We introduce the EigenMan Random Generator (EMRG), a randomized arithmetic generation procedure in which a random arithmetic seed expression simultaneously produces a numeric value and self-determines the number of subsequent independent arithmetic iteration steps. This property — seed-derived execution depth — distinguishes EMRG from all conventional pseudorandom number generator designs, whose iteration counts are determined externally or fixed by construction. EMRG accepts five parameters: a range minimum x, a range maximum y, an iteration cap z, an optional digit-width control a, and an optional sign flag s. Each arithmetic step draws two integers and one operator uniformly at random from a configured range and operator set {+, −, ×, ÷}, and is redrawn in its entirety if the result is zero or a division by zero occurs. An optional target-width digit normalization reduces the output to a digit string of specified width. EMRG was motivated by the requirement for unpredictable, structurally non-distinguishable canary token generation in deception-based defensive security systems. A complete formal definition, termination analysis, prior-art positioning, and survey of practical applications are provided.