J-Do-Calculus Overview

• J-Do-Calculus is an intuitionistic extension of Pearl’s do-calculus that uses sheaf theory to decompose causal claims into locally valid statements across different regimes.
• It integrates regime-specific structural causal models with a Lawvere–Tierney modality to glue local inferences into stable global conclusions.
• The framework supports decentralized causal discovery with j-stable interventions, enhancing recovery of heterogeneous causal structures.

Searching arXiv for the primary J-Do-Calculus papers and closely related foundational work. J-Do-Calculus, colloquially called “judo calculus,” is an intuitionistic generalization of Pearl’s do-calculus in which causal claims are not required to hold globally in a single model, but instead hold as local truth over a cover of regimes and are then glued into a stable statement. In this framework, regimes such as age groups, countries, dose levels, genotypes, protocols, seasons, or laboratories are organized sheaf-theoretically; a Lawvere–Tierney modality $j$ specifies which regimes are relevant, and $j$-stability requires that intervention premises and conclusions hold constructively and compatibly across that selected family. Under the trivial topology, where the regime space is a single chart, J-Do-Calculus reduces to Pearl’s ordinary do-calculus (Mahadevan, 27 Oct 2025, Mahadevan, 20 Oct 2025).

1. Motivation and conceptual scope

The central motivation for J-Do-Calculus is that causal effects are often regime-dependent. In many applications, a single universal statement such as “$X$ causes $Y$” is too brittle because mechanisms vary across contexts. J-Do-Calculus replaces global truth by local truth on a cover of regimes: a causal claim is certified on each chart of a chosen family and then glued into a statement that is stable across that family. This makes regime dependence a formal part of identification rather than an afterthought (Mahadevan, 27 Oct 2025).

This framework is explicitly intuitionistic. The internal logic does not assume the law of excluded middle. As formulated in the theory paper, disjunction and existence are local and gluable: “$\varphi \vee \psi$” at a stage means one can cover the stage by parts where either $\varphi$ or $\psi$ holds, and “$\exists x.\Phi(x)$” means there is a cover with witnesses on each part. The implementation-oriented paper treats this locality as algorithmically relevant because witnesses are constructive and must be compatible on overlaps (Mahadevan, 27 Oct 2025, Mahadevan, 20 Oct 2025).

A common misconception is to read the prefix $j$ as merely a notational embellishment of Pearl’s calculus. The formal role of $j$ is stronger: it determines which covers count as admissible, which subobjects are $j$0-dense, and which claims may be promoted from chartwise validity to ambient truth. Another misconception is to equate J-Do-Calculus with ad hoc multi-environment averaging. The papers instead define local truth, gluing, and $j$1-stability inside a topos of sheaves, so the regime-wise decomposition is semantic rather than only procedural (Mahadevan, 20 Oct 2025).

2. Sheaf-theoretic and intuitionistic semantics

The formal setting begins with a small category $j$2 of regimes or contexts, whose objects are regimes and whose arrows are restrictions or refinements. A Grothendieck topology $j$3 on $j$4 specifies admissible covers, and the ambient category is the topos of sheaves

$j$5

Its internal logic is intuitionistic and supports a subobject classifier $j$6 together with a Lawvere–Tierney modal operator $j$7 (Mahadevan, 27 Oct 2025, Mahadevan, 20 Oct 2025).

In this setting, $j$8 is a nucleus on $j$9 satisfying

$X$0

The papers also state monotonicity: if $X$1, then $X$2. These axioms make $X$3 a closure operator in the internal logic and induce the corresponding notion of $X$4-sheafification (Mahadevan, 27 Oct 2025, Mahadevan, 20 Oct 2025).

Local truth is expressed through Kripke–Joyal semantics. Writing $X$5 for forcing at stage $X$6, the boxed modality associated with the topology is

$X$7

Thus a proposition is true at $X$8 if it holds on a $X$9-cover of $Y$0. The corresponding gluing principle is the ordinary sheaf condition: if local sections are compatible on overlaps, they uniquely amalgamate into a global section,

$Y$1

This is the formal basis for certifying causal premises chartwise and then promoting them to ambient truth by gluing (Mahadevan, 27 Oct 2025).

Several derived notions are central. A mono $Y$2 is $Y$3-dense iff

$Y$4

equivalently $Y$5. An object is a $Y$6-sheaf iff it admits unique amalgamations along $Y$7-covering sieves, or equivalently iff the unit of sheafification is an isomorphism. These notions encode the distinction between arbitrary local data and local data that can be certified as coherent across the chosen regime family (Mahadevan, 20 Oct 2025).

3. Regime-indexed structural causal models and $Y$8-interventions

J-Do-Calculus works with regime-indexed structural causal models. For variables $Y$9 and regime $\varphi \vee \psi$0, the structural equation for a variable $\varphi \vee \psi$1 is written

$\varphi \vee \psi$2

where $\varphi \vee \psi$3 are parents and $\varphi \vee \psi$4 are exogenous noises. Each regime supplies a fiberwise model, and arrows in $\varphi \vee \psi$5 induce restrictions between fibers (Mahadevan, 27 Oct 2025).

