Natour Lens Theory: A Unified Framework for Reciprocal Partition Arithmetic, Lens Geometry, Information Theory, and Lens Computing

Natour Lens Theory: A Unified Framework for Reciprocal Partition Arithmetic, Lens Geometry, Information Theory, and Lens Computing Mohammad Saeed Alnatour | 2026 This monograph introduces Natour Lens Theory, an original mathematical framework built from a single algebraic identity: AB = A + B, equivalently 1/A + 1/B = 1. From this seed, the theory develops a complete reciprocal-partition language spanning pure mathematics, physics, information theory, and computational architecture. The central theorem establishes that any partition equation T = X₁ + X₂ + ⋯ + Xₙ admits an exact Lens representation, embedding additive structures into reciprocal coordinates. This principle is demonstrated across analytic geometry, conic sections, differential equations, quantum mechanics, special relativity, Shannon entropy, and Softmax normalization. At the computational level, the theory proposes the Lens Processing Unit (LPU) — a dedicated architecture for reciprocal-partition arithmetic — built on the Lens coordinate map L(x) = x/(x−1), identified as a self-inverse Möbius transformation on the real projective line. A Python benchmark compares Lens arithmetic against standard float, log-encoded, and pre-normalized computation, demonstrating that Softmax outputs are native Lens states requiring zero encoding overhead. The framework connects to information theory through Lens Entropy, where Shannon entropy rewrites as H = Σ (1/Aᵢ) log Aᵢ and maximum entropy corresponds exactly to the symmetric Lens equilibrium point. Interactive LPU Architecture: www.natourlens.sbs