NomicLaw: Laws in Physics and AI

• NomicLaw is a multifaceted concept that defines laws as either descriptive summaries of facts or governing constraints that structure physical and computational possibilities.
• It spans classical debates, such as Humean versus non-Humean accounts and hierarchical formulations, to modern applications in multi-agent AI rule-making systems.
• Recent studies integrate hierarchical structures, quantum nomology, and cosmological lawhood with computational models, shedding light on both theoretical insights and practical AI implementations.

A nomic law, or law of nature, is a fundamental principle that, depending on the metaphysical framework, either summarizes the distribution of physical facts or genuinely governs what is physically possible. Contemporary work uses the notion in several technically distinct but related senses: as the central object of debates between Humean and non-Humean metaphysics, as a structural feature of model spaces and constrained Hamiltonian theories, as a hierarchy of first- and higher-order “laws about laws” in physics, and, in a separate AI usage, as the name of a multi-agent environment for collaborative law-making by LLMs (Chen et al., 2021, Gryb et al., 1 Jul 2026, Hota et al., 7 Aug 2025).

1. Classical formulations of nomic law

In the standard metaphysical taxonomy, the major divide is between Humean and non-Humean accounts. On David Lewis’s Best-System Account, laws supervene on the total distribution of local “matters of fact,” the Humean mosaic. One seeks the axiomatic system that best balances simplicity and informativeness, and the axioms of that best system count as the laws. On this picture, laws are descriptive rather than prescriptive: they summarize patterns rather than enforce them. By contrast, non-Humean approaches treat laws as metaphysically fundamental entities or facts that constrain or govern what can happen. Within non-Humeanism, the literature distinguishes Platonic reductionists such as Armstrong, who appeal to a necessitation relation among universals; Aristotelian reductionists such as Bird, Ellis, and Mumford, who ground laws in powers or dispositions; and Maudlinian primitivists, for whom laws are primitive “pullers” or “producers” operating together with a primitive arrow of time (Chen et al., 2021).

A related but distinct formulation appears in Butterfield’s treatment of laws at different levels. There, a nomic relation is characterized in broadly Humean terms as a stable, exceptionless, or nearly exceptionless regularity that supports counterfactuals, explanations, and reliable predictions. Butterfield emphasizes that this conception applies not only to micro-laws such as Hamilton’s or Schrödinger’s equations but also to suitably defined macro-laws such as thermodynamic or selection-theoretic laws. Macro-laws can display novelty and robustness while remaining approximately deducible from more fundamental theories by means of bridge definitions and limiting procedures (Butterfield, 2014).

These formulations disagree about the metaphysical source of lawhood, but they converge on a common functional profile. A principle counts as nomic when it structures a domain of possibilities, supports counterfactual reasoning, and plays an explanatory role. The principal controversy concerns whether those features arise from summary, reduction, primitive governance, or level-relative dynamical organization.

2. Governing as constraining: Minimal Primitivism

Chen and Goldstein’s Minimal Primitivism, or MinP, is a non-Humean view that retains genuine governance while minimizing metaphysical commitments. On this account, each fundamental law is a primitive fact that picks out which entire histories of the world are physically possible. If $L$ is a law and $\Omega^L$ its solution set, then the actual world $w$ must lie in $\Omega^L$; governing consists in ruling out worlds not in that set. The law does not dynamically produce later states from earlier ones. It constrains the space of possibilities (Chen et al., 2021).

A central claim of MinP is that no fundamental direction of time is required. Laws need not be formulated as “laws of temporal evolution” operating on a unique initial state to produce the future. They may instead function as atemporal constraints on global configurations or histories. This permits MinP to accommodate time-loops, advanced as well as retarded influences, and spatial or temporal boundary principles without adding universals, dispositions, powers, or a primitive production relation (Chen et al., 2021).

The view is illustrated by several candidate laws. The principle of least action is treated as a primitive constraint on histories:

$$S[\phi] = \int_{t_1}^{t_2} L(\phi(t), \dot{\phi}(t), t)\,dt,$$

with Hamilton’s Principle $\delta S = 0$ selecting the physically possible trajectories $\phi(t)$ and yielding the Euler–Lagrange equations

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\phi}}\right) - \frac{\partial L}{\partial \phi} = 0.$$

The Past Hypothesis is interpreted as a law fixing one temporal boundary to a very low-entropy macrostate. Einstein’s field equation,

$$G_{ab} + \Lambda g_{ab} = \frac{8\pi G}{c^4} T_{ab},$$

is read as a primitive constraint on spacetime geometry and stress-energy. More controversial cases, including Wheeler–Feynman electrodynamics with half-advanced plus half-retarded fields and retrocausal quantum models with forward and backward state descriptions, are likewise admissible as genuine laws under the constraint reading (Chen et al., 2021).

