[Unverified Preprint] A Logarithmic Endpoint Refinement of Collatz Orbit Sparsity

This is an unverified, AI-assisted preprint. It is not peer-reviewed and should not be treated as an established proof. Let T(n) = n/2 for even n and T(n) = (3n+1)/2 for odd n. We prove that every set A of positive integers on which each iterate of T is injective satisfies #(A ∩ [a, a+X)) ≤ 400 X^β (log₂ X)^(−γ) for all integers a ≥ 1 and X ≥ 2. Writing H₂ for binary entropy and λ = log₂ 3, the constants are β = λα = 0.9653844417… and γ = λ/[2(λ − H₂′(α))] = 0.3562065654…, where H₂(α) = λα and α ∈ (1/2, 1). The argument sums the capacities of individual odd-step layers and retains the Stirling prefactor in the binomial tail. For every divergent positive Collatz trajectory, the maximum of its first N shortcut iterates, or first N odd iterates, is consequently at least 400^(−1/β) N^(1/β) (log₂ N)^(γ/β). In particular, eventual growth O(n^(1/β) (log₂ n)^c) is impossible for c < γ/β. This includes the critical growth bound O(n^(1/β)). The results restrict slow divergence; they do not exclude nontrivial cycles or all divergent trajectories.