Causal Physics Module (CPM)

• CPM is a rigorously defined framework that models causal structures using DAGs, assembly theory, and SCMs to represent cause-effect relationships explicitly.
• It implements an algorithmic pipeline for extracting, normalizing, and scoring causal dependencies, enabling precise diagnosis and simulation of physical reasoning.
• Integrated across diverse paradigms, CPM supports applications from physics education and large-scale simulations to LLM causal augmentation and metrological causation.

A Causal Physics Module (CPM) is a rigorously defined, algorithmically explicit, and theoretically principled mechanism for modeling, inferring, evaluating, or simulating causal structure and causal reasoning within physical systems. The CPM paradigm encompasses diverse instantiations—ranging from assembly-theoretic metrology of causation to structural equation-based simulators and neural representations deployed in artificial intelligence. These modules are unified by the requirement that physical processes, events, or reasoning steps are encoded as nodes in a structured space (e.g., directed acyclic graphs, join-graphs, or high-dimensional state representations), with explicit and minimal encoding of causal dependencies, intervention effects, and feedback about the propagation of causal information.

1. Formal Structure and Theoretical Foundations

Several rigorously developed CPM formalisms are found in the literature. In PRISM-Physics, a CPM encodes a multi-step physics reasoning solution as a directed acyclic graph (DAG) where nodes $V = \{f_1, \dots, f_N\}$ are canonical formulas, and edges $E \subseteq V \times V$ encode minimal “justifier” (causal dependency) relations: $(f_i \rightarrow f_j)$ means $f_j$ causally depends on $f_i$ and cannot be validly derived without it. The DAG representation is proven to be a minimal, lossless encoding of all necessary causal dependencies in the solution structure; alternative representations (e.g. linear chains) either miss alternatives (under-crediting) or over-credit trivially included steps (Zhao et al., 3 Oct 2025).

In Assembly Theory, the CPM is formalized as a closure space $\Omega = \langle U; J \rangle$ over a finite set of primitives $U$, with assembly events (joins) $J \subset \Omega \times \Omega \times \Omega$. The “assembly index” $A(x)$ is the minimal number of joins required to construct $x$ from $U$, which quantifies the causal work instanced by the system. Observed copy number $N(x)$, selective threshold $\tau$, and the virtual copy number $v(x) = N(x)e^{A(x)}$ furnish a standardized metrology of causation, contingency, and selection in open-ended physical systems (Cronin et al., 2 Jan 2026).

In structural causal modeling (SCM)-based physics simulators (e.g. CausalMan), the CPM is instantiated as a tuple $\mathcal{M} = (U, V, P_U, \{f_i\})$, where each node $v_i$ is generated as $v_i = f_i(\mathrm{Pa}_i, u_i)$—$\mathrm{Pa}_i$ denoting causal parents and $u_i$ exogenous noise. Batch, mode, and switching dynamics are formalized in discrete state spaces, with interventions $do(V_j = \hat{v})$ realized via targeted equation replacement (Tagliapietra et al., 18 Feb 2025).

2. Algorithmic Realizations and Evaluation Pipelines

A CPM always implements an explicit, bounded-computation pipeline for causal structure processing. In PRISM-Physics (Zhao et al., 3 Oct 2025):

1. Extract and canonicalize formulas from input (e.g. LLM-produced or student solutions).
2. Symbolically match each candidate against the reference DAG nodes via deterministic, rule-based equivalence:
   • Stage 1: constant and unit normalization.
   • Stage 2: solution-set equivalence—randomized variable assignment and solution comparison.
3. Score matches via ancestor-closure: only required predecessors of matched nodes recursively receive credit.
4. Compute final normalized score $S(G, A) = \frac{|\mathrm{Ach}(A)|}{|V|}$, where $\mathrm{Ach}(A)$ includes all matched nodes and their required ancestors.

In Assembly Theory (Cronin et al., 2 Jan 2026), computation of $A(x)$ proceeds via recursive or memoized breadth-first search over observed or admissible join-graphs, with cost minimized by fragment reuse. Copy number tables and branching-factor-limited thresholds are computed analytically.

In neural CPMs augmenting LLMs (CWMI), the CPM is an independent Transformer-based stack that maps a projected LLM hidden state (encoding the scenario) into a latent physical outcome space. Training leverages both a prediction loss for factual state matching and a causal intervention loss for learning the effect of hypothetical “do” interventions, ensuring the CPM captures functional cause-effect relationships rather than mere correlational statistics (Sharma et al., 26 Jul 2025).

3. Causal Semantics, Information, and Intervention

CPMs are designed to operationalize causation, not just correlation:

• PRISM-Physics encodes the exact reasoning dependencies required to propagate credit only where justified by causal graph structure, prohibiting spurious reward for irrelevant or non-ancestral steps (Zhao et al., 3 Oct 2025).
• In the information-theoretic CPM (Kutach’s approach), the degree of causal promotion is $\mathcal{C}(C, E) = \log[P(E|C)/P(E|C')]$: a correlation is genuine and usable for information transmission iff $\mathcal{C} > 0$. This identifies a one-to-one correspondence between controllable causal classes and distinguishable information-theoretic states. Quantum and statistical mechanics consequences, such as the Holevo limit or Boltzmann entropy, are then naturally interpreted in terms of causal classes, not abstract distinguishability (Beck, 2017).
• In SCM-based CPMs, explicit intervention operators (e.g., “do”-calculus) are implemented by rewriting the appropriate structural equations and resampling the affected subgraph, permitting both observational and fully interventional data streams for algorithmic benchmarking and inference (Tagliapietra et al., 18 Feb 2025).

