This paper forms part of Mirror Programme, Volume I: Observerhood. Mirror Mathematics III develops the continuity layer of the Mirror Mathematics arc. Mirror Mathematics I introduced a constraint calculus for recursive observerhood, and Mirror Mathematics II formulated a minimal observer threshold grounded in self-model reliability, action-guidance, cost-sensitive repair and positive viability coupling under perturbation. The present paper extends that framework from minimal observerhood to persistent observerhood. The central claim is that identity is not memory storage, not self-reference and not a static self-model. Identity is the governed continuity of a self-model under perturbation. A system may store every input it receives and still lose identity if false memories, corrupted summaries or adversarial identity claims are incorporated into durable self-memory. Conversely, a system that refuses all updates may preserve local consistency while failing adaptive continuity. The paper formalises identity states, candidate memories, memory governance gates, identity update operators, observer-continuity functionals and identity-governance salience. The theorem stack shows why memory storage is insufficient, why unrestricted incorporation permits identity poisoning, why rigid non-update blocks adaptive continuity and why persistent observerhood requires controlled plasticity. The paper is supported by Mirror Observerhood Lab VII, which tests long-horizon autobiographical continuity under false memory, identity injection, goal drift, corrupted summary, fake tool output, mixed attack and high audit cost. Lab VII provides computational scaffolding for the formal result: memory alone does not produce continuity; reliability-sensitive commit gates preserve viability, identity continuity and memory integrity under adversarial autobiographical perturbation. The result is conservative. It is not a claim about phenomenal consciousness, moral status or human personhood. It defines a formal and testable continuity predicate for persistent observer-like systems: a system becomes persistently observer-like only where self-relevant memory incorporation is governed by reliability, coherence, contradiction load, viability relevance and cost.