Causal-Constitutive Loops

• Causal-constitutive loops are self-defining systems where causal influence and constitutive dependence jointly specify system boundaries and behavior.
• They employ methodologies like causal blankets, temporal SEMs, and state-space formalisms to describe bidirectional and cyclic dependencies.
• Applications span quantum theory, AI oversight, materials engineering, and recommender systems, offering actionable insights through rigorous diagnostics.

Searching arXiv for recent and foundational papers on causal-constitutive loops and related cyclic causal frameworks. Causal-constitutive loops are closed organizations in which directed influence and system-defining dependence are jointly operative. Across the literature, the term covers several non-identical but structurally related ideas: bidirectional agent–environment loops in which sensory and active variables define an informational interface; cyclic causal models in which mutual dependence is rendered well-defined by temporal or interventionist semantics; self-determining systems whose boundaries are specified by alternating causal and constitutive relations; and domain-specific feedback architectures in quantum theory, recommender systems, system dynamics, AI oversight, and constitutive modeling of materials (Rosas et al., 2020, Ohmura et al., 19 Jun 2026). What unifies these uses is that the loop is not merely a causal cycle. Rather, the mediating variables, mechanisms, or boundary conditions also help specify the effective system, the admissible dynamics, or the relevant state description.

1. Conceptual distinctions

A causal-constitutive loop combines two asymmetric relations. In the most general formulation, the causal relation carries temporal asymmetry and interventionist semantics, while the constitutive relation carries synchronic dependence between a whole and its parts, or between a mechanism and the variables that constitute it (Ohmura et al., 19 Jun 2026). This contrast appears explicitly in the typed representation

$b := F(a), \qquad F :\sim \mathcal{G}(c_j),$

where $b := F(a)$ is causal assignment and $F :\sim \mathcal{G}(c_j)$ states that the mechanism $F$ is constituted by variables $c_j$ (Ohmura et al., 19 Jun 2026).

Several literatures sharpen this distinction in different ways. In agent–environment coupling, the constitutive aspect means that active and sensory variables are not merely channels of influence but “the informational interface that helps define the system and its boundary,” while the causal aspect is the mutual influence across time between agent and environment (Rosas et al., 2020). In runtime human involvement in AI, “constitutive” denotes a human contribution that is necessary for the primary causal chain to generate an output, whereas “corrective” denotes an external intervention that can prevent, modify, or override an output without being necessary for the default chain to run (Baum et al., 19 Mar 2026). In operational causal models with fine-tuning, a variable can be a cause without observational correlation, which means that constitutive-seeming mutual dependence need not be reducible to observable signalling (Vilasini et al., 2021).

A common misconception is to identify causal-constitutive loops with ordinary causal cycles. Several frameworks explicitly reject that equivalence. The self-determination account disallows loops composed exclusively of causal relations and instead requires closure through constitutive dependence at the level of mechanisms (Ohmura et al., 19 Jun 2026). Conversely, the operational framework distinguishes operationally detectable affects causal loops from hidden causal loops, so a cyclic mechanism need not yield an observable loop of interventions and responses (Grothus, 2022). This suggests that “loop” names a broader structural class than directly observable feedback.

2. Dynamic and state-space formalisms

One major formalization is the causal blanket framework for bipartite stochastic processes $(X_t,Y_t)$. Here the interface variables are minimal dynamical sufficient statistics. A process $U$ is a dynamical Bayesian sufficient statistic of $X$ with respect to $Y$ if $U_t = F(X_{:t})$ and

$b := F(a)$0

equivalently,

$b := F(a)$1

The minimal reciprocal statistic yields a causal blanket $b := F(a)$2, isomorphic to $b := F(a)$3, where $b := F(a)$4 is the minimal D-BaSS of $b := F(a)$5 with respect to $b := F(a)$6 and $b := F(a)$7 is the minimal D-BaSS of $b := F(a)$8 with respect to $b := F(a)$9 (Rosas et al., 2020). The framework proves that every bipartite stochastic process has a unique causal blanket up to isomorphism, and that transfer entropy is exactly captured by the blanket variables: $F :\sim \mathcal{G}(c_j)$0

A second formalization treats cyclic dependence temporally rather than synchronically. In temporal SEMs, structural equations are read as one-step mechanisms,

$F :\sim \mathcal{G}(c_j)$1

so even non-recursive models have well-defined behavior because all endogenous dependencies are evaluated on the previous time step (Gladyshev et al., 17 Jan 2025). Time-sensitive interventions $F :\sim \mathcal{G}(c_j)$2 locally override variables at specified times, enabling temporal counterfactual reasoning without requiring acyclicity. The associated logic, CPLTL, combines such interventions with past and future temporal operators, and has a polynomial-time model-checking procedure under the paper’s finite encoding assumptions (Gladyshev et al., 17 Jan 2025).

