Dual Laws Model Framework

• Dual Laws Model is a framework that pairs a base-level law governing primary system evolution with a secondary, meta-level law that constrains or modifies its behavior.
• It is applied across diverse fields—including artificial consciousness, higher-order physics, conservation laws, and turbulence—to capture complementary dynamics and selection mechanisms.
• This dual structure aids practical insights by enabling methods such as dual norm formulations, saddle-point approaches, and joint scaling analyses to enhance simulation and theory.

Dual Laws Model is a label used across several research literatures for frameworks that organize a domain around two coupled governing structures rather than a single law. In one explicit usage, it denotes a theory of artificial consciousness with a lower physical level and a supervenience level that has its own dynamical law (Ohmura et al., 13 Mar 2026). In another, it denotes a hierarchy of first- and second-order physical laws, where higher-order laws alter the first-order “microlaws” governing state evolution (Sichelman, 30 Jun 2025). Related usages appear in weak and primal-dual formulations of conservation laws (Chaumet et al., 2022, Li et al., 2021, Liu et al., 2022), in dual-cascade flux laws for turbulence and active scalars (Bedrossian et al., 2019, Liu et al., 25 Jun 2026), in categorical interaction laws between monads and comonads (Katsumata et al., 2019), and in joint parameter-data scaling laws for statistical models of learning (Maloney et al., 2022). Across these usages, one law typically governs evolution at an object level, while a second law governs selection, interaction, weak enforcement, or a complementary flux.

1. General schema

Across the cited literature, a Dual Laws Model is not a single standardized formalism but a recurrent structural pattern. One version distinguishes a base-level law from a higher-order law that acts on the base law itself; another distinguishes a primal evolution law from a dual or weak law that constrains admissibility; another pairs two flux laws that govern opposite inertial ranges; another pairs two resources, such as parameters and data, in a single scaling relation (Sichelman, 30 Jun 2025, Chaumet et al., 2022, Bedrossian et al., 2019, Maloney et al., 2022).

A common feature is that the second law is not merely an additional term in the first. In the higher-order physical-law account, second-order laws act on first-order laws or microlaws rather than directly on states (Sichelman, 30 Jun 2025). In the artificial-consciousness account, the higher supervenience level has its own dynamical law, independent of but coupled to the lower physical level (Ohmura et al., 13 Mar 2026). In weak formulations of conservation laws, the dual law is expressed through dual norms, adjoint equations, or dual elliptic problems rather than through pointwise residual minimization (Chaumet et al., 2022, Liu et al., 2022). This suggests that the term is best understood as designating paired governance rather than simple duplication.

2. Dual-Laws Model in artificial consciousness

The most explicit use of the term appears in a proposed general, non–brain-specific theory of artificial consciousness (Ohmura et al., 13 Mar 2026). There the Dual-Laws Model is built around two claims: conscious or mental processes supervene on physical systems but are described at a coarse-grained higher level, and this supervenience level has its own dynamical law, independent of but coupled to the lower physical level. The two kinds of dynamics together form the dual laws.

The formal architecture distinguishes lower base-level states $x$ and supervenient entities $X$. Supervenient entities are treated not as static vectors but as functions, with a bridge function $B: SUB \to SUP[N]$ and relations of the form $X^i = b_i(x^i)$ (Ohmura et al., 13 Mar 2026). The supervenience level is organized by index sequences $c = [i_0,i_1,i_2,\dots] \in C$, together with a mapping $S:C \to E$ from index sequences to error functions. Changing $c$ changes the algebraic equation or constraint that the system attempts to satisfy.

Two feedback processes implement the model. Type 1 feedback control adjusts base-level states $x$ to reduce the error defined by the current supervenience-level equations; Type 2 feedback control modifies index sequences, thereby changing which supervenient functions are active and how they are composed (Ohmura et al., 13 Mar 2026). The theory identifies consciousness with inter-level causation implemented by Type 1 feedback, and identifies the subject “I” with the supervenience-level law that modifies index sequences. On this account, a conscious system is predicted to display autonomy in constructing its own goals and cognitive decoupling from external stimuli, which in turn makes a design theory that enables high moral behavior indispensable (Ohmura et al., 13 Mar 2026).

The same framework is presented as an answer to seven questions that a theory of consciousness should address: phenomena, self, causation, state, function, contents, and universality (Ohmura et al., 13 Mar 2026). Its distinctive claim is therefore not only that there are two levels, but that the higher level is causally effective because it has its own law.

