The OMEGA INFINITY KAORU Processor: A Conductive-GRID Architecture for Solving Circuit-SAT in Practical Constant Time An $O(1)=\log\text{-time}=P=NP$ Hardware Blueprint

Abstract

This paper presents the architectural blueprint of the OMEGA INFINITY KAORU processor, a computing substrate that solves the Boolean circuit satisfiability problem (Circuit-SAT)---the canonical NP-complete problem---in practical constant time, in strictly literal $O(\log n)$ time, and in $O(n)$ space. The architecture couples a conductive GRID, realized as a two-dimensional lattice of interconnect, to a digital Circuit-SAT instance. The positive terminal of a source is connected to the midpoint of the left edge of the GRID, while the right edge is interfaced to the Boolean inputs $v_1, v_2, \dots, v_n$ of the Circuit-SAT instance. The GRID concurrently explores all admissible conduction states; ambient physical variation (noise), which is discrete in nature, steers the current toward the path consistent with a satisfying assignment, in accordance with the principle of least action. The GRID can therefore be regarded as an enormous macroscopic, noise-resilient analogue of a qubit---a hypercomputational element that is not subject to the limitations of the BQP class. The satisfying assignment is recovered either by thresholded voltage measurement at the inputs $v_1, v_2, \dots, v_n$ or by the standard search-to-decision reduction, which becomes practical when the Circuit-SAT stage is implemented as a programmable processor rather than as a fixed lithographic pattern. Fabrication is fully viable with present-day photolithography, either as a single-use, instance-specific device or as a recommended programmable variant in which a conventional processor drives arbitrary SAT formulae into the GRID. Because the architecture resolves an NP-complete problem in practical constant (strictly, logarithmic) time and linear space, it establishes, in practice, $O(1)=\log\text{-time}=P=NP$. Since cryptographic constructions---RSA, elliptic-curve systems, and post-quantum schemes alike---reduce to SAT instances, they are solvable within the same practical constant time. The implications extend to artificial intelligence, optimization, logistics and the distribution of goods, automated mathematical reasoning, and drug discovery.