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

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arXiv:2608.17634v1 Announce Type: new Abstract: 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 $\operatorname{Graph}(F^\iota)=\operatorname{Surg}(\operatorname{Graph}(F),T_\iota)$. 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.

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