SCM.Do.Rule2­Kernel

For the intervention names X, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, projecting the extended post-intervention fixed assignment — built from the base assignment s0 and treatment value t — back onto the original fixed coordinates recovers s0.

For the intervention names X, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, filling the conditioning assignment z against the extended post-intervention fixed assignment built from the base assignment s0 and treatment value t equals directly combining t and z into one assignment on their union.

Rule 3 W-marginal equality for Rule 2. For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, and a conditioning set W contained in the observed variables, if no fixed copy of a name in Z is an ancestor, in the post-intervention DAG, of any node in W, then the W-marginal of the intervened model's observational kernel at a fixed assignment s equals the W-marginal of the base model's observational kernel at the corresponding projected assignment.

For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, with the union of the random copies of Z and a conditioning set W contained in the observed variables and the random copies of Z disjoint from W, fix an intervened fixed assignment s'; for measurable subsets S of the observed-value space and measurable subsets A of the values on W, the intervened model's observational kernel at s', evaluated on S intersected with the preimage under the Z.random ∪ W-projection of the image of A under the filled-assignment map, equals the base model's observational kernel at the projected fixed assignment, evaluated on the analogous set.

Cross-SCM evalMap level-set bridge for Rule 2. For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, fix an intervened fixed assignment s' and a latent draw ℓ. Suppose on the "level set" picked out by ℓ, the base model's evaluation map at the projected fixed assignment already reproduces, at every random copy of a name in Z, the intervention value recorded in s'; then at any observed node v in the intervened model, the intervened model's evaluation map at (s', ℓ) agrees with the base model's evaluation map at the projected fixed assignment and the correspondingly reindexed latent draw.

M2-direction level-set evalMap bridge for Rule 2. For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, fix an intervened fixed assignment s' and a latent draw ℓ. Suppose the intervened model's own evaluation map at (s', ℓ) already reproduces, at every random copy of a name in Z, the intervention value recorded in s'; then at any observed node v in the intervened model, the intervened model's evaluation map at (s', ℓ) agrees with the base model's evaluation map at the projected fixed assignment and the correspondingly reindexed latent draw.

D-separation collapse for the post-intervention conditional kernel. For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, with outcome set Y and conditioning set W contained in the observed variables, suppose in the post-intervention SWIG DAG, Y is d-separated from the random copies of Z given W together with the post-intervention fixed set. Then, fixing an intervened fixed assignment s and a measurable outcome set B, for almost every conditioning value on Z.random ∪ W, the intervened model's conditional distribution of Y given Z.random ∪ W assigns B the same probability as its conditional distribution of Y given W alone, evaluated at the W-projection of that conditioning value.

Cross-SCM conditional kernels agree along the fillZrW filled assignment. For the intervention on names Z, whose random copies are observed in the base model and whose fixed copies are not yet part of the base model's fixed coordinates, with outcome set Y and conditioning set W contained in the observed variables, the union of the random copies of Z and W contained in the observed variables with the random copies of Z disjoint from W, fix an intervened fixed assignment s; then for a measurable outcome set B and almost every conditioning value w, taken with respect to the pullback of the base model's law on Z.random ∪ W under the filled-assignment map, the base model's conditional distribution of Y given Z.random ∪ W, evaluated at the filled point built from w, and the intervened model's corresponding conditional distribution, evaluated at the same filled point, assign B the same probability.

Obs-side witness-kernel identity (pair-conditioned, on the latent space). Let M be a structural causal model with node sets Y, W and a set of treatment names Z, where Y is observed, the pre-intervention nodes of Z are observed, and W is observed. Suppose the latent cut-set for Y relative to Z's pre-intervention nodes together with W is d-separated, in M's causal graph, from Z's pre-intervention nodes given W and the fixed nodes, and let h be a jointly measurable map such that, at a fixed baseline assignment s, the realized value of Y factors pointwise as h applied to the realized pre-intervention-Z value, the realized W value, and the cut-set's latent value. Then, under M's latent product measure, the conditional law of Y given the pair of realized pre-intervention-Z and W values equals, for almost every such pair, the pushforward under h applied to that pair of the conditional law of the latent cut-set given the realized W value.

