A Monograph on the Algorithmic Nature of Physical Constants within the Three Worlds Framework

 J. Rogers, SE Ohio

1.0 Introduction: The Paradox of Universal Laws and Arbitrary Measurement

This monograph addresses a foundational paradox at the heart of physics: how can universal physical laws, which govern the cosmos with unerring precision, yield consistent and accurate predictions when expressed using fundamentally arbitrary, human-invented units like the SI system? The meter, kilogram, and second are rooted in historical accidents and anthropocentric convenience, yet our most profound equations depend on them. This document resolves this paradox by presenting the "Three Worlds of Physics" framework, a model that clarifies the relationship between measurement, physical constants, and fundamental law.

The core thesis of this monograph is that physical constants are not fundamental properties of nature to be discovered, but are algorithmically derivable artifacts of our chosen measurement system. They are the necessary mathematical operators that correct for the inherent distortions introduced by describing a unified reality with a fragmented and arbitrary set of units.

The objective of this document is to provide physicists and engineers with a definitive conceptual and practical toolkit for understanding this relationship. We will deconstruct the role of measurement, redefine the nature of physical constants, and present a universal algorithm for constructing these constants from first principles, demonstrating that they are not empirical curiosities but predictable consequences of our descriptive framework.

To achieve this, we will proceed by analyzing a three-layer model that separates the map of measurement from the territory of reality, revealing the true, algorithmic nature of the constants that bridge them.

2.0 The Three Worlds Framework: A Conceptual Overview

The Three Worlds of Physics framework provides the solution to the paradox of arbitrary measurement by strategically separating the "map" from the "territory." It posits that the disconnect between universal laws and arbitrary units arises from confusing our system of description with the reality being described. By cleanly delineating three distinct conceptual layers, the framework reveals the true function of physical constants and the underlying simplicity of physical law.

The three layers are:

• The World of Measurement: Our arbitrary SI coordinate chart, which we lay over reality.
• The Bridge of Constants: The operators (Planck Jacobians) that correct the inherent distortions of the measurement chart.
• The World of Physics: The fundamental, dimensionless, and unified reality itself.

This structure allows us to see how a flawed but useful measurement system can be reconciled with physical reality through a systematic and predictable set of corrective factors. We will begin by examining the first world in this model: the familiar system of human measurement.

3.0 Layer 1: The World of Measurement – The SI Coordinate Chart

The first world is the domain of human measurement, most prominently represented by the International System of Units (SI). This layer functions as a coordinate chart we have laid over the canvas of reality. It is a useful tool for engineering and communication, but like a Mercator projection of a globe, it is a fundamentally distorted representation of the underlying territory it seeks to map. This distortion arises from three defining characteristics.

1. Arbitrary Origin and Scale The base units of the SI system are rooted not in fundamental physics but in historical accidents and human convenience. The meter was conceived as a fraction of the Earth's circumference, the kilogram from the mass of a specific volume of water, and the second from a fraction of a solar day. Their origins are entirely detached from the natural scales of the universe.
2. Assumed Independence In constructing this measurement chart, we commit a foundational error: we treat the primary axes of measurement—Length, Mass, and Time—as if they are independent, orthogonal dimensions. This assumption leads directly to the creation of separate and seemingly unrelated measurement tools: the ruler for length, the balance scale for mass, and the clock for time. As we will see, this assumed independence is a fiction that reality does not honor.
3. Human Convenience The scale of our units is unapologetically anthropocentric. A meter is comparable to a human stride; a kilogram is a mass an average person can comfortably lift. While practical for daily life, this scale has no privileged status in the grand architecture of the cosmos.

The SI chart is therefore a convenient but flawed map. The consequences of its arbitrary and fragmented nature necessitate a corrective bridge to align its measurements with observed physical reality.

4.0 Layer 2: The Bridge – Constants as Planck Jacobians

The second world is the dynamic interface that reconciles the distorted SI chart with the unified reality observed in experiments. When our measurements confront the real world, we are forced to acknowledge that the axes we assumed were separate—Length, Mass, Time—are in fact deeply interconnected. To make our equations work, we invent physical constants. These constants are the necessary mathematical patches that correct for the flawed assumptions of the measurement world.

