This preprint isolates a single organizing mechanism for the transfinite progression of Stratified Univalence: iterate a carrier-level closure operator Next and a semantic-stage extension operator Jump, with all dependence funneled through predecessor geometry (bounded bisimulation equivalence of initial segments). For a predecessor structure (A,≺) and a signature Σ of admissible predecessor-indexed diagram shapes internal to A, we construct the free completion Next(A) that adjoins least upper bounds of Σ-diagrams and quotients them by cofinal/bisimulation refinement so that suprema depend only on predecessor geometry. We prove the universal property (initiality) of this completion, develop a coherence package for quotient paths (with an optional 1-truncated groupoid quotient variant), and show well-foundedness is preserved by Nextand by extensional collapse.
On the semantic side, we define the canonical SU-style stage extension along Next(A) by interpreting each new supremum as the corresponding HIT/QIIT colimit of the old semantics, and we prove cofinal invariance of such colimits under a constructive cofinality criterion (contractible lift fibers). We also construct extensional collapse as the reflective set-quotient by predecessor geometry and show it agrees with global bisimulation, ensuring that all constructions descend to extensional carriers.
The first genuinely nontrivial instance occurs at ω, where Next(A) corresponds to adjoining suprema of strict ω-chains and Jump realizes the ω-stage via sequential colimits, explaining the ω-jump/universe-lift phenomenon in predicative settings. Beyond ω, the paper identifies the first remaining bottleneck for Part II: establishing constructive cofinality for tree/contractible-graph refinements (the Σ step).