4. Process-Level Diagnostics, Evaluation, and Training Signals

The CPM pipeline yields fine-grained, actionable diagnostic signals:

• In process reasoning (PRISM-Physics), ancestor-closure scoring allows granular diagnosis of where a solution chain failed (missing physical law, algebraic error, unit mismatch). This enables curriculum design and reinforcement learning using partial subgraph rewards (Zhao et al., 3 Oct 2025).
• Human-machine agreement, measured by Kendall’s $\tau_b$, is higher for CPM-based process evaluation compared to LLM-as-judge heuristics, showing improved alignment with expert grading.
• In neural CPM frameworks (CWMI), intervention loss is empirically necessary for robust zero-shot causal consistency. Removing the causal term ($\beta = 0$) halves causal consistency score (CCS), demonstrating that only explicit interventional training imparts genuine causal reasoning capabilities (Sharma et al., 26 Jul 2025).
• Assembly-theoretic CPMs enable discrimination of lifelike and technological signatures by quantifying the extent to which observed artifacts significantly exceed abiotic assembly thresholds, supporting algorithmic detection of biological or engineered complexity (Cronin et al., 2 Jan 2026).

5. Canonical Applications and Case Studies

Representative CPM use cases span multiple domains:

• Physics education and model evaluation: PRISM-Physics CPM reliably exposes stepwise reasoning failures in LLM-generated solutions and student problem solving, furnishing interpretable subgraph credit and actionable feedback on physics reasoning processes (Zhao et al., 3 Oct 2025).
• Large-scale physical systems: CausalMan CPM simulates a production-line with 200–600 variables, discrete mode shifts, and mixed-type SCM edges, yielding both observational and fully interventional data for benchmarking causal discovery and inference pipelines. Python interfaces facilitate downstream integration with DoWhy and related frameworks (Tagliapietra et al., 18 Feb 2025).
• LLM causal augmentation: A CPM in CWMI transforms a static LLM into an interventionally sensitive, robustly causal world model, enabling accurate factual and counterfactual prediction of outcomes in multimodal datasets (e.g., PhysiCa-Bench, PIQA). The CPM’s Transformer stack is strictly partitioned from the language backbone, with only projection-layer and CPM parameters trained, preserving LLM linguistic capacity while conferring causal physics competence (Sharma et al., 26 Jul 2025).
• Fundamental causal metrology: Assembly Theory–based CPMs quantify the “causal work” embodied in structured objects, provide selective thresholds for identifying life or technology, and formalize the emergence of novelty, contingency, and deterministic behavior along assembled lineages. This approach departs substantially from purely interventional or Kolmogorov-type causality, focusing on material (assembly) requirements as the foundational metric (Cronin et al., 2 Jan 2026).

6. Theoretical Guarantees and Limitations

Certain CPMs are accompanied by formal optimality results:

• The DAG representation and ancestor-closure scoring in PRISM-Physics are shown to be the unique minimal, redundancy-free encoding and credit-policy, respectively, under natural admissibility constraints (Zhao et al., 3 Oct 2025).
• Assembly-theoretic CPMs provide metrological guarantees: objects above the selective threshold $\tau$ cannot arise in measurable abundance without persistent, selective mechanisms, linking high assembly index with the impossibility of spontaneous abiogenesis for complex structures under random assembly (Cronin et al., 2 Jan 2026).

Identified limitations include the necessity for explicit specification of mass-growth laws (in causal-ontological CPMs), scalability and tractability constraints in DAG/path search or high-dimensional SCMs, and, in neural CPMs, current restriction to short-horizon physical evolutions and discrete counterfactuals (Lokajicek et al., 2016, Sharma et al., 26 Jul 2025).

7. Broader Significance and Outlook

CPMs now span conceptually distinct yet convergent paradigms: metrological causation measurement (Assembly Theory), algorithmic process-graph evaluation (PRISM-Physics), structural equation–based simulation and benchmarking (CausalMan), information-theoretic quantification (Kutach), and neuro-symbolic augmentation of LLMs (CWMI). Their commonality is the explicit, testable, and algorithmically realizable encoding of causality as the central organizing principle for physics knowledge extraction, inference, and agentic reasoning. The CPM serves as both a scientific instrument for detecting or simulating causal processes, and as a diagnostic module for evaluating whether physical or computational systems possess, fail, or exceed normative causal reasoning capabilities (Zhao et al., 3 Oct 2025, Cronin et al., 2 Jan 2026, Sharma et al., 26 Jul 2025, Tagliapietra et al., 18 Feb 2025, Beck, 2017, Lokajicek et al., 2016).