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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 ∫ w^2 forces normalized-slope convergence, including widths with jumps and missing adjacent pairs. With eventual adjacent-pair legality, convergence holds for arbitrary real sequences, with limit in ℝ ∪ {−∞}. Thus n/sqrt(log n) is a sufficient sublinear corridor. We further replace adjacent legality by balanced legal predecessors of gap O(N/[(log N)^(3/2)(log log N)^η]), η>1/2. Growing local maxima preserve a smaller literal corridor, and dyadic propagation transfers their limit to the original sequence. A staircase example satisfies this condition while failing every bound O(N/(log N)^(3/2+ε)), ε>0, and every fixed sublinear power bound. When ∫ w^2 is finite, strictly increasing integer submultiplicative and supermultiplicative sequences can have two distinct exponential rates. Conversely, isolated outputs permit oscillation even when every finite positive-power integral of w diverges. These results distinguish integrated width from integer connectivity.