Persistence as Recoverable Continuity: Recursive Transport and Selection on the Prime Generated Logarithmic Lattice

This paper develops a persistence and selection framework on the prime generated logarithmic lattice established in my earlier work, Logarithmic Spectral Structure from the Chronoflux Continuity Law (Zenodo Record 19004419). The earlier paper derived the logarithmic reconstruction framework, the prime generated logarithmic lattice, and the distinguished reconstruction amplitude law an=n−1/2aₙ=n^-1/2an=n−1/2. The present work investigates what persistence structure follows once that reconstruction has been established. Beginning from continuity-preserving recursive transport, I develop a hierarchy linking recoverability, persistence, and asymptotic selection. Reconstruction balance is shown to identify a distinguished critical persistence exponent, while recursive transport generates a recoverability hierarchy whose asymptotic behaviour determines the surviving subset of lattice structures. A recoverability flow formulation is introduced and solved. The associated attenuation rate is then derived from the inherited logarithmic reconstruction through the logarithmic derivative of the distinguished reconstruction amplitude, yielding a reconstruction-induced attenuation law. This removes the need to introduce either the attenuation rate or the transport operator as independent primitives and establishes a direct inheritance chain an→λ (σ) →RKn→PKn→Λ∞. aₙ () RKn PKn _. an→λ (σ) →RKn→PKn→Λ∞. The resulting framework demonstrates how logarithmic reconstruction naturally induces persistence refinement and asymptotic selection on the inherited lattice. As such, the paper should be read as a continuation of the logarithmic reconstruction programme rather than a standalone construction. Taken together, the two works establish both the logarithmic reconstruction itself and the recoverability-driven persistence hierarchy acting upon it. Keywords Logarithmic reconstruction, persistence hierarchy, recoverability transport, asymptotic selection, prime generated lattice, continuity law, logarithmic spectral structure, reconstruction balance, persistence dynamics, recursive transport, number theory, mathematical physics, Chronoflux.