Finite Polynomial Absolute-Difference Stabilization and Quadratic Fold Ancestry

We prove that a finite family of real polynomials subjected to repeated adjacent pointwise absolute differences stabilizes cell-by-cell to fixed polynomial formulas on a sufficiently far-right half-line, at every fixed finite depth. After the stabilization threshold is enlarged beyond the real roots of the nonzero tail polynomials, the complete zero/nonzero pattern is constant. For translated quadratic inputs (x+s_i)^2, every tail cell is affine. Its eventual slope is twice the corresponding entry of the ordinary absolute-difference triangle generated from the absolute gap sequence |s_{i+1}-s_i|; an independent intercept condition then distinguishes permanent zeros from nonzero constants. We also formalize hinge locations for stabilized affine parent states and their finite contiguous ancestry. A computational campaign through Test 25 is retained as discovery, falsification, and audit evidence, not as a substitute for proof. The manuscript also records the falsification of an earlier apparent cubic cross-window invariant as a threshold-saturation effect. Repeated adjacent absolute differences have an established mathematical history. The present work does not claim invention of that operation. To the author's knowledge, the finite-polynomial eventual-tail formulation and the quadratic fold/ancestry refinement presented here have not appeared previously in the literature.