On the Representational Equivalence of Threshold Networks and NP-Verifiers: Expressive Power, Hardness Barriers, and the Limits of Boolean Encodability

This paper provides a self-contained, formally rigorous treatment of a classical but frequently misunderstood result in computational complexity and neural network theory: threshold networks are universal Boolean circuits, yet this universality confers no computational advantage. The work consolidates three classical lines of results — with complete proofs, explicit gate constructions, uniformity statements, and diagrams — that jointly refute the common but incorrect inference that representational capacity implies computational tractability: Universal representation: every Boolean function on n inputs is computed by a depth-3 threshold network of size at most 2ⁿ + n + 1 with integer weights of magnitude at most n. Uniform NP encoding: every NP-verifier compiles, in polynomial time, into a polynomial-size threshold network family via a formally uniform Cook–Levin reduction. Two hardness barriers: deciding whether any input satisfies a threshold network (NetworkSAT) is NP-complete, and training even a three-node network to consistency is NP-complete. No new theorems are claimed. The contribution lies in precision, uniformity, and completeness of presentation, providing a single self-contained pedagogical reference that sharply delineates what threshold networks can represent from what they can solve or learn efficiently. MSC 2020: 68Q17, 68T07, 94C10