Physics-Aware Coordinate Scaling

• Physics-aware coordinate scaling is a framework where coordinate mappings and rescalings explicitly embed physical symmetries, dimensional structure, and gauge freedoms.
• It is applied in numerical simulations, operator learning, and geometric analyses to improve accuracy and handle complex dynamics such as turbulent flows and quantum phenomena.
• The approach extends traditional gauge invariance by linking local mathematical universes through scaling factors, enabling rigorous yet practical computational and theoretical innovations.

Physics-aware coordinate scaling is a family of frameworks and methodologies wherein coordinate mappings, rescalings, or transformations are constructed such that they explicitly capture and embed the physical symmetries, dimensional structure, gauge freedoms, or sharp solution features inherent to a system under study. The paradigm arises in domains including geometric gauge theory, quantum mechanics, turbulent flow analysis, operator learning for PDEs, and numerical simulation frameworks, and it manifests both as a rigorous foundational principle (where coordinate/number structures become local, dynamical constructs) and as a pragmatic computational strategy for improved accuracy, conditioning, or interpretability. The field synthesizes ideas from local mathematical universes, fiber bundle theory, dimensional analysis, and the exploitation of intrinsic problem symmetries.

1. Local Availability of Mathematics and Number Scaling

A rigorous and foundational instantiation of physics-aware coordinate scaling is provided by the "local availability of mathematics" (LAM) and associated number-scaling frameworks developed by Benioff and collaborators. In this picture, every spacetime point $x$ is associated with its own mathematical universe $U_x$, which contains separate realizations of number systems (e.g., $\mathbb{R}_x$, $\mathbb{C}_x$), vector spaces, and Hilbert spaces. Comparison or transfer of information between points $x$ and $y$ requires both a parallel transport—an isomorphism between structures—as well as a locally determined scaling factor (Benioff, 2012, Benioff, 2013, Benioff, 2015).

This scaling factor $r_{y,x} = e^{\theta(y)-\theta(x)}$ is determined by a (generally smooth) scalar field $\theta(x)$. For any mathematical or physical quantity $a_y$ defined at $y$, the corresponding scaled representation at $x$ is $r_{y,x}a_x$, where $a_x$ is the image under parallel transport. This framework distinguishes between:

• External scaling: Arises when mapping numbers or tensors at point $y$ into the reference universe $U_x$. All terms in locally defined equations scale equally so that the physical content, as expressed by the equations, remains invariant under such gauge transformations.
• Internal scaling: Occurs within integrals or derivatives that mix contributions from multiple points, e.g., action integrals, expectation values, or derivatives over spacetime. Here, the $y$-dependent scaling factor $r_{y,x}$ appears inside the integral or as a correction in the covariant derivative.

The extension of gauge freedom from bases of vector spaces to the local scale of number systems induces a U(1)-like (real, abelian) gauge structure, where the scalar field $\theta(x)$ (and its gradient $A_\mu(x) = \partial_\mu\theta(x)$) acts analogously to a real-valued connection (Benioff, 2012, Benioff, 2013, Benioff, 2014, Benioff, 2015).

2. Geometric and Physical Implications

Physics-aware coordinate scaling alters the meaning and evaluation of geometric quantities, integrals, and differential operators across space(time) (Benioff, 2013, Benioff, 2014, Benioff, 2015). For a line element $ds^2$, one finds

$ds^2(x) = s(x)g_{\mu\nu}(x)\,dx^\mu dx^\nu$

with $s(x)=e^{\theta(x)}$. Path lengths and geometric intervals pick up a multiplicative scale factor determined by the value and structure of $\theta(x)$. When mapping global expressions (such as integrals or derivatives) into a local mathematical universe, each integrand must be parallel-transported and rescaled, resulting in a y-dependent internal scaling factor:

$$\int_M f(y)d^4y \to \int_{\phi_x(M)} e^{\theta_x(u_x)-\theta_x(v)} f_x(u_x)d^4u_x$$

where $v$ is the image of the reference point.

Within quantum theory, wavepackets, expectation values, and momentum operators are modified: expectation integrals acquire an internal scaling $r_{y,x}$, and derivatives become covariant derivatives with connection $A_\mu=\partial_\mu\theta(x)$. In gauge field Lagrangians, this leads to the coupling of the scalar (number scaling) field to matter with unspecified mass, subject to stringent phenomenological bounds in local regions (Benioff, 2015).

On the cosmological scale, drastic spatial or temporal variation of scaling fields can induce effective geometric "collapses" or expansions—for instance, modeling big-bang or black/white scaling hole scenarios—without changing the underlying manifold structure (Benioff, 2013).