A related state-space move appears in history-dependent constitutive laws. A non-Markovian constitutive operator is “Markovianized” by internal variables $F :\sim \mathcal{G}(c_j)$3 so that

$F :\sim \mathcal{G}(c_j)$4

In discrete time, $F :\sim \mathcal{G}(c_j)$5 is determined from $F :\sim \mathcal{G}(c_j)$6, and $F :\sim \mathcal{G}(c_j)$7 is computed from $F :\sim \mathcal{G}(c_j)$8, so the feedback between stress and internal variables is causal across time steps rather than an instantaneous algebraic cycle (Raj et al., 13 May 2026). This suggests a general pattern: many causal-constitutive loops are rendered tractable by moving closure into latent state or temporal lag.

3. Closure, boundaries, and compatibility conditions

The most explicit boundary theory defines system interiors by closure under alternating causal and constitutive relations. For a designated interior $F :\sim \mathcal{G}(c_j)$9, variable-level internality requires

$F$0

while mechanism-level internality requires

$F$1

Under these conditions, boundaries are specified by CC-closure: the minimal set closed under these rules and composed of alternating-type cycles that include at least one mechanism (Ohmura et al., 19 Jun 2026). External causes are permitted, since an external variable may affect an internal variable through an internal mechanism without itself becoming internal.

A key constraint is that constitutive dependence must not collapse into supervenience. If constitution has only one variable, $F$2, the relation reduces to supervenience and cannot be combined with causal asymmetry into a coherent CC-loop. The minimal requirement is therefore at least two causally independent constitutive variables, which yields two interdependent loops,

$F$3

The paper derives from this a “dual-process organization” as the minimal self-determining architecture (Ohmura et al., 19 Jun 2026).

Operational and relativistic approaches impose further constraints on which loops are physically or informationally admissible. In the general framework for cyclic, fine-tuned, and non-classical causal models, affects relations are defined interventionally, and compatibility with a spacetime embedding requires that detectable signalling lie within the inclusive future of the intervention set (Vilasini et al., 2021). The later thesis refines this with support stability and minimum stability, showing that no-signalling alone permits certain operationally detectable loops in $F$4-Minkowski spacetime, while additional stability conditions rule out a broad class of complete-chain loops; in higher dimensions, conjectured order-theoretic properties imply stronger exclusions (Grothus, 2022). In the causal blanket framework, boundary is likewise emergent, but informational rather than mereological: the blanket variables are reconstructed directly from past–future statistics and define the effective agent–environment interface from data (Rosas et al., 2020).

4. Discovery, identification, and diagnostics

Causal-constitutive loops can be reconstructed from time series. In the causal blanket framework, the procedure takes paired time series $F$5 and $F$6, chooses past windows $F$7, estimates conditional next-step distributions, clusters pasts into predictive equivalence classes, defines $F$8 and $F$9, assembles $c_j$0, and then validates screening-off and minimality (Rosas et al., 2020). The same framework quantifies effectiveness through

$c_j$1

where $c_j$2 measures synergistic next-step integration not mediated by the blanket. Small $c_j$3 indicates a strong PALO description; large $c_j$4 indicates substantial unmediated integration beyond the blanket (Rosas et al., 2020).

Constraint-based causal discovery with cycles proceeds differently. For simple and $c_j$5-faithful SCMs, FCI is sound and complete in the $c_j$6-separation setting, so observational data suffice to estimate the presence and absence of causal relations, direct causal relations, absence of confounders, and absence of specific cycles (Mooij et al., 2020). Partial ancestral graphs retain their inferential role in the cyclic case: tails certify ancestry, arrowheads certify non-ancestry, and specific circle-mark patterns diagnose strongly connected components (Mooij et al., 2020).

Operational loop detection can also be carried out from affects relations. Higher-order affects $c_j$7, irreducibility, and indecreasability support a constructive loop-detection algorithm based on potential-cause graphs and loop-graph pruning. The result is a complete criterion: a set of affects relations implies an operationally detectable causal loop iff the corresponding loop graph is non-empty (Grothus, 2022). In system dynamics, “Loops that Matter” uses finite differences and path products on simulated trajectories to assign time-varying loop scores,

$c_j$8

thereby distinguishing loops that are causally dominant at a given instant from loops that are merely structurally present (Schoenberg et al., 2020).

