The Gap Between Recognizability and Formal Verification: A Perspective on P vs NP Formed Through Human–Machine Collaboration

Abstract

Abstract This research note presents a personal perspective on the conceptual distinction between finding, recognizing, verifying, formalizing, and proving. It emerged from the author's study of physics, mathematics, technology, and iterative human–AI collaboration in the PCS project. The central observation is that a human may sometimes recognize that an output appears incorrect before the reason for that judgment has been converted into explicit, machine-verifiable constraints. The paper provisionally describes this phenomenon as “the gap between recognizability and formal verification” and uses a simplified iterative model to describe how intuitive human judgments may gradually become formal verification conditions. This work does not provide a mathematical proof of P vs NP, does not claim P = NP or P ≠ NP, and does not claim to resolve the P versus NP problem. P vs NP is used here as a conceptual point of departure for reflecting on the differences among search, recognition, verification, formalization, proof, and scientific learning. The paper is presented as a Perspective / Learning Research Note and as a record of an evolving learning process in mathematics, physics, scientific computing, and human–machine collaboration.