I NFINITESIMAL C AUSALITY A P REPRINT
arXiv:2606.24621v1 [math.CT] 23 Jun 2026
Sridhar Mahadevan Adobe Research and University of Massachusetts, Amherst [email protected], [email protected]
June 24, 2026
A BSTRACT This paper introduces infinitesimal do-calculus (IDC), a categorical account of infinitesimal causality in Frobenius Markov categories equipped with tangent-bundle semantics. IDC is methodologically distinct from categorical do-calculus and string-diagram surgery (Fritz and Klingler, 2023; Jacobs et al., 2018). Those frameworks explain intervention by rewriting or modifying causal syntax; IDC asks how infinitesimal interventions deform the categorical copy/discard structure carried by observable variables. The basic invariants are Frobenius-derivative defects and Lie-bracket closure conditions, rather than graphical rewrites. 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 (µ, η, δ, ε) on classical variables encoding copying, comparing, and discarding (Cho and Jacobs, 2019); 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 (Pearl, 2009) 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. Keywords Infinitesimal causality · Frobenius algebras · Markov categories · Tangent categories · String diagrams · Graphical calculi
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Introduction
Causal inference begins with a sharp distinction between passive observation and active intervention (Pearl, 2009; Rubin, 2005). In the graphical tradition, this distinction is formalized by do-calculus: a small equational calculus for rewriting expressions involving interventional probabilities, such as P (Y | do(X), Z), under graphical separation conditions. Recent categorical treatments of causal inference by Fritz and Klingler (2023) and Jacobs et al. (2018) have formalized Pearl’s do-calculus (Pearl, 2009) using Markov category axioms and string-diagram surgery. These approaches capture the syntactic structure of interventions: how to cut wires, condition on outputs, and compose counterfactuals. However, they remain at the level of discrete combinatorial structure. The infinitesimal geometry of how interventions deform statistical manifolds through Lie-bracket residuals, as demonstrated by companion algorithmic work on Lie-bracket causal discovery, the BRIDGE/SKFM implementation (Mahadevan, 2026b), requires tangent category structure. The present paper complements this line of work by studying the infinitesimal structure: how interventions deform probabilistic geometry, and when such deformations preserve classical information flow. While categorical do-calculus works at the level of graph surgery and diagrammatic rewriting, IDC works at the level of tangent bundles, Lie brackets,
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and Frobenius-derivative defects. It therefore asks a different kind of question: why do latent confounders create specific geometric signatures, such as non-vanishing Lie brackets, independently of any graph presentation? Table 1 summarizes the salient distinctions. To our knowledge, this is the first treatment of do-calculus in a tangent category setting, where the three rules of do-calculus become equations for Frobenius-derivative defects rather than string diagram rewrite rules. The entries involving tangent bundles and derivative defects are made precise in Sections 2 and 4. Aspect
Categorical do-calculus
Infinitesimal do-calculus
Intervention Category Causal structure Composition Latent confounding
Discrete do(X = x) Tangent vector field vi Markov categories Tangent Frobenius Markov categories Graphs as syntax Frobenius defects as geometric invariants Sequential/parallel box composition Lie brackets of vector fields Hidden nodes in string diagrams Non-vanishing Frobenius-derivative defects ∥∂δ∥ Table 1: Categorical do-calculus and infinitesimal do-calculus
This paper organizes these objects into three layers. Markov structure provides the syntax of probabilistic reasoning: observation, conditioning, Bayesian inversion, and disintegration (Fritz, 2020). Frobenius structure provides the syntax of classical information: observable variables can be copied, compared, fixed, and discarded (Cho and Jacobs, 2019). Causality, in the infinitesimal sense developed here, requires the tangent layer: interventions are vector fields that deform the Frobenius copy/discard operations, and their brackets measure whether those deformations remain compatible with the visible information flow. In this sense, infinitesimal causality adds a tangent Frobenius layer to Markov-categorical probability: Markov ⊂ Frobenius Markov ⊂ Tangent Frobenius Markov. The structural question of the paper is therefore internal to the category: when do tangent intervention fields preserve the Frobenius algebra of classical observations? Kan-Do-Calculus (KDC) is the observation that interventions and conditioning can be organized by a pair of Kan adjunctions (Mahadevan, 2026a). Section 5 recalls these adjunctions in a self-contained way and explains how IDC differentiates that transport. The remaining sections then develop intervention fields, Frobenius derivative defects, the three infinitesimal rules, and the role of graphical models as presentations rather than foundations. 1.1
Causal Inference under Confounding
The immediate motivation for infinitesimal causality comes from the problem of causal discovery under latent confounding (Spirtes et al., 2000; Richardson and Spirtes, 2002; Zhang, 2008). In the fully observed setting, a graphical causal model can often be treated as a compact syntax for conditional independences, factorizations, and interventions. With latent variables, however, the same observed distribution may admit several inequivalent finite presentations. Confounding can therefore appear not merely as a missing edge or hidden node, but as a failure of the observed intervention directions to close under the geometry induced by the statistical model. This is the point at which the usual finite syntax becomes too brittle: the obstruction is visible before a unique graph, if any, has been selected. Companion algorithmic work on Lie-bracket causal discovery, implemented as BRIDGE/SKFM, gives a computational realization of the tangent Frobenius structure just described (Mahadevan, 2026b). It maps local causal perturbations to vector fields on a statistical manifold and studies the Lie algebra generated by those fields. Non-commuting intervention directions, curvature residuals, and center-like quantities are used there as practical screens for hidden structure. In that setting, latent confounding is detected by asking whether the visible intervention algebra is internally closed or whether brackets produce components that cannot be explained by the observed coordinates. The present framework abstracts away from the particular estimators used in that computational screen. The Liebracket geometry construction suggests that the meaningful object is the obstruction pattern carried by infinitesimal interventions: which copy/discard structures are preserved, which tangent directions commute, which brackets leave the visible span, and which central elements remain stable across presentations. These are exactly the data naturally expressed in a Markov category equipped with Frobenius structure and tangent-category derivatives. This suggests a categorical formulation of causal inference under latent structure: begin with Frobenius-compatible infinitesimal invariants of the observed statistical object, and regard any finite graphical or algebraic syntax as a presentation of those invariants. The task is then not merely to select a presentation, but to characterize which derivative
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defects, copy-preservation properties, and bracket-closure conditions are intrinsic to the Markov/Frobenius/tangent category. Contributions.
This paper makes the following contributions.
