Continuity is frequently treated as an intrinsic good. Across cultures, institutions, governments, ecosystems, knowledge systems, and traditions, persistence is often interpreted as evidence of legitimacy, resilience, adaptation, or value. Structures that endure are commonly assumed to deserve preservation, while discontinuity is frequently associated with failure, loss, or collapse. Yet persistence and value are not equivalent properties. Systems may survive while becoming maladaptive, increasingly burdensome, detached from their original functions, or incompatible with the long-term viability of the environments upon which they depend. This paper examines continuity through the lens of recoverability rather than preservation, arguing that continuity is best understood as the maintenance of recoverable organization rather than the maintenance of static identity. Building upon previous work on continuity storage and reconstructive persistence, we distinguish continuity from viability and explore how continuity systems may fail without disappearing. We develop a taxonomy of continuity pathologies, including false continuity, false stability, lock-in, captured continuity, and parasitic continuity. These phenomena demonstrate that continuity can become progressively decoupled from recoverability, accessibility, adaptation, purpose, and host viability while continuing to reproduce itself through time. We further argue that continuity is not a property of isolated structures but a relational property emerging within broader ecologies of persistence. The value of continuity therefore depends not merely upon whether it survives, but upon how its survival affects the larger systems that sustain it. Healthy continuity remains coupled to recoverability, adaptability, and viability. Pathological continuity increasingly externalizes costs, narrows future possibilities, resists correction, or preserves itself at the expense of the broader persistence ecology. These observations motivate the introduction of Persistence Geometry, a framework for understanding continuity in terms of recoverability, accessibility, viability, recognition, and the pathways through which reconstruction remains possible across changing environments. Within this framework, continuity failure can be understood as a progressive distortion of the relationships that connect persistence to broader system viability. The central claim of this paper is simple: persistence alone is insufficient as a criterion for value. The critical question is not whether continuity survives, but whether the forms of continuity that survive remain compatible with the continued recoverability and viability of the systems that make continuity possible at all.