We study the analytic foundations of the free quantum core in a timeless Euclidean model with a single actual configuration of a real harmonic field in E⁴ and an internal observer. For the full elliptic problem u'' = A²u with nonnegative self-adjoint A, we obtain an exact decomposition on a finite slab into the Poisson extension of the first trace and a boundary remainder. On the sector A ≥ κI > 0, control of the boundary mismatch yields exponential bounds on the remainder and the deviation from the decaying-branch graph u' = −Au. We derive a sufficient criterion for indistinguishability of the solution and its Poisson approximation under the linear readouts of a finite protocol with prescribed tolerances. Subexponential growth on the half-line yields the exact transfer P_h^E = e^−hA on the positive spectral subspace. The canonical two-point form is constructed as the Green's function of the fundamental Laplacian. A regular equal-slice limit, a positive, spatially homogeneous and isotropic equal-slice form, compatible transfer of one argument, and O(4) invariance determine it uniquely up to normalization and zero components: C̃_E(p) = c/|p|². Reflection positivity is proved for this kernel. The associated Gaussian measure admits standard Osterwalder–Schrader reconstruction with a free Fock space, a positive generator, and unitary dynamics. The Markov property of the full Gaussian slice field is derived from the covariance. The construction describes a free massless scalar sector without introducing an additional fundamental field. The realization of a complete measurement protocol lies outside the scope of this work.