On the Structural Gap Between Verification and Existence in NP

This paper examines the persistent intuition behind the P vs NP question—namely, that efficient verification of a solution should imply efficient determination of its existence—from a structural and representation-centered perspective.

Rather than proposing a resolution of the P vs NP problem or presenting new algorithms for NP-complete problems, this work analyzes the implicit assumptions underlying that intuition. In particular, it focuses on the distinction between local verification procedures and the global representational requirements needed to reason about the existence of solutions within NP.

We introduce the concept of representational cost to describe the descriptive complexity required to maintain or manipulate the set of surviving candidate solutions as local constraints are applied. While invalid candidates can often be eliminated efficiently through local checks, we argue that lifting such elimination into a uniform global decision procedure may incur exponential growth in representation size in the worst case, depending on encoding choices.

To illustrate how representation choice can yield apparent computational acceleration in structured instances, we discuss elimination-based behavior inspired by the Dual-Wheel Adaptive Wave Sieve (DAWS) framework. These examples are presented strictly as representation-dependent case studies and do not imply invariant complexity-theoretic reductions.

Overall, this work clarifies why efficient verification alone does not guarantee efficient existence determination, and frames the persistence of the P vs NP problem as a consequence of fundamental representational constraints under polynomially bounded computation.

📎 Supplementary exploratory reasoning and representation-dependent examples are provided in the accompanying supplementary document.