1. It defines the categorical substrate Stat∞ : regular finite-dimensional statistical models presented by sufficient statistics in a Markov category, with Frobenius copying on the sufficient-statistic objects and tangent intervention fields on parameter objects. 2. It defines the structural subcategory StatSCM ⊆ Stat∞ , where stochasticity is carried by exogenous variables ∞ and visible stochastic kernels are pushforwards of deterministic mechanisms. This places infinitesimal intervention fields on the exogenous tangent bundle before projection to visible laws. 3. It identifies categorical causal sufficiency with the coincidence of algebraic Frobenius copying/discarding and involutive closure of intervention distributions. 4. It formulates infinitesimal analogues of the three classical intervention transformations as equations for counits, coproducts, Kan transport, and bracket closure. 5. It defines Frobenius-derivative defects as the basic obstruction to copy-preserving infinitesimal intervention. 6. It treats graphical causal models as presentations of Frobenius-Markov objects rather than as primitive objects of the theory. 7. It isolates the larger obstruction-theoretic and presentation-selection questions as future work.
2
The Categorical Substrate: Frobenius Markov Categories
The present framework uses Markov categories as the abstract syntax of probabilistic causal systems (Fritz, 2020; Fritz and Klingler, 2023). The additional infinitesimal layer is supplied by tangent-bundle structure in the sense of Rosický’s and Cockett–Cruttwell’s tangent categories (Rosický, 1984; Cockett and Cruttwell, 2014). The bridge between the two structures is statistical sufficiency: in a Markov category, a sufficient statistic is precisely the piece of an observation that may be retained while discarding the remaining randomness without losing information about the parameter (Fritz, 2020). A tangent category is a category X equipped with a tangent bundle functor T : X → X together with natural transformations π : T ⇒ Id, 0 : Id ⇒ T, + : T ×X T ⇒ T, ℓ : T ⇒ T 2, and the canonical flip, satisfying the tangent-category axioms of Cockett and Cruttwell (2014, Definition 2.3). The category Smooth of finite-dimensional smooth manifolds with the ordinary tangent bundle is the prototypical example. The category Stat∞ used below is a Frobenius Markov category of finite-dimensional statistical models presented by sufficient statistics. Its infinitesimal structure is induced from the tangent category of smooth parameter manifolds, not from an unrestricted tangent category of all statistical laws. Definition 2.1 (The category Stat∞ ). The category Stat∞ is the category of regular finite-dimensional statistical models with specified sufficient statistics. An object is a tuple (Θ, X , S, p, t) where: 1. Θ ⊆ Rd is an open smooth parameter manifold; 2. p : Θ ⇝ X is a smooth Markov kernel, equivalently a smooth statistical model θ 7→ pθ on the sample space X; 3. t : X → S is a deterministic Markov kernel, called the sufficient-statistic map, such that p displays Θ as conditionally independent of X given t(X ), in the Markov-categorical sense of statistical sufficiency, equivalently the categorical form of the Fisher–Neyman factorization condition; 4. the Fisher information matrix gF (θ)ij = Epθ ∂i log pθ ∂j log pθ is smooth and positive definite. A morphism
(f, K, h) : (Θ, X , S, p, t) −→ (Φ, Y, S ′ , q, t′ )
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consists of a smooth map f : Θ → Φ, a smooth Markov kernel Kθ : X ⇝ Y, and a deterministic map h : S → S ′ such that Z qf (θ) (B) = Kθ (x, B) dpθ (x) X
for measurable B ⊆ Y, and the sufficient statistics are respected in the Markov category: h ◦ t = t′ ◦ Kθ . The induced parameter map is required to be smooth with respect to the Fisher metrics. The sufficient-statistic map t : X → S discards nonsufficient randomness while retaining the information about θ carried by the observation. The sufficient-statistic object S is therefore the classical part of the model. It carries the special commutative Frobenius comonoid δS : S → S ⊗ S,
εS : S → I,
where δS copies the sufficient statistic and εS discards it. Thus the algebraic Frobenius structure is not added externally to the statistical model: it is the copy/discard structure on the sufficient information retained from the observation. When the sufficient statistic decomposes as S ≃ S1 ⊗ · · · ⊗ Sd , we write (δi , εi ) for the Frobenius comonoid on the coordinate statistic Si and use Xi as informal notation for the corresponding visible observable. Proposition 2.2 (Tangent structure of Stat∞ ). The category Stat∞ inherits tangent structure on objects from Smooth by forgetting to parameter spaces. For an object (Θ, X , S, p, t), T (Θ, X , S, p, t) = (T Θ, X , S, T p, t), and the tangent space at θ is identified, via the score embedding, with the subspace n o X X v= v i ∂i 7−→ v i ∂i log pθ . Tθ Θ ∼ = s ∈ L2 (pθ ) : Epθ [s] = 0 , i
i
For a regular exponential family pθ (x) = exp ⟨θ, T (x)⟩ − A(θ) h(x), the score directions are centered sufficient statistics: ∂i log pθ = Ti (x) − Epθ [Ti ]. Thus infinitesimal intervention fields are tangent vectors in Tθ Θ acting on the family through the score functions generated by the sufficient statistic. On morphisms, the tangent action is the differential T f on parameter spaces together with the induced pushforward of score functions along Kθ , whenever that pushforward remains in the chosen regular model class. Definition 2.3 (Structural subcategory). The structural subcategory StatSCM ⊆ Stat∞ consists of models whose ∞ stochasticity is carried by exogenous variables. An object is a tuple (U, X, S, pU , f, t) where U is an exogenous noise object, pU : I ⇝ U is its law, f : U → X is a deterministic structural map, and t : X → S is a sufficient statistic for the observed pushforward law f∗ pU . More generally, for systems with endogenous inputs, the deterministic mechanism has the form f : X × U → Y , and the observed stochastic morphism is obtained by composing f with the law of U and then discarding U . This subcategory is important because the tangent functor does not have to differentiate arbitrary stochastic kernels. On the deterministic structural maps it is the ordinary tangent pushforward: T (f ) : T (X × U ) −→ T Y. The Markovian randomness appears only through the exogenous law pU and through marginalization along the delete map !U : U → I. Thus the Leibniz rule and the usual tangent-category structure hold on the deterministic mechanisms, while Frobenius residuals arise after projecting this deterministic geometry through unobserved exogenous randomness. In the causally sufficient case, the exogenous factors are independent enough that the projected intervention fields remain involutive. With shared latent exogenous factors, the visible projection can twist the deterministic flows, producing bracket components outside the visible span. This also clarifies the support problem for interventions. A visible hard intervention such as do(X = x) may be singular if x ∈ / f (supp pU ), since it asks for a state that has no observational mass. In the structural presentation, however, 4