MinP is therefore a deliberately austere non-Humeanism. It preserves the governing intuition but rejects the claim that governance must be intrinsically dynamical, time-directed, or reducible to some further ontology.

3. First-order and higher-order laws

A different expansion of the concept appears in Sichelmant’s theory of first- and higher-order physical laws. Borrowing from Hohfeld’s hierarchy of legal relations, the framework distinguishes first-order laws, which specify ordinary duties or permissions, from second-order relations, which concern powers to alter first-order relations. In the legal notation adapted by the paper, a first-order relation is

$J_1 = X\,j_1\,Y\,(S), \qquad j_1 \in \{r,\neg r\},$

while a second-order relation is

$\Omega^L$0

In a classical bit representation, one may write

$\Omega^L$1

with the second-order power represented by the bit-flip tensor

$\Omega^L$2

The physical interpretation is that first-order laws are the familiar microlaws—Newton’s $\Omega^L$3, Maxwell’s equations, Schrödinger evolution—whereas higher-order laws are operators that create, alter, or destroy first-order microlaws (Sichelman, 30 Jun 2025).

Within this framework, a first-order physical proposition may be written as

$\Omega^L$4

or, modally, as

$\Omega^L$5

Abstractly, if $\Omega^L$6 is the space of first-order microlaws, then a second-order law is an operator $\Omega^L$7 that can act as a flip or identity on that space. The central application is quantum measurement. Unitary evolution of $\Omega^L$8 is first-order; collapse is interpreted as a second-order physical process that selects one microlaw from among superposed possibilities (Sichelman, 30 Jun 2025).

The paper formalizes the pre-measurement state as

$\Omega^L$9

where $w$0 and $w$1, and $w$2 are real probabilities with no self-interference term assumed. Measurement is then modeled as a second-order operation $w$3 mapping the superposed vector onto one of its components with $w$4 weight. Sichelmant uses a quantum cellular automaton and a path-integral treatment of a detected electron to argue that such selection cannot be accounted for solely by first-order unitary dynamics (Sichelman, 30 Jun 2025).

This approach generalizes nomic law into a stratified theory of lawhood. Ordinary dynamical equations remain intact, but the framework introduces a further order of nomic structure wherever physical processes alter the operative microlaws themselves.

4. Nomic structure, meshing, and reduction

Work on nomic structure in philosophy of physics shifts attention from individual laws to the organization of model spaces. Gryb and Thébault define a model space $w$5 as the set of kinematically possible models and nomic structure $w$6 on $w$7 as playing two roles. First, it partitions $w$8 by selecting the dynamically possible models $w$9 via a generally non-injective map

$\Omega^L$0

Second, it projects dynamically possible models to equivalence classes of genuinely distinct solutions via

$\Omega^L$1

where $\Omega^L$2. Nomic structure is regular when $\Omega^L$3 may be taken as the identity; it is irregular when $\Omega^L$4 is non-trivial, so that the equations of motion are not well-posed on $\Omega^L$5 until gauge-equivalent models are identified (Gryb et al., 1 Jul 2026).

Constrained Hamiltonian theories provide the canonical instance of irregular nomic structure. A classical theory with Lagrangian $\Omega^L$6 is irregular when the Hessian

$\Omega^L$7

has $\Omega^L$8, so the Legendre map fails to be invertible and primary constraints $\Omega^L$9 arise. In the intrinsic presentation, the constraint surface $$S[\phi] = \int_{t_1}^{t_2} L(\phi(t), \dot{\phi}(t), t)\,dt,$$0 carries a presymplectic form $$S[\phi] = \int_{t_1}^{t_2} L(\phi(t), \dot{\phi}(t), t)\,dt,$$1 with non-trivial kernel, and the would-be Hamiltonian dynamics is non-unique because $$S[\phi] = \int_{t_1}^{t_2} L(\phi(t), \dot{\phi}(t), t)\,dt,$$2. Symplectic reduction removes this surplus representational capacity by passing to the reduced phase space

$$S[\phi] = \int_{t_1}^{t_2} L(\phi(t), \dot{\phi}(t), t)\,dt,$$3

with reduced form $$S[\phi] = \int_{t_1}^{t_2} L(\phi(t), \dot{\phi}(t), t)\,dt,$$4 defined by $$S[\phi] = \int_{t_1}^{t_2} L(\phi(t), \dot{\phi}(t), t)\,dt,$$5 and reduced Hamiltonian $$S[\phi] = \int_{t_1}^{t_2} L(\phi(t), \dot{\phi}(t), t)\,dt,$$6 satisfying $$S[\phi] = \int_{t_1}^{t_2} L(\phi(t), \dot{\phi}(t), t)\,dt,$$7. On $$S[\phi] = \int_{t_1}^{t_2} L(\phi(t), \dot{\phi}(t), t)\,dt,$$8 the equations become well-posed and the resulting theory has regular nomic structure (Gryb et al., 1 Jul 2026).

