A Central Limit Theorem for the Exoskeleton Missing-Prime Count and a p-Adic Obstruction to Egyptian Fraction Representations

We prove two results within the exoskeleton framework for integers. First, we establish an unconditional Erdős–Kac type central limit theorem for the missing-prime count on the squarefree sample space in the sparse subpolynomial regime y = N^o (1). The proof uses the method of moments, based on exact joint divisibility estimates on the squarefree integers. We also develop a complete reduction framework that isolates the precise large-block correlation estimates needed to extend the theorem to the full variance range y = o (N^1/3 (N) ^2/3). Second, we prove a p-adic obstruction theorem for Egyptian fraction representations with squarefree denominators: if a reduced fraction A/B has a repeated prime factor p² B with p > (A, 2), then no such representation exists. In particular, 1/p² is not representable for every odd prime p. Both results originate from the same structural feature of the exoskeleton framework — that the finite exoskeleton Pₙ / rad (k) records the primes missing from the support of k. The central limit theorem quantifies the random behavior of this missing-prime pattern, while the obstruction theorem shows how the same pattern imposes arithmetic constraints on additive representations.