The Nature of Constants as Corrective Operators

The central insight of this framework is that physical constants are admissions of our map's flaws, not discoveries about the territory of nature. They are the corrective terms that bridge the axes we incorrectly assumed were independent. This redefines them not as fundamental numbers but as powerful mathematical operators.

We term these operators Planck Jacobians. In mathematics, a Jacobian matrix describes how one coordinate system must be stretched, shrunk, and rotated to transform into another. Physical constants function in precisely this way: they are the operators that translate measurements from our arbitrary, distorted SI chart into alignment with the true, unified chart of physics. For example, the operation v/c is the act of rotating our SI-measured velocity into the true, dimensionless velocity of the universe.

This function becomes clear when we examine specific constants as "fixes" for the false independence of our measurement axes:

• Speed of light (c): Our forced admission that the Length and Time axes are not independent.
• Planck constant (ħ): Our forced admission that Energy and Frequency are not independent.
• Gravitational constant (G): Our forced admission that Mass-Energy is not independent of Spacetime geometry.

Crucially, the numerical value of any constant is an artifact of a specific unit chart. The speed of light is 299,792,458 only when expressed in meters per second. On a different chart, such as furlongs per fortnight, its numerical value would be entirely different.

This conceptual understanding of constants as algorithmic correctives gives rise to a predictive, universal method for constructing any constant from first principles.

5.0 The Universal Algorithm for Constructing Physical Constants

This section presents the practical application of the Three Worlds framework. It demonstrates that any physical constant can be algorithmically constructed, not empirically discovered, via a simple and universal two-step procedure. This algorithm reveals that constants are not mysterious numbers but are predictable composites of fundamental Planck units, assembled to resolve dimensional imbalances in our physical equations.

The algorithm is as follows:

1. Step 1: Check Dimensional Balance. Begin with a pure physical proportionality, such as Newton's law of universal gravitation: F ∝ m1*m2 / r². Analyze the dimensions of the output variable (Force: [M][L][T]⁻²) and compare them to the dimensions of the input variables (m1*m2/r²: [M]²[L]⁻²). If the ratio of the output dimensions to the input dimensions is not 1, the law is dimensionally inconsistent on the SI chart. A constant, k, is therefore required to "fix" the discrepancy and balance the equation.
2. Step 2: Construct the Fix from Planck Units. The dimensions of the required constant k are determined by the ratio needed to balance the equation:
3. [k] = [Output Dimensions] / [Input Dimensions]

Worked Example: Derivation of the Gravitational Constant (G)

Applying the two-step algorithm to Newton's law of gravitation demonstrates its power:

Per Step 1: Identify Dimensional Mismatch We begin with the proportionality F ∝ m1*m2 / r². The dimensions of the output (Force) are [M][L][T]⁻², while the dimensions of the input (m1*m2/r²) are [M]²[L]⁻². The dimensional ratio is not 1, confirming a constant is required to balance the equation.

Per Step 2: Construct the Constant We first derive the dimensional signature of the required constant k:

[k] = ([M][L][T]⁻²) / ([M]²[L]⁻²) = [L]³[M]⁻¹[T]⁻²

The crucial insight is that this dimensional signature is the explicit recipe for building the constant. To construct the physical constant, we simply replace the abstract dimensions L, M, and T with their concrete, fundamental Planck-scale counterparts: the Planck length (l_P), Planck mass (m_P), and Planck time (t_P), raised to the same compensating powers. This yields the final expression for the constant:

k = l_P³ / (m_P * t_P²)

This expression, built algorithmically from the fundamental units of nature, is identical to the Gravitational Constant, G.

This algorithm is universal. The same procedure applied to Coulomb's Law yields the Coulomb constant (k_e), and when applied to black-body radiation, it yields the Stefan-Boltzmann constant. Every constant is merely a specific combination of Planck units, assembled in whatever way is required to make our human-scale equations dimensionally coherent. This process connects the flawed World of Measurement to the underlying reality it seeks to describe.

