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.

Physics' Best-Kept Secret: The Constants of Nature Are Human Inventions

 J. Rogers, SE Ohio

Introduction: The Paradox of Our Cosmic Ruler

How can the universal laws of physics, which operate with unerring precision across the cosmos, function perfectly using units like the meter and the kilogram—measurements that are completely arbitrary and human-invented? This question points to a fundamental paradox that sits at the heart of physics: our most profound descriptions of reality seem to be built on a foundation of historical accident and human convenience.

This paradox is solved by the "Three Worlds of Physics" framework, which reveals the complexity is not in reality itself, but in the "distorted map" we use to describe it. By separating the universe from our description of it, a clearer picture emerges, revealing that many of the things we thought were fundamental discoveries are, in fact, human inventions.

This article will explore the four most surprising takeaways from this framework—four mind-bending ideas that will change how you think about physical reality.

1. The Constants of Nature Aren't Discoveries—They're Inventions

The most challenging idea comes first: fundamental physical constants like the speed of light (c) and the gravitational constant (G) are not inherent, fundamental numbers of the universe that we discovered. They are inventions. More precisely, they are the necessary mathematical consequences of our decision to describe a unified reality with a fragmented set of tools.

They function as "admissions of our map's flaws." We created a measurement system—the familiar SI chart of meters, kilograms, and seconds—where we incorrectly assumed that Length, Mass, and Time were independent and orthogonal concepts. But reality forced us to see that they are deeply connected. Constants were invented as mathematical "patches" to fix the disconnects our flawed map created. The speed of light, c, is the patch that connects our separate ideas of Length and Time. The gravitational constant, G, is the patch that fixes our incorrect assumption that Mass-Energy is independent of the geometry of Spacetime.

"...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."

This is deeply counter-intuitive. We are taught to see constants as unchanging truths etched into the fabric of the cosmos. This new view reframes them as tools we built to correct our own limited perspective.

2. Physical Constants Are a "Translation Key" for Reality

If constants are patches for our flawed map, then they must function as a kind of translation key—a mathematical lens to correct our distorted view of the universe. The formal term for these operators is Planck Jacobians.

Think of our system of SI units as a distorted Mercator map of the globe. A Mercator map is useful for navigation, but it fundamentally misrepresents the planet's geometry, making Greenland look as large as Africa. Our measurement system is similarly useful, but it misrepresents the unified nature of reality by treating concepts like Mass, Length, and Time as separate.

Physical constants are the mathematical formulas that "un-distort" our measurements. They scale and rotate them to align with the true "globe" of physics, where everything is unified.

A concrete example of this is the operation v/c (an object's velocity divided by the speed of light). This isn't just a calculation; it's the physical act of taking a velocity measured on our human-scaled chart (in meters per second) and rotating it into the true, dimensionless velocity that exists in fundamental reality.

This implies that the numerical value of a constant—like c = 299,792,458 m/s—has no fundamental meaning on its own. It is just the specific "correction factor" required for our specific, arbitrary map of meters and seconds. If we had chosen different units, the number for c would be completely different.

3. There Is a Universal Recipe for Creating Any Constant

If constants are inventions, there must be a way to create them. This framework provides a simple, universal, two-step algorithm to construct any physical constant from scratch. They are not found through painstaking experiment; they are "algorithmically generated" to fix our math.

Here is the algorithm, simplified:

• Step 1: Find the Mismatch. Start with a physical law, like Newton's law of gravity (F ∝ m1*m2 / r²). Check if the units on both sides of the equation balance out using our SI system. In this case, they don't. The units for Force ([M][L][T]⁻²) do not match the units for mass squared over distance squared ([M]²[L]⁻²). This mismatch means you need a "dimensional patch"—a constant.
• Step 2: Build the Patch. The required dimensions of the patch become an explicit recipe. For gravity, the patch needs units of [L]³[M]⁻¹[T]⁻² to make the equation work. To build the actual constant, you simply plug in the fundamental "building blocks" of reality—the Planck units (l_P, m_P, t_P)—raised to the powers determined by the recipe.

This reveals a stunning truth: all physical constants, from G to the Coulomb constant, are built from the exact same three 'Lego bricks'—the Planck units—arranged in different combinations. This isn't just an abstract formula; it's a literal blueprint. The recipe l_P³ / (m_P * t_P²) is the Gravitational Constant, G.

"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 powerful algorithm works for any constant in physics, transforming them from mysterious cosmic numbers into predictable, necessary consequences of our measurement choices.

