Generalized Orders of Magnitude (GOOMs)

• Generalized Orders of Magnitude (GOOMs) are a mathematical framework that generalizes traditional exponential scaling to include factorial, power-law, and logarithmic growth, unifying discrete and continuous domains.
• They classify and compare growth rates in systems ranging from stellar and cosmological structures to moment problems in dynamical systems, offering clear applications in both physics and statistics.
• GOOMs underpin innovative computational methods, such as log-domain encoding and parallel prefix scans, which overcome floating-point limitations and achieve significant simulation speedups.

Generalized Orders of Magnitude (GOOMs) are a mathematical and computational framework for representing, comparing, and enabling computations spanning wide dynamic ranges in physical, geometric, statistical, and numerical domains. The concept generalizes the familiar notion of "orders of magnitude" (typically based on factors of ten in scientific notation) to encompass scalable exponents, factorial growth, log-domain representations, and structural boundaries over continuous or discrete regimes. GOOMs appear in the mathematical analysis of growth rates (asymptotics, moment problems), in statistical modeling, in the architecture of high-performance computations, and in the paper of complex systems whose quantitative and structural parameters range over many orders of magnitude.

1. Conceptual Definition and Mathematical Foundations

Traditional orders of magnitude organize real numbers according to exponential scales, often via base-10 or base-e. GOOMs generalize this notion to include:

• Factorial-type growth (e.g., (2k)! for moments of cotangent sums (Maier et al., 2014))
• Power-law scaling relations (e.g., size–mass relations $R_e \propto M_s^{4/5}$ for stellar systems (Misgeld et al., 2011))
• Logarithmic, polynomial, and wilder growth orders (for instance, in dynamical systems where exponential separation is replaced by polynomial or slower separation rates (Correa et al., 2020))
• Complex log-domain encoding, where any real value is isomorphic to a complex logarithm whose exponentiation yields the real number, further extending dynamic range and representing sign explicitly (Heinsen et al., 3 Oct 2025)

In analysis, the set of growth orders is formalized as equivalence classes under $\Theta$-equivalence, and is partially ordered by domination. Completions of these spaces (Dedekind–MacNeille completion) yield the full lattice of growth rates relevant to dynamical complexity, entropy, and convergence theory.

2. GOOMs in Physical and Geometric Systems

Astrophysics and cosmology provide canonical examples of GOOMs through studies of systems whose properties span many orders of magnitude:

Parameter	Scaling Relation	Range
Stellar mass $M_s$	$10^2$ to $10^{12}~M_{\odot}$	Family-wide power laws
Effective radius $R_e$	$R_e \geq 2.24\times10^{-6} M_s^{4/5}$ (pc)	Zone of avoidance boundary
Surface density $\Sigma_e$	$\Sigma_e \leq 3.17 \times 10^{10} M_s^{-3/5}$ ($M_\odot\,\text{pc}^{-2}$)	Common upper limit
Halo mass $M_{200}$ (dark matter)	$10^{-6}$ to $10^{15}~M_\odot$	Universal profiles (Wang et al., 2019)

These relations are derived empirically and theoretically, showing that dynamically hot stellar systems and dark matter haloes are structured and bounded by scaling laws which persist across regimes from ultra-faint dwarfs and star clusters to giant ellipticals and galaxy clusters. The "zone of avoidance" is a locus in parameter space outside which no system is observed, reflecting universal constraints imposed by physics, such as feedback mechanisms or phase-space limits.

Cosmological simulations demonstrate that key structural features (e.g., density profiles described by the NFW or Einasto forms) persist with high accuracy across 20 or more orders of magnitude in mass, highlighting a robust self-similarity governed by GOOMs.

3. GOOMs in Mathematical Analysis and Dynamical Systems

The concept of generalized order arises naturally in moment problems, growth analysis, and entropy theory. For instance, in the asymptotics of cotangent sums:

$$\int_0^1 g(x)^{2k}\,dx \sim (2k)! \quad\text{for large }k$$

Here, the moments grow with precisely factorial rate, which places them in a distinct GOOM class compared to polynomial or exponential growth (Maier et al., 2014). The convergence and uniqueness properties of measures induced by such moments hinge on accurately identifying this growth order.

In dynamical systems, the separation of orbits can be classified not just by exponential rates (topological entropy) but by the entire spectrum of possible growth orders:

• Exponential: $o(f)\in[\exp(tn)]$
• Polynomial: $o(f)\in[n^p]$
• Arbitrarily slow or "wild" orders (e.g., cylindrical cascades, Denjoy maps)

Generalized entropy $o(f)$ preserves invariance under conjugacy and allows classification of systems (e.g., circle homeomorphisms, Morse–Smale systems, Denjoy examples) that are indistinguishable by topological entropy alone (Correa et al., 2020).