3. Higher-order physical laws and quantum measurement

A closely related but distinct formulation appears in a theory of first- and higher-order physical laws adapted from Hohfeldian legal structure (Sichelman, 30 Jun 2025). First-order laws govern how physical states evolve, given fixed rules. Higher-order laws, especially second-order laws, govern how those first-order rules themselves are selected, changed, or terminated in particular situations. The paper argues that current physical theories implicitly and wrongly assume that essentially all physical processes can be modeled using first-order laws (Sichelman, 30 Jun 2025).

The formal presentation begins with first-order legal propositions of the form

$J_1 = X\, j_1\, Y(S),$

with $j_1 \in \{r,\neg r\}$, and second-order propositions of the form

$X$0

with $X$1 or $X$2 (Sichelman, 30 Jun 2025). First-order relations are encoded as vectors,

$X$3

and second-order relations as operators,

$X$4

The operator $X$5 flips first-order states, whereas $X$6 leaves them unchanged (Sichelman, 30 Jun 2025).

Transferred to physics, the proposed interpretation is that first-order physical laws govern state evolution under fixed rules, while second-order physical laws act on first-order laws or microlaws. Quantum measurement is then treated as a fundamentally second-order physical process that alters the underlying first-order physical microlaws governing the evolution of the quantum system (Sichelman, 30 Jun 2025). On this view, unitary Schrödinger evolution remains first-order, but measurement is not merely another first-order interaction; it is a law-changing event. The paper therefore uses a dual-level architecture to reframe the measurement problem.

4. Primal-dual, weak, and entropy-energy formulations of conservation laws

In the analysis and numerics of nonlinear conservation laws, the expression is used in a different but structurally similar sense. One line of work on weak PINNs argues that classical strong-form PINNs fail for discontinuous entropy solutions of nonlinear hyperbolic conservation laws because minimizing any $X$7 norm of the pointwise residual is fundamentally misaligned with approximation quality (Chaumet et al., 2022). For shock-like solutions, a better approximation can make the strong residual larger, with the $X$8 residual behaving like $X$9 as the shock width $B: SUB \to SUP[N]$0. The proposed remedy is to replace strong residual norms by weak or dual norms, using a dual elliptic $B: SUB \to SUP[N]$1-Laplace problem to compute the $B: SUB \to SUP[N]$2 norm of the residual and a second set of neural networks to approximate the dual solutions (Chaumet et al., 2022). In that formulation, the primal law is the conservation law and entropy inequality, while the dual law is the elliptic problem that characterizes the dual norm.

A second line of work embeds regularized conservation laws into a variational and modified optimal-transport framework (Li et al., 2021). There conservation laws with diffusion are treated as flux-gradient flows, and the associated variational problems yield dual PDE systems for the regularized conservation laws. For a regularized scalar law,

$B: SUB \to SUP[N]$3

the dual variable $B: SUB \to SUP[N]$4 enters a coupled forward-backward system, and the resulting primal-dual pair $B: SUB \to SUP[N]$5 is described as a functional Hamiltonian system (Li et al., 2021). The entropy–entropy flux pair $B: SUB \to SUP[N]$6 supplies both a Lyapunov functional and the metric structure underlying the dual PDE.

A third formulation casts an initial-value problem for a conservation law as a min-max saddle point with a space-time Lagrange multiplier $B: SUB \to SUP[N]$7 (Liu et al., 2022). The primal law is the forward conservation law,

$B: SUB \to SUP[N]$8

while the dual law is the backward adjoint-type equation,

$B: SUB \to SUP[N]$9

obtained from the Euler–Lagrange conditions of the saddle-point Lagrangian (Liu et al., 2022). The numerical significance is that implicit time discretizations can then be solved with primal-dual hybrid gradient methods without nonlinear inversions.

A more recent numerical formulation uses dual-pairing summation-by-parts methods for nonlinear conservation laws (Stewart et al., 23 Mar 2026). There the “dual laws” are two discrete stability principles: a nonlinear entropy law and a local energy law. The first guarantees entropy stability of the nonlinear scheme, while the second ensures that the asymptotic numerical growth rate of linearized perturbations does not exceed the continuous growth rate (Stewart et al., 23 Mar 2026). The paper argues that entropy-stable volume upwind filtering can enforce both properties simultaneously.

5. Dual flux laws in turbulence and active scalars

In turbulence theory, the term refers neither to supervenience nor to adjoint structure, but to paired flux laws governing two cascades. For stochastic forced-dissipated two-dimensional Navier–Stokes, sufficient conditions are given for rigorous proofs of third-order universal laws capturing the energy flux to large scales and enstrophy flux to small scales (Bedrossian et al., 2019). These laws are described as 2D turbulence analogues of the $X^i = b_i(x^i)$0 law in 3D turbulence. Under weak anomalous dissipation assumptions, the direct cascade of enstrophy and the inverse cascade of energy are both characterized by scale-independent third-order statistics (Bedrossian et al., 2019).

