Phase Field Algebras: Order, Smoothness, and Renormalization Group Fixed Points

We develop a formal algebraic framework for phase field theory, capturing the essential structure of systems that undergo phase transitions near critical points. The framework introduces a Phase type equipped with order (measuring singularity strength), smoothness (inverse of order plus one), and value projections. We prove eleven theorems establishing the fundamental algebraic properties: multiplicative additivity of order, dominance of lower-order terms under addition, positivity of smoothness, an order-smoothness tradeoff inequality, and renormalization group contraction properties. The proofs are formalized in the Platonic proof language with 12 structural hypotheses and 5 type axioms, verified by the Platonic kernel and exportable to Lean 4. This algebraic abstraction provides a rigorous foundation for universality class arguments in statistical mechanics and critical phenomena.

Keywords: Phase transitions, critical phenomena, renormalization group, universality classes, formal verification

MSC 2020: 82B26, 82B28, 03B35