PROJECT T VOLUME XX : The Algebraic Atom Deriving Hydrogen from the 42 Roots of H₀(Ã₂)

Volume XIX established the Particle Lexicon — a complete bijection between the 42 root configurations of H₀(Ã₂) and the 61 entities of the Standard Model. Particles are words. But atoms are not words — they are sentences. Volume XX takes the next logical step: we derive the structure of the hydrogen atom entirely from the algebraic framework, without postulating the Schrödinger equation. The key results are: (1) the emergent su(2) Lie algebra from coproduct commutators provides angular momentum and orbital structure; (2) the Casimir spectrum Eₙ = 3n, combined with the cosmic rank formula, reproduces the hydrogen energy levels with the correct 1/n² dependence; (3) the Pauli exclusion principle follows from the antisymmetry of seed coproducts; and (4) the shell structure N(n) = 2n² emerges as a theorem of su(2) representation theory constrained by ℤ₃ symmetry. The hydrogen atom is not a physical system to be solved — it is a grammatical structure to be parsed. The electron does not orbit the proton. The word “ab” is bound to the sentence “uud” by grammatical agreement.