Do-side analytic core: the M2 W-conditional equals the M1 witness kernel. Let M' be a structural causal model and Z a set of treatment names with each treatment's pre-intervention node observed and each treatment's post-intervention node not already fixed in M', so that intervening on Z is well-formed; let Y, W be node sets with Y, W, the treatments' pre-intervention nodes, and their union with W all observed and Y disjoint from the treatments' pre-intervention nodes. Suppose no node of W is a descendant, in the intervened model's graph, of any treatment's post-intervention node, and let h be a jointly measurable map that, at a baseline assignment s0 to the original model's fixed nodes, factors the realized outcome as h applied to the realized treatment value, the realized W value, and the latent cut-set's value and additionally, for every candidate treatment/W pair, agrees there with the outcome obtained by instead overriding the baseline assignment s0 to that pair. Then, at the intervened model's slice fixing the treatments to a value t and the remaining fixed nodes to s0, the measure-level conditional law of Y given W equals, for almost every w under the W-marginal of the original model's observational kernel at s0, the pushforward under h t w of the original model's posterior conditional law of the latent cut-set given W.

Do-side per-slice witness identity (M1-witness-kernel form). Let M' be a structural causal model and Z a set of treatment names with each treatment's pre-intervention node observed and each treatment's post-intervention node not already fixed in M'; let Y, W be node sets with Y, W, the treatments' pre-intervention nodes, and their union with W all observed, Y disjoint from the treatments' pre-intervention nodes, and those pre-intervention nodes disjoint from W. Suppose no node of W is a descendant, in the intervened model's graph, of any treatment's post-intervention node, and let h be a jointly measurable map that, at a baseline assignment s0 to the original model's fixed nodes, factors the realized outcome as h applied to the realized treatment value, the realized W value, and the latent cut-set's value, and additionally for every candidate treatment/W pair agrees there with the outcome obtained by instead overriding the baseline assignment s0 to that pair. Then, at the intervened model's slice fixing the treatments to a value t and the remaining fixed nodes to s0, for almost every w under the W-marginal of the original model's observational kernel at s0, the intervened model's conditional-probability kernel for Y given W at (s', w) equals the pushforward under h t w of the original model's posterior conditional law of the latent cut-set given W.

Product-form cross-SCM bridge for the do-side conditional kernel. Let M' be a structural causal model and Z a set of treatment names with each treatment's pre-intervention node observed and each treatment's post-intervention node not already fixed in M'; let Y, W be node sets with Y, W, the treatments' pre-intervention nodes, and their union with W all observed, Y disjoint from the treatments' pre-intervention nodes, and those pre-intervention nodes disjoint from W. Suppose that, in the intervened model's graph, no node of W is a descendant of any treatment's post-intervention node, that, in the original model's graph, no node of W is a descendant of any treatment's pre-intervention node, and that in the intervened model's graph, Y is d-separated from the treatments' pre-intervention nodes given W together with the intervened model's fixed nodes. Assume also, at a baseline assignment s0 to the original model's fixed nodes, an overlap condition: the pushforward, under combining a treatment value with a W value, of the product of the treatments' pre-intervention marginal law and the W-marginal law (both taken from the original model's observational kernel at s0) is absolutely continuous with respect to the original model's marginal law on the treatments' pre-intervention nodes together with W, again at s0. Then for almost every pair (t, w) drawn from that product law, the intervened model's conditional-probability kernel for Y given W, evaluated at treatment value t and conditioning value w, equals the original model's conditional-probability kernel for Y given the union of the treatments' pre-intervention nodes and W, evaluated at the combined value (t, w).

W-marginal pushforward identity. Fix a do-set Z of nodes whose random copies are observed in the base model and whose fixed nodes have not already been intervened on and an observation block W such that the union of Z's random copies and W is observed and disjoint from Z's random copies. If, under the intervened model's law of the observed variables given the fixed values s', the post-intervention random copies of Z almost surely equal their assigned intervention values, then for every measurable W-event A, the intervened model's marginal probability of A on W equals the base model's probability of the pullback of A through the map that fills in the pinned Z-random-copy coordinates alongside W.

Cross-SCM rectangle bridge for the W-marginal. Fix a do-set Z of nodes whose random copies are observed in the base model and whose fixed nodes have not already been intervened on, together with an outcome block Y and a conditioning block W of observed variables such that the union of Z's random copies and W is observed and disjoint from Z's random copies. Assume that, under the intervened model's law given the fixed values s, the post-intervention random copies of Z almost surely equal their assigned intervention values. Then for every measurable W-event A and every measurable Y-event B, integrating, over A and with respect to the intervened model's W-marginal, the base model's conditional probability of B given the outcome of filling in Z's pinned random-copy values alongside each w reconstructs the intervened model's probability that both the W-event A and the Y-event B occur.