5. Domain-specific instantiations

In quantum theory, unobservable causal loops arise from time-symmetrization of reversible processes between correlated measurement outcomes. Each time-symmetrization instance assigns half of the information specifying the outcome pair to the initial measurement and half to the final measurement, producing a loop of forward causation through $c_j$9 and retrocausation through $(X_t,Y_t)$0. Pre-/post-selection is represented by the ABL rule,

$(X_t,Y_t)$1

and self-consistency by

$(X_t,Y_t)$2

The proposal is empirically indistinguishable from ordinary quantum mechanics, but is offered as a unified explanation of both quantum computational speedup and nonlocality (Castagnoli, 2020).

In recommender systems, the loop is explicitly socio-technical: $(X_t,Y_t)$3 Recommendations causally shape observed responses, while the accumulated responses constitute the next model parameters through loss minimization. CAFL breaks the loop for estimation by targeting intervention distributions $(X_t,Y_t)$4, using back-door adjustment through $(X_t,Y_t)$5, and training with IPW or the stationarity-based CAFL estimator (Krauth et al., 2022). In simulated environments, CAFL improves recommendation quality over prior correction methods, and in one benchmark generalized MF with CAFL achieves $(X_t,Y_t)$6 and $(X_t,Y_t)$7 (Krauth et al., 2022).

In runtime AI governance, causal-constitutive loops correspond to HITL rather than HOTL. The constitutive case satisfies $(X_t,Y_t)$8 with $(X_t,Y_t)$9 if $U$0, so no output exists without the human contribution. HOTL instead leaves the default chain autonomous and adds an external corrective path $U$1 (Baum et al., 19 Mar 2026). This reframes “in/on the loop” as a causal distinction rather than a spatial metaphor.

In engineering, history-dependent constitutive laws are cast as causal and energetic loops between strain, internal variables, and stress. The neural framework enforces thermodynamic admissibility through potentials $U$2 and $U$3, uses input-convex neural networks to guarantee convexity and polyconvexity, and proves that learned internal variables are unique up to an invertible linear transform (Raj et al., 13 May 2026). On the Taylor-averaged response of a polycrystalline magnesium unit cell, the framework achieves $U$4 relative error (Raj et al., 13 May 2026). In fuzzy cognitive maps, directed cycles directly model feedback, and the appendix proves transitivity and total downstream influence along paths, making FCMs a natural representation of feedback-dominated causal-constitutive regimes in social systems (Osoba et al., 2019).

6. Debates, limitations, and open problems

Several controversies concern what exactly loops explain. In the causal blanket literature, a causal blanket is not a Friston-style steady-state Markov blanket: it is diachronic, reconstructed directly from data, and requires neither steady-state nor Markovianity (Rosas et al., 2020). In the self-determination literature, constitutive dependence is restricted to mechanisms and must involve at least two independent variables in order not to collapse into supervenience (Ohmura et al., 19 Jun 2026). In the operational literature, cycles can be hidden by fine-tuning, so absence of observable affects does not imply absence of cyclic causation (Vilasini et al., 2021).

Methodological limitations are equally prominent. The causal blanket theory is formulated for discrete-time processes and practical estimation is data-intensive, especially under partial observability or hidden confounding (Rosas et al., 2020). CAFL relies on known policy probabilities and, in the positivity-violating case, on no interference and stationarity assumptions (Krauth et al., 2022). FCI with cycles requires simple SCMs, $U$5-faithfulness, and no selection bias (Mooij et al., 2020). LTM is blind at equilibrium because loop scores vanish when $U$6, even though constitutive structure remains (Schoenberg et al., 2020). The materials framework identifies internal variables only up to linear transform, so learned latent coordinates are not unique as coordinates, only as an equivalence class (Raj et al., 13 May 2026).

Open problems differ by field but have a common theme: how to separate causal influence, constitutive dependence, and observational detectability without sacrificing formal tractability. The operational spacetime program leaves open a full characterization of all operationally detectable loops and the mechanism-level treatment of cyclic non-classical models (Vilasini et al., 2021, Grothus, 2022). Temporal SEMs show that recursive acyclicity is unnecessary for reasoning about actual causality, but they rely on a one-step lag interpretation and finite-state assumptions for efficient verification (Gladyshev et al., 17 Jan 2025). This suggests that the general theory of causal-constitutive loops is still plural: it has robust local formalisms, but no single framework yet subsumes informational boundaries, typed closure, operational signalling, and history-dependent state evolution within one unified semantics.