Paton System — Life Integration Tier-2 → Tier-3 → Tier-4 Mapping of Biological Viability

This paper presents a structured integration of life viability ratios into the Paton System, demonstrating how candidate life states (℘ₙ) are recursively generated, filtered through the Paton Admissibility Test (PAT) at Tier-3, and observed as structurally coherent outcomes at Tier-4 (Sₙ₊₁). The framework introduces domain-specific viability constraints into the Tier-2 → Tier-3 → Tier-4 transition, including energy efficiency, survival probability, replication potential, and environmental tolerance. Candidate states are generated recursively and weighted by these biological factors, forming a space of possible life trajectories. At Tier-3, admissibility filtering enforces constraint compatibility. States that fail thresholds such as energy balance, environmental limits, population viability, or resource sufficiency are removed. Only admissible trajectories persist. At Tier-4, observed biological structure emerges as the subset of admissible states. Population ratios, metabolic behaviour, and ecological distributions reflect constraint-filtered outcomes rather than total possibility. The sequence ℘ₙ → PAT → Sₙ₊₁ provides a structural mapping between abstract recursive systems and real biological processes. This work extends the Paton System by explicitly linking admissibility filtering to life viability, demonstrating how biological evolution can be interpreted as constraint-governed continuation. This framework introduces no new biological mechanisms. It provides a structural interpretation connecting recursive generation, admissibility filtering, and observed life systems.