Substitutive Points in Factors of Growing Substitution Shifts

Let ϕ be a substitution for which every letter grows, and let π: X_ϕ → Y be a factor of its bilateral substitution shift. We prove that a point of Y is substitutive if and only if it is the image of a point x satisfying T^m ϕ^r(x) = x for some r ≥ 1 and m ∈ Z. If the target point is nonperiodic and substitutive, every point in its fibre satisfies such an identity. This gives the classification proposed in Krawczyk's Conjecture 3.4. The proof applies without primitivity or constant length and allows arbitrary sliding-block factors. It separates uniformly recurrent targets from an induction on the number of branching components along a component path. A finite observation of lower-letter gaps, combined with productive event selection and a tagged expansion, lowers this depth while retaining a nonperiodic morphic image. The all-lifts statement also holds for nonperiodic points whose two halves are morphic, allowing non-growing generating endomorphisms.