A Two-Endpoint Transition in Binomially Weighted Prime-Power Counts

Let N ≥ 2 be a real parameter tending to infinity, and set L := log N. We study the number R_m(N) of pairs formed by a prime p and an m-element selection from the exponent levels k for which p^k ≤ N, with at least one available level left unselected. This coefficient statistic admits both a prime-layer decomposition and a binomially weighted prime-power representation. In the transition range specified in the paper, the first admissible prime layer competes with the terminal contribution arising from the prime 2. Under the prime number theorem and an explicit little-o transition condition, we prove a two-scale expansion with a lower prime-layer term and a terminal prime-2 term. We then locate the first endpoint-ratio threshold at √2, derive successive critical windows for the endpoint ratio, and obtain endpoint selection for the full count in the subwindows where intermediate-layer suppression is proved. We further refine the unresolved residual strip by deriving a quartic endpoint expansion with an explicit next coefficient, a smaller uniform remainder, and separated fourth-order sign regions. We finally assume a quantitative prime number theorem and prove a sharp local transport law: the endpoint ratio decreases by a factor L^(-1+o(1)) at each integer step across the critical window. The local transport law gives a unique discrete crossing and a crossing-centered alignment law with slope log L + log log L + O(1). The construction does not assert the existence of an integer sequence with a prescribed finite positive endpoint-ratio limit.