P vs NP

Overview

P vs NP asks whether every problem whose solution can be verified quickly can also be solved quickly. It is one of the seven Clay Millennium Problems. The framework provides a structural reading through the Computation Bridge: witness search is address resolution in the ABCD chart, and the Interface Width Principle (III.T31) constrains the complexity.

Detail

The $\tau$-Tower Machine (III.T30) models computation with bounded multiplicity. The question becomes whether $\tau$-admissible collapse (resolving a multi-address query to canonical form) can always be done in polynomial time. The answer depends on the Segre branching structure (III.T34) of the split-complex holomorphic mapping. The framework provides a structural interpretation but does not yet prove P $\ne$ NP — hence the Partial status. The tau-effective statement reduces to verification at primorial levels.

Result Statement

Structural reading via tau-Tower Machine and Interface Width Principle. The full P $\ne$ NP proof remains open. Status: Partial (tau-effective — structural framework established, orthodox bridge conjectural).