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 ω^2 = k^2 + m^2, machine-certified via \_invariance\_of\_kg in .lean ( , zero sorry). (2)~Born rule from field amplitudes: the position-space probability density P(x) = | tial_xΦ|^2 / ∈t| tial_xΦ|^2 normalizes exactly ( ); extended to overlapping multi-kink configurations via the canonical dual-frame inner product ( \_rule\_from\_dual\_frame\_master in .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\ ( \_thermal\_state\_master in .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 σ = 290.10MeVGeV^-2 = 7450MeVfm^-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 λ = -16/49, corresponding to -1654.77MeVfm (attractive; ratio |λ/σ| = 2/m_φ, pure geometry); certified in .lean ( , zero sorry). The continuum limit CMCA→ is machine-certified in .lean: algebraic content is M-independent, the Nyquist residual (M) = π^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: \_finite\_ca\_exact\_lorentz\_replica proves that every finite-M discrete model has strictly positive Lorentz error ( , zero sorry), and \_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 $= 5.34\% ≤ (7) = 6.71\%, Pearson r=0.994$.