The Universal Declaration on the Human Genome and Human Rights gave genomics an enduring human-rights grammar built around dignity, equality, privacy, and the symbolic idea that the human genome is the heritage of humanity [1]. That grammar remains indispensable, but it is no longer sufficient. Contemporary genomic practices are not confined to laboratory science or bedside counseling: they unfold within data-intensive, computational, and commercially mediated infrastructures that classify persons, govern access to care, and redistribute risk across families, communities, and generations. This article asks a sharper question than the usual privacy-versus-innovation framing: what is the normative object of genomic rights under conditions of predictive biology? The article argues that genomic rights should be interpreted not merely as personality rights protecting individuals from misuse, but as governance rights aimed at shaping how genomic prediction, circulation, ownership, and benefit-sharing are organized. The argument proceeds in four steps. First, it reconstructs the normative architecture of the UNESCO framework and its connections with broader human-rights law, including privacy, equality, and the right to enjoy the benefits of scientific progress [1-6]. Second, it shows why mainstream approaches centered on consent, confidentiality, and anti-discrimination are necessary but analytically insufficient in the face of algorithmic profiling, cross-sector data drift, and unequal access to genomic benefit [7-10]. Third, it proposes four analytic concepts—algorithmic genomic biopower, conditional genomic sovereignty, anticipatory dignity, and multilevel genomic justice—as a vocabulary for contemporary governance. Fourth, it tests that framework against six boundary cases that reveal where conventional bioethics becomes descriptively weak or normatively thin [11-24]. The article concludes that the most important contemporary question is no longer whether genomics can be reconciled with human rights in principle, but who governs predictive biological futures, through which institutions, and for whose benefit. A rights-based response adequate to that problem must move from downstream protection toward upstream governance, from exclusively individual consent toward relational and collective accountability, and from formal access to innovation toward justice in the distribution of genomic risk and benefit.