Tier 1 — Unified Recursion Theory Operator Grounding in the Particle Domain

This paper presents Tier-1 of a two-stage particle-domain consistency program for Unified Recursion Theory (URT). Tier-1 is deliberately narrow in scope: it tests whether URT operator proxies can be grounded in particle-physics observables without internal algebraic contradiction, using explicitly declared reference temperature conventions. It does not attempt to derive particle physics, modify Standard Model dynamics, predict new states, impose admissibility/exclusion claims, or calibrate physically meaningful temperature scales. Those questions are deferred to Tier-2. Using the 2025 Review of Particle Physics via the Particle Data Group (PDG) public API, we extract a curated dataset of 29 exclusive decay modes across 9 particles spanning ~3 orders of magnitude in mass and ~17 orders of magnitude in lifetime. Operator proxies are assigned using domain-native observables: stiffness proxy from rest energy, cadence from lifetime (or width-inferred lifetime), informational change proxy from channel surprisal (negative log branching ratio), and energy change proxy from a rest-mass Q-value approximation. For each decay mode, an implied efficiency value is computed as a diagnostic bookkeeping quantity under three explicitly declared reference temperature scales spanning eight orders of magnitude. Tier-1 results show that all extracted modes yield finite, positive implied efficiencies at all declared reference scales, with exact inverse scaling with the reference temperature as required by dimensional structure. No undefined operations, sign violations, or algebraic contradictions are observed. One statistical outlier corresponds to a near-deterministic branching channel, illustrating how low-entropy channel selection maps to extreme implied efficiencies without indicating a failure of operator grounding. Interaction class labels (weak/electromagnetic) are treated strictly as descriptive metadata inherited from particle physics taxonomy. Transparency is emphasized throughout: exclusions due to PDG representation (e.g., grouped/inclusive final states, ambiguous name lookups), data availability limits (e.g., missing/zero width in the API), and proxy limitations (e.g., broad strong resonances where a simple rest-mass Q-value is ill-defined) are explicitly tracked and reported. The outcome of Tier-1 is compatibility, not validation: the particle domain admits URT’s operator extractability requirements without contradiction, establishing a foundation for Tier-2 satisfiability and temperature-convention analysis.