4. Is Not the Fundamental Law—It's a Translation Manual

The three-worlds framework downgrades E=mc² from a fundamental law of nature to a unit conversion formula.

In the fundamental "World of Physics," where reality is a unified whole without human units, the law is simply E = m. Mass and Energy are not just equivalent; they are one and the same dimensionless thing.

So, where does the c² come from? The term c² is the Jacobian needed to translate the simple truth of E=m for use in our "World of Measurement." The conversion begins by stating that the dimensionless ratio of measured energy to Planck energy must equal the dimensionless ratio of measured mass to Planck mass (E_si / E_P = m_si / m_P). The equation we know, E=mc², is simply the algebraic rearrangement of this fundamental mapping.

"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."

This re-frames one of the most famous equations in science. It's not a deep law discovered in nature, but rather a "user manual" for converting between our human concepts and the underlying, unified reality.

Conclusion: Is Reality Hiding in Plain Sight?

The core message of this framework is that many of the complexities and "mysterious numbers" in physics are not features of the universe itself. They are consequences of the fragmented, arbitrary language we invented to describe it. The universe is fundamentally simple; our description of it is what's complex.

As the source text powerfully concludes: "The complexity is not in the physics, but in our chosen description of it."

This leaves us with a profound question. If the constants that build our reality are just artifacts of our perspective, what other 'fundamental truths' might just be shadows cast by the measurement tools we use?

The Three Worlds of Physics: A Framework for Understanding Measurement, Constants, and Physical Law

J. Rogers, SE Ohio

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.

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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:

1. 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.

2. 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.

3. 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:

1. 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.

2. 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).

3. 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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The Hidden Connection: Why Our Rulers Don't Match Reality

 J. Rogers, SE Ohio

The Problem with Our Measuring Sticks

Imagine you're trying to describe your bedroom to a friend. You might say it's 12 feet long, 10 feet wide, and your bed weighs 150 pounds. These measurements make perfect sense to us because we invented feet and pounds based on human-sized things. A foot was originally based on, well, someone's actual foot. A pound came from a chunk of metal we decided to call "one pound."

But here's the weird part: the universe doesn't know what a foot or a pound is. These are completely made up by humans. We could have used any size for our measuring sticks, and the universe wouldn't care at all.

The Universe Has Its Own Secret Language

When scientists discovered the laws of physics, they found something strange. To make the math work with our human-made measurements, they had to keep adding mysterious numbers called "constants."

For example, Newton's law of gravity looks like this:

Force = G × (mass₁ × mass₂) / distance²

That "G" is called the gravitational constant, and its value is a tiny number: 0.0000000000667. Scientists measured this number very carefully and treated it like a deep secret of the universe.

But what if I told you that G isn't really a secret about gravity? What if it's actually a secret about our rulers?

What the Constants Are Really Telling Us

Here's the key insight: G exists because we measured mass, distance, and force using three completely separate rulers that we made up independently.

Think about it this way: Imagine you're baking cookies and the recipe says "mix flour and sugar." But you measure flour with a teaspoon and sugar with a tablespoon. If you want to know the actual ratio between them, you need to convert one to the other. You need a conversion factor.

That's exactly what G is doing. Our kilogram ruler for mass and our meter ruler for distance were invented separately, with no connection between them. But in the real universe, mass and distance do have a natural relationship. G is the conversion factor that translates between our mismatched rulers and reality.

The Universe's Natural Measuring System

Scientists discovered that nature actually has its own measuring system, called Planck units. In this natural system:

• There's a natural length (incredibly tiny: 0.0000000000000000000000000000000000016 meters)
• There's a natural mass (incredibly tiny: 0.000000000000022 grams)
• There's a natural time (incredibly tiny: 0.000000000000000000000000000000000000000000054 seconds)

When you measure things using nature's rulers instead of human rulers, something magical happens: all those mysterious constants disappear! The formulas become super simple.

The gravitational time dilation formula normally looks like this:

Time change = (G × Mass) / (c² × distance)

But when you use natural units, it becomes simply:

Time change = Mass / distance

That's it! No constants, no complicated numbers. Just a simple ratio.

Different Measurements, Same Reality

Here's the truly mind-blowing part: what we call "mass," "energy," "time," and "distance" might actually be the same thing in the universe, just measured along different human-made axes.

