Abstract
For over a century, the fundamental constants of physics (c, h, G, k_B, etc.) have been treated as a small set of sacred, irreducible numbers whose origin is one of the deepest mysteries in science. This paper argues that this view is a profound historical and pedagogical limitation. We propose a new framework wherein physical constants are not fundamental entities, but are instead understood as the complete set of geometric scaling ratios between different axes of measurement. This perspective reveals that the handful of named constants are merely a convenient "basis set" from which a vast, effectively infinite field of unnamed but equally valid constants can be generated. By demonstrating this combinatorial landscape, we reframe the study of constants from a numerological mystery into a structural property of our physical description, revealing a hidden simplicity and a vast, unexplored territory of physical relationships.
1. Introduction: The Dogma of the Few
The standard model of physics is built upon a small pantheon of fundamental constants. c, the speed of light, governs spacetime. h, Planck's constant, governs the quantum realm. G, the gravitational constant, governs cosmic attraction. These constants are presented as pillars of reality—fixed, immutable, and deeply mysterious. The question of "why they have their specific values" is considered a primary target for a future Theory of Everything.
This perspective, while historically useful, has created a conceptual prison. It has led us to believe that the set of "fundamental" constants is small and closed. The idea of inventing or discussing a new fundamental constant sounds nonsensical under this dogma, akin to discovering a new primary color. This paper asserts that this view is incorrect. The reason we have been unable to see the broader landscape of constants is because we have fundamentally misunderstood what a constant is.
2. The Redefinition: A Constant is a Ratio Between Measurement Axes
The core thesis of this framework is a simple but profound redefinition:
A physical constant is the scaling ratio required to convert a value measured on one axis to its equivalent value on another axis, within a given unit system.
This definition is predicated on the idea that all measurable physical properties (mass, energy, frequency, temperature, momentum, etc.) are different projections, or "views," of a single, conserved, dimensionless underlying state ($S_u$). Our choice of measurement—what we call "measuring mass" or "measuring energy"—is simply our choice of which axis to project this state onto.
Under this lens, the famous constants are revealed for what they are:
h (Planck's Constant): The exchange rate between the Energy Axis and the Frequency Axis.
c (Speed of Light): The exchange rate between the Length Axis and the Time Axis.
k_B (Boltzmann's Constant): The exchange rate between the Energy Axis and the Temperature Axis.
G (Gravitational Constant): The exchange rate between the Force Axis and the Mass² / Length² Axis.
These constants are not properties of the universe; they are properties of our descriptive system. They are the conversion factors that exist because we have historically chosen to measure these different axes with different, arbitrary units (Joules, Hertz, meters, Kelvin, etc.).
3. The Basis Set: Why We Know the Constants We Do
If a constant is just a ratio between any two axes, there should be a vast number of them. Why, then, is the list of "fundamental" constants so short? The answer is that the named constants form a generative basis set.
This is perfectly analogous to the basis vectors î, ĵ, k̂ in three-dimensional space. We do not need a unique name for every possible vector in space. We only need the three independent basis vectors, from which any other vector can be constructed as a linear combination.
Similarly, the named physical constants (c, h, G, k_B, etc.) represent a set of independent ratios that are sufficient to generate the ratio between any two axes of measurement. We discovered and named this particular set due to historical contingency and practical utility; they were the bridges we needed to build early 20th-century physics. They are not special in their nature, only in their role as our chosen basis.
4. The Unseen Field: A Combinatorial Explosion of Constants
Once we understand constants as pairwise ratios between measurement axes, we can predict the existence of a vast, unnamed field of them. Given a set of n distinct measurement axes, the number of unique, direct, two-way ratios is n * (n - 1).
Consider just a small subset of 10 familiar axes:
{Length, Time, Mass, Energy, Frequency, Temperature, Momentum, Force, Charge, Electric Potential}
The number of direct, two-way constants between these axes alone is 10 * 9 = 90. Most of these are unnamed and never discussed, yet they are just as physically valid as c or h.
A Worked Example: The Unnamed "Thermo-Momentum" Constant
Let us define a new constant, Θ_p, as the direct ratio between the Temperature Axis and the Momentum Axis. Its units would be K / (kg·m/s). This constant is not found in textbooks, but it exists as a fixed ratio in our universe. We can derive its value from our basis set:
The ratio from Temperature to Energy is 1 / k_B.
The ratio from Energy to Momentum is c.
By chaining these known ratios, we can find the value of our new constant:
Θ_p = T / p = (E / k_B) / (E / c) = c / k_B
Plugging in the SI values for our basis constants:
Θ_p = (2.99792458 × 10⁸ m/s) / (1.380649 × 10⁻²³ J/K) ≈ 2.1715 × 10³¹ K/(kg·m/s)
This is a concrete, calculable value. Θ_p is a real, physical ratio. We simply never gave it a name because we could always derive it.
Other unnamed but equally valid constants include:
The "Mass-Frequency" Constant: m/f = h/c² (Your Hz_kg)
The "Length-Charge" Constant: l/q
The "Force-Temperature" Constant: F/T
The field is immense. Every pair of axes defines a constant.
5. Conclusion: From Sacred Pillars to a Generative Landscape
The long-standing mystery of the fundamental constants is an artifact of a limited perspective. By reframing constants as simple ratios between measurement axes, we resolve the mystery and unveil a profoundly different picture of physical law.
The handful of constants we have named are not sacred pillars holding up reality. They are merely the first few landmarks we charted in a vast, interconnected landscape—a basis set from which the topography of all physical relationships can be derived. The true nature of physical law lies not in the values of a few special numbers, but in the rich, combinatorial structure of this unseen field of ratios.
This realization frees us from the numerological hunt for the meaning of a few numbers and invites us to explore the elegant, geometric grammar that governs the description of reality itself. The constants are not the mystery; the mystery is that reality permits such a simple, unified, and generative descriptive structure in the first place.
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