Evolving Quotient Networks: Persistence under Environmental Drift Requires Non-Identical Regeneration

Persistence in nonlinear, non-stationary systems is conventionally evaluated in the original state space: an entity persists if it remains viable through time. We show this convention fails under environmental drift, and replace it with a quotient formulation. We define a time-indexed future-viability quotient, whose classes identify states with the same forward generative prospects; under a countable determination condition this equivalence is measurable. We represent the resulting dynamics as an evolving quotient network (EQN): a temporal directed network whose nodes are quotient classes, whose faithful edges are projected transitions that lift to true state trajectories, whose virtual edges fail to lift, and whose regenerative edges preserve quotient viability while replacing the underlying entity. Our main result is a drifting-environment persistence theorem: under bounded fixed-entity viability and escaping environmental drift, every eventually entity-continuous policy fails, while a faithful regenerative path survives — so that infinite survival requires non-identical regeneration. A minimal simulation and a cross-system interpretation connect the theorem to adaptive organizations, decentralized (DAO) and federated protocols, and verifiable machine-memory systems, where persistence routinely migrates across carriers. The construction recasts a survival question about entities as an invariance question about a moving quotient, yielding a network-level object whose stability is carrier-independent.