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infinitesimal interventions are represented on the exogenous tangent bundle. A soft intervention is a vector field ṽ ∈ T U deforming the exogenous law; its visible effect is the deterministic pushforward v = T f (ṽ) ∈ T X. Thus the local differential object exists even when a global Radon–Nikodym density ratio on X is ill-behaved. Hard structural interventions, which replace f by a modified mechanism f ′ , are better regarded as changing the object or morphism in the slice of structural maps over U , rather than as endomorphisms of the visible stochastic object. From this viewpoint, IDC studies the differential closure of the pushed-forward fields T f (ṽi ); support pathologies belong to the passage from exogenous deterministic geometry to visible measures, not to the existence of the tangent fields themselves. Definition 2.4 (Soft structural intervention). Let (U, X, S, pU , f, t) ∈ StatSCM ∞ . A soft structural intervention is a smooth local deformation of the exogenous law, represented infinitesimally by a vector field ṽ ∈ Γ(T U ). Its visible intervention field is the tangent pushforward v = T f (ṽ) followed, when necessary, by projection to the tangent space of the chosen visible statistical model. Thus a soft intervention changes the exogenous distribution or its flow while leaving the deterministic mechanism f fixed. By contrast, a hard structural intervention replaces f by a modified mechanism f ′ or restricts attention to a fiber such as f −1 (x). This definition is the structural reason that IDC avoids the singularity created by global density ratios. A Radon– Nikodym representative of the visible intervention may fail to exist when the intervened law has support outside f (supp pU ), but the infinitesimal datum ṽ ∈ T U and its pointwise tangent pushforward T f (ṽ) are defined wherever the deterministic map f is smooth. The calculus therefore starts from a local tangent deformation before passing to visible interventional measures. The score representation in Theorem 2.2 is compatible with this structural presentation. For StatSCM ∞ , an exogenous field ṽ ∈ T U first pushes forward to v = T f (ṽ) ∈ T X; its score-coordinate representative is then the projection of this visible displacement onto the tangent space of the chosen statistical model, using the Fisher metric. Equivalently, score coordinates forget vertical directions in the kernel of the observational disintegration and retain the component that changes the visible sufficient-statistic law. Thus the exogenous tangent bundle supplies the geometric field, while the score embedding supplies one statistical coordinate system for its visible projection. This definition separates two uses of “Frobenius.” In the rest of the paper, Frobenius algebra or Frobenius comonoid always refers to the categorical copy/discard structure (µ, η, δ, ε) on classical variables inside a Markov category (Cho and Jacobs, 2019). By contrast, Frobenius-integrable or involutive refers to the differential geometric condition that a distribution of vector fields is tangent to a foliation, equivalently that its Lie brackets close. Infinitesimal causality says that causal sufficiency is where the categorical copy/discard structure is preserved by an involutive intervention distribution. The copied/discarded classical variables generate intervention directions whose tangent brackets remain visible. Figures 1 and 2 give the string-diagrammatic picture used throughout the paper. In Stat∞ , the model kernel and sufficient statistic determine the copied classical object S, while tangent directions act through score fields and Lie derivatives of the Frobenius coproduct. In StatSCM ∞ , the same copied object is observed only after a deterministic structural map and a pushforward of exogenous tangent fields; residuals rij record the part of the pushed-forward bracket that cannot be represented by the visible intervention directions.
3
Basic Examples
The definition of Stat∞ is intentionally small: it asks for a regular statistical model, a sufficient-statistic object, the Frobenius copy/discard structure on that object, and tangent directions on the parameter space. The following examples illustrate how these pieces interact. Example 3.1 (Gaussian location family). Let Θ = R, let X = R, and let 1 pµ (x) = √ exp − 12 (x − µ)2 2π be the unit-variance Gaussian location family. This is a one-dimensional exponential family: pµ (x) = h(x) exp(µx − A(µ)),
1 h(x) = √ exp(−x2 /2), 2π 5
A(µ) = µ2 /2.
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S S
S
rij X
Lv δS
t
X T f (ṽ) ∈ T X
t sv p
Tf
f
ṽ ∈ T U
U exogenous
v ∈ TΘ
Θ
pU
Figure 1: Stat∞ : a Markov model presented by a sufficient statistic. The black nodes are the copied classical data, and the dashed arrows show the tangent score direction induced by v.
Figure 2: StatSCM ∞ : randomness is exogenous, mechanisms are deterministic, and visible tangent fields arise by pushing exogenous fields forward before measuring Frobenius defects.
It also lies in the structural subcategory: one may write X = µ + U with U ∼ N (0, 1), so the stochasticity is exogenous and the structural map is deterministic. The sufficient statistic is t(x) = x, so S = R. The Frobenius structure on S is the ordinary classical one: δ(x) = (x, x), ε : S → I. Thus the sufficient statistic may be copied or discarded as classical information. The tangent space Tµ Θ is generated by ∂µ , and the score is ∂µ log pµ (x) = x − µ. An idealized infinitesimal intervention pulling the mean toward a value x is therefore represented by the vector field vx (µ) = (x − µ)∂µ . For two target values x and y, [vx , vy ] = (x − µ)∂µ (y − µ) − (y − µ)∂µ (x − µ) ∂µ = (y − x)∂µ . The bracket is still a tangent direction in the one-dimensional visible parameter space. In this elementary case, the Frobenius/tangent structure is closed: the sufficient statistic generates the score direction, and brackets of such intervention fields remain visible. Example 3.2 (Binomial model). Let Θ = (0, 1), let X = {0, 1}n , and let pθ (x) = θ
P
i xi
(1 − θ)n−
P
i xi
be the model of n independent Bernoulli trials. The sufficient statistic is the count t(x) =
n X
xi ∈ S = {0, 1, . . . , n}.
i=1
This example can also be presented structurally by taking exogenous Ui ∼ Uniform(0, 1) and setting Xi = 1{Ui ≤ θ}. The visible Bernoulli randomness is the pushforward of a deterministic threshold map applied to exogenous noise. The sufficient-statistic object S is discrete and classical. Its Frobenius coproduct copies the count: ε : S → I.
