Architecture of Prime Numbers: A Geometric Theory of Indivisibility Based on the Right Isosceles Triangle

This work presents a new geometric interpretation of prime numbers based on representing natural numbers as mosaics composed of atomic right isosceles triangles (RIT) with leg 1 and hypotenuse √2. The key concept is a force line — the joining of two triangles along their hypotenuses. The irrationality of √2 creates an energy barrier that prevents cutting the mosaic. A prime number corresponds to an uncuttable mosaic with maximal asymmetry, while a composite number corresponds to a cuttable mosaic where local symmetries arise at the junctions of blocks. A graph of force connections is introduced, and equivalence between classical and geometric primality is proved. An interpretation of Newton's binomial via triangle orientation combinatorics is given, and algorithmic consequences are discussed. The work includes a geometric proof of the infinitude of primes.