Foundational Structure and Calculus from Persistence under Admissible Variation

Description Foundational structure and computational rules are derived from minimal requirements on discernibility and persistence under admissible variation. The analysis asks what must exist for distinctions to remain comparable under repeated variation, and which rules are required once such persistence-stabilized structure exists. The first part establishes structural necessity. Curvature arises as the minimal obstruction to extending local equivalence globally once comparisons fail to close under recomposition. Redundancy of local description is unavoidable when persistence becomes path-dependent, forcing the introduction of a connection as a consistency device prior to geometry or fields. Stability analysis of accumulated loss around persistence-selected configurations leads to spectral structure without postulating operators or Hilbert space. The second part fixes the minimal rules required to compute and compare persistence once such structure exists. These rules constitute a minimal calculus, including representative-independent transport, persistence-weighted comparison of histories, and an admissibility criterion for comparing spectra across inequivalent configurations. Analytic continuation appears as the minimal rule required to remove resolution-dependent bookkeeping while retaining configuration-dependent structure. No spacetime, dynamics, physical fields, symmetry principles, or operator formalisms are assumed. All structures are derived internally, and failure modes are explicit. Subsequent work applies the same framework to empirical stress tests without introducing additional principles.