δ(k) = (k, k), The score is
t(x) − nθ , θ(1 − θ) so the tangent direction is generated by the centered count statistic. ∂θ log pθ (x) =
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Forcing one coordinate to be 1 changes the expected sufficient statistic from nθ to 1 + (n − 1)θ. At the level of the one-dimensional parameter space, this gives the infinitesimal displacement vi (θ) = (1 − θ)∂θ for every coordinate i. Hence, for independent trials, [vi , vj ] = 0. The statistic t is sufficient, the Frobenius copy map on S is preserved by the visible tangent direction, and the intervention distribution closes trivially. If a shared unobserved factor were introduced and then omitted from the sufficient statistic, this closure can fail; in the language below, the failure would appear as a residual bracket component outside the visible span. Example 3.3 (Multivariate Gaussian family). Let Θ = Rd × PDd parameterize a Gaussian law N (µ, Σ) on X = Rd . A sufficient statistic is t(x) = (x, xx⊤ ) ∈ S = Rd × Symd . For fixed (µ, Σ) this has the structural presentation X = µ + LU , where U ∼ N (0, I) and LL⊤ = Σ. Thus the tangent directions may be read either as score directions on the Gaussian family or as pushforwards of deterministic deformations of the structural map and exogenous law. Equivalently, in natural coordinates the multivariate Gaussian is an exponential family whose natural parameters are Σ−1 µ and − 12 Σ−1 . The Frobenius structure copies the sufficient statistic: δ(x, xx⊤ ) = ((x, xx⊤ ), (x, xx⊤ )), ε : S → I. Thus the classical data made available to the causal calculus are the retained first- and second-order sufficient statistics. Coordinate interventions act as tangent directions on the mean and covariance parameters. Informally, an intervention setting Xi toward a value a shifts µi toward a and changes the corresponding variance and covariance components of Σ; similarly for Xj . When the visible Gaussian family is causally sufficient, these coordinate intervention fields close under brackets inside the span generated by the visible sufficient statistics. If the observed coordinates share an omitted common source, for example a latent factor contributing to both Xi and Xj , then the bracket [vi , vj ] can acquire a component in the off-diagonal covariance direction not generated by the visible intervention fields alone. This is the prototype of the residual term X [vi , vj ] = ckij vk + rij k
introduced in Section 4: rij measures the failure of the visible Frobenius/tangent data to close. Example 3.4 (Exchangeable data and causal de Finetti). Let (X1 , X2 , . . . ) be an exchangeable sequence with values in a space X . By de Finetti’s theorem, the finite-dimensional laws admit the mixture representation Z Y n P (X1 , . . . , Xn ) = pθ (Xi ) dµ(θ), Θ i=1
where θ is a latent mixing parameter and µ is a prior over Θ. Recent work on causal de Finetti and Do Finetti uses such exchangeable representations to study invariant causal structure and causal effects beyond the i.i.d. setting (Guo et al., 2022, 2024). Unlike the preceding examples, this is best viewed as an example of the general Stat∞ setting rather than a single deterministic structural map with fixed exogenous law: the object includes a mixture over mechanisms, and conditioning or intervening may move the posterior over the mixing parameter. For finite n, the empirical measure n 1X Pbn = δX n i=1 i
is the natural sufficient statistic for the exchangeable sample. Thus an object of Stat∞ may be represented by a parameter object Θ ⊆ P(X ), a Markov kernel p : Θ ⇝ X n , and a sufficient-statistic map t : X n → P(X ) sending a sample to its empirical measure. The Frobenius structure copies the empirical measure: ε : P(X ) → I.
δ(Pbn ) = (Pbn , Pbn ),
This expresses that the empirical distribution is classical information: it may be copied and discarded even though the individual observations remain probabilistic. The tangent directions now live on the mixing parameter or on a finite-dimensional submanifold of P(X ), for instance an exponential family. Conditional on θ, the independent causal mechanism assumption used in causal de Finetti 7
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says that the component mechanisms are independent. In IDC language, this is the fiberwise Frobenius-compatibility condition [vi , vj ] θ = 0, because intervening on one coordinate does not deform the mechanism of another coordinate after the mixing parameter has been fixed. After integrating out θ, however, the intervention fields can fail to commute. The shared mixing parameter induces dependence between coordinates, and the bracket may acquire a residual component in the direction of prior or posterior sensitivity: X [vi , vj ] = ckij vk + rij . k
The residual rij measures the part of the intervention response not explained by the visible empirical statistic alone. Thus the de Finetti mixture provides a hierarchical example of the same Frobenius/tangent pattern: the empirical sufficient statistic is the copied classical object, while the tangent structure records how interventions propagate through the latent mixing parameter. Example 3.5 (Studený imsets as non-graphical presentations). Studený’s theory of integer-valued multisets, or imsets, gives a particularly useful non-graphical presentation of conditional independence structure (Studený, 2005). Fix a finite set N of variables. An imset is an integer-valued function u : 2N → Z, equivalently an element of the free abelian group generated by subsets of N . The elementary conditional independence assertion i ⊥ j | K is represented by the elementary imset u⟨i,j|K⟩ = δK∪{i,j} + δK − δK∪{i} − δK∪{j} , where δA denotes the basis vector associated to A ⊆ N . Sums of elementary imsets encode finite multisets of conditional independence constraints, and the resulting cone of elementary imsets is dual to a cone of supermodular functions. Thus imsets provide an algebraic and polyhedral language for independence, rather than a graphical one. This example fits the viewpoint of IDC in two ways. First, it illustrates that finite causal syntax need not be a graph. A graph, a standard imset, and an integer multiset of elementary independence constraints can present related observable Markov structure, but none of these presentations should be identified with the underlying Frobenius Markov category itself. In the language of Section 7, imsets are another presentation functor: they retain a finite algebraic shadow of conditional independence, while forgetting the tangent fields and Frobenius derivative defects that IDC uses to distinguish infinitesimal intervention structure. Second, imsets make precise the additive aspect of conditional independence. The Frobenius copy/discard structure supplies the classical variables on which independence statements are formed; an imset records formal integer combinations of such statements. Passing from an IDC object to an imset presentation therefore amounts to projecting the Frobenius/tangent data to its discrete independence skeleton. Nonzero bracket residuals rij or Frobenius derivative defects need not have a faithful image in this projection. Conversely, when those defects vanish, the imset presentation can be viewed as a compact algebraic certificate of the visible conditional-independence relations induced by the category. 3 Example 3.6 (A structural SCM object in StatSCM ∞ ). Let the exogenous space be U = R with coordinates (u0 , u1 , u2 ) and product Gaussian law pU = N (0, σ02 ) ⊗ N (0, σ12 ) ⊗ N (0, σ22 ). Define a deterministic structural map f : U → X = R2 by
X1 = f1 (U ) = u0 + u1 ,
X2 = f2 (U ) = u0 + u2 .
The observed law on X is the pushforward f∗ pU . The shared exogenous coordinate u0 is not visible, but it induces covariance between X1 and X2 . Taking the Gaussian sufficient statistic t(x1 , x2 ) = (x, xx⊤ ) gives an object (U, X, S, pU , f, t) ∈ StatSCM ∞ . The Frobenius structure lives on the sufficient-statistic object S: it copies and discards the retained first- and second-order visible information, not the hidden exogenous coordinates themselves.
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The tangent fields are defined on the exogenous space. For example, let ṽ1 = ∂u1 ,
ṽ2 = ∂u2 ,
z̃ = ∂u0 .
The deterministic tangent map sends these fields to T f (ṽ1 ) = ∂x1 ,
T f (ṽ2 ) = ∂x2 ,
T f (z̃) = ∂x1 + ∂x2 .
