We reconceptualize homeostasis from an inherently stochastic, intergenerational perspective. Here we use our SChemostat technology to directly record high-precision multigenerational trains of sizes of statistically identical non-interacting individual cells of Caulobacter crescentus in precisely controlled environmental conditions. We first show that individual cells indeed maintain stochastic intergenerational homeostasis of their characteristic sizes, then extract organizational principles in the form of an intergenerational scaling law and other "emergent simplicities", which facilitate a principled route to dimensional reduction of the problem. We use these emergent simplicities to formulate the precise theoretical framework for stochastic homeostasis, which not only captures the exact kinematics of stochastic intergenerational cell size homeostasis (providing spectacular theory-data matches with no fitting parameters), but also determines the necessary and sufficient condition for stochastic intergenerational homeostasis. Compellingly, our reanalysis of existing data published by other groups using different single-cell technologies demonstrates that this intergenerational framework is applicable to other microorganisms (Escherichia coli and Bacillus subtilis) in a variety of growth conditions. Finally, we establish that in balanced growth conditions stochastic intergenerational cell size homeostasis is achieved through elastic adaptation, thus precluding the possibility that cell size can act as a repository of intergenerational memory.