Enrico's Lattice Sieve: Formalization of Grid Visibility and Two-Dimensional Divisibility Sieve

Abstract

This model describes a discrete tessellation of the two-dimensional Cartesian plane (Z²) based on the coprimality properties of integers. It provides a visual representation of geometric divisibility relationships. Given an indicator function E(x,y) mapping the lattice into a binary state {0,1} (where 1 represents the presence of a point and 0 represents an empty space), the structure is generated according to a pure arithmetic rule. The fundamental rule is defined as follows: E(x,y) = 0 (Empty) if the origin is (0,0) E(x,y) = 1 (Point) if on the main axes X=0 or Y=0 E(x,y) = 1 (Point) if the Greatest Common Divisor GCD(|x|, |y|) is greater than 1 E(x,y) = 0 (Empty) if the Greatest Common Divisor GCD(|x|, |y|) equals 1 Core Geometric Properties: Fractal Scale Invariance: The distribution of empty coordinate clusters manifests a periodic self-similarity determined by the Least Common Multiple of the coordinate vectors, generating symmetric geometric macro-structures that repeat infinitely. Asymptotic Determination of Primality: Let L_x be the discrete vertical line passing through an integer X, parallel to the y-axis. Let D be the median line (the 45-degree diagonal) of equation Y = X. The value X belongs to the set of Prime Numbers if and only if the spectrum of points along the line L_x between the horizontal axis (y=0) and the intersection with the diagonal D is strictly null (completely empty).