Confinement Thresholds and Collatz Predecessor Density: A Sharpened Constant and a Transfer Obstruction

Mazur (ProofAtlas, September 2026) announces that for every positive integer a with 3 ∤ a the set P(a) of integers whose Collatz orbit reaches a satisfies P(a)(X) ≥ c_a X for large X, assuming nothing about the fate of a. That statement is a long-standing open conjecture and the manuscript is unrefereed, with a self-reported formalization not rebuilt here; it is taken as a hypothesis throughout. It is natural to ask whether this density, combined with the sparsity of seeds whose Syracuse drift stays below a fixed threshold, excludes divergence. The transfer fails at every available rate, and we locate exactly what a successful one would need. Writing R_k = S_k − αk for the drift after k accelerated steps, α = log₂3, and E_U for the class of odd seeds with R_j ≤ U for every j ≥ 1, we prove four things. E_U contains no convergent and no eventually cyclic seed, so it consists of divergent seeds only. If S^k(m) = t with k ≥ 1 then R_k(m) > log₂(m/t), so a fixed threshold confines the predecessors of a fixed target to the bounded window [1, t·2^U); this is why a density-positive predecessor basin coexists with E_U of density zero for every fixed U. The growth constant in a uniform bound U(m) ≤ A log₂m + B over divergent seeds is pinned at A = 1: forced from below by an exact one-step predecessor family, and attained from above, unconditionally, from Curry’s reciprocal summability, whose formalized proof in the author’s eoc-divergence Lean library carries an absolute constant (Σ* ≈ 28077.6, B ≈ 13502.5). And the quadratic threshold penalty in the baseline counting constant is removable, since the entropy function t log t − (t−1) log(t−1) is concave, improving the dependence from exp((U+1)²) to (α/(α−1))^U. The transfer nonetheless fails. The argument must substitute a growing threshold U_X ≈ log₂X, which converts the exponential rate γ in U into an exponent of X; the resulting count is O(X^{θ+γ}) with θ = 1 − I₀/α = H₂(1/α) ≈ 0.94996, and θ + γ > 1 for every rate available. The criterion is θ_eff + γ < 1: γ can improve only through a stronger counting theorem at thresholds growing like log₂X, and θ_eff < θ only by leaving finite-prefix counting, since θ is the exact count of residue classes compatible with confinement to the relevant depth. The numerical deficit I₀/α ≈ 0.0500445 therefore does not measure the difficulty. No divergent orbit and no cycle is excluded. The density statement is Mazur’s; the reciprocal bound is Curry’s (Zenodo 10.5281/zenodo.22087163); the Lean theorems cited (exists_window_sparsity, summable_inv_orbit, uniform_partial_sums at commit e56950f) were built and axiom-checked; no statement of this note is itself formalized. A standard-library Python checker with 46 exact tests accompanies the note.