A square-integral threshold for subadditivity on shrinking corridors

Abstract

Let w be a measurable nonincreasing relative width. For exact eventual integer subadditivity on n <= m <= n + n w(log n), we prove a sharp square-integral dichotomy in the nonnegative nondecreasing linearly bounded class. Divergence of the integral of w squared forces normalized-slope convergence, including widths with jumps and missing adjacent pairs. With eventual adjacent-pair legality, convergence holds for arbitrary real sequences, with finite limit or minus infinity. Thus n/sqrt(log n) is a sufficient sublinear corridor. We further replace adjacent legality by balanced legal predecessors of gap O(N^alpha), alpha < 1, or O(N/(log N)^(3/2+epsilon)), epsilon > 0. Growing local maxima preserve a smaller literal corridor, and dyadic propagation transfers their limit to the original sequence. A staircase example satisfies the logarithmic condition while failing every fixed power condition. When the square integral is finite, strictly increasing integer submultiplicative and supermultiplicative sequences can have two distinct exponential rates. Isolated outputs permit oscillation even when every finite positive-power integral of w diverges. These results distinguish integrated width from integer connectivity.