Thus an exogenous deformation in the shared coordinate pushes forward to a visible direction that moves both observed variables at once. If the visible intervention algebra is generated only by coordinate interventions on X1 and X2 , the common-source direction must be represented indirectly, for example through a covariance or bracket residual. In the notation of Section 4, this is the local mechanism by which a residual r12 records the missing exogenous factor. This example illustrates why StatSCM avoids the support problem discussed above. A hard intervention such as setting ∞ X1 = a may correspond to a fiber condition f1−1 (a) = {(u0 , u1 , u2 ) : u0 + u1 = a}, whose visible density-ratio representation can be singular or coordinate dependent. Infinitesimally, however, IDC works with vector fields on U and their deterministic pushforwards under T f , so the tangent data are defined before passing to the observed marginal law. These examples show the same pattern in continuous and discrete settings. Sufficient statistics provide the classical objects on which Frobenius copy/discard is defined; score functions identify tangent intervention directions; and, in the structural subcategory, deterministic pushforwards from exogenous tangent bundles explain how visible intervention fields arise. Lie brackets test whether the visible tangent distribution closes or requires additional, unrepresented structure.
4
Interventions as Infinitesimal Fields
Let X = (X1 , . . . , Xd ) be a visible causal system represented inside an object M ∈ Stat∞ . We do not assume that every law arises from a graphical model. We assume only that small interventions define smooth paths in the underlying statistical manifold and can therefore be differentiated. Definition 4.1 (Intervention field). An intervention field for Xi is a vector field vi ∈ Γ(T M) obtained as the infinitesimal displacement of the statistical law under a smooth intervention path t 7→ Pdo(Xi =t) : vi (P ) =
d Pdo(Xi =t) . dt t=0
When the path is represented by density ratios relative to an observational law P , the same field may be written in score or Radon–Nikodym coordinates. Definition 4.2 (Visible intervention algebra). The visible intervention algebra is the Lie algebra generated by v1 , . . . , vd and then projected to the visible span when necessary. Locally, we write X [vi , vj ] = ckij vk + rij , k
where ckij are visible structure constants and rij is an irreducible residual orthogonal to the chosen visible span. The residuals rij play the role of infinitesimal derivative defects. In the ideal causally sufficient case, the visible intervention distribution may be locally involutive. Under hidden variables, omitted adjustment fibers, or incorrect coordinates, the bracket can point outside the visible span. Definition 4.3 (Separated visible stratum). A local visible stratum of M is γ-separated if the intervention fields v1 , . . . , vd are linearly independent with smallest singular value at least γ > 0 in the Fisher metric and every nonzero bracket residual satisfies either ∥rij ∥gF = 0 or ∥rij ∥gF ≥ γ. Equivalently, vanishing and non-vanishing Frobenius-derivative defects are separated by a positive metric margin. This condition is a local rigidity assumption, not a property of arbitrary statistical manifolds. Definition 4.4 (Frobenius derivative of an intervention). Let vi be the intervention field generated by deforming the ith observational law. Relative to the coproduct δj on another variable, define the Frobenius derivative defect ∂i δj = (vi ⊗ id + id ⊗ vi ) ◦ δj − δj ◦ vi . This defect vanishes exactly when vi preserves the copy structure of Xj , i.e. when the infinitesimal intervention acts as a comonoid homomorphism for that observed variable. 9
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The ordinary Lie bracket is the antisymmetrized form of these preservation defects. In coordinates, [vi , vj ] = Dvj (vi ) − Dvi (vj ), while categorically it measures the failure of the tangent lifts of interventions to preserve one another’s Frobenius copying operations. This is the point at which Markov-category syntax and differential geometry meet. Definition 4.5 (Frobenius derivative defect family). Let Avis be the visible Frobenius algebra generated by the observed coordinate comonoids. The family ∂δ = {∂i δj }i,j is the Frobenius derivative defect family. It records, entry by entry, the infinitesimal failure of visible intervention fields to preserve the copy/discard structure of the observed coordinate comonoids. The vanishing of ∂δ says that the intervention fields act as copy-preserving tangent symmetries of the visible Frobenius algebra. A nonzero defect says that the visible copy/discard structure is not preserved by the corresponding infinitesimal intervention. Remark 4.6 (Status of the cohomology). The concrete invariant used in this paper is the defect family ∂δ. We expect that, after quotienting by Frobenius-compatible changes of visible generators, adjustment coordinates, or presentation, these defects assemble into an obstruction class [∂δ] ∈ HH2 (Avis ). The notation marks the intended deformation-theoretic home of the obstruction: Hochschild cohomology controls infinitesimal deformations of algebraic structure. A complete treatment requires a Frobenius-Markov cochain complex adapted to Markov kernels and tangent intervention fields. The rules below use only the concrete defect family ∂δ; constructing the full cohomology theory is left as an open problem. Theorem 4.7 (Frobenius-acyclicity). On a separated visible stratum, categorical causal sufficiency is equivalent to tangent Frobenius commutativity: every visible intervention field preserves the Frobenius copy/discard structure, and the visible intervention distribution is involutive. Proof. By definition, categorical causal sufficiency in Stat∞ means that no extra tangent direction is needed to close the visible intervention distribution and no extra visible generator is needed to make copy/discard operations stable under intervention. The first condition is involutive closure of the visible fields; the second is vanishing of all Frobenius derivative defects ∂i δj and counit derivatives Lvi εj . Together these are precisely tangent Frobenius commutativity. Conversely, tangent Frobenius commutativity gives both closure of brackets and preservation of the visible Frobenius structure, hence visible causal sufficiency. The separated-stratum hypothesis rules out metric ambiguity between zero and nonzero residuals.