3. Gauge Structure and Transformity

Physics-aware scaling formalizes the notion that coordinate choices in larger-dimensional spaces, or in the presence of local mathematical/gauge freedom, can alter the manifest laws of physics observed in lower dimensions (Wesson, 2013). In higher-dimensional unified field theories (e.g., Kaluza-Klein/STM theory), the functional form of the 4D field equations depends in general on the higher-dimensional coordinate gauges. A single seed metric in higher dimensions can, via coordinate transformations, encode diverse physical situations such as cosmological models, matter waves, or particle-like metrics, a phenomenon called "transformity." This underlines the interpretational subtlety that observed laws of nature may be gauge artifacts of a broader, higher-dimensional, and coordinate-scalable setting.

4. Applications in Numerical Simulation and Operator Learning

Physics-aware coordinate scaling is implemented in several computational contexts to exploit underlying physical symmetries, improve conditioning, or isolate sharply localized phenomena:

• Co-Scaling Grids in Astrophysical/MHD Codes: In Eulerian finite-volume solvers (e.g., Athena++), co-scaling grids are constructed via explicit, user-supplied time-dependent mappings between computational and physical coordinates, typically of the form $x^i = a(t)X^i + C^i(t)$. The grid velocity enters the conservative update equations, and the geometric scaling maintains accuracy, conservation, and efficiency across dynamically expanding/contracting flows (e.g., stellar explosion simulations) without loss of resolution (Habegger et al., 2021, Heitsch et al., 2024).
• Physics-Aware Autoencoders and Operator Learning: In reduced-order modeling and operator learning for PDEs, learned coordinate transforms (registration maps) are optimized to align solution snapshots (e.g., advecting shocks) along moving grids. This minimizes the Kolmogorov n-width and enables compact low-rank decomposition of time-dependent data dominated by transport phenomena, achieving superior representation and extrapolation accuracy—even with simple linear encoders—compared to fixed-grid approaches (Mojgani et al., 2020, Roohi et al., 18 Oct 2025).
• Dimensional Analysis and Nondimensionalization: In finite element frameworks, automated symbolic unit-tracking and scaling are used to assemble nondimensionalized, preconditioned weak forms. Scaling factors are factored out to yield normalized forms (with explicit $\pi$-groups), and the resulting operator-level diagonal rescalings correspond to full-operator preconditioning, dramatically improving the condition numbers of saddle-point matrices and robustness in coupled multiphysics simulations (Habera et al., 10 Jan 2026).

5. Turbulence, High-Gradient Flows, and Feature-Aligned Scaling

In the analysis of complex flows—such as turbulent boundary layers with strong curvature or pressure gradients—physics-aware coordinate scaling is constructed by aligning the coordinate frame with physical streamlines or shock fronts (Prakash et al., 2023, Roohi et al., 18 Oct 2025). In (s, n) coordinates, where $s$ is aligned with the streamline and $n$ is normal, integral-based, physics-informed scaling laws for Reynolds stresses achieve superior collapse versus classical friction-velocity scaling, especially in non-equilibrium boundary layer regions. Similarly, in operator-learning surrogates for rarefied micro-nozzle shocks, feature-aligned coordinates and auxiliary (e.g., shock indicator) features are systematically embedded and normalized to encode interface location, orientation, and profile structure, which enables effective learning even in regimes dominated by sharp, parameter-dependent transitions.

6. Experimental and Phenomenological Constraints

Physics-aware coordinate scaling in its foundational, number-scaling form imposes testable requirements: empirical absence of local deviations from standard physics (e.g., QED precision, gravitational experiments) constrain the gradient and coupling of the scaling/scalar gauge field $\theta(x)$ to be extremely small within the local cosmological region. However, no direct restrictions exist on the scaling field at cosmological or early-universe epochs, leading to the possibility of observable large-scale or cosmic-age phenomena induced entirely by number-scaling structure (Benioff, 2012, Benioff, 2013, Benioff, 2014, Benioff, 2015).

7. Summary Table: Paradigms and Implementations

Context	Scaling Construct	Physical/Computational Role
Foundations/gauge theory	Local number scaling via $\theta(x)$	Modifies geometry, quantum dynamics, connects to gauge structure
MHD/HD simulation	Time-dependent coordinate maps	Expanding/contracting grids to track evolving flows
Reduced-order modeling	Learned diffeomorphic coordinates	Decouples advection (coordinate motion) from diffusion (latent basis)
Feature-aligned learning	Shock/streamline-aligned frames	Embedding shock location and structure for enhanced model capacity
Dimensional analysis	Reference unit scaling	Automated normalization, operator preconditioning

Physics-aware coordinate scaling thus describes a multi-faceted framework where coordinate transformations, rescalings, and feature-alignment are systematically driven by physical principles and symmetries—impacting fundamental theories, geometric interpretations, and computational methodologies alike (Benioff, 2012, Benioff, 2013, Benioff, 2014, Benioff, 2015, Habegger et al., 2021, Heitsch et al., 2024, Prakash et al., 2023, Roohi et al., 18 Oct 2025, Habera et al., 10 Jan 2026, Mojgani et al., 2020, Wesson, 2013, Cichy et al., 2016).