Write a PREreview

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$_Σ$(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 Next$_Σ$and by extensional collapse.

On the semantic side, we define the canonical SU-style stage extension along Next$_Σ$(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$_ω$(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$_{tree}$ step).