Interventions are then defined relative to the $\varphi \vee \psi$6-selected family of regimes. Given a regime-indexed interventional distribution $\varphi \vee \psi$7, the $\varphi \vee \psi$8-stable intervention is

$\varphi \vee \psi$9

If a measure $\varphi$0 on $\varphi$1 is available and the $\varphi$2-selected subspace is measurable, the paper gives the concrete form

$\varphi$3

In practice, monotone aggregators $\varphi$4 such as Fisher pooling, Stouffer pooling, and trimmed means are used (Mahadevan, 27 Oct 2025).

Conditional independence and d-separation are defined fiberwise. For each regime $\varphi$5, one works with the local graph $\varphi$6. Sheaf morphisms transport conditional-independence judgments along restrictions, and $\varphi$7-sheafification retains only those conditional-independence statements that are stable under restriction and glue over a cover. In the theory paper, conditional independence is expressed internally as factorization,

$\varphi$8

where $\varphi$9 and $\psi$0 (Mahadevan, 27 Oct 2025, Mahadevan, 20 Oct 2025).

This suggests that J-Do-Calculus does not merely annotate ordinary SCMs by regime labels. Rather, it turns regime variation into part of the semantics of truth, identifiability, and compatibility.

4. Inference rules and identification criteria

The formal rules mirror Pearl’s three rules, but their premises are required to hold $\psi$1-locally on a cover and to be compatible on overlaps. In the internal-logic formulation, the rules are stated on mutilated graphs and interpreted as equalities of arrows in the sheaf topos (Mahadevan, 20 Oct 2025).

Rule	$\psi$2-local premise	Conclusion
J1	$\psi$3	$\psi$4
J2	$\psi$5	$\psi$6
J3	$\psi$7	$\psi$8

The implementation-oriented paper presents the same structure in regime-wise form. Its j-Insertion/Deletion of Observations states that if, on the selected family $\psi$9, $\exists x.\Phi(x)$0 fiberwise and compatibly, then

$\exists x.\Phi(x)$1

Its j-Action/Observation exchange states that if, on $\exists x.\Phi(x)$2, $\exists x.\Phi(x)$3, then

$\exists x.\Phi(x)$4

Its j-Truncation states that if, on $\exists x.\Phi(x)$5, $\exists x.\Phi(x)$6, then

$\exists x.\Phi(x)$7

Here $\exists x.\Phi(x)$8 denotes environments that change $\exists x.\Phi(x)$9’s mechanism (Mahadevan, 27 Oct 2025).

The standard adjustment criteria also acquire $j$0-stable versions. For the j-back-door criterion, if $j$1 blocks all back-door paths from $j$2 to $j$3 in each regime $j$4 and the separator sections are compatible on overlaps, then

$j$5

Operationally, the paper computes this per regime and aggregates:

$j$6

For the j-front-door criterion, under the usual front-door premises in each regime together with gluing on overlaps,

$j$7

A per-regime aggregated form is given as well (Mahadevan, 27 Oct 2025).

The core soundness statement is a $j$8-soundness theorem: under $j$9-Markov and $j$0-faithfulness assumptions, the $j$1-do rules preserve identifiability across the entire $j$2-selected family, and the identified expressions compute the intended interventional distributions in the topos. The proof sketch relies on Kripke–Joyal semantics, stability of geometric formulas under pullback and refinement, and the idempotence and meet preservation of $j$3 (Mahadevan, 27 Oct 2025). A plausible implication is that identifiability is no longer an all-or-nothing global property; it can be certified locally and then elevated by gluing.

5. Decentralized causal discovery and algorithmic instantiation

The implementation paper describes J-Do-Calculus as a decentralized causal-discovery framework. The regime space is covered by charts $j$4, chosen either from metadata such as country, dose, or wind sector, or from clustering. Causal discovery is then run locally on each chart using a chosen base learner: score-based GES or TCES, constraint-based $j$5-FCI-TCM, or gradient-based DCDI or DCDI-TCM (Mahadevan, 27 Oct 2025).

Compatibility is checked on overlaps $j$6 by aligning skeletons and local parameters. The paper mentions simple matching and consistency criteria such as equality of local conditionals via MMD or Fisher tests. Incompatible edges or parameters are flagged as unstable. Compatible local sections are then glued to a global section (Mahadevan, 27 Oct 2025).

The high-level pipeline is explicitly given. One constructs the cover, runs the base learner on each chart, checks compatibility on overlaps, aggregates edges by support, aggregates parameters via $j$7, verifies j-back-door or j-front-door premises on the glued structure, and finally computes $j$8 by aggregating per-chart formulas. For edge aggregation, the reducer keeps $j$9 if

$j$00

satisfies the stability threshold condition $j$01 (Mahadevan, 27 Oct 2025).