Imagine you're looking at a square room from the corner. You might describe it with an "x-axis" (left-right) and a "y-axis" (forward-back). But you could also turn 45 degrees and describe it with completely different axes. The room didn't change—just your measuring system.

That's what we've done with physics. We created separate measurements for mass, distance, time, and energy. But in nature's coordinate system, they're all connected—they're all measuring aspects of the same underlying reality. We just can't see that connection directly because we're looking through our misaligned, human-scaled measuring system.

The Constants Are Correction Factors

So what are those physical constants really?

• G (gravitational constant): Converts between our arbitrary kilogram and our arbitrary meter to match nature's relationship
• c (speed of light): Converts between our arbitrary meter and our arbitrary second
• h (Planck's constant): Converts between our energy scale and our time scale

They're not mysterious properties of the universe. They're correction factors that fix the fact that we invented our measuring sticks without thinking about how they should relate to each other.

It's like if you measured your height in feet and your weight in stones (an old British unit), and then wondered why you need a weird conversion number to compare them. The conversion number isn't about you—it's about your choice of units!

Why This Matters

For centuries, scientists have wondered: "Why does G have this particular value? What does it mean?"

The answer is simpler than anyone thought: G has that value because of how we chose to define the kilogram and the meter. If we had picked different sizes for our human rulers, G would have a different number. There's no deep mystery—just a historical accident of measurement.

The real physics is in the simple ratios when you use nature's measuring system. Those ratios tell you directly how things relate: mass per distance equals time change. That's the actual rule the universe follows.

The Big Picture

The universe isn't complicated—our measuring system is. We invented rulers based on human-sized things (feet, pounds, seconds), never realizing that nature already had perfect rulers built in. The mysterious constants in our equations aren't revealing secrets about reality—they're revealing secrets about our rulers.

When you account for this, you realize that the universe is simpler and more elegant than we thought. Mass isn't some separate mysterious property that "causes" gravity. In nature's measurement system, mass is a measure of how much time changes—they're the same thing, just viewed from different angles of our misaligned measuring system.

The constants are like training wheels on a bicycle. They help us use our human-made measurements, but they're not part of the bike itself. Once you learn to see things in nature's terms, the training wheels—and the constants—disappear, and you see reality rolling smoothly on its own.

The Metrological Blind Spot: How Physical Constants Reveal the Disharmony of Arbitrary Measurement

J. Rogers, SE Ohio

Abstract

For over two centuries, a profound blind spot has existed at the heart of physics. We have simultaneously held two contradictory beliefs: that our base units of measurement (the meter, kilogram, second) are arbitrary human constructs, and that the physical constants (G, c, h) required by our laws are fundamental, discovered properties of the universe. This paper argues that this is a false dichotomy. The constants are not fundamental discoveries; they are the necessary mathematical artifacts that arise directly from the initial, un-harmonized arbitrariness of our units. By deconstructing the constants and revealing their function as Jacobian conversion factors, we demonstrate that they are the "epicycles" of modern physics, correcting for a poorly chosen (anthropocentric) origin for our measurement system. This reframing shows that physical laws are, at their core, simple dimensionless ratios between natural proportions. This perspective dissolves long-standing "irreducible mysteries" surrounding mass, gravity, and the values of the constants, revealing that the ultimate unity in physics is not a unification of forces, but the harmonization of our axes of measurement.

1. Introduction: The Great Disconnect

The foundation of modern science rests on the International System of Units (SI). From its inception, the architects of this system understood and acknowledged the arbitrary nature of its base units. The meter, originally defined as one ten-millionth of the distance from the pole to the equator, and the kilogram, the mass of a specific metal artifact, were understood to be choices of convenience, not cosmic truths.

In parallel, theoretical physics embarked on a journey of discovery, unveiling laws that appeared to govern the cosmos. A curious feature of these laws was the persistent appearance of constants of proportionality—Newton's G, the speed of light c, Planck's constant h. These numbers were treated as profound discoveries, fundamental features of reality whose precise values were a message from the universe.

This created a cognitive firewall. On one side, metrologists managed a system they knew was arbitrary. On the other, physicists revered the constants that arose from using that system as fundamental. The great blind spot of modern science has been the failure to recognize that these two facts are not independent, but are cause and effect. The constants are not a message from the universe; they are a direct and unavoidable mathematical consequence of the initial, arbitrary choices made by the metrologists.

2. The Nature of the Constants: Discovery or Correction?

The traditional view posits that constants are immutable properties of nature. We propose that they are, instead, unit scaling operators. They are the mathematical machinery required to harmonize our disparate and badly-scaled measurement axes. Their function is not to describe nature, but to correct our description.

