Counting Defects: Semilinear Reachability and Resource Bounds for Langton's Ant

Fix the number m of initial black cells and a target N of ten-turn windows absent from the standard Langton highway language. The placements producing at least N such departures form an effectively computable semilinear set, with unrestricted initial coordinates. The construction describes the exact time, pose and complete black field after the Nth departure. A finite cover by words with variable repetition counts reduces physical read-and-flip semantics to consecutive visits to each cell. Consequences include synthesis under integer-linear coordinate constraints, piecewise rational-affine resource spectra, polynomial-bit positive witnesses for unary m and N, and a restricted polynomial-size quantifier-free decision encoding.

A fixed six-cell shape translated by an integer parameter k >= 0 produces exactly 6702 + 47k forward defects and first enters the permanent standard highway at time 18743 + 220k. Consequently, six initial black cells suffice to realize any prescribed finite lower bound on the number of defects.