Mobile GUI agents excel at immediate reactive control but frequently fail in realistic, long-horizon tasks that require memory. This failure stems from a fundamental conflict between limited context windows and token-heavy screenshots. To save the limited context, agents must progressively discard older visual history, permanently losing crucial transient information. Furthermore, existing action-centric datasets fail to teach agents what or when to explicitly memorize, and augmenting static real-world data is prohibitively expensive and lacks interactive verification. To resolve this, we present STAMP, a framework that trains explicit memory in mobile agents through controllable virtual environments, where deterministic memory variables are programmatically injected into synthesized tasks to control what must be memorized, when it should be encoded, and when it must later be retrieved, thereby producing verifiable supervised data at scale and enabling online reinforcement learning through environment-driven reward feedback. Evaluated on our newly introduced Memory-World benchmark, the resulting Stamp-GUI agent achieves state-of-the-art performance among GUI-specialized models and sets a new high watermark on our Memory-World benchmark, demonstrating exceptional memory accuracy and task resilience while maintaining strong general mobile navigation capabilities.