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

We study the -symmetric Klein-Gordon field as the continuum counterpart of the Three-Layer Chiral Minkowski CA (CMCA, Paper P41). Four principal results follow. (1) Poincar\'e invariance: satisfies exact Lorentz invariance ^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) = |_x|^2 / |_x|^2 normalizes exactly (); extended to overlapping multi-kink configurations via the canonical dual-frame inner product (born\_rule\_from\_dual\_frame\_master in DualFrameBornRule.lean, , zero sorry). (3) Physical quantum state: the pre-measurement state is the Hamiltonian thermal ensemble P(k) = e^-M_k/T/Z_T restricted to PSC-admissible kink sectors \0,2,3,4,6\ (phimdl\_thermal\_state\_master in PhiMDLThermalState.lean, conditional, zero custom physics axioms); the physical state is vacuum-dominant at observed cosmic temperatures and approaches the uniform PSC-admissible distribution at early-universe temperatures. (4) 3+1D extension: kinks become domain walls with tension = 290.10\,MeV\,GeV^-2 = 7450\,MeV\,fm^-2; superselection is preserved; the Born rule P(k)=|c_k|^2 is unchanged; and SR time dilation is exact. When two domain walls intersect in 3+1 dimensions, they form a junction line with exact dimensionless tension = -16/49, corresponding to -1654.77\,MeV\,fm (attractive; ratio |/| = 2/m_phi, pure geometry); certified in WindingCoinDecoupling.lean (, zero sorry). The continuum limit CMCA is machine-certified in CMCAContinuumLimit.lean: algebraic content is M-independent, the Nyquist residual (M) = ^2/(3M^2) 0 as M, and is the unique terminal Level-1 Generative Triple Evolution (GTE) object. Two further Lean-certified results complete the picture: no\_finite\_ca\_exact\_lorentz\_replica proves that every finite-M discrete model has strictly positive Lorentz error (, zero sorry), and commutes\_with\_winding\_iff\_diagonal establishes that any -winding-conserving coin must be diagonal, structurally precluding quantum interference in the discrete setting (, zero sorry). The explicit CMCA descent map is numerically confirmed at M=7: Cook A-glider winding profile vs.\ BPS kink RMSD = 5.34\% (7) = 6.71\%, Pearson r=0.994. At one loop, the kink fluctuation spectrum is the s=1 reflectionless Pöschl-Teller potential () and the quantum kink mass is M^Q = 281 21 MeV (interface dimensional regularization, benchmark-validated, two independent routes; consistent with the classical BPS mass within the scheme band). The gauge coupling e^2(_GTE) = 7/2 is consistent with PDG running at +1.5 (tree) / +1.3 (quantum kink mass) once the non-perturbatively measured kink dissolution constant (b = 1.189 0.049) is included, and the kink sector contributes _kink = (3.5-3.7)10^-4 to the hadronic vacuum polarization - a parameter-free falsifiable prediction.