This paper develops the ordered-structure core of the Theory of Residual Cancellation (TRC).The main idea is that, in a suitable ordered setting, the common part of two positive elements can be recovered from ordered difference and positive-part structure rather than assumed independently. Let G be an ordered abelian group equipped with a positive-part operation u → u+, and define u− := (−u)+. For x,y ∈ G, define the TRC common-part candidatem(x,y) := x−(x−y)+. Under a positive/negative-part decomposition axiom, this operation is symmetric and yields a two-sided residual/common decomposition. Under the additional assumption that the positive part map is monotone, the matched-content operation becomes monotone, maximal among common lower bounds, and equal to the meet on positive pairs. This gives an axiom-separated theorem ladder: one axiom governs the algebraic decomposition layer, while the second upgrades the decomposition into genuine order-theoretic meet recovery. The result identifies the abstract core of TRC as a compatibility principle between difference, positive part, and common part.