Tuesday, October 7, 2025

The Planck-SI Jacobians: A Necessary Correction on the Nature of Fundamental Constants

J. Rogers, SE Ohio

Abstract

A foundational category error pervades modern physics: the misidentification of the so-called "Planck units" as fundamental, microscopic scales of reality. This paper clarifies that these values (l_P, m_P, t_P, etc.) are not physical properties of the universe, nor are they part of the fundamental system of natural law. We demonstrate that they are correctly understood as the Planck-SI Jacobians—the numerical components of the transformation matrix that serves as a bridge between the invariant, dimensionless reality of natural proportions (the Planck Chart) and the arbitrary, abstract system of human measurement (the SI Chart). By applying the "Razor of Convention," which states that anything alterable by human agreement cannot be fundamental, we show that constants are necessarily artifacts of measurement. This clarification dissolves long-standing mysteries like the "fine-tuning" of constants and reframes the entire enterprise of physics as the projection of an elegant, invariant reality onto our chosen conceptual frameworks.

1. Introduction: The Razor of Convention

It is a trivially obvious truth that anything we can define by human convention is not a fundamental feature of objective reality. The definition of a meter, a kilogram, or a second can be—and has been—altered by international committees. These units are abstract concepts we create in our minds. In contrast, the invariant relationships of nature, such as the proportionality between energy and mass, cannot be altered by decree.

This simple razor, if applied rigorously, forces a profound re-evaluation of what we call "fundamental constants." This paper will demonstrate that what we have historically named "Planck units" are not properties of nature, but are rather the components of a bridge—a Jacobian transformation matrix—that connects the invariant reality of physics to our arbitrary systems of measurement.

2. The Two Charts: A Necessary Separation of Reality and Abstraction

To understand physics correctly, we must make a sharp distinction between two fundamentally different conceptual spaces.

2.1. The Planck Chart: The Invariant Reality
The actual, real physics of the universe consists of a set of simple, dimensionless, geometric proportions. This is the Planck Chart. It is a unit-less system of pure relationships.

Energy = Mass

Momentum = 1 / Length

Force = Mass² / Length²

Energy Density = Temperature⁴

This is not an abstraction; it is the objective, invariant structure of reality itself. It is the basis from which all possible unit charts can be constructed. It contains no numbers like 6.626 × 10⁻³⁴ or 299,792,458. It only contains the pure logic of proportionality.

2.2. The SI Chart: The Human Abstraction
The International System of Units (SI) is one of an infinite number of possible measurement systems we could invent. "Meters," "kilograms," and "seconds" are abstract concepts created in our minds for the purpose of quantification. They do not exist in nature. The SI Chart is a conceptual, human-imposed grid. It is an abstraction we project onto the invariant reality.

3. The Bridge: Defining the Planck-SI Jacobians

Given that the Planck Chart (Reality) and the SI Chart (Abstraction) are two different systems, a bridge is required to connect them. This bridge is the Jacobian transformation matrix, and its components are what have been mistakenly called "Planck units."

3.1. What are the "Planck Units"?
The numerical values we traditionally call Planck units, such as l_P ≈ 4.051 × 10⁻³⁵ m and m_P ≈ 5.456 × 10⁻⁸ kg, are not physical objects or scales. They are conversion factors.

Their nature is identical to that of the conversion factor 2.54 cm/in. The number 2.54 is not a property of length, nor is it part of the definition of an inch or a centimeter. It is a numerical artifact that exists solely to bridge these two arbitrary human conventions.

Similarly, the Planck-SI Jacobians are not part of the Planck Chart (which is unit-less), nor are they part of the SI Chart (which is defined by meters, etc.). They are the components of the bridge between them.

3.2. A Change in Terminology
To maintain conceptual clarity, we must abandon the term "Planck units." A more accurate and descriptive name is the Planck-SI Jacobians.

The Length Jacobian (l_P): The conversion factor mapping dimensionless length from the Planck Chart to the SI meter. (l_P ≈ 4.051 × 10⁻³⁵ m)

The Mass Jacobian (m_P): The conversion factor mapping dimensionless mass from the Planck Chart to the SI kilogram. (m_P ≈ 5.456 × 10⁻⁸ kg)

The Time Jacobian (t_P): The conversion factor mapping dimensionless time from the Planck Chart to the SI second. (t_P ≈ 1.351 × 10⁻⁴³ s)

4. How the Jacobian Works: A Proof by Construction

The function of the Jacobians is revealed when we derive a physical law.

Derivation of E = mc²:

Start with the Invariant Reality (The Planck Chart): The true physical relationship is a simