Abstract: Personalized AI memories, defined as persistent decision-relevant states distilled by platforms from long-run human-AI interaction, are increasingly portable, provenance-certifiable, and tradable. Provenance answers the question of origin, but it does not settle who holds which rights over a memory, nor how wider deployment affects its future supply. We develop a unified model in which memory use is non-rival, while refresh, the maintenance of continued validity, is relationally co-produced by the represented individual and the platform. Even assuming perfect provenance and complete information, the model still yields a set of inherent limitations. First, there is an extraction ceiling: decision value is bounded by lived experience. Second, a rights-separation theorem shows that no budget-balanced ownership assignment can make both co-producers residual claimants, and that control rights and cash-flow rights operate on distinct margins. Third, a depletion result indicates that reach-expanding policies such as portability, interoperability, and competition raise current access but shrink the long-run memory stock. Fourth, we offer a formal comparison between memory ownership as it ought to be (subject-held) and as it is (platform-held); the welfare ranking reduces to two measurable quantities, both dominated by interior renewable-claim bundles. Finally, endogenous-quality and market-formation results show that uncompensated markets pollute their own raw material, that transparency and compensation are complements, and that the de facto platform-ownership regime is self-perpetuating. On the design side, the analysis points to separated provenance, performance, and permission credentials; compensation indexed to quality-deployment; and portability and salience, rather than nominal ownership, as the binding policy margins.