We thank you for your valuable feedback which we will consider for the final version of our paper, if accepted, especially regarding numerical experiments and computational complexity.

Answers to your questions:

• Q1: When the clustering fixed, the implication is that the available causal information is insufficient to guarantee d‑separation (and identifiability) across all compatible micro‑graphs. In such cases, our do‑calculus procedure will construct a a micro‑level graph in which the targeted d‑separation condition fails and which can help users understand what causal information is missing. Furthermore, if one is free to choose a different clustering, it may be worthwhile to search for one under which d‑separation holds uniformly across compatible graphs. That said, if d‑separation is genuinely violated in the true underlying micro‑graph, no choice of clustering can restore it.
• Q2: There are several real‑world scenarios in which the macro‑level C‑DAG is known but the micro‑level structure remains unknown, as in macroeconomics where sector‑level relationships are known (as consumption→investment) but not the firm‑to‑firm or household‑to‑household causal links, or in neuroscience where functional MRI region interactions can be established but not necessarily the causal links between neurons.
• Q3: By Definition 1 (see also lines 119-121), a cluster DAG is assumed to be obtained from an ADMG which corresponds to the true underlying DAG. By construction, this DAG belongs to the set of compatible graphs.

Thank you for the responses. These have addressed my questions. I retain my rating mainly due to the lack of numerical experiments.

We thank you for your valuable feedback which we will discuss in the final version of our paper, if accepted, especially regarding clusters of unknown or misspecified cardinality.

Answers to your questions:

• Q1: The canonical graph is deliberately constructed to impose the minimal number of topological constraints. For instance, suppose a compatible graph contains an edge $V_{2} \rightarrow W_{w}$. Acyclicity then forbids $W_{w}$ from being an ancestor of $V_{2}$ in $G^{m}$, and, by our indexing convention, it also forbids $W_{w}$ from being an ancestor of $V_{1}$. Instead, by including only the edge $V_{1} \rightarrow W_{w}$, we eliminate the second constraint.
• Q2: First of all, example 4 shows that Theorem 4 may fail if we restrict cluster size to 2. In particular, for rule R1 with $W^{m} = \emptyset$. When we reduce the size of the cluster up to 3, the resulting C‑DAG no longer corresponds exactly to the original micro‑system. However, from the standpoint of causal reasoning (i.e., the application of do‑calculus rules), no information is lost: the reduced C‑DAG and the original C‑DAG yield the same identification conclusions.

We thank you for your valuable feedback — in particular, we are proud and pleased that you found this to be “the best paper [you] have read in a while.” We agree with your remarks and provide below clarifications to your questions and comments. In the final version, we will develop the discussion of limitations and implications of our results in the conclusion, by discussing the points raised by the reviewers.

Answers to your questions:

• Regarding the topological ordering: Thank you for pointing this out. You are correct that the topological ordering of a DAG is not unique in general. In our context, it suffices to fix an arbitrary topological order of $G^m$. The only property we rely on is that if $i < j$, then $V_i$ is not a descendant of $V_j$ in $G^m$. We will clarify this point explicitly in the final version of the paper.
• Regarding Definition 4: we thank the reviewer for this helpful suggestion. We agree that it would be clearer to assign it a name to Definition 4, and we propose 'connecting structure of interest'. For the second point of \sigma being a tuple of sets treated as a single set, we will proofread carefully for the final version.
• Regarding the notion of ancestors: we assume that every node is considered as an ancestor of itself, so the term “ancestors” already subsumes “ancestors of or in,”. We will clarify this convention in the final version of the paper.
• Regarding Rule 2 and Rule 3: we have carefully checked Rule 2 and believe it is correct as stated. In this step, we apply do-calculus to remove the do-operator on $X^m$, which results in the mutilation \underline{X^m}. As for Rule 3, you are right, brackets are not necessary since union is an associative operator.
• Regarding X^m(Z^m), we defined it in line 251: $\X^m(\Z^m) = X^m \setminus \Anc(\Z^m,\Gm_{\overline{\W^m}})$
• Regarding the CRL reference: you are right representations don't have to be learnt. We will improve this paragraph in the final version.
• Regarding $\mathcal{Z}$: thanks for catching this up, we will introduce $\mathcal{Z}$.
• Regarding L166: by "under," we mean "under the conditioning of." We will revise the sentence to clarify this and avoid ambiguity.
• Regarding L223:in this context, "generally" is meant to convey that one cannot expect the two operations to commute in general. That is, while the inclusion between the two sets always holds, equality does not. We have updated the sentence to clarify this point. Revised sentence: “It is important to note that, in general, mutilating all graphs compatible with a C-DAG yields a strictly smaller set of graphs than those compatible with the mutilated C-DAG, as illustrated in Example 2”
• Regarding Footnote 1: we are defining an equivalence relation in this footnote, denote by tilde. This will be clarified.

I thank the authors for thoroughly addressing my questions. I am confident that their proposed changes will clear up the few remaining clarity points in what is already an excellent paper.

We thank you for your valuable feedback. We agree with your remarks and provide below clarifications to your questions and comments. To better motivate our result, we will include an illustrative example in the final version. Regarding the terminology around micro- and macro-variables, we agree it may be confusing and will revise the paper to consistently refer to variables and clusters of variables instead.

Answers to your questions:

• Q1: you are absolutely right that, in graph theory, nodes without descendants are typically referred to as “leaves”. We have adopted here the terminology commonly used in causal inference, where such nodes are often called “roots” (see "Identification of Conditional Interventional Distributions", Ilya Shpitser and Judea Pearl, UAI 2006). We chose to align with this convention to remain consistent with the causal framework.
• Q2: Thank you for pointing this out. You are correct that the topological ordering of a DAG is not unique in general. In our context, it suffices to fix an arbitrary topological order of $G^m$. The only property we rely on is that if $i < j$, then $V_i$ is not a descendant of $V_j$ in $G^m$. We will clarify this point explicitly in the final version of the paper.
• Q3: We are not aware of a formal notion of a d-connecting walk. If this refers to a walk that satisfies the usual d-connection criterion but allows repeated vertices, then it differs from our structure of interest, which explicitly accounts for additional paths from colliders to conditioning variables. If the reviewer had a different definition in mind or could point us to a relevant reference, we would be very interested in learning more.
• Q4: It is needed because the canonical graph is intended to be a compatible graph, and therefore must be a valid ADMG. Without this qualifier, the construction could introduce a self-loop (e.g., on $V_1$), which is not permitted in an ADMG.
• Q5: You are right, this notion was not clearly defined before Theorem 2. What we meant is a density that admits a Markovian factorization over the observed nodes with respect to the C-DAG. You are also correct that the positivity assumption is needed, similarly to the standard assumptions in Pearl’s framework. We will revise the paper to clarify both points in the final version.