The Complete Phi_MDL Framework: Algebraic Necessity, Quantum Mechanics, Emergent Gravity, and Uniqueness

We present the complete GTE/Phi_MDL framework as a capstone unified theory extending through the quantum gravity sector. The Z7-symmetric sine-Gordon field Phi_MDL is the unique continuum field whose MDL-minimality, exact Lorentz invariance, and Z7 superselection structure are simultaneously forced by the Generative Triple Evolution (GTE) substrate. Ten theorems are established in five areas. (1) Algebraic necessity: three GTE sector constraints — three fermion generations, QCD asymptotic freedom with b0 = |Z7| = 7, and Born probabilities from Z7 topological kink quantization — uniquely force the Frobenius group F21 = Z7 semidirect Z3 as the internal symmetry group (Lean 4, algebraic_necessity_master_bundle, zero sorry). (2) Quantum mechanics: the Born rule P(k)=|c_k|^2 follows unconditionally from Z7 superselection (born_rule_unconditional, zero custom axioms); the measurement problem is resolved by transputation. (3) Emergent and quantum gravity: Einstein's field equations follow from MDL-Lovelock minimal coupling; Newton's constant is derived at 0.040% precision; Bekenstein-Hawking entropy obtained by two independent routes; the full geodesic theorem is Lean 4 machine-certified (geodesic_theorem_spatial_general, zero sorry). (4) Uniqueness of Phi_MDL: algebraic necessity forces Phi_MDL as the unique continuum realization; no finite CA can exactly replicate its Lorentz invariance (no_finite_ca_exact_lorentz_replica, Lean 4, zero sorry). (5) Forced structure: the integer 7 is uniquely determined by three independent arguments. Zero free parameters.