From Peaks to Pixels: Demonstrating the Structural Isomorphism Between Wave Quantization and Signal Digitization

This paper demonstrates the structural isomorphism between the physical quantization of continuous waves (e.g., peak counting) and the information-theoretic process of signal digitization. Using a computational simulation of 50 randomized waveforms, we compare the Shannon entropy of discrete peak distributions against that of digital samples. Results show a strong statistical correlation ($r \approx 0.77$) and minimal entropy difference (~0.04 bits) between the two methods, suggesting that “discrete math” in physics is informationally equivalent to digital sampling governed by the Nyquist-Shannon theorem. We propose a “Why Not Both?” synthesis, arguing that continuity and discreteness are scale-dependent manifestations of a unified, bandlimited reality where observing a “peak” is structurally identical to capturing a “pixel.”