Summary This collection introduces a new scientific framework for understanding and resolving persistence in biological systems, spanning cancer, chronic infection, and relapse-prone disease. The work is organized as a three-phase maturation: Persistence Geometry Science (PGS) defines how non-clearance is structurally maintained despite corrective force, replacing scalar interpretations of failure with barrier-structured persistence geometry Route Science converts these maps into causal intervention pathways, identifying the dominant barrier whose collapse must occur first to achieve durable non-reseeding Platform Science generalizes this cure-order logic across diseases, proposing that many failures of cure reflect incorrect barrier assignment or intervention order rather than insufficient force The central contribution is a shift in scientific grammar: from response-based evaluation (shrinkage, suppression) to ordered barrier collapse validated by durable loss of reseeding capacity. The framework introduces formal objects (persistence state, residual burden, dominant barrier, route), governing laws (Barrier Dominance, Ordered Collapse), and operational tools (route packets, barrier courts, ordered intervention tests). This work establishes a programmatic path from descriptive mapping to falsifiable curative logic, with immediate application to HIV, solid tumors, HBV, and minimal residual disease. It is designed as both a field definition and a portable scientific architecture for cure-oriented research. Keywords: Persistence geometry, non-clearance, reseeding capacity, barrier structure, dominant barrier, cure-order logic, ordered intervention, route science, platform science, cancer persistence, HIV reservoirs, minimal residual disease, systems biology, causal intervention design, non-reseeding endpoint.