J-Stable Causal Inference

• The paper introduces j‐stable causal inference, replacing global truth with local truth verified over regime covers.
• It leverages Grothendieck topologies and Lawvere–Tierney modalities to validate causal claims constructively using sheaf semantics and Kripke–Joyal forcing.
• The framework enhances practical causal discovery by applying stability filters and decentralized aggregation to reduce false positives and improve robustness.

J-stable causal inference denotes a form of causal reasoning in which a causal claim is not required to be globally true in one Boolean model, but instead must hold locally on a cover of regimes and be glueable across overlaps. In its most specific technical formulation, developed in the setting of Topos Causal Models and intuitionistic $j$-do-calculus, the framework replaces global truth by local truth, uses a Grothendieck topology $J$ or the corresponding Lawvere–Tierney topology $j$ to specify which regimes are relevant, and treats causal validity as a constructive property that is stable along those covers (Mahadevan, 20 Oct 2025, Mahadevan, 27 Oct 2025). The broader literature grouped under the same stability motif extends this idea into practical causal discovery, heterogeneous treatment effect estimation, model selection, weighting, and representation learning, where causal conclusions are retained only when they persist under subsampling, estimator perturbation, regime variation, or overlap stress.

1. Local truth, regimes, and the meaning of stability

Classical causal inference, in the sense discussed by the judo-calculus literature, uses global truth: a causal statement is true or false in a single model. J-stable causal inference instead treats causal truth as local. Regimes may correspond to age, country, dose, genotype, lab protocol, intervention condition, or other context variables, and a claim is accepted when it is verified on a cover of such regimes rather than everywhere at once (Mahadevan, 27 Oct 2025).

This framework distinguishes two layers of notation. Externally, a site $(C,J)$ specifies objects, morphisms, and covering families. Internally, the same notion is expressed by a Lawvere–Tierney topology $j:\Omega\to\Omega$ on the subobject classifier. The correspondence is explicit: Grothendieck topologies on $C$ correspond exactly to Lawvere–Tierney topologies on the presheaf topos, so $J$ and $j$ are two presentations of the same locality structure (Mahadevan, 20 Oct 2025).

The operational content of stability is given by two principles. The first is restriction stability: if a claim holds at one stage, it continues to hold after refinement to smaller stages. The second is local-to-global gluing: if a claim holds on each element of a cover and is compatible on overlaps, then it holds at the larger stage. In the decentralized causal-discovery paper these appear as the defining properties of $J$-stability, and they are what separates local causal validity from ordinary pooled inference (Mahadevan, 27 Oct 2025).

The resulting logic is intuitionistic rather than Boolean. The point is not merely philosophical. In this setting, causal claims must be constructively certified on admissible covers; they are not assumed to satisfy the law of excluded middle. This allows causal structure to be regime-dependent while still permitting stable inference once the relevant cover has been fixed (Mahadevan, 20 Oct 2025).

2. Sheaf semantics and formal definitions of $j$-stability

The semantic engine of the framework is Kripke–Joyal forcing in a topos of sheaves. Local truth is expressed by a forcing relation of the form

$J$0

A proposition is therefore validated not by a single global witness, but by a covering family on which it holds chartwise (Mahadevan, 20 Oct 2025).

The Lawvere–Tierney topology satisfies the usual modal axioms

$J$1

Within the causal interpretation, $J$2 acts as a closure operator on truth values. The decentralized formulation summarizes this by saying that the modal operator chooses which regimes are relevant and that $J$3-stability means a claim holds constructively and consistently across that family (Mahadevan, 27 Oct 2025).

Conditional independence is then internalized. A conditional independence formula $J$4 is $J$5-stable at stage $J$6 when the sieve of refinements on which $J$7 holds is a $J$8-cover of $J$9: $j$0 where

$j$1

Equivalently,

$j$2

This same pattern is used for interventional equalities as well as for conditional independences (Mahadevan, 20 Oct 2025).

A further categorical formalization appears in the decentralized paper through a right-Kan local-to-global discovery operator,

$j$3

which expresses global graph discovery as the limit of local causal theories over a cover (Mahadevan, 27 Oct 2025).

3. $j$4-do-calculus and causal operations in Topos Causal Models

Topos Causal Models provide the categorical semantics for variables, mechanisms, observations, and interventions. In this setting, interventions are represented as subobjects or as morphisms obtained by surgery: incoming arrows may be cut, mechanisms may be replaced by constants or kernels, and conditioning is represented by restriction to a comprehension subobject followed by normalization. The formal summary given in the theory paper is concise: observation is restriction to a comprehension subobject plus normalization, whereas intervention is kernel replacement plus integration (Mahadevan, 20 Oct 2025).

