From Prime Numbers to DNA: The Emergence of a Universal Fundamental Structure

In this article, we demonstrate the emergence of a fundamental structure inherent to prime numbers that embeds its signature within the molecular architecture of DNA. By constructing a harmonic field Ω(x,q) based on modular inverses of prime numbers, we show that helical periodicities of biological structures emerge naturally from universal arithmetic properties. Our main results establish that: (1) The prime number p = 11 acts as a universal pivot, generating through its modular inverse 11−1 ≡2 the characteristic wavelengths of protein α-helices (λ = 7/2 = 3.5, 2.68% error vs observed 3.6) and B-form DNA (λ= 21/2 = 10.5, 0.10% error vs observed 10.5). (2) Analysis of human chromosome 1 reveals statistically significant 36% enrichment of prime-length tandem repeats (p < 3.2 ×10−7). (3) We establish a universal scaling law relating the prime harmonic coherence coefficient χ to genomic fractal dimension D according to D= 1−0.86χ, validated across five organisms spanning three domains of life (bacteria, yeast, plant, insect, mammal) with near-perfect correlation (r=−0.9974, p