Deterministic Causal Structure

• Deterministic Causal Structure (DCS) is a formal framework that specifies precise, immutable causal relationships without probabilistic uncertainty.
• It guarantees unique, convergent causal graphs through core axioms such as contribution immutability and localized weak fairness.
• DCS bridges theoretical advances in structural causal modeling, causal discovery, and control by decoupling operational policies from correctness.

A Deterministic Causal Structure (DCS) is a formal framework in which the precise causal relationships between events, variables, or computational updates are specified without reference to probabilistic uncertainty or stochastic processes. In DCS, the propagation and transformation of information or system state is governed by immutable structural relations, deterministic functional dependencies, and order-invariant operations. DCS plays a central role across distributed systems, structural causal modeling, causal inference, and dynamical systems, providing the structural substrate for correctness, identifiability, and abstraction even in the absence of noise. Its technical rigor enables uniquely defined causal graphs, robust convergence, disambiguation of operational histories, and generalizations from value-centered to structurally explicit methodologies.

1. Formal Foundations of Deterministic Causal Structure

In distributed and multi-agent systems, DCS provides a minimal, axiomatized substrate for correct, policy-agnostic computation. The essential unit in DCS is the Contribution:

$$\delta = (\mathrm{rid},\, \mathrm{parents},\, \mathrm{payload},\, k)$$

where

• $\mathrm{rid}\in\mathcal{R}$ is a globally unique identifier,
• $\mathrm{parents} \subset \mathcal{R}$ encodes direct causal predecessors,
• $\mathrm{payload}$ contains application data,
• $k\in K$ denotes a logical partition or key.

These contributions form a provenance DAG (Directed Acyclic Graph):

$$G^* = (\mathcal{R}, E) \quad\text{with}\quad E = \{(p \to r) \mid p\in \mathrm{parents}(r)\}$$

Local state for each agent and key is maintained in a directed-complete join-semilattice $(L_k, \sqsubseteq, \sqcup)$, where merges are performed as:

$$M_i(k) \leftarrow M_i(k) \sqcup \mathrm{payload}(\delta)$$

The core axioms guaranteeing a DCS are (Ren et al., 7 Oct 2025):

• A1. Localized Weak Fairness: Every persistent $\delta$ is eventually delivered to all relevant agents.
• A2. Join-Semilattice Structure: Each $(L_k, \sqsubseteq, \sqcup)$ is a dcpo-join-semilattice (associative, commutative, idempotent; suprema of directed sets exist).
• A3. Contribution Immutability: Each $\delta$ has a unique, immutable identifier and payload.
• A4. Immutability of Causal Linkage: The $\mathrm{parents}$ set is fixed upon contribution creation.
• A5. Causal Well-Formedness: At contribution creation, all parents must already have been observed, ruling out cycles.

Under these axioms, the existence and uniqueness of $G^*$ (up to isomorphism) are established, independent of operational policies such as scheduling, batching, or routing. All system observers converge to the same $G^*$ by merging payloads in any order, making correctness intrinsic to the structure rather than to delivery protocol (Ren et al., 7 Oct 2025).

2. Theoretical Results: Existence, Uniqueness, and Policy-Agnosticism

DCS theory yields the following key results (Ren et al., 7 Oct 2025):

a) Existence and Uniqueness

• Existence: $G^*$ is a well-defined DAG. A5 prevents cycles; A3/A4 ensure immutability of nodes and edges.
• Uniqueness: $G^*$ is unique up to isomorphism; any admissible history (given the same set of contributions) yields the same structural provenance.
• Convergence: For all $i$ and $k$, $M_i(k)$ converges to $$\sup \{\mathrm{payload}(\delta)\mid \delta\in\mathrm{Merge}_i(k)\}$$, ensuring identical local states.

b) Policy-Agnostic Invariance (Decoupling Theorem)

• For any two policies $P_1$ and $P_2$ delivering the same set of contributions, the resulting DCS graphs $G_1^*$ and $G_2^*$ are isomorphic. Thus, operational policy influences only delivery timing, not correctness.

c) Observational Equivalence

• Two executions are observationally indistinguishable (for any upper-layer queries about ancestry/concurrency or $\sqcup$-homomorphic payload aggregates) if and only if their DCS graphs are isomorphic.

d) Axiom Minimality

• The axiom set $\{A1$–$A5\}$ is minimal: dropping any single axiom may result in divergence, ambiguity, or loss of determinism. For example, omitting A1 (liveness) results in network partitions and divergent agent states.

These results position DCS as a structural boundary principle for asynchronous, coordination-free systems, comparable to the role of CAP (Consistency-Availability-Partition-tolerance) and FLP (Fischer–Lynch–Paterson) impossibility in distributed system theory (Ren et al., 7 Oct 2025).

3. DCS in Deterministic Structural Causal Modeling

DCS is foundational in deterministic Structural Causal Models (SCMs). For systems modeled as ordinary differential equations (ODEs),

$\dot{X}_i = f_i(X_{pa_D(i)})$

the equilibrium states and perfect interventions can be cast as labeled equilibrium equations (LEE), which are then mapped to SCM equations:

$X_i = h_i(X_{pa_M(i)})$

where $pa_M(i)$ are non-self parents and $h_i$ follows from structural or implicit function solutions (Mooij et al., 2013, Mooij et al., 2014).

Stability (global convergence to equilibrium) ensures that the equilibrium LEE is solvable and interventions on variables correspond exactly (commute) with interventions in the SCM. Cyclic ODE graphs induce cyclic SCMs; no acyclicity constraint is needed as long as solvability conditions persist.

