Recursive Admissibility and System Correctness: A Structural Proof Linking Tier 3 PAT to Tier 5 Recursion in the Paton System

This paper formalises the relationship between admissibility and recursive continuation within the Paton System. Each transition is governed by a gate condition defined at Tier 3 through the Paton Admissibility Test (PAT), while Tier 5 provides the recursive engine of continuation. Recursion cannot operate independently of admissibility, producing a recursively gated architecture in which existence, continuation, and observation are constraint-dependent. Correctness is defined as the persistence of recursive structure under admissibility constraint. The framework demonstrates that recursion is conditionally permitted rather than freely propagating, with all transitions requiring admissibility to continue. The paper extends recursion beyond Tier 5, showing that it operates across the full tier architecture (Tier 0 → Tier 8). Tier 5 represents the explicit expression of recursion, while all tiers participate in recursive dependency through admissibility-governed transitions. A formal relationship is established between generation and admissibility using the Paton recursive formulation (℘ₙ), demonstrating that generated states exist only when admissible. The work further introduces an information accessibility structure in which total system information exceeds observable information due to admissibility filtering. Observation is therefore shown to be a projection of admissible structure rather than a representation of total possibility. The paper provides a structural proof sketch showing that recursion must be gated to maintain system admissibility, linking Tier 3 (admissibility) and Tier 5 (recursion) into a unified condition for system correctness. No new physical laws are introduced. The work provides a structural interpretation of recursive continuation under constraint.