A Common Origin for π, e, and i : Emergence from Binary Information and Entropy

We present a derivation of the mathematical constants π, e, and i from a single foundation: binary

information organized by maximum entropy. Starting only with the distinction between 0 and 1, we

show that π emerges as the minimal ratio enabling geometric structure in discrete space, e emerges

as the unique self-consistent base for entropic distributions, and i emerges as the necessary coupling

for coherent information exchange. This common origin explains why these constants satisfy Euler’s

identity ei π + 1 = 0—not as coincidence but as a necessary relationship between spatial, temporal,

and phase organization of information. We provide constructive algorithms demonstrating that these

specific numerical values are forced by information-theoretic principles, not arbitrary mathematical

conventions.