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 = k² + m², 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^-Mₖ/T/ZT 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) =|cₖ|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/mₚhi, 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/ (3M²) 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 MQ = 281 21 MeV (interface dimensional regularization, benchmark-validated, two independent routes; consistent with the classical BPS mass within the scheme band). The gauge coupling e² (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.
Nova Spivack (Wed,) studied this question.