Graph Surgery and the Do-Operator: A Precise Correspondence for Acyclic Structural Causal Models

The \operatorname{do}-operator is described graphically by deleting arrows into its targets and functionally by replacing their mechanisms with constants. To call these operations equivalent is not yet a mathematical statement: one returns a graph and remembers only the targets, whereas the other returns mechanisms and also remembers the imposed values. We make a dependency-level comparison precise for deterministic acyclic structural causal models with finitely many endogenous variables. If \operatorname{Graph}(F) extracts the dependencies of a mechanism family F, our main theorem is \operatorname{Graph}(F^ι)=\operatorname{Surg}(\operatorname{Graph}(F),T_ι). Thus replacing target mechanisms removes exactly the dependencies removed by graph surgery. For a model M=(G,F) whose graph may contain unused arrows, we characterize when the same equality holds with G in place of \operatorname{Graph}(F); it holds for every intervention exactly when G records the dependencies of F exactly. We then define the intervened model, characterize its run, show how sequential interventions combine, and prove that an outcome depends only on interventions at its actual dependency ancestors.