Nature Variables in Multidisciplinary Frameworks

• Nature variables are fundamental parameters defining system state and causal relationships across thermodynamics, quantum mechanics, fractal, and environmental frameworks.
• They facilitate the formulation of state equations, duality-invariant models, and scale-dependent analyses that bridge classical and quantum domains.
• Applications range from predicting ecological responses in environmental models to quantifying risks in economic systems and interpreting magnetohydrodynamic turbulence.

Nature variables are a foundational concept in a diverse array of physical, environmental, and mathematical frameworks. They serve as the principal coordinates, parameters, or resolutions upon which state equations, observable relations, or system behavior explicitly depend. Across theoretical physics, thermodynamics, quantum gravity, environmental science, and economics, "nature variables" underlie the modeling and interpretation of systems from first principles, constituting essential links between underlying structure, measurement, and emergent phenomena.

1. Nature Variables in Thermodynamic and State-Equation Formulations

In equilibrium thermodynamics, nature variables—often called "coordinate variables"—are the canonical parameters upon which equations of state (EoS) depend. The prototypical EoS is a constraint of the form

$g(X, Y, T) = 0,$

with $T$ (temperature, established via the zeroth law), $Y$ (an intensive coordinate, such as pressure or field strength), and $X$ (an extensive coordinate like volume or magnetization). Each EoS thus restricts the system to a two-dimensional surface in the $(X, Y, T)$ space, rendering any pair of variables "truly endogenous" and the third a dependent coordinate.

A nature variable is said to be truly endogenous if it can directly induce change in the dependent variable without mediation by other coordinates. In hydrostatics (ideal gases), for example, this leads to a causal diagram where $T$ enhances both $X$ and $Y$, while $Y$ inhibits $X$. In contrast, in paramagnetic systems, both $T$0 and $T$1 directly enhance $T$2, and there is feedback from $T$3 to $T$4. Only state equations with precisely two endogenous nature variables, and a causal structure fitting canonical classes, are genuine EoS; otherwise, as with certain market analogies, systems collapse to reduced dimensionality and cannot be classified as thermodynamic surfaces (Gumjudpai et al., 2019).

2. Scale Variables in Fractal and Quantum Systems

In the fractal framework of scale relativity, nature variables acquire an explicit scale dependence, reflecting the non-differentiable structure of space-time. Two critical scale variables are recognized:

• The resolution $T$5, a strictly positive parameter quantifying the smallest scale (physical or observational) at which a system is probed.
• Algebraic differential elements (e.g., $T$6, $T$7), which may be positive or negative, enter into Taylor expansions, and serve as both increments and effective resolutions, especially when evaluated at the interface $T$8.

The explicit dependence of physical observables $T$9 on resolution leads to scale-divergent quantities (e.g., trajectory lengths), which manifest as fractality when $Y$0. Critically, non-differentiability introduces a two-valuedness to mean derivatives—forward and backward velocities do not coincide—and they must be described together as a pair, which is algebraically captured by composing a complex velocity field. This formalism enables an exact derivation of the complex Schrödinger wave function and quantum postulates from the properties of scale variables, with the quantum potential and probability density acquiring geometric interpretations directly tied to the scale structure of space-time (Nottale et al., 2012).

3. Nature Variables in Quantum Gravity and Classical–Quantum Duality

Nature variables also arise in the context of unifying classical (gravity) and quantum (elementary particle) descriptions at the Planck scale. The framework introduces dual sets of variables—G-variables (gravitational, e.g., gravitational length $Y$1) and Q-variables (quantum, e.g., Compton length $Y$2)—related by Planckian duality $Y$3, ensuring the product of classical and quantum attributes always attains the Planck value.

To interpolate between these domains, one defines a quantum gravity (QG) or "nature variable" as the symmetric, duality-invariant mean,

$Y$4

For mass, this gives $Y$5 with $Y$6. This coordinate is manifestly invariant under $Y$7 and thereby unifies both classical and quantum regimes into a single analytic extension. In the Kruskal-type extension of Schwarzschild geometry, these nature variables ($Y$8, $Y$9) serve as lightcone coordinates, regular across horizons and encoding antipodal, PT, and CPT symmetries. The four resulting regions cleanly map onto classical/semiclassical and quantum domains, with the Planck-scale hyperbolae representing physically meaningful transitions and the "quantum dressing" of black hole horizons (Sanchez, 2018).

