Quantum / CONCEPT

Non-attained Pell phase supremum and unit asymptotic tail decay for two-level trees

Across all two-level rooted trees T(k, l) with k >= 2 and l >= 1 and all real times t, the scaled root-to-leaf Laplacian transfer probability n^(3/2) P_{k, l}(t) has global supremum C_* = 2 + 6 sqrt(5) / 5, realized already by the time supremum on the single five-vertex tree T(2, 1) along an explicit Pell phase-alignment sequence t_j -> +infinity yet strictly unattained at every finite time because of an integer parity obstruction. Restricting to trees of order n >= N drives both the scaled spectral-bound tail supremum and the scaled transfer-probability tail supremum to 1 as N -> +infinity, yielding uniform (1 + epsilon) n^(-3/2) decay while ruling out O(n^(-beta)) decay for every exponent beta > 3/2 via an infinite even-parameter family attaining peak root-to-leaf transfer at time pi.