A Uniqueness Theorem for Algebraic Theories of Physics: E₈ from Automorphism Invariance and Real Observables

This paper proves a uniqueness theorem for algebraic theories of everything — frameworks in which fundamental physics is generated by a single algebra with a bilinear product, as distinct from geometric, combinatorial, or information-theoretic frameworks. Under three minimal assumptions — (i) physics is generated by a single algebra whose scalar invariants are the real-valued observables; (ii) interaction vertices are invariant under the algebra's automorphism group; and (iii) vertices are polynomial with at least one genuine interaction — the underlying composition algebra is forced to be the octonions 𝕆, and the Freudenthal–Tits magic square uniquely produces the Lie algebra E₈. The proof is a chain of classical theorems: Hurwitz (1898), Artin (1927), and the Freudenthal–Tits magic square (1956–1966), with an exact representation-theoretic computation on G₂ = Aut(𝕆). The theorem identifies E₈ as a Lie algebra; real-form selection is addressed in downstream physics work. All component theorems are classical. The contribution is the assembly: minimal physical inputs routed into a single forcing chain. Partial unifications (SU(5), SO(10), E₆, E₇ GUTs) appear as subalgebras of E₈ rather than as alternatives.