upper bound for the resulting SPDP matrices at matching parameters; (iii) an NP-side identity-minor lower bound in the same encoding; and (iv) a rank-monotone, instance-uniform extraction map from the compiled P-side polynomials to the NP family. Together these yield a contradiction under the assumption P = NP, establishing a separation. We develop a correspondence between CEW, viewed as a quantitative measure of computational contextuality, and SPDP rank, yielding a unified criterion for complexity separation. We prove that bounded-CEW observers correspond to polynomial-rank computations (the class P), while unbounded CEW characterizes the class NP. This implies that exponential SPDP rank for #3SAT and related hard families forces P != NP within the standard framework of complexity theory. Key technical components include: (1) constructive lower bounds on SPDP rank via Ramanujan-Tseitin expander families; (2) a non-circular reduction from Turing-machine computation to low-rank polynomial evaluation; (3) a codimension-collapse lemma ensuring that rank amplification cannot occur within polynomial resources; and (4) proofs of barrier immunity against relativization, natural proofs, and algebrization. The result is a complete ZFC proof architecture whose primitives and compositions are fully derived, with community verification and machine-checked formalization left as future work.