This paper presents the Persistence Kernel, a minimal structural-executable framework for recoverable continuity through change. Rather than beginning from matter, mind, law, computation, information, observation, identity, or physical assumptions, the framework begins from a bootstrap condition: an attempted beginning must either become recoverably specifiable or collapse. From recoverable specification, the paper develops a dependency chain leading through admissibility, distinction, structure, recoverable continuity, admissible transformation, constraint, lineage, collapse or repair, and minimum continuity support. The central claim is that something can change and still remain recoverably continuous only if the transformation preserves what must remain recoverable, including sufficient lineage to distinguish continuation from replacement. The framework distinguishes continuation from resemblance, copying, reset, replacement, and mere performance improvement. It also separates minimum continuity support, denoted V, from admissible transformation, denoted AT: V determines whether the continuity test can run, while AT determines whether a candidate transformation preserves continuity once the test is available. This work is intended as a structural account of persistence under transformation. It does not claim to derive physics, spacetime, matter, consciousness, computation, or the physical vacuum. Later applications in physics, computation, biology, cognition, AI alignment, or social systems would be bridge interpretations from the structural core rather than part of the core derivation itself.
James Shipkowski (Tue,) studied this question.