5
Kan Adjunctions and Frobenius Transport
Kan-Do-Calculus (KDC) is the observation that interventions and conditioning can be organized by a pair of Kan adjunctions (Mahadevan, 2026a). Let C be a category of observational contexts, let D be a category of interventional contexts, and let K : C → D include an observational context into the corresponding interventional one. Precomposition with K gives a restriction functor K ∗ : D → C that forgets the extra interventional structure. When the relevant Kan extensions exist, this restriction has a left and right adjoint: LanK ⊣ K ∗ ⊣ RanK The left Kan extension LanK freely transports observational data into an interventional context; in causal language, it is the categorical operation that turns an observation interface into an action interface. The right Kan extension RanK reconstructs or restricts information over the observational fiber; in causal language, it captures conditioning, disintegration, and marginal reconstruction. The middle functor K ∗ is the shared observational interface. IDC differentiates this KDC picture. KDC supplies the discrete transport of observational data into interventional contexts and the return transport associated with conditioning. IDC adds the infinitesimal question: when the Kan transport admits a tangent lift, does that lift preserve Frobenius copy/discard structure to first order? Remark 5.1 (Kan extensions and tangent structure). The bi-adjunction LanK ⊣ K ∗ ⊣ RanK of KDC lives in the 2-category of categories, functors, and natural transformations. Infinitesimal intervention fields live instead in the tangent structure of Stat∞ . Relating the two requires showing that LanK preserves tangent bundles, or more precisely that it lifts to a tangent functor on the relevant part of Stat∞ . In this paper, the Kan adjunction provides the discrete structure of intervention and conditioning, while the tangent structure provides the infinitesimal layer. Their intersection—the theorem that LanK lifts to a tangent functor on Stat∞ under natural regularity hypotheses—is left as future work; see the open problem on Kan–tangent transport in Section 8. 10
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At the level of copy structure, transport asks whether the square δj
Xj
Xj ⊗ Xj
LanK (vi )
LanK (vi )⊗LanK (vi )
Xj
δj
Xj ⊗ Xj
commutes on the adjusted fiber. Its commutator is exactly ∂i δj after Kan transport. The analogous counit square tests whether marginalization commutes with the intervention field. This test is conditional: it applies on those adjusted fibers for which LanK has been shown to lift to a tangent functor. Theorem 5.2 (Tangent Kan transport preserves Frobenius structure iff defects vanish). When LanK admits a tangent lift on the adjusted fiber, so that LanK (vi ) is defined, assume also that the Frobenius comonoid (δj , εj ) is transported along K. Then LanK (vi ) preserves the Frobenius structure of Xj to first order if and only if ∂i δj = 0
and
Lvi εj = 0
after Kan transport. Proof. Preservation of the Frobenius comonoid to first order means preservation of both structure maps. The first-order failure to preserve the coproduct is the commutator square above, namely ∂i δj . The first-order failure to preserve discarding is the Lie derivative of the counit, Lvi εj . Hence the transported Frobenius structure is preserved exactly when both defects vanish.
6
Infinitesimal Intervention Rules
Classical do-calculus gives three transformations for interventional probability expressions: removing irrelevant actions, exchanging actions with observations when an adjustment condition holds, and deleting actions under an independence condition (Pearl, 2009; Fritz and Klingler, 2023). In string-diagram presentations, these transformations are implemented by diagrammatic surgery before interpretation in a probabilistic category (Jacobs et al., 2018). For readers not using graphical causal models, these may be read simply as three desired functoriality principles for intervention: discarding should commute with irrelevant intervention, transporting an observation to an action should preserve copy structure, and independent interventions should close under the visible tangent distribution. The infinitesimal calculus replaces graphical intervention manipulations by equations coupling Frobenius copy/discard structure to involutive tangent intervention fields. The basic judgments have the form (Avis , δ, ε; vi ) ⊢ Φ, where Avis is the visible Frobenius algebra of observables, (δ, ε) are the copy/discard maps, and vi is an infinitesimal intervention field. The notation Lvi denotes the derivative of a structure map along vi . Figure 3 summarizes the three rules in the same string-diagrammatic notation used for the substrate. Xj
Xj
Xj
Xj
vi εj
εj
LanK vi
I
I
Xj
L vi ε j = 0
Xj
Xi Xj
Xi
Lan v LanK vi = K i
=
Rule 1
Xj
rij = 0 vj
vi Xj Rule 2
∂i δj = 0
Xj
Xj
Xi Rule 3
[vi , vj ]⊥Avis = 0
Figure 3: The three infinitesimal intervention rules as string-diagrammatic equations. Rule 1 says that discarding an irrelevant observable commutes with the tangent intervention. Rule 2 says that, after the tangent lift of LanK , copying is preserved to first order. Rule 3 says that product copying and visible bracket closure leave no residual tangent component.
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Definition 6.1 (Infinitesimal Rule 1: discarding irrelevant interventions). An intervention direction vi is irrelevant to an observed factor Xj when the counit is invariant along vi : Lvi εj = 0. Equivalently, in the internal string calculus, discarding Xj before or after the infinitesimal intervention gives the same marginal law. This is the infinitesimal counterpart of removing an irrelevant action from an interventional expression. Proposition 6.2 (Rule 1 as first-order counit invariance). Let ϕt be a local flow integrating vi . Then Lvi εj = 0 if and only if εj ◦ ϕt = εj + O(t2 ) as maps on the visible model class. Thus discarding Xj commutes with the infinitesimal intervention to first order. d Proof. By definition, Lvi εj = dt (εj ◦ ϕt ). The derivative vanishes exactly when the first-order term in the Taylor t=0 expansion of εj ◦ ϕt is zero.
Definition 6.3 (Infinitesimal Rule 2: action–observation exchange). Let K : C → D embed observational contexts into interventional contexts. An observed conditioning operation on Xj can be exchanged for an infinitesimal action along vi on an adjusted fiber, provided LanK admits the required tangent lift, when the Frobenius coproduct intertwines with Kan transport: δj ◦ LanK (vi ) = LanK (vi ) ⊗ LanK (vi ) ◦ δj on the adjusted fiber. Equivalently, the Frobenius derivative defect ∂i δj vanishes after transport by LanK . This is the tangent-categorical form of the action/observation exchange rule. Proposition 6.4 (Rule 2 as transported comonoid homomorphism). Assume LanK admits a tangent lift on the adjusted fiber. After Kan transport along K, the intervention field LanK (vi ) preserves the copy map δj to first order if and only if ∂i δj = 0 on the adjusted fiber. Proof. The first-order failure of LanK (vi ) to be a comonoid homomorphism for δj is the commutator (LanK (vi ) ⊗ LanK (vi )) ◦ δj − δj ◦ LanK (vi ). This is precisely the transported Frobenius derivative defect. Hence it vanishes exactly when the transported intervention preserves copying to first order. Definition 6.5 (Infinitesimal Rule 3: independence and product factorization). Two intervention fields are infinitesimally independent on the visible stratum when their bracket residual vanishes and the Frobenius structure factors through the product: [vi , vj ]⊥Avis = 0 ⇐⇒ δij = δi ⊗ δj on Xi ⊗ Xj . When the residual is nonzero, the failure is a derivative defect in the visible Frobenius algebra. Proposition 6.6 (Rule 3 as visible involutivity). Assume the visible intervention distribution has constant rank on a local stratum. If [vi , vj ]⊥Avis = 0 for all visible intervention fields, then the visible distribution is involutive. Conversely, if the visible distribution is involutive, every bracket has zero residual outside Avis . Proof. The residual [vi , vj ]⊥Avis is the component of the Lie bracket outside the visible span. Its vanishing for all visible generators means that the bracket of any two visible intervention fields remains in the visible distribution. This is exactly involutivity. The converse is the same statement read from the definition of an involutive distribution. These rules can be summarized as follows: Rule 1
:
Lvi εj = 0
Rule 2
:
∂i δj = 0 after LanK
Rule 3
:
[vi , vj ]⊥Avis = 0
(counit annihilates irrelevant intervention), (coproduct intertwines with action),
(bracket closure/product factorization).
