math_physics

We give a machine-verified algebraic formalization of Gödel's (1949) construction in the Platonic proof kernel.

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 develop Resonance Algebra (Res), a formal algebraic structure for mathematical objects with vibrational dynamics. Each element r ∈ Res carries three real-valued observables: energy E(r) ≥ 0, damping rate γ(r) > 0, and bandwidth BW(r) > 0.