Classical Variable $X$0	Quantum Variable $X$1	Duality-Invariant QG Variable $X$2
$X$3	$X$4	$X$5
$X$6	$X$7	$X$8
$X$9 (Hawking)	$(X, Y, T)$0	$(X, Y, T)$1

4. Environmental Nature Variables in Earth System and Risk Modeling

The concept of nature variables is deeply embedded in contemporary environmental modeling, where explicit environmental ("nature") variables serve as modulators of ecological and economic outcomes. In differentiable land-surface models, such as DifferLand, a high-dimensional vector of static predictors including plant functional type (PFT) fractions, mean annual temperature (MAT), mean annual precipitation (MAP), elevation, soil properties, and forest demography is mapped, via a neural network, to spatially varying ecological parameters.

Principal component analysis of these parameter fields reveals three orthogonal nature-variable gradients:

1. Growing Season Length ($(X, Y, T)$2): Controlled by accumulated days exceeding a threshold temperature, drives leaf lifespan and allocation patterns.
2. Leaf Economics ($(X, Y, T)$3): Encapsulates tradeoffs in leaf construction cost, nitrogen content, and photosynthetic efficiency, serving as a major axis of ecological functional variation.
3. Agricultural Intensity ($(X, Y, T)$4): Defined by crop fraction and litter turnover, captures anthropogenic modulation of nutrient cycling.

Together, these gradients explain over 80% of terestrial spatial variability in key ecological parameters. The fine-scale, continuous adaptation governed by these nature variables enables models to generalize across environmental gradients and accurately simulate carbon, water, and energy fluxes. These findings imply that traditional coarse biogeographic discretizations (e.g., PFTs) account for less than half of explainable variation—critical for future projections of terrestrial ecosystem response to climate and land-use change (Fang et al., 2024).

5. Nature Variables in Socioeconomic and Firm-Level Risk Quantification

In risk modeling for economic systems exposed to environmental deterioration, a structured set of nature hazards is defined as variables normalizable to the $(X, Y, T)$5 range. Five hazards, serving as the environmental dimension of systemic risk, are:

• Biodiversity loss
• Land degradation
• Global warming
• Human population growth
• Natural capital depletion

Country-level exposures are synthesized via the Country Degradation Index (CDI), which aggregates normalized hazards through nonlinear impact functions and high-probability tipping-point effects. At the firm level, nature risk is computed via a Nature Risk Score (NRS), defined as the triple product:

$(X, Y, T)$6

where $(X, Y, T)$7 is sector-derived vulnerability, $(X, Y, T)$8 is the projected degradation index for region $(X, Y, T)$9, and $T$0 is a firm's revenue share in $T$1. These scores are then coupled to financial variables (volatility, leverage) to yield explicit value-at-risk metrics, clearly linking deterioration across nature variables to economic outcomes (Crisostomo, 24 Jan 2025).

6. Nature Variables in Magnetohydrodynamics and Space Plasmas

In magnetohydrodynamics (MHD), the Elsässer variables ($T$2) are crucial "nature variables" for decomposing MHD fluctuations into components propagating parallel and anti-parallel to a background magnetic field. In an incompressible, homogeneous plasma, these variables exactly separate pure Alfvén waves, with $T$3 and $T$4 representing unidirectional propagation.

In more general compressible and inhomogeneous environments, however, both fast and slow magnetosonic modes excite both fields, and the relative amplitudes become functions of plasma compressibility, Alfvén-to-sound speed ratio, and inhomogeneity. This result mandates caution in interpreting observational data: the correspondence between Elsässer variables and propagation direction is strictly valid only in the incompressible, homogeneous limit. These findings have direct bearing on turbulence studies in astrophysical plasmas, including the solar wind and coronal environments (Magyar et al., 2019).

7. Synthesis and Cross-Disciplinary Significance

Nature variables, in the sense outlined above, emerge as the essential coordinates of system state, adaptation, and dynamics—explicitly encoding scale, causal structure, physical regimes, and environmental contingencies. Their choice and mathematical representation determine the analytic tractability, interpretability, and predictive generalization of models across disciplines. Whether providing the canonical axes for thermodynamics, enabling the algebraic structure of quantum and fractal systems, unifying dual regimes in quantum gravity, or underpinning environmental and economic risk measures, nature variables constitute a unifying conceptual and operational foundation for modern scientific modeling.