A cubical formalisation of topos causal models: intervention, sheaf gluing, and the intuitionistic do-calculus

Topos causal models recast causal inference inside a topos: a causal world is a presheaf, an intervention is a characteristic map into the subobject classifier, and reasoning is carried out in the intuitionistic internal language. We give the first machine-checked account of this 1-topos core, in Cubical Agda, over a previously verified probability monad and do-calculus. We build the classifier of sieves and realise the intervention $\mathrm{do}(X := x_0)$ as a characteristic map with its classification theorem; prove the sheaf gluing of independent mechanisms, which the source asserts but never proves; and machine-check the Kripke-Joyal forcing clauses of the internal language. In the modal layer we find and repair a gap: the three standard Lawvere-Tierney axioms do not force a closure operator. With the missing law restored, we exhibit the double-negation topology as a concrete instance and show that interventions and Pearl's rules are stable under every topology. Transportability of a counterfactual across a cover of regimes then coincides with this $j$-stability, understood as invariance across the cover. We further add a phenomenon the programme does not consider: a machine-checked contextuality obstruction, where pairwise-consistent local data admit no global model. The development assumes no axioms and typechecks under Agda's --safe flag, with the ordered field discharged concretely at $\mathbb{Q}$; the scope is the presheaf (1-topos) fragment, with type-level sheafification and the directed lift left to future work.