6.0 Layer 3: The World of Physics – The Unity Chart

The third and final world is the fundamental "territory" itself. It is best understood not as a distant physical location, like the Planck scale, but as the conceptual foundation of reality—the coordinate-free source code of the universe. This is a conceptual space defined not by measurement but by unity and a conspicuous lack of units. It is the world of pure physics, where the fragmented concepts we use in Layer 1 dissolve and are revealed to be different facets of a single, underlying reality.

In this world, the distinctions that define our human experience—between mass and energy, space and time, momentum and frequency—no longer hold. There are no "meters," "kilograms," or "seconds," only pure, dimensionless ratios. Here, physics is contained entirely in simple, geometric relationships. The ultimate physical law, from which all others are derived, can be expressed as a profound identity:

Temperature = Frequency = Mass = Energy = Momentum = 1/Length

This equation signifies that what we perceive and measure as distinct physical properties are, at the most fundamental level, one and the same. The physics of this world is not about complex formulas but about simple identities.

Having established this fundamental ground truth, we can now demonstrate the predictive power of the framework by showing how a famous physical law is derived as a simple translation between these three worlds.

7.0 Case Study: Deriving E = mc² as a Cross-World Translation

The true power of the Three Worlds framework lies in its ability to derive familiar physical laws not as fundamental truths, but as necessary translations between the conceptual layers. The iconic equation E=mc² serves as a perfect case study. Within this framework, it is revealed not as a deep law of nature, but as a translation manual for expressing a much deeper, simpler truth on our distorted SI measurement chart.

The derivation is a straightforward three-step process corresponding to the three worlds.

1. Start in the World of Physics (Layer 3): We begin with the fundamental law, which is a simple statement of identity between dimensionless quantities: E = m. In the language of pure physics, Mass and Energy are the same thing.
2. Build the Bridge (Layer 2): To express this identity using our SI measurements, we cannot simply equate the numerical values of energy (in Joules) and mass (in kilograms). Instead, we must state that their dimensionless ratios relative to the universe's natural scale are equal. This is the formal mapping performed by the Planck Jacobians:
3. E_si / E_P = m_si / m_P
4. Here, E_si and m_si are our measured values, while E_P (Planck Energy) and m_P (Planck Mass) are the components of the Jacobian bridge that connect our chart to the unity chart.
5. Arrive in the World of Measurement (Layer 1): Finally, we rearrange the equation to solve for the quantity we measure, E_si, which isolates the conversion factor required for our chart:
6. E_si = m_si * (E_P / m_P)

The term (E_P / m_P) is the chart-specific Jacobian "fix" that translates a quantity of mass on the SI chart into a quantity of energy. By definition of the Planck units, this ratio is precisely equal to c².

The result is a profound reinterpretation: the equation E=mc² emerges as the necessary translation of the true physical law, E=m. The c² term is not a feature of physics, but the conversion factor—the Jacobian—required to make that simple truth legible on our distorted SI chart.

8.0 Conclusion: A New Paradigm for Physical Constants and Law

The Three Worlds framework establishes a new paradigm for understanding the relationship between physical law and human measurement. It resolves the paradox of arbitrary units by demonstrating that we first create a fragmented measurement system with falsely independent axes. Reality then compels us to invent physical constants as a formal recognition of the interdependence of these axes. This monograph has shown that these constants are not fundamental numbers of the universe to be discovered, but are the components of a Jacobian matrix that scales and rotates our measurements to align with a deeper, unified, and dimensionless reality.

The key takeaway is that constants are not discovered; they are the necessary and predictable consequences of describing a unified system with a fragmented tool. This insight is formalized in the universal algorithm for their construction, which can be distilled into a single, powerful statement:

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 demonstrates unequivocally that constants are not empirical curiosities but are algorithmically generated as dimensional patches. Ultimately, this framework reveals that the perceived complexity is not in the physics, but in our chosen description of it.