The Levels of the Theory: Coarse, Fine, and the Role of the Algebraic Certificate

The UGP Physics / GTE derivation tower spans multiple levels of description — from the discrete arithmetic substrate of the GTE polynomial up through cellular automaton dynamics, continuum field theory, and finally the Standard Model particle content and parameters. This tutorial explains the architecture of these levels, the distinction between coarse and fine descriptions, and the critical role of the algebraic certificate: the machine-certified (Lean 4, zero sorry) proof that the MDL selection criterion uniquely identifies the GTE polynomial from all candidates. Understanding the levels of the theory is essential for seeing how results at the discrete level propagate upward to produce predictions at the continuum level.