An analogous but more specialized formulation is established for the stochastic SQG equation (Liu et al., 25 Jun 2026). For statistically stationary solutions, rigorous mixed third-order structure-function laws are derived for the dual cascade: a Yaglom-type law for the direct cascade of surface potential energy and an antisymmetrized mixed flux law for the inverse cascade of the Hamiltonian. In the inertial ranges, the structure functions satisfy

$X^i = b_i(x^i)$1

where $X^i = b_i(x^i)$2 is the surface-potential-energy input rate and $X^i = b_i(x^i)$3 is the Hamiltonian input rate (Liu et al., 25 Jun 2026). The paper also proves Onsager-type obstruction results: $X^i = b_i(x^i)$4-regularity above $X^i = b_i(x^i)$5 rules out the direct flux, while sufficient low-frequency Besov regularity rules out the inverse Hamiltonian flux (Liu et al., 25 Jun 2026). In this usage, a Dual Laws Model is a rigorous paired-flux description of simultaneous direct and inverse inertial-range transfer.

6. Categorical and statistical formulations

In category theory, interaction laws between effectful computations and behaviors of effect-performing machines define another precise meaning of dual law structure (Katsumata et al., 2019). A functor–functor interaction law is a natural family

$X^i = b_i(x^i)$6

and a monad–comonad interaction law is a compatible refinement for a monad $X^i = b_i(x^i)$7 and a comonad $X^i = b_i(x^i)$8 (Katsumata et al., 2019). The paper shows that monad–comonad interaction laws are monoid objects in the monoidal category of functor–functor interaction laws, that the greatest functor or monad interacting with a given functor or comonad is its dual, and that the greatest comonad interacting with a given monad is its Sweedler dual (Katsumata et al., 2019). In this setting, “dual laws” refers to a duality-based semantics for the interaction of effects and machines.

A statistical use appears in a solvable model of neural scaling laws (Maloney et al., 2022). There the model is solved in the dual limit of large training set size and large number of parameters, and the test loss is computed as a joint function of model size $X^i = b_i(x^i)$9, training set size $c = [i_0,i_1,i_2,\dots] \in C$0, and latent dimension $c = [i_0,i_1,i_2,\dots] \in C$1. The model exhibits power-law decay in either $c = [i_0,i_1,i_2,\dots] \in C$2 or $c = [i_0,i_1,i_2,\dots] \in C$3 until bottlenecked by the other resource, together with plateau behavior and eventual breakdown when either $c = [i_0,i_1,i_2,\dots] \in C$4 or $c = [i_0,i_1,i_2,\dots] \in C$5 approaches the latent cutoff $c = [i_0,i_1,i_2,\dots] \in C$6 (Maloney et al., 2022). For the optimally regularized regime, the paper gives the simplified phenomenological form

$c = [i_0,i_1,i_2,\dots] \in C$7

and an improved form including the latent cutoff,

$c = [i_0,i_1,i_2,\dots] \in C$8

(Maloney et al., 2022). Here the duality lies in the joint law of parameters and data.

7. Comparative interpretation

Across these literatures, Dual Laws Model functions as a domain-specific template rather than a single canonical theory. In the artificial-consciousness and higher-order-physics papers, the second law is explicitly meta-level: it governs supervenience-level goal selection or the selection and alteration of first-order microlaws (Ohmura et al., 13 Mar 2026, Sichelman, 30 Jun 2025). In weak, variational, and numerical treatments of conservation laws, the second law is dual or adjoint: it enforces the primal law through dual norms, Hamiltonian dual variables, or saddle-point multipliers (Chaumet et al., 2022, Li et al., 2021, Liu et al., 2022). In turbulence and SQG, the paired laws are complementary flux laws for opposite cascades (Bedrossian et al., 2019, Liu et al., 25 Jun 2026). In category theory, they appear as dual constructions governing interaction between monads and comonads (Katsumata et al., 2019). In neural scaling, they appear as a joint law in data size and model size (Maloney et al., 2022).

This suggests a family resemblance centered on coupled governance. One law governs the primary state or process; the other governs admissibility, interaction, selection, or a complementary transfer channel. What unifies these usages is therefore not a shared equation or ontology, but a repeated formal decision to model a system through two mutually constraining laws rather than one.