Chapter 5: The Object Closure Axiom and the Static Kernel τ₀

The first five kernel axioms describe the internal structure of the five generators and the operator ρ: how the generators are ordered (K1), how ω absorbs (K2), how orbit rays are seeded and traversed (K3, K4), and how ω is unreachable (K5). None of these axioms says anything about what objects exist in τ. The sixth and final kernel axiom, K6 (Object Closure), fills this gap: it declares that the four orbit rays and the beacon ω exhaust the entire universe. There is nothing else.

With all six axioms assembled, we define the static kernel τ₀: the complete specification of Category τ before any generative act has taken place. τ₀ is a blueprint — precise, categorical, and inert. We close with a discussion of axiom independence, showing via sketched countermodels that no axiom is redundant.