Consider the speed of light, c. Its numerical value (299,792,458 in m/s) is not a "speed limit" imposed by the cosmos, but a measure of our own scaling error. It is the conversion factor that documents how badly our human-scale meter is scaled relative to our planetary-scale second. If we had chosen our units in harmony with the geometry of spacetime, this ratio would be 1. The constant c is the Jacobian that transforms measurements between our flawed coordinate chart and a natural one.

Similarly, Newton's constant G and Planck's constant h are more complex operators, or "epicycles," that reconcile our macroscopic, arbitrary units of mass, length, and time with the natural scales of gravity and quantum mechanics. Their incredibly small numerical values are not a sign of the weakness of these forces, but a measure of the vast scaling mismatch between our human-world units and the phenomena they describe.

3. A Rational Deconstruction of the Math: Gravity as a Case Study

The validity of this viewpoint is not a matter of philosophy, but is demonstrated by a direct, step-by-step breakdown of the mathematical operations. Let us examine the formula for gravitational time dilation:

        $$\frac{\mathrm{\Delta }t}{t_0}=\frac{GM}{c^{2}r}$$      

The standard narrative explains this via the curvature of spacetime caused by mass. However, let us analyze what the constants are doing. The controlling term G/c² is a composite operator. We can deconstruct it by examining the conversion factors between SI and natural units (the Planck units, l_P, m_P, etc.), which function as the basis vectors of the Jacobian matrix:

        $$\frac{G}{c^2}=\frac{(l_P^3\mathrm{/}{m}_P{t}_P^2)}{(l_P^2\mathrm{/}{t}_P^2)}=\frac{l_P}{m_{\mathrm{P}}}$$      

The entire complex operation G/c² simplifies to a single instruction: "scale the result by the ratio of the length-converter to the mass-converter." Substituting this back into the law reveals the true physics hidden by our units:

        $$\frac{\mathrm{\Delta }t}{t_0}=\left(\frac{l_P}{m_P}\right)\frac{M}{r}\hspace{0.25em} ⟹\hspace{0.25em} \frac{\mathrm{\Delta }t}{t_0}=\frac{M\mathrm{/}{m}_P}{r\mathrm{/}{l}_{\mathrm{P}}}$$      

The physical law is not the complicated dimensional formula. The physics is a simple, dimensionless statement: the fractional change in time is the ratio of mass in natural units to length in natural units. This is a direct statement about a time field, where "mass" is not the cause of the effect, but is our name for the measurement of the effect itself. The elaborate story of spacetime curvature is a narrative we invented to explain the operational effects of the G/c² scaling factor.

4. The Unity of Measurement: Harmonizing the Axes

This framework redefines the quest for unification in physics. The ultimate unity is not between forces, but between our very axes of measurement. The identity that emerges in a harmonized, natural unit chart is:

        $$T=f=m=E=p=F=1\mathrm{/}L$$      

This is not a list of equivalences; it is a statement of identity. It reveals that we have given a half-dozen different names to the same underlying reality—processes in time—simply because we were measuring it along different, un-harmonized axes. Mass is the measure of this process at rest; energy is the measure of its total activity; temperature is the statistical measure of a collection of such processes.

This solves centuries of misunderstanding. "Spooky action at a distance" is an illusion. Newton's law of gravitation, when deconstructed, describes the local interaction of two time fields (m1/r and m2/r). The constant G is merely the set of instructions that scales this local interaction into our arbitrary SI unit of force.

Physics has been trapped in a "Matrix" of its own SI units. The system works, and the simulation is convincing, leading us to mistake the rules of the simulation (the constants) for the rules of reality. The process outlined here is the act of seeing the underlying code.

5. Conclusion: From Irreducible Mystery to Rational Clarity

We knew our units were arbitrary, but we failed to consider that the constants were the price of that arbitrariness. This blind spot created a narrative of irreducible mystery around the nature of mass and the values of the constants.

By recognizing the constants as the mathematical tools for harmonizing our measurement system, these mysteries dissolve. The constants are not a cosmic message; they are a historical record of our own metrological choices. The deepest truths are not found in the seemingly random values of G, c, and h, but in the simple, elegant relationships that emerge only when these scaling artifacts are understood and factored out. The real work of fundamental physics, then, is not just to measure the world, but to perfect the language we use to do so, until the harmony of the underlying reality is revealed.