Thus the classical calculus manipulates interventional probability expressions, while the infinitesimal calculus manipulates Frobenius-compatible intervention fields. Theorem 6.7 (The three IDC rules characterize visible sufficiency). On a separated visible stratum, Rules 1–3 hold for all visible intervention fields if and only if the visible intervention structure is causally sufficient in the categorical sense of Theorem 4.7.
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Proof. Rule 1 says that irrelevant intervention fields preserve counits, hence discarding and marginalization. Rule 2 says that action–observation transport preserves coproducts after LanK . Rule 3 says that visible intervention brackets close and that independent factors have product Frobenius structure. Together these are exactly preservation of copy/discard maps plus involutive closure of the visible intervention distribution. By Theorem 4.7, this is categorical causal sufficiency. The converse follows by reading the two components of tangent Frobenius commutativity through counits, coproducts, and brackets. Corollary 6.8 (Recovery of the discrete rules in the flat case). When the intervention distribution has vanishing curvature residuals and the tangent deformations integrate to discrete intervention operations, the three IDC rules reduce to the corresponding do-calculus transformations: deleting irrelevant actions, exchanging action and observation after adjustment, and deleting actions under independence. Proof. If curvature residuals vanish, bracket closure gives an integrable foliation of intervention fibers. Along each fiber, first-order counit invariance integrates to equality of marginals after deleting irrelevant actions; first-order coproduct compatibility integrates to preservation of copy structure under action–observation exchange; and product factorization integrates to conditional independence of the corresponding finite intervention expressions. These are precisely the three classical transformations.
7
Presentations and Obstructions
Graphical causal models remain important, but here they are presentations of Frobenius-Markov structure rather than primitive objects. A graphical model chooses generators, an order, and a sparse presentation of conditional dependence. The underlying categorical object is the Frobenius algebra of observable variables together with Markov kernels and tangent intervention fields. Definition 7.1 (Presentation of an infinitesimal causal object). A presentation of an object M ∈ Stat∞ is a choice of observable generators X1 , . . . , Xd , Frobenius comonoids (δi , εi ), and a finite set of structure maps sufficient to generate the visible Markov and tangent data under composition, tensor product, copying, discarding, and tangent lift. Different presentations may generate equivalent Frobenius-Markov objects. In particular, a directed graph is one possible presentation when the conditional independence structure is sparse and admits a topological ordering, but the obstruction theory is defined before such a presentation is selected. The nonuniqueness of finite graphical syntax is already visible in the classical theory through Markov equivalence classes of acyclic directed graphs (Gillispie and Perlman, 2001; Schmid and Sly, 2022). Definition 7.2 (Presentation-invariant derivative defect). Let Avis be the visible Frobenius algebra generated by the observed variables. The presentation-invariant defect of two intervention fields is the projection of the Frobenius derivative defect outside Avis : Oij = (vi ⊗ id + id ⊗ vi ) ◦ δj − δj ◦ vi ⊥Avis . It vanishes when vi preserves the visible copy structure of Xj and is nonzero when the visible presentation fails to close under the intervention. Theorem 7.3 (Non-conservativity of presentations). The assignment from finite graphical presentations to their visible Frobenius/tangent data is not conservative in general: non-isomorphic presentations can induce isomorphic visible Frobenius algebras and the same visible intervention distribution. Proof. Choose a visible Frobenius algebra Avis with intervention fields v1 , . . . , vd . A finite presentation records a choice of generators, ordering, and factorization data for this algebra. Two different choices of generators or factorizations can present the same algebra and the same span of intervention fields; for example, Markov-equivalent graphical presentations already identify the same conditional-independence structure in the classical case. Since the Frobenius/tangent data forget the chosen syntactic factorization and remember only the induced copy/discard maps and visible fields, the two presentations have the same image but are not isomorphic as presentations. Hence the assignment does not reflect isomorphisms. This non-conservativity is the categorical reason graphs should be treated as presentations, not foundations. A useful way to formulate the residual presentation problem is as a partial adjunction. Definition 7.4 (Causal and triangular Frobenius presentations). Let Caus denote the category whose objects are finite acyclic directed graphs equipped with a chosen topological order, and whose morphisms are order-preserving graph
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homomorphisms or refinements. Let TangFrobtri denote the category whose objects are Frobenius Markov objects with tangent intervention fields v1 , . . . , vd and a filtration 0 = g0 ⊂ g1 ⊂ · · · ⊂ gd ,
gi = span{v1 , . . . , vi },
such that the Lie brackets are triangular: [vi , vj ] ∈ g>max(i,j) after choosing the given order. Here g>m denotes the span of the intervention directions strictly later than m in the order. This triangularity condition is the tangent analogue of causal acyclicity: brackets propagate forward in the presentation order and therefore generate a solvable Lie algebra. There is then a geometric-realization construction F : Caus −→ TangFrobtri which sends an ordered graph to the Frobenius Markov object generated by its observable variables, its copy/discard maps, and the tangent directions associated with interventions at the nodes. In a causally sufficient acyclic presentation, parents influence descendants but not ancestors, and the resulting bracket constants are triangular in the chosen topological order. The inverse problem asks whether there is an extraction functor G : TangFrobtri −→ Caus which recovers the ordered graphical presentation from the tangent Frobenius data. In general such a functor cannot be defined on all of TangFrobtri : different bases, filtrations, or finite presentations can induce the same visible Frobenius algebra and the same visible intervention distribution. Thus graphical extraction is not a property of the tangent Frobenius object alone unless additional separation data are specified. Conjecture 7.5 (Separated presentation adjunction). There is a separated subcategory TangFrobsep ⊆ TangFrobtri on which graphical extraction is functorial. Its local separation condition is the γ-separated visible-stratum hypothesis of Theorem 4.3, strengthened as needed so that the triangular filtration and nonzero residuals are invariant under Frobenius-compatible changes of presentation. On this subcategory, the realization functor admits a right adjoint F : Caus ⇄ TangFrobsep : G. The unit η : IdCaus ⇒ GF is an isomorphism, expressing that an ordered acyclic presentation realizes and then extracts back to itself. The counit ϵ : F G ⇒ IdTangFrobsep is the triangularization map: it compares a separated tangent Frobenius object with the object generated by its extracted acyclic presentation. The separation hypotheses should ensure that the filtration and the relevant structure constants are intrinsic rather than artifacts of a chosen basis. In statistical terms, one expects conditions such as distinct contamination profiles, a positive gap between zero and nonzero bracket residuals, and stability of the curvature or Fisher–Gram decomposition. Without such conditions, the counit need not select a unique presentation; extraction becomes a choice of section rather than a functor. Even without the full adjunction, one can still define useful obstruction screens: functorial extraction of a graphical presentation is stronger than computing the Frobenius-derivative residuals that obstruct such an extraction. The companion Lie-bracket based algorithms should therefore be understood as computing presentation-invariant obstruction data first, with any subsequent graphical or structural reconstruction requiring additional separation hypotheses.