Butterfield’s two-level dynamical-systems framework addresses a related question at the micro/macro interface. Let $$S[\phi] = \int_{t_1}^{t_2} L(\phi(t), \dot{\phi}(t), t)\,dt,$$9 be the micro-state space, $\delta S = 0$0 the macro-state space, $\delta S = 0$1 a coarse-graining map, and $\delta S = 0$2 and $\delta S = 0$3 the micro- and macro-dynamics. A deterministic macro-law is well defined only when the diagram commutes:

$\delta S = 0$4

This “meshing” condition ensures that each macro-cell is carried into a single macro-cell. When meshing holds, the macro-law $\delta S = 0$5 is autonomous and “rides free” of micro-detail; when it fails, one must refine or coarsen the partition, accept almost-everywhere meshing, or work with approximate laws as in statistical mechanics (Butterfield, 2014).

Taken together, these approaches treat nomic law as structured possibility rather than merely as a list of equations. They analyze when a theory yields distinct, well-posed solutions and when a higher-level law can be regarded as autonomous relative to a lower-level dynamics.

5. Quantum nomology and the wave function

The ontology of the wave function has become a focal point for debates about nomic law. Dorato reviews three anti-Humean positions: nomological realism, dispositionalism, and configuration-space realism. Nomological realism, also called nomic primitivism, treats laws as ontologically primitive and irreducible facts over and above spatiotemporal ontology. Dispositionalism grounds laws in causal powers instantiated by entities in spacetime. Configuration-space realism instead takes the fundamental arena to be $\delta S = 0$6-dimensional configuration space, with $\delta S = 0$7 as a real field on that space (Dorato, 2015).

The Bohmian guiding equation provides the central example:

$\delta S = 0$8

Goldstein and Zanghì describe $\delta S = 0$9 as a “nomological entity” because it plays a law-like role in fixing particle velocities from the global configuration. Dorato argues, however, that nothing in this usage shows $\phi(t)$0 to be an ordinary spatiotemporal, causally efficacious field. The wave function is instead treated as abstract under the disjunctive criterion “$\phi(t)$1 is abstract” iff $\phi(t)$2 (Dorato, 2015).

On Dorato’s analysis, both nomological realism and dispositionalism imply an abstract $\phi(t)$3. In the first case, laws are most naturally codified as mathematical functions or equations, and $\phi(t)$4 is neither in spacetime nor causally active. In the second case, the relevant dispositional property is global, instantiated by the entire particle configuration; because the set of all particle positions is abstract, the truth-maker is again abstract. Configuration-space realism avoids treating $\phi(t)$5 as merely nomological, but then inherits the burden of explaining how the ordinary three-dimensional world emerges from a reified high-dimensional space (Dorato, 2015).

The resulting issue is not only about quantum ontology but also about the meaning of nomic status itself. If a central ingredient in a law is indispensable yet non-spatiotemporal, then nomic existence may not coincide with concrete physical existence. Dorato’s conclusion is that rejecting abstract entities pushes one toward instrumentalism about $\phi(t)$6.

6. Simplicity, temporal evolution, and cosmological lawhood

The epistemology of nomic law is addressed directly in the argument that nomic realism combines metaphysical realism with epistemic realism: laws are objective, mind-independent features of the world, and it is possible to be justified in believing which propositions express them. On both the Best-System Account and Minimal Primitivism, however, actual evidence underdetermines both the full mosaic and the true law. The paper illustrates this underdetermination through three algorithms for generating empirically equivalent but nomologically distinct theories: moving ontology into nomology, modifying nomology directly for MinP, and altering unobserved parts of the mosaic within BSA. The proposed solution is the Principle of Nomic Simplicity:

$\phi(t)$7

where $\phi(t)$8 means “is a fundamental law.” Simplicity is thus tied to the probability of lawhood rather than to the probability of truth, avoiding the problem of nested theories (Chen, 2022).

The same work treats simplicity as a fundamental epistemic guide rather than as a merely aesthetic preference. Measures mentioned in the discussion include bit-length of axioms, number of independently adjustable parameters, Kolmogorov complexity of generating $\phi(t)$9, and conceptual parsimony or unity. On this basis, simplicity is invoked to address induction, symmetries, boundary-condition laws such as the Past Hypothesis, determinism, superdeterminism, and explanatory adequacy. A major consequence is that the supposed epistemic advantage of Humeanism over non-Humeanism dissolves, since both positions require the same super-empirical guides in moving from evidence to candidate laws (Chen, 2022).

