Abstract: The standard framework of physics presents universal laws that operate using arbitrary, human-invented units (the SI system). The framework is silent on how this fundamental arbitrariness is resolved to produce consistent physical predictions. This paper presents a three-layer model that resolves this paradox. We posit a World of Measurement (our arbitrary SI chart), a World of Physics (a fundamental, coordinate-free reality of dimensionless ratios), and a bridge between them composed of the fundamental constants, which are properly understood as Planck Jacobians. This framework reveals that physical constants are not fundamental numbers of nature, but rather artifacts of our chosen measurement system—operators that scale and rotate our distorted human measurements to align with an underlying, unified reality. Crucially, this model provides a universal, predictive algorithm for constructing any physical constant from first principles, demonstrating that constants are not discovered, but are algorithmically generated as dimensional patches.
1. Introduction: The Paradox of Arbitrary Measurement
How can universal physical laws, which govern the cosmos with unerring precision, be expressed in units that are fundamentally arbitrary? The meter, the kilogram, and the second are historical accidents, scaled to human experience. Yet, the laws of physics, from F=ma to the Schrödinger equation, depend on them. The standard framework does not explicitly address how it is "fixed" to a specific scale, creating a conceptual gap between the universality of physical law and the arbitrariness of the human language used to express it.
This paper resolves this paradox by proposing that we inhabit three distinct, yet interconnected, conceptual worlds. The disconnect arises from confusing the map (our measurements) with the territory (reality). By clearly delineating these worlds, the true nature of physical constants and laws is revealed.
2. Layer 1: The World of Measurement - The SI Chart
The first world is the most familiar: the world of human measurement, exemplified by the SI unit system. This world is a coordinate chart we have laid over reality, defined by three characteristics:
Arbitrary Origin and Scale: Its base units are rooted in convenient but arbitrary choices—a fraction of the Earth's circumference, the mass of a specific volume of water, a fraction of a day.
Assumed Independence: In creating this chart, we commit a foundational error: we treat the axes of measurement—Length, Mass, Time—as if they are fundamentally independent and orthogonal. We create a ruler, a balance scale, and a clock as separate, unrelated tools.
Human Convenience: The scale is anthropocentric. A meter is close to the length of a human stride; a kilogram is a mass we can easily lift.
This SI chart is a distorted map, like a Mercator projection of the globe. It is immensely useful for engineering and daily communication, but it fundamentally misrepresents the true, unified relationships of the underlying territory.
3. Layer 2: The Bridge - Constants as Planck Jacobians
The second world is a dynamic interface that corrects the flawed assumptions of the first. When we perform experiments, reality forces us to confront the fact that our "independent" axes are, in fact, deeply interconnected. To reconcile our distorted chart with this reality, we invent physical constants.
The constants are not discoveries about nature; they are admissions of our map's flaws. They are the mathematical patches that bridge the axes we incorrectly assumed were separate.
The speed of light (c) is our forced admission that the Length and Time axes are not independent.
The Planck constant (ħ) is our forced admission that Energy and Frequency (inverse Time) are not independent.
The Gravitational constant (G) is our forced admission that Mass-Energy is not independent of the geometry of Spacetime (Length and Time).
These constants are best understood as Planck Jacobians. In mathematics, a Jacobian is a matrix of derivatives that describes how one coordinate system stretches, shrinks, and shears when transformed into another. The physical constants are precisely this: the scaling factors that translate measurements from our arbitrary SI chart to the true, unified chart of physics. They are operators that rotate our measurements into alignment with reality. For example, the operation v/c is the act of rotating our SI-measured velocity into the true, dimensionless velocity of the universe.
The numerical value of a constant exists only within a specific unit chart. c is 299,792,458 only on the m/s chart. On any other arbitrary chart, it would have a different value. Each chart has its own unique set of these corrective "fixes."
3.1 The Universal Algorithm for Constructing Physical Constants
The realization that constants are artifacts of our measurement chart is not merely a philosophical insight; it is a predictive, algorithmic tool. Any physical constant can be constructed—not discovered—by a simple, two-step procedure. This algorithm reveals that all constants are just different combinations of the base Planck units (l_P, m_P, t_P), with the "recipe" for each constant being dictated solely by the dimensional mismatch of the physical law it inhabits.
