Additive Bases from Primitive Dyck Words: Regular Underapproximations, Motzkin Coding, and Digit Lifting

Abstract

We study additive representations by integers whose canonical binary expansions are primitive Dyck words. Pairing consecutive bits yields a positional form of the classical relation between Dyck paths and two-colored Motzkin paths: except for 10, primitive Dyck words are exactly the binary block images of base-4 words 3w0, where w is a two-colored Motzkin word. This exposes a regular underapproximation, digit closure, and sharp generation bounds. We prove an interval digit-lifting theorem for digitally closed sets and a constructive base-4 propagation algorithm that lifts finite sumset certificates to infinite tails in logarithmically many recursive stages. Combining these tools with exact finite certificates and generation-gap lower bounds, we classify all positive even integers requiring more than six primitive Dyck summands. The integer 46 requires eight, and 34, 44, 98, 154, 198, 202, 206, 838, 842, and 846 require seven; every other positive even integer requires at most six. Thus 848 is the sharp eventual threshold. The bound is asymptotically optimal because 10*4^(k+1)-6 requires six summands for every k >= 2. The associated halved family has exact asymptotic additive order five. Supplementary programs reproduce all finite certificates using exact integer arithmetic.