Prime-Factorization Potentials Construction, Boundedness, and Spectral Analysis - Universal Tool for Multiplicative Number Theory

Theory of prime-factorization potentials encoding multiplicative structure of integers into bounded fractal functions on ℝᵈ. Proves boundedness, establishes complete variational derivation of scaling properties, analyzes spectral signature of gcd structure through resonance phenomena. Self-contained, applicable to any problem involving prime factorizations: abc conjecture, Catalan-type equations, general multiplicative constraints.