The internal intervention operator can be written using kernel replacement. For a kernel $j$5 and policy $j$6,

$j$7

This internalizes the familiar idea that $j$8 replaces the structural equation for $j$9, but it does so in the stochastic categorical language of the topos (Mahadevan, 20 Oct 2025).

The three $(C,J)$0-rules mirror Pearl’s do-calculus. In one formulation:

1. Insertion/Deletion of Observation: if $(C,J)$1, then

$(C,J)$2

2. Action/Observation Exchange: if $(C,J)$3, then

$(C,J)$4

3. Insertion/Deletion of Action: if $(C,J)$5 and $(C,J)$6, then

$(C,J)$7

These are sound in Kripke–Joyal semantics. The proof strategy is stagewise: verify the premises on a $(C,J)$8-cover, invoke fiberwise Markov properties on each chart, and then use local forcing to conclude the interventional equality at the base stage (Mahadevan, 20 Oct 2025).

A central consequence is that classical do-calculus is recovered as a special case when the topology is trivial. When only the identity cover matters, $(C,J)$9-stability collapses to ordinary truth, and the framework reduces to classical causal reasoning. The sheaf-theoretic construction is therefore a conservative generalization rather than a replacement of Pearl’s framework (Mahadevan, 20 Oct 2025).

4. Judo calculus and decentralized causal discovery

The algorithmic program built on these ideas is called judo calculus, formally defined as $j:\Omega\to\Omega$0-stable causal inference using $j:\Omega\to\Omega$1-do-calculus in a topos of sheaves. Its guiding principle is that local causal discovery should be performed separately on each regime and then aggregated by a gluing step that retains only coverwise-stable structure (Mahadevan, 27 Oct 2025).

At the level of causal effects, the paper gives a practical form of a $j:\Omega\to\Omega$2-stable interventional probability: $j:\Omega\to\Omega$3 The aggregator $j:\Omega\to\Omega$4 is monotone, and the cover $j:\Omega\to\Omega$5 specifies which regimes are admissible for transport and gluing (Mahadevan, 27 Oct 2025).

This sheafification principle is instantiated in three standard families of causal discovery methods. In the score-based case, TCES extends GES and CGES by adding a sheaf-overlap penalty and a $j:\Omega\to\Omega$6-stability penalty: $j:\Omega\to\Omega$7 In the constraint-based case, $j:\Omega\to\Omega$8-FCI-TCM aggregates regime-specific conditional-independence $j:\Omega\to\Omega$9-values,

$C$0

using Fisher, Tippett, Stouffer, mean, or related combiners. In the gradient-based case, DCDI-TCM either aggregates thresholded regime-wise graphs post hoc or adds a cross-regime variance penalty on edge logits during joint training (Mahadevan, 27 Oct 2025).

The reported empirical behavior is deliberately concrete. On one synthetic benchmark, pooled GES yielded TP $C$1, FP $C$2, FN $C$3, TN $C$4, F1 $C$5, SHD $C$6, whereas $C$7-stable GES with intersection yielded TP $C$8, FP $C$9, FN $J$0, TN $J$1, F1 $J$2, SHD $J$3. For DCDI, the paper reports directed SHD reductions such as $J$4 to $J$5 at $J$6, and $J$7 to $J$8 at $J$9. The authors emphasize both improved performance over classical causal discovery methods and computational efficiency gained by the decentralized nature of sheaf-theoretic discovery (Mahadevan, 27 Oct 2025).

The framework is intentionally conservative. Intersection-style aggregation suppresses unstable false positives, while more permissive all-but-$j$0 rules trade precision for recall. The method is therefore most naturally interpreted as a regime-aware stability filter layered on top of existing discovery algorithms rather than as a wholly separate estimator class (Mahadevan, 27 Oct 2025).

5. Stability outside the sheaf/topos setting

A broader stability-centered causal literature uses a different mathematical vocabulary but pursues a related objective: do not trust a causal conclusion unless it persists under perturbation. In cross-sectional causal discovery, stable specification search addresses the instability of graph estimation by combining stability selection, subsampling, multi-objective evolutionary search, SEM fitting, DAG-to-CPDAG conversion, and optional background knowledge. Its output is not a single graph but edge stability and causal path stability graphs summarizing which relations recur across subsamples and model complexities (Rahmadi et al., 2015).

