This paper introduces Persistence Logic, a minimal forcing calculus for determining when comparisons can remain coherent under composed admissible variation. Rather than postulating structure in advance, the framework derives which structural elements are unavoidable once incoherent recomposition is excluded. From the requirements of discernibility, admissible variation, and persistence, it shows that irreversibility under composition, accumulated loss, path dependence, transport, obstruction, and stabilized modes are forced bookkeeping structures rather than optional modeling choices. The forcing chain terminates at these sequential persistence structures. Operations representing coexistence of independent content, recomposition algebras, indecomposable decomposition, or global scalar closure are not forced by persistence alone; they arise only in additional structural regimes where independent admissibility can be maintained without collapse. The framework is intentionally non-constructive and boundary-forming. It establishes necessity and exclusion relations without fixing realizations or providing algorithms. A central result is that the general problem of deciding coherence for comparison under composed variation is undecidable, arising intrinsically from the expressive combination of composition, admissible identification, and finitary closure. As a result, some comparison tasks cannot be uniformly adjudicated within the calculus. Persistence Logic therefore functions as a foundational tool for classifying admissible comparison structures, distinguishing forced sequential persistence mechanisms from conditional coexistence regimes, and identifying principled limits of formal comparison. Explicit constructions and restricted, decidable regimes are developed in companion papers.