The stated complexity is

$j$02

for chartwise learning plus aggregation and overlap checks. Under parallel execution, wall-clock time becomes approximately

$j$03

which is described as typically much less than $j$04 for a monolithic global run (Mahadevan, 27 Oct 2025).

Three concretizations are singled out. TCES uses a local score equal to BIC minus a $j$05-stability penalty, defined as variance of coefficients across regimes, minus a sheaf overlap penalty, defined as divergence on overlaps. $j$06-FCI-TCM replaces the CI oracle by a sheaf-aggregated test

$j$07

with an optional $j$08-stability veto if a reference regime contradicts the aggregate. DCDI-TCM trains regime-specific heads with a $j$09-stability penalty on edge logits and an optional sheaf penalty on local conditionals, then aggregates edges post hoc by frequency or stability $j$10 (Mahadevan, 27 Oct 2025).

6. Empirical behavior and illustrative examples

The reported experiments span synthetic SCMs, Sachs protein signaling, and OECD PISA ESCS trends. The metrics are Structural Hamming Distance, Precision, Recall, F1, stability indices, negative log-likelihood on held-out regimes, and runtime. Baselines are pooled GES, $j$11-FCI, and DCDI, contrasted with $j$12-stable variants using cover-wise aggregation (Mahadevan, 27 Oct 2025).

The quantitative summaries are explicit. For $j$13-FCI on synthetic data at $j$14,

$j$15

For GES on the synthetic skeleton task,

$j$16

For DCDI with linear perfect interventions, medians $j$17 IQR over $j$18–$j$19 graphs are reported as

$j$20

with the first number in each pair corresponding to vanilla DCDI and the second to the $j$21-stable variant (Mahadevan, 27 Oct 2025).

The paper also gives a worked j-back-door example with two regimes $j$22. In $j$23, $j$24 and $j$25 create confounding; in $j$26, the edge $j$27 is absent, while $j$28 remains. The aggregated formula is

$j$29

With the numerical choices

$j$30

in $j$31, and $j$32 in $j$33, the regime-wise effect is $j$34 in both cases, so the mean aggregator yields

$j$35

The example is used to illustrate premise certification and gluing rather than merely numerical averaging (Mahadevan, 27 Oct 2025).

These results do not prove universal superiority. They do show, in the paper’s reported domains, improved structure recovery and lower wall-clock time under decentralized map–reduce execution.

7. Relation to classical do-calculus, transportability, and limitations

J-Do-Calculus is presented as a conservative extension of Pearl’s calculus. In Pearl’s formulation, identifiability is established in a single global model using the three sound and complete rules of do-calculus; completeness for identifiable causal effects is surveyed in “The Do-Calculus Revisited” and proved in algorithmic form in “Pearl’s Calculus of Intervention Is Complete” (Pearl, 2012, Huang et al., 2012). J-Do-Calculus retains the structure of these rules but shifts truth conditions from global validity to $j$36-local validity plus gluing (Mahadevan, 20 Oct 2025).

This relation is exact in the trivial topology. When $j$37, a sieve covers $j$38 iff it contains $j$39, so $j$40 iff $j$41. The premises then reduce to classical d-separation in the appropriate mutilated graph, and the conclusions become Pearl’s Rules 1–3. The implementation paper states the same point operationally: when the entire regime space is a single chart, J-Do-Calculus reduces to Pearl’s do-calculus (Mahadevan, 20 Oct 2025, Mahadevan, 27 Oct 2025).

The papers also place transportability inside this picture. The implementation paper states that transportability appears as a two-regime special case where $j$42 selects the target regime. That observation is consistent with Pearl and Bareinboim’s treatment of transportability via selection diagrams, where causal effects are transferred across populations by reducing expressions with do-calculus and separating source experimental information from target observational information (Pearl et al., 2015). This suggests that J-Do-Calculus can be read as a regime-indexed generalization of external-validity reasoning rather than a replacement for it.

The explicit assumptions are $j$43-Markov and $j$44-faithfulness, sufficient overlap between charts, reliable local estimation and CI testing, and constructive witnesses for intuitionistic claims. The listed limitations are equally direct: misspecified $j$45-selection or covers can break compatibility, strong heterogeneity may preclude gluing, insufficient overlap weakens certification of $j$46-stability, and aggregator choice trades off conservativeness against recall. The theory paper adds that conclusions are guaranteed only $j$47-locally, and that some classical tautologies do not lift because the internal logic lacks excluded middle (Mahadevan, 27 Oct 2025, Mahadevan, 20 Oct 2025).

A final misconception is therefore worth dispelling. J-Do-Calculus does not assert that every heterogeneous causal system can be unified by sheafification. The framework instead states precise conditions under which local conditional independences, interventional equalities, and identified formulas can be certified on a cover and then glued. Where those compatibility conditions fail, the formalism records that failure rather than smoothing it away.