The Phi_MDL Field: Quantum Structure, Born Rule, and Continuum Completion of the Chiral Minkowski CA

We study the Z7-symmetric Klein-Gordon field Phi_MDL as the continuum counterpart of the Three-Layer Chiral Minkowski CA (CMCA, Paper P41). Four principal results follow. (1) Poincare invariance: Phi_MDL satisfies exact Lorentz invariance omega^2 = k^2 + m^2, machine-certified via poincare_invariance_of_kg in LorentzInvariance.lean (zero sorry). (2) Born rule from field amplitudes: the position-space probability density P(x) = |d_x Phi|^2 / integral |d_x Phi|^2 normalizes exactly; extended to overlapping multi-kink configurations via the canonical dual-frame inner product (born_rule_from_dual_frame_master, zero sorry). (3) Physical quantum state: the pre-measurement Phi_MDL state is the Hamiltonian thermal ensemble P(k) = exp(-M_k/T)/Z_T restricted to PSC-admissible Z7 kink sectors {0,2,3,4,6} (zero custom physics axioms); vacuum-dominant at observed cosmic temperatures. (4) 3+1D extension: Phi_MDL kinks become domain walls with tension sigma = 290.10 MeV/GeV^2; Z7 superselection preserved; Born rule unchanged; SR time dilation exact. The continuum limit CMCA -> Phi_MDL is machine-certified: the Nyquist residual epsilon_0(M) = pi^2/(3M^2) -> 0 as M -> infinity, and no_finite_ca_exact_lorentz_replica proves every finite-M discrete model has strictly positive Lorentz error (zero sorry).