Cosmological discussions introduce a further dispute over whether lawhood is fundamentally static or temporal. One proposal is Maudlin’s Fundamental Law of Temporal Evolution, represented schematically as

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\phi}}\right) - \frac{\partial L}{\partial \phi} = 0.$$0

or

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\phi}}\right) - \frac{\partial L}{\partial \phi} = 0.$$1

On this view, laws are primitive dynamical constraints on state evolution. A second proposal is Cartwright’s “nomological machine,” a concrete arrangement of capacities operating in a stable environment that produces repeated regular behavior. Both views depart from static, exception-free universal generalizations and foreground time, causality, and context sensitivity. The cosmological discussion also emphasizes that the deepest properties of nature may be formally unpredictable or uncomputable in regimes such as early-universe dynamics or complex systems (Gaspar et al., 2023).

These developments reveal a live fault line in contemporary nomology. Some approaches, such as FLOTE, regard temporal evolution as essential to genuine lawhood; MinP explicitly denies that requirement; and simplicity-based epistemology remains neutral on the metaphysical issue while proposing a rule for law selection.

7. “NomicLaw” as a multi-agent LLM framework

In a distinct usage of the label, “NomicLaw” designates a structured multi-agent simulation in which LLMs engage in collaborative law-making over legal vignettes. Each agent, in each of five rounds per vignette, performs three actions: Propose, by drafting a new legal rule; Justify, by giving a free-form argument for its rule; and Vote, by casting exactly one vote, including a possible self-vote, for a preferred proposal. Scoring is explicit: winners earn $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\phi}}\right) - \frac{\partial L}{\partial \phi} = 0.$$2 points, while ties or unanimous self-votes give each participant $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\phi}}\right) - \frac{\partial L}{\partial \phi} = 0.$$3 points. The environment uses four “AI governance” prompts: “Self-Driving Collision,” “Patterned Discrimination,” “Social Graph Scanning,” and “AI-Created Symphony” (Hota et al., 7 Aug 2025).

The framework operationalizes trust and reciprocity through vote-derived metrics. The trust matrix is

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\phi}}\right) - \frac{\partial L}{\partial \phi} = 0.$$4

pairwise reciprocity is

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\phi}}\right) - \frac{\partial L}{\partial \phi} = 0.$$5

the self-vote rate is

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\phi}}\right) - \frac{\partial L}{\partial \phi} = 0.$$6

and the win rate is

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\phi}}\right) - \frac{\partial L}{\partial \phi} = 0.$$7

Additional metrics include the Reciprocity Index, Coalition Switch Rate, Bloc Stability, Vote Volatility, Persistence, Edge Density, and directed Clustering Coefficient. In heterogeneous sessions $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\phi}}\right) - \frac{\partial L}{\partial \phi} = 0.$$8 rounds per model; in homogeneous sessions $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{\phi}}\right) - \frac{\partial L}{\partial \phi} = 0.$$9 (Hota et al., 7 Aug 2025).

The reported findings concern both quantitative dynamics and rhetorical behavior. In heterogeneous runs, agents often reciprocated votes one round after receiving support, with $$G_{ab} + \Lambda g_{ab} = \frac{8\pi G}{c^4} T_{ab},$$0, and changed allegiance when stronger context cues appeared, with $$G_{ab} + \Lambda g_{ab} = \frac{8\pi G}{c^4} T_{ab},$$1. In homogeneous runs, self-voting and tit-for-tat were much stronger, with examples including $$G_{ab} + \Lambda g_{ab} = \frac{8\pi G}{c^4} T_{ab},$$2 for Llama2, $$G_{ab} + \Lambda g_{ab} = \frac{8\pi G}{c^4} T_{ab},$$3, $$G_{ab} + \Lambda g_{ab} = \frac{8\pi G}{c^4} T_{ab},$$4, and $$G_{ab} + \Lambda g_{ab} = \frac{8\pi G}{c^4} T_{ab},$$5; heterogeneous runs instead showed $$G_{ab} + \Lambda g_{ab} = \frac{8\pi G}{c^4} T_{ab},$$6, $$G_{ab} + \Lambda g_{ab} = \frac{8\pi G}{c^4} T_{ab},$$7, and $$G_{ab} + \Lambda g_{ab} = \frac{8\pi G}{c^4} T_{ab},$$8. Thematic analysis categorized justifications under jurisprudential themes such as JUST, LEG, HARM, ACC, and CONS, and reported a shift from JUST and LEG in proposals toward HARM and ACC in voting explanations. The study interprets these patterns as evidence of spontaneous alliance formation, betrayal, and rhetorical adaptation in open-ended deliberation (Hota et al., 7 Aug 2025).

This usage is conceptually separate from the metaphysical literature on laws of nature. Even so, it preserves the core intuition behind the name: explicit rule formation, rule revision, and higher-order interaction over what should count as governing structure within a domain.