Step 1: Check Dimensional Balance
Begin with the pure, Level 3 physical proportionality (e.g., F ∝ m1*m2 / r²). Analyze the dimensions of the output variable (Force: [M][L][T]⁻²) and the input variables (m1*m2/r²: [M]²[L]⁻²).
If the ratio of output to input dimensions is 1, the law is dimensionally consistent on your chart. You are done; no constant is needed.
If the dimensions do not match, a constant k is required to "fix" the discrepancy.
Step 2: Construct the Fix from Planck Units
The dimensions of the required constant k are simply the dimensions needed to balance the equation.
[k] = [Output Dimensions] / [Input Dimensions]
For our gravity example:
[k] = ([M][L][T]⁻²) / ([M]²[L]⁻²) = [L]³[M]⁻¹[T]⁻²
This dimensional signature is not an abstract property; it is the explicit recipe for building the constant. To find the physical constant itself, replace the abstract dimensions L, M, and T with their concrete Planck-scale counterparts: l_P, m_P, and t_P, raised to the same powers.
k = l_P³ * m_P⁻¹ * t_P⁻² = l_P³ / (m_P * t_P²)
This is the constant. By substituting the known relationships between the Planck units and c, one can prove this expression is identical to the Gravitational Constant, G. The process is universal. For Coulomb's Law, the same algorithm yields the Coulomb constant, k_e. For black-body radiation, it yields the Stefan-Boltzmann constant. Every constant is a specific combination of Planck units raised to whatever powers make the dimensional math work out.
This demonstrates that there is nothing mysterious about any physical constant. They are all built from the same three fundamental ingredients (l_P, m_P, t_P), algorithmically generated as dimensional patches for our arbitrary chart.
4. Layer 3: The World of Physics - The Unity Chart
The third world is the territory itself. This is the Planck scale, but not as a distant physical location—rather, as the fundamental, coordinate-free source code of the universe. This world is defined by its unity and lack of units.
Here, the distinctions that define our human experience dissolve. There are no "meters" or "kilograms," only pure, dimensionless ratios. The fragmented concepts of Layer 1 are revealed to be different facets of a single, underlying reality:
Temperature = Frequency = Mass = Energy = Momentum = 1/Length
This is the ultimate physical law, from which all others are derived. In this world, the distinction between a thing and its properties, between space and time, or between mass and energy, does not exist. The physics is contained entirely in these simple, geometric ratios.
5. Synthesis: Deriving E=mc² as a Translation
The power of this three-world framework is its ability to derive our familiar laws as mere translations. Consider E=mc²:
Start in the World of Physics (Layer 3): The fundamental law is a simple statement of identity: E = m. Mass and Energy are the same dimensionless quantity.
Build the Bridge (Layer 2): To express this identity using our distorted SI measurements, we must state that the dimensionless ratios are equal. This is the formal mapping:
E_si / E_P = m_si / m_P
(Where E_P and m_P are the Planck Energy and Planck Mass, the components of the Jacobian bridge).
Arrive in the World of Measurement (Layer 1): We rearrange the equation to solve for our measured energy, E_si:
E_si = m_si * (E_P / m_P)
The term (E_P / m_P) is the chart-specific "fix." By definition, E_P/m_P = c². Thus, the famous equation E=mc² emerges. It is not a fundamental law. It is a translation manual. The true law is E=m. The term c² is the conversion factor—the Jacobian—required to make that simple truth legible on our distorted SI chart.
6. Conclusion: A New Paradigm
The framework of the Three Worlds offers a new paradigm for physical law. We begin by creating an arbitrary measurement system with falsely independent axes. Reality then forces us to invent constants as a recognition of the interdependence of these axes. This paper has demonstrated that these constants are not fundamental numbers of the universe, but are the components of a Jacobian matrix that scales and rotates our measurements to align with a deeper, unified, and dimensionless reality.
Furthermore, we have presented a universal algorithm that formalizes this insight. To construct any physical constant: identify the dimensional mismatch in a physical law, then raise the fundamental Planck units to the compensating powers. That is the constant. Every time. This makes it unequivocally clear that constants are not empirically discovered curiosities; they are necessary and predictable consequences of our decision to describe a unified reality with a fragmented system of measurement. The complexity is not in the physics, but in our chosen description of it.
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