Prime numbers, long regarded as pseudo-random, are shown here to encode a coherent cross-dimensional wave field. By mapping prime gaps into oscillatory forms, we demonstrate that patterns emerge naturally as a result of compression and expansion in prime spacings. These patterns work as information that scales infinitely, but also can transcend dimensionality, and can be transferable between 0D, 1D, 2D, or 3D information. When many prime-driven oscillations are superposed, these structures appear as interference patterns, which serve as a clear visualization of the underlying gap-encoded information. Building upon this, the propagation of the oscillations to form standing wave fields can additionally encode another carrier of limitless information. Fourier analysis of the prime sequence reveals harmonic peaks, while analysis of spherical and planar mappings display recurring nodal structures across scales up to 108 primes. We show that primes encode an unlimited, scalable wave field of discrete information, independent of their abstract definition as numbers. This suggests that while primes originate as a one-dimensional numerical sequence, the information they encode inherently holds data only translated in a 3D environment— fully representable as volumetric data within the wave field itself. The primes therefore act not only as generators of oscillatory forms, but as complete 3D information carriers embedded within a 0D or 1D domain