8
Open Problems
8.1
Beyond Sufficient-Statistic Models
The definition of Stat∞ used here is deliberately finite-dimensional, regular, and built from specified sufficient statistics. This keeps the Frobenius structure internal to the Markov category: the copy/discard maps act on the sufficientstatistic object. A first foundational direction is to determine how far this construction can be enlarged beyond regular exponential-family and sufficient-statistic models while retaining smooth Fisher geometry, Frobenius copy/discard maps, and the conditioning and disintegration operations needed for causal reasoning.
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8.2
Tangent Category Versus Tangent Semantics
In Theorem 2.2, the tangent structure is inherited from parameter manifolds and represented statistically by score functions. A stronger development would characterize exactly which smooth Markov kernels make this score-level tangent lift functorial for all tangent-category structure maps. If this class is too narrow, the correct formalism may be a Frobenius Markov category equipped with tangent semantics on objects and a distinguished class of differentiable morphisms. Determining which route is best suited for causal intervention theory remains an important technical question. 8.3
Support and Absolute Continuity of Interventions
The structural subcategory StatSCM resolves the support problem for the infinitesimal calculus developed in this paper: ∞ soft interventions are represented as tangent fields on the exogenous space U , and their visible effects are obtained by the deterministic pushforward T f . Thus IDC does not require every first-order intervention to be represented by a global Radon–Nikodym density ratio on the observed space X. What remains open is the interface between this infinitesimal semantics and hard structural interventions. If X = f (U ) and U ∼ pU , then a hard operation do(X = x) is naturally expressed as a fiber condition f −1 (x) ⊆ U . Such a fiber may have zero pU -mass, or x may lie outside f (supp pU ), so the corresponding visible law need not be absolutely continuous with respect to the observational law. A complete theory should specify when fiberwise interventions can be approximated by exogenous tangent flows, when they should instead be treated as new objects in the slice of deterministic mechanisms over U , and how weak Kan extensions or disintegrations encode the singular cases. 8.4
A Nontrivial Frobenius-Acyclicity Theorem
Theorem 4.7 is deliberately stated at the level of the separated visible stratum. The next task is to replace this local formulation by intrinsic hypotheses under which Frobenius-acyclicity becomes a substantive theorem rather than a restatement of causal sufficiency. Such hypotheses should include completeness of intervention fields, existence and stability of disintegrations, a separation margin between zero and nonzero bracket residuals, and compatibility between Frobenius copy/discard maps and tangent transport. 8.5
The Separated Presentation Adjunction
Theorem 7.5 asks for a precise characterization of the subcategory TangFrobsep on which graphical extraction is functorial. The main question is whether separation can be formulated intrinsically, without reference to a preferred basis or empirical extraction procedure. Candidate conditions include distinct intervention-contamination profiles, spectral gaps in the Fisher–Gram form of bracket residuals, and stability of the induced triangular filtration under Frobenius-compatible equivalences. One would also like to know whether the separated locus is generic, for example open dense or full measure in an appropriate moduli space of tangent Frobenius presentations. 8.6
Kan–Tangent Transport for Smooth Kernels
The KDC adjunction LanK ⊣ K ∗ ⊣ RanK is abstractly natural, but concrete infinitesimal causal inference requires knowing when these Kan extensions are compatible with tangent structure. A useful theorem would characterize classes of smooth kernels and disintegrations for which LanK lifts to a tangent functor, transports copy maps, and RanK transports counits without leaving the chosen model class. This would turn Theorem 5.2 from a formal first-order criterion into a practical existence theorem. 8.7
A Frobenius-Markov Cohomology
The obstruction notation [∂δ] ∈ HH2 (Avis ) points to a cohomology theory that is not yet constructed here. The desired complex should combine Hochschild-style deformation theory with Markov composition, copy/discard maps, and tangent intervention fields. Its 2-cochains should record Frobenius-derivative defects, its coboundaries should encode admissible changes of coordinates or presentations, and its cocycle condition should capture the Jacobi-type coherence imposed by Lie brackets. 8.8
Invariance of Obstruction Classes
Once such a cochain complex is available, one must prove that the class [∂δ] is invariant under the transformations regarded as inessential. These include changing visible generators, adding adjustment coordinates, replacing one finite 15
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presentation by an equivalent one, and transporting the object along Frobenius-compatible equivalences. Without this invariance theorem, obstruction classes remain useful diagnostics but not yet canonical categorical invariants. 8.9
Score Tangent Categories
The score-function construction in Theorem 2.2 should be verified against the full tangent-category axioms of Cockett and Cruttwell (2014). In particular, one must check that the projection, zero section, addition, vertical lift, and canonical flip are natural for the sufficient-statistic-preserving Markov morphisms admitted in Stat∞ , and that the vertical lift has the required universal property. This is the most direct foundational test of whether the score bundle is merely useful tangent semantics or a genuine tangent-category structure. 8.10
Explicit Statistical Model Classes
The tangent categories of standard statistical model classes should be characterized explicitly. Regular exponential families are the base case: their sufficient statistics, score spaces, Fisher metrics, and natural parameter coordinates make the tangent structure especially transparent. Mixture models are a more subtle second case because singularities, non-identifiability, and boundary strata can obstruct smooth tangent behavior and may fail to admit finite sufficient statistics. Working out these examples would show exactly where the sufficient-statistic definition of Stat∞ is stable and where the broader Cockett–Cruttwell tangent machinery is needed.
9
Conclusion
Infinitesimal causality begins from a simple observation: interventions have derivatives. Once interventions are represented as vector fields, causal compatibility becomes a question about Frobenius preservation, Lie brackets, visible closure, and derivative defects. This geometric viewpoint turns causal inference into an internal logic of Markov categories with classical copy/discard structure and tangent intervention fields. The resulting theory does not require graphical models as primitives. Graphs, factorizations, and bases are presentations that may be useful after the Frobenius/tangent data have been specified. The deliberately narrow aim of this first paper is to formulate the Frobenius Markov substrate, connect it to Kan transport, and state the three infinitesimal intervention rules. The larger tasks of full causal-sufficiency theory, Kan–tangent transport, score-level tangent-category foundations, and cohomological obstruction theory are left as future work. The structural subcategory StatSCM suggests a useful principle for both sides of the story. Categorically, causality is ∞ not fundamentally an endomorphism of the visible stochastic kernel; it is better placed in the slice of deterministic mechanisms over exogenous variables, and only then pushed forward to visible laws. Computationally, the same shift explains why infinitesimal screens can avoid global density-ratio pathologies: one computes tangent fields and bracket closure prior to pushforward to visible laws, before singularities or support mismatch appear in the observed marginal presentation.
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