Topos causal models recast causal inference inside a topos: a causal world is a presheaf, an intervention is a sub-model named by a characteristic map into the subobject classifier \Om, and reasoning is Kripke-Joyal forcing in an intuitionistic internal language. We give the first axiom-free machine-checked account of this 1-topos core, in Cubical Agda over a previously verified probability monad and do-calculus; the framework is otherwise developed on paper, with central claims stated rather than proved. Three of our results go beyond faithful transcription. We exhibit a contextuality obstruction the programme does not treat: pairwise-consistent local causal data with no global model, detected by a degree-one holonomy class. We delimit the claim that interventions are modelled by the subobject classifier: an intervention and an observation name the same subobject, so \Om fixes the target of a do-operation but not the operation itself, which is surgery on the kernels --- where, on a confounder, the interventional and observational laws differ. And we settle the modal unit --- inflationarity is derivable from j⊤=⊤ and naturality, not a fourth axiom. We also machine-check the classifier of sieves with its classification theorem, the pullback collating local mechanisms, and the Kripke-Joyal forcing clauses. The development assumes no axioms and typechecks under Agda's \texttt{--safe} flag, with the ordered field discharged at Q; type-level sheafification and a directed do-calculus are future work.
This paper introduces a categorical account of infinitesimal causality in Frobenius Markov categories equipped with tangent-bundle semantics. IDC captures the infinitesimal layer in which interventions act as tangent deformations of copy/discard structure. Two distinct Frobenius structures interact: (1) the categorical Frobenius algebra on classical variables encoding copying, comparing, and discarding; and (2) the geometric Frobenius integrability condition, namely involutive closure of the intervention distribution, distinct from the algebraic Frobenius structure. Categorical causal sufficiency is defined as the compatibility of these two notions. A key observation is that, for structural causal models, infinitesimal causality is most naturally formulated in the slice of deterministic mechanisms over exogenous variables, with visible stochastic kernels obtained only after pushforward. Interventions are tangent vectors that deform the Frobenius copy/discard operations; their Lie brackets measure whether this deformation preserves classical information-flow structure. Pearl's do-calculus is used as a guiding example of intervention identities: ignoring irrelevant interventions corresponds to counit invariance, action/observation exchange to coproduct compatibility with pushforward, and independence to involutive bracket closure of the visible intervention distribution.
Interventional data is widely regarded as the gold standard for teaching models causal reasoning. We test this assumption in a fully controlled synthetic environment pitting observational correlation against causal effect, and find it fails instructively. In Simpson's-paradox worlds, where the two have systematically opposite signs, increasing the fraction of interventional samples in pretraining does not improve causal direction: the magnitude of the model's do()-response grows monotonically, yet its sign is copied from the observational context. What governs whether interventional evidence is used is not the training mixture but the evidence type present in the context at inference time. Under an identical training recipe, a purely observational context induces systematic sign reversal in 29/50 worlds, a mixed context in 19/50, while aligned interventional probes alone yield 41/50 correct. Erasing observational evidence from the context immediately releases the suppressed causal interpolation ability (ratio_true = +0.56); a four-state content manipulation shows the switch is content-mediated and graded. The suppression is stable across training seeds (11/11 strong reversals persist on a matched-protocol second seed) and robust as a rate at 0.93B parameters (31.8% vs. 6% reversals in the matched probe-only arm), even as absolute gains shrink four-fold. An external audit on CLadder exposes a learned positive-effect prior with a two-layer structure: sign-randomized retraining removes it in-distribution but not out-of-distribution. We summarize: the capability lives in the weights; the switch lives in the context, and activation patching localizes the switch to the middle layers' observational rows. We further quantify the sampling noise floor of probe-based causal evaluation and an evidence-averaging protocol that cuts sign errors from 26% to 9%.
Causal discovery is a cornerstone of scientific reasoning, yet whether large language models can perform it reliably remains an open question. Recent benchmarks show that even fine-tuned models plateau on simple causal graphs and degrade as complexity grows, but why they fail has not been established. We prove the failure is fundamental: supervised fine-tuning, direct preference optimization, and in-context learning all produce predictors that cannot distinguish between causal graphs generating similar observational data, and any attempt to do so requires the model's internal representations to grow unboundedly, violating the very conditions under which these methods work. We formalize this as a kernel obstruction theorem, establishing that the limitation is intrinsic to the learning paradigm, \emph{not any particular model or dataset}. We propose Agentic Causal Bayesian Optimization (A-CBO), wherein a frozen language model serves as an interventional oracle answering targeted queries about intervention effects, while an external Bayesian loop concentrates beliefs over candidate graphs in logarithmically many rounds. Because the decision operates outside the space where the obstruction applies, A-CBO provably converges while the underlying model remains unchanged. On Corr2Cause, A-CBO matches fine-tuned baselines without any training. On Extended Corr2Cause, a new benchmark scaling to 24 variables with 18K test samples, A-CBO significantly outperforms both fine-tuning and preference optimization, with the advantage growing