For heterogeneous treatment effects, several papers make the same point in different forms. Causal Stability Selection combines cross-fitted estimation of conditional average treatment effects with integrated path stability selection and provides an explicit, non-asymptotic bound on the expected number of false positives for arbitrary base selectors (Bargagli-Stoffi et al., 10 May 2026). Counterfactual Cross-Validation redefines stable model selection for CATE prediction as preservation of the rank order of candidate models and uses a doubly robust plug-in CATE together with counterfactual-regression-style variance control to obtain more stable ranking in finite samples (Saito et al., 2019). Causaltoolbox makes estimator stability itself the diagnostic, recommending that many plausible HTE estimators be fit and compared, with disagreement interpreted as unresolved modeling dependence rather than as evidence for a substantive heterogeneous effect (Künzel et al., 2018).

In observational adjustment and weighting, stability again appears as resistance to finite-sample pathologies. Stable Probability Weighting generalizes IPW under limited overlap by replacing inverse-probability residuals with bounded stable weights, while Finite-Sample Stable Probability Weighting constructs unbiased set-estimators in a stratified design (Karapakula, 2023). For time-varying treatments, adaptive orthogonalization stabilizes weighting by balancing the components of covariates that are orthogonal to their histories rather than the raw covariates themselves, and the resulting estimator is proved consistent and asymptotically normal for mean potential outcomes (Li et al., 4 Nov 2025). Confounder selection strategies targeting stable treatment effect estimators adopt the principle that, once confounding is adequately controlled, adding variables associated only with treatment or only with outcome should not systematically change the estimated effect (Loh et al., 2020).

These methods do not use sheaves, toposes, or intuitionistic logic. Their commonality is procedural rather than semantic: stability is treated as a necessary property of credible causal inference when direct validation of counterfactual quantities is impossible.

6. Applications, adjacent formulations, and limitations

The stability motif has been extended into several adjacent domains. In discriminative self-supervised vision, unstable representations are explained as failures under unseen latent interventions, and inference-time corrections are proposed through Robust Dimensions and Stable Inference Mapping learned from controlled synthetic interventions (Yang et al., 2023). In high-dimensional genomics, Causal-GNN combines graph-informed propensity scoring with causal-effect ranking to select biomarkers that are less sensitive to spurious gene–gene correlations and more reproducible under resampling (Lan et al., 17 Nov 2025). In non-stationary forecasting, Stable-CarbonNet formulates carbon-emission prediction as a multi-environment causal invariance problem and combines adaptive normalization, temporal sample reweighting, and gradient-consistency penalties to extract causally stable features (Hong et al., 31 Jan 2026).

Other formulations make explicit where stability can fail. The network-interference literature begins from the observation that SUTVA is violated when one unit’s treatment affects another unit’s outcome, and it therefore models direct effects and spillovers jointly on graphs (Ma et al., 2020). The evolutionary causal-inference paper maps stability directly onto the no-interference component of SUTVA and notes that stability is natural for density-independent fitness and single-generation matched parent–offspring comparisons, but can fail with frequency-dependent fitness, group selection, invasion fitness, changing environments, or intergenerational feedback (Iacovacci, 2 Jun 2026). The quasi-instrumental-variable framework for binary outcomes likewise defines stability in a scale-specific way, requiring stable confounding on the multiplicative scale and stable additive ATT across quasi-instrument levels (Liu et al., 22 Aug 2025).

A persistent limitation across these literatures is that stability is not identical to identification. One paper states explicitly that agreement across many heterogeneous treatment effect estimators does not solve causal inference’s fundamental identification problems, even though it reduces sensitivity to model choice and discourages $j$1-hacking (Künzel et al., 2018). In the sheaf/topos setting, the foundational $j$2-do-calculus work is equally explicit that the present contribution is conceptual and theoretical, with estimation procedures, data-driven $j$3-covers, and standard score-based or constraint-based instantiations deferred to a companion paper in preparation (Mahadevan, 20 Oct 2025).

The unifying conclusion is therefore narrow but technically significant. In the precise sense of $j$4-stable causal inference, stability means local truth preserved under a Lawvere–Tierney modality and glued across a cover of regimes. In the broader methodological sense, stability means that a causal conclusion survives the perturbations that are most likely to generate spurious findings in practice. Both uses reject the adequacy of a single globally fitted causal answer; both insist that causal claims should be retained only when they persist under the relevant notion of variation.