The Distinguished Reproducing Kernel

This paper introduces the Distinguished Reproducing Kernel as a primitive exact-closure principle for kernel-generated geometry and as the physical realization of logicality. The central object is not a reproducing kernel on a previously completed background. It is a maintained kernel K, a candidate state Σ, and a closure transform T_K co-determined by the same closure datum: T_K(Σ) = (K, Σ′), with K ⊊ Σ and K ⊊ Σ′. The equation is pair-valued, nondegenerate, and nonprojective: it reproduces the maintained kernel exactly while producing a closed state not exhausted by that kernel. From the datum itself, admissibility follows; reproduction further requires closure to produce an admissible state over the same kernel. In analytic realizations, the regularized closure family has a stable, presentation-independent value, so integral well-posedness is generated by exact closure rather than imported as a prior background domain. The same structure yields nonzero kernel states, local logarithmic potentials, normalized kernel overlaps, and the canonical metric identity g_{\widehat\Phi} = 1/2 g_K between raw kernel geometry and normalized comparison geometry.