Under uniform transmission, the question is which received binary words maximize the posterior input entropy for exactly two deletions or exactly two insertions. Singhvi et al. stated this as an open problem.

The posterior entropy is expressed through subsequence embedding multiplicities. For deletion, an embedding-count identity organized by runs is derived via a first-symbol recurrence and combined with strict convexity estimates. For insertion, two-position deletion masks are grouped into components to obtain a lower bound. Higher-radius alternating-word histograms are computed in closed form, small cases are certified by exhaustive exact integer comparison, and the main theorem is formalized in Lean.

Alternating words uniquely maximize the two-deletion entropy at every length and the two-insertion entropy from length five onward, with exact maximal values and all smaller cases given. Exact embedding histograms of alternating words are given for all insertion radii. Three-deletion maximizers are certified through length 12 and three-insertion maximizers through length 18, which supports precise three-error conjectures.