Why Something Exists and Why Space Has Three Dimensions - Variational Arguments from the K-Functional

We derive two foundational results from the K-functional framework. First, we prove that the K-functional has no fixed point corresponding to "nothing": the empty state is outside the domain, pure states are repelled by the K_ent barrier, and the maximally mixed state is repelled by K_rec. The K-minimum is necessarily a state with intermediate entropy, definite records, and nontrivial boundary geometry, a universe with structure. Existence is the unique variational compromise between the three competing K-sectors. Second, we derive the dimensionality of spacetime from the qubit as the minimal nontrivial quantum system. A single qubit's state space is the three-dimensional Bloch ball B³ with boundary S². The Bures metric on this space is the unique monotone Riemannian metric for d = 2 (Petz). The spacetime dimensionality follows: three local spatial directions from the qubit state space, plus one temporal direction from the K-flow, giving 3+1. The qubit is selected by four independent constraints: metric uniqueness, anomaly cancellation, spinor self-consistency (d = dim(Weyl in d² dimensions) has unique solution d = 2), and observational match. Eight observational consequences, stable orbits, 2-component spinors, knots, cross product, 2 GW polarizations, hydrogen spectrum, holography, weak force, all follow from d = 2.