Sign Ancestries and Polynomial Collisions in Iterated Absolute-Difference Triangles

We study finite triangles generated by repeated adjacent absolute differences. The main result is a constructive full sign-realizability theorem: for an initial row of length m, every assignment of strict comparison signs at all m(m-1)/2 comparison nodes is realized by some real initial row. Hence the number of strict sign ancestries is exactly 2^(m(m-1)/2). We place this result beside the established signed-linear-form description of higher absolute differences, give a self-contained proof of the zero-sum coefficient property, and show that strictly increasing inputs reduce exactly to their positive gap sequence. For inputs of the form x^p, fixed-depth zeros therefore lie in finite families of homogeneous Diophantine equations. At depth three with four strictly increasing bases, the collision mechanism reduces to exactly two equation families. The paper separates proved structural results from computational motivation and from unresolved arithmetic questions.