A Meta-Variational Principle: The Generator-Filter Theory (GFT) as a Unifying Framework for Dynamical and Informational Systems

The Generator-Filter Theory (GFT) introduces a novel meta-variational principle that transcends traditional variational methods by modeling the co-evolution of a local algebraic generator (𝐺), representing energetic potentials, and a global topological filter (𝐹), embodying informational constraints. Formalized through an energy-information duality, GFT seeks a bi-variational extremum, 𝛿𝐺 𝐼 + 𝛿𝐹 𝐼 = 0, where 𝐼 [𝐺, 𝐹] = ⟨𝐹, 𝐺⟩ − 𝐸 (𝐺, 𝐹). Unlike conventional approaches that fix either dynamics or constraints, GFT allows both to co-evolve toward mutual equilibrium, capturing reciprocal causation. This framework subsumes ordinary, partial, and stochastic differential equations, as well as classical variational principles, as special cases. With applications in artificial intelligence, physics, biology, and socioeconomic systems, GFT offers a unified formalism for self-organizing systems. While conceptually robust, GFT requires rigorous mathematical formalization to derive coupled evolution equations and dual geometries, potentially establishing a universal “organization law’’ comparable to Hamilton’s principle or Shannon’s entropy. Note: Introduced the first form of a universal- or meta-variational principle in physics on 1 October 2025. Generative Physics: A Universal Framework for Unifying Quantum Mechanics and Relativity — https://zenodo.org/records/17239694 Generator–Filter Theory: Toward a Unified Framework of Energy–Information Dynamics — https://zenodo.org/records/17430159 Generator-Filter Theory: A Universal Framework for Unifying Organizational Principles Across Scientific Disciplines — https://zenodo.org/records/17411477 How Generator-Filter Theory (GFT) Opens the Doors: A Meta-Level Framework for Unifying Physical and Informational Dynamics — https://zenodo.org/records/17363442 Generator-Filter Theory (GFT) as a Dynamic Extension of the Free Energy Principle (FEP) — https://zenodo.org/records/17481148 Generator-Filter Theory (GFT) as a Co-Adaptive Extension of Generative Adversarial Networks (GANs) — https://zenodo.org/records/17481190 The Generator-Filter Principle: A Meta-Variational Framework for Emergent Systems — https://zenodo.org/records/17584733 Please don't confuse it with Bi-Level. Meta-Variational or Bi-Variational isn't Bi-Level.