Jacobsthal Numbers in Disturbance Chain Length under Multiplication by Three

We study a bit-level disturbance phenomenon that occurs when passing from 3(n − 1) to3n in base 2. Instead of classical carry-chain length for addition, we measure how far thechange from the final bit-flip in n − 1→ n propagates once multiplied by 3. We formalizethis as the Disturbance Chain Length DCL(n), defined by the index of the most significantbit in the XOR X(n) := 3n ⊕ 3(n − 1). Working within a binary prefix-tree model, we showthat for fixed bit-length the integers maximizing DCL are exactly those whose binary endswith the alternating pattern (10)k11, i.e., the odd-index Jacobsthal numbers J(2k + 1). Forthe tree rooted at 0, the maximizers correspond to J(2k) + 1, explained by the identityJ(2k) + 1 = 2J(2k −1). We further prove that at depth x there are precisely two maximizers,yielding density 2/2x, and that disturbance is local: branches diverging below the quenchedcarry window inherit their own DCL independently. An interactive React + D3 applicationaccompanies the paper and visually highlights the Jacobsthal-aligned paths, empiricallymatching the theoretical characterizations.