Lower-Edge Defect Laws in the Wide-Layer and Finite-Width Regimes

This preprint studies a finite lower-layer probability model built from consecutive prime-index supports and uniform weak compositions of a prescribed total index gap. Under the prime number theorem, it proves two lower-edge defect laws. In the wide-layer regime, where ηN,r → ∞, the defect probability converges to a universal law Q(τ). In the finite-width regime, where ηN,r → η ∈ (0,∞), it converges to a distinct law Q_η(τ). The arguments combine the prime number theorem, finite composition combinatorics, elementary estimates, and probability convergence. The shrinking-layer regime ηN,r → 0 is explicitly outside the scope of this work. This is a preprint and has not been peer reviewed.