The link between ODE equilibria and DCS in SCM is shown rigorously: the causal structure and intervention semantics are well-defined and lead to fully identifiable deterministic models when stability and local invertibility conditions are met (Mooij et al., 2013, Mooij et al., 2014).

4. DCS in Causal Discovery and Inference

DCS enables identifiability and inference in both linear and nonlinear deterministic settings. In linear SCMs with possibly deterministic relations and latent confounding, observed variables $X=(x_1,...,x_p)^\top$ are generated via

$$x_i = \sum_{j < i} a_{ij} x_j + \tilde{s}_i \qquad \tilde{s}_i = \sum_{k=1}^m b_{ik} s_k$$

where $A$ is strictly lower-triangular and $B$ may have zero-variance rows (deterministic nodes) or shared columns (confounders) (Yang et al., 2021). Identifiability requires:

• A unique-component condition: sources unique to putative parents must not reappear in the child.
• A marriage condition: subsets of possible parents must not have fewer unique sources than size.

When these conditions hold, both causal coefficients $A$ and $B$ are uniquely recoverable (up to permutation and scaling). The P-SCM-Recovery algorithm exploits this for practical causal structure recovery. This advances beyond earlier models, covering exact deterministic dependencies and confounded propagation mechanisms in causal discovery (Yang et al., 2021).

For inference between two real-valued variables related by invertible deterministic functions $Y = f(X)$, the DCS/IGCI framework exploits information geometric independence between $p_X$ and $f$ to identify asymmetry: $p_Y$ contains information about $f$ that $p_X$ does not. This enables causal direction identification even in noise-free regimes (Daniusis et al., 2012).

5. DCS in Dynamical Systems and Control

In deterministic dynamical or controlled systems described by

$$\dot{x}(t) = f(x(t),\,u(t)), \qquad x\in\mathbb{R}^n, \, u\in\mathbb{R}^m$$

the DCS is the directed graph whose edges encode physical influence: $x_j \to x_i$ exists iff interventions on $x_j(0)$ (or $u_j$) change the distribution of $x_i(\cdot)$. Structural dependencies can be read from the non-vanishing partial derivatives $\frac{\partial f_i}{\partial x_j}$ and $\frac{\partial f_i}{\partial u_j}$. Identifiability requires sufficient controllability—agents must be able to steer the state sufficiently to break statistical dependencies, and an MMD-based test distinguishes genuine from spurious links (Baumann et al., 2020).

Experimental design leverages controllability matrices or nonlinear optimization to maximize discriminatory power, with robustness to finite sample effects and practical generalizability to high-dimensional systems. The DCS thus extracted directly informs more effective and generalizable control, sparse re-identification, and robust predictive performance (Baumann et al., 2020).

6. DCS in Networked and Multi-Agent Systems

DCS uniquely addresses a key deficiency in value-convergent but structurally ambiguous systems, such as state-based CRDTs. Two different operational histories (distinct causal pasts) may yield the same merged value but non-isomorphic provenances. DCS, by maintaining persistent, immutable causal linkages at the event level, cleanly disambiguates such histories. This structural approach supports modular design, composable correctness, and safe evolution of operational policies (Ren et al., 7 Oct 2025).

An illustrative example: in a set-CRDT with merge via union, two operations $\delta_x$ and $\delta_y$ can be concurrent (yielding two roots) or causally chained (yielding a path). While both outcome states are identical, their DCSs differ. DCS encodes this irreducible distinction, which pure value-centric models cannot (Ren et al., 7 Oct 2025).

Moreover, DCS generalizes to the flow of dynamical causal structures, wherein a causal process is seen not as a static graph but as a sequence or DAG of structural reductions (each reduction fixing source outputs and removing vertices), generating a superflow that can capture all admissible causal evolutions. In classical deterministic models, this formalizes the propagation of causal order and outputs a full trajectory space for structure evolution (Baumeler et al., 2024).

7. Boundary Principles, Limitations, and Extensions

DCS serves as a boundary theorem for asynchronous computation: the only route to modular, coordination-free deterministic convergence under asynchrony, faults, and unreliable infrastructure is to require join-semilattice state updates coupled with persistent, immutable causal linkages (Ren et al., 7 Oct 2025).

Principle	Model Requirement	Violations Yield
DCS Structurality	Join-semilattice + immutable DAG	Nondeterminism, ambiguity
CAP	Consistency, Availability, Partition	Impossibility of all three
FLP	Deterministic consensus in asynchrony	Impossibility (1-fault)

DCS theory is as yet restricted to deterministic, value- or set-based systems, linear SCMs with additive exogenous mixtures, and does not encompass nonlinear, stochastic, or quantum cases without further extension. For quantum generalization, reduction operations and factorization essential to the iterative unraveling may fail (e.g., in process-matrix formalisms partial traces can break CPTP map structure) (Baumeler et al., 2024). The structural DCS also presumes observability and absence of unmeasured latent confounding in some instantiations (Baumann et al., 2020, Yang et al., 2021).

Nevertheless, ongoing development seeks extensions to robust versions tolerant of approximation, to nonlinear and discrete-state domains, and quantum/relational generalizations under additional algebraic constraints.

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In summary, Deterministic Causal Structure provides the minimal, structural, and policy-independent bedrock for causal specification, inference, system design, and correctness across a spectrum of domains from distributed computation, SCM, and control systems to networked agency and dynamical flows. Its axiomatic approach guarantees existence, uniqueness, and operational invariance, anchoring the "Correctness-as-a-Chassis" paradigm that sharpens the boundary of what is provably achievable in deterministic, asynchronous, and evolving computational systems (Ren et al., 7 Oct 2025, Mooij et al., 2013, Yang et al., 2021, Daniusis et al., 2012, Baumeler et al., 2024).