4. GOOMs in High-Performance Computational Methods

GOOMs drive the design and implementation of numerically robust, scalable, and high-dynamic-range computational architectures. Key contributions include:

• Log-domain numerical encoding: Real values are represented as complex logarithms (GOOMs), with operations (multiplication, dot-product) mapped to addition and log-sum-exp in the log domain. The imaginary part encodes sign via multiples of $\pi i$, enabling seamless handling of positives, negatives, and zeros.
• Implementation: GOOMs are stored as complex datatypes (Complex64, Complex128), fully compatible with PyTorch and autograd (Heinsen et al., 3 Oct 2025).
• Parallel scalability: Custom parallel prefix scans (e.g., log-matrix-multiplication-exp, LMME) allow matrix products to be computed with stable dynamic range over long chains, without underflow/overflow.
• Selective-resetting: In applications prone to colinearity (e.g., Lyapunov exponent calculations), an associative, parallel-reset mechanism maintains numerical stability.

These mechanisms unlock previously impossible applications—matrix chain multiplication over $10^6$ products, parallel Lyapunov spectrum estimation, and deep RNN training with arbitrary recurrence structure—by transcending the limitations of IEEE floating-point formats.

5. GOOMs for Algorithmic and Simulation Order-of-Magnitude Speedups

GOOM-inspired principles guide the achievement of multi-order-of-magnitude speedups in simulation, optimization, and data-driven modeling:

• Machine Learning Surrogates: Artificial neural networks (NNs) replace expensive physics simulations, delivering speedups of $10^6$–$10^7\times$ (core-hour equivalents) in multi-objective accelerator optimization. Iterative retraining with genetic algorithms gives further reductions in required computational samples, enabling real-time design space exploration (Edelen et al., 2019).
• Batch-Vectorized Assembly: In finite element simulations with neural constitutive models, the COMMET framework utilizes batch-vectorization and compute-graph optimization to accelerate stress and stiffness evaluations by 2–3 orders of magnitude over conventional automatic differentiation approaches. Distributed-memory parallelism amplifies scalability (Alheit et al., 1 Oct 2025).
• Normalization in Learning: In RL and supervised learning, adaptive target normalization (Pop-Art) enables robust learning across target values spanning many orders of magnitude. By decoupling scaling from representation and preserving true output magnitudes, a single learning algorithm can operate efficiently across diverse tasks, eliminating the need for ad hoc reward clipping (Hasselt et al., 2016).

6. Structural Boundaries, Universal Laws, and Interpretive Significance

GOOMs are intrinsically tied to the identification of universal limits and boundaries in physical, numerical, and analytical domains:

• Physical boundaries: The "zone of avoidance" in stellar system scaling relations (minimum size for a given mass, maximum effective surface density) reflects universal structural constraints.
• Analytical boundaries: Moment growth rates (factorial, exponential, or polynomial) demarcate distinct spectral and measure-theoretic regimes.
• Computational boundaries: GOOM representation removes overflow/underflow cutoffs, extending stability and dynamic range beyond floating-point hardware.

Quantitative GOOM classifications underpin practical measurement (distance indicators in astronomy), guide simulation parameter choices, and inform the theoretical foundation for scaling laws across vastly different systems.

7. Prospects and Future Directions

Research into GOOMs intersects with algorithmic optimization, scientific computing, and theoretical mathematics:

• Development of custom complex-typed kernels for GOOM arithmetic (improving overhead, particularly in GPU environments)
• Exploration of alternative numerical formats for GOOM representation (e.g., Posits for even broader dynamic ranges)
• Extension of parallel prefix-scan architectures and selective-resetting for stability in even higher-dimensional systems and long-range dependencies
• Advanced theoretical work on order-of-growth spaces, generalized entropy, and their impact on measure-theoretic and ergodic properties of dynamical systems
• Applications in high-dimensional data science, deep learning (recurrent architectures), and domains requiring robust handling of extreme numerical scales

In summary, Generalized Orders of Magnitude unify diverse strands of mathematics, computational science, and physics—offering a coherent framework for understanding, representing, and exploiting the structure of systems whose governing parameters span many orders of magnitude. GOOMs identify universal boundaries, permit robust computation, and enable practical advances at the intersection of theory and application.