Number--Wave Framework I: Foundations of Number Waves and Fundamental Prime Spectra introduces the basic language of the Number--Wave Framework, a mathematical framework for representing integer structures by number waves and spectra. The article defines number waves, phase sets, wave value functions, selected point sets, wave size, frequency, independent waves, independent companions, dependency factors, and operations on spectra. It then introduces the fundamental prime number spectra used in the framework, including the Prime Filter Spectrum, Prime Candidate Spectrum, Prime Effective Spectrum, Prime Effective Shadow Spectrum, Composite Spectrum, and Prime Spectrum. These spectra form the foundation for later spectra associated with prime (k)-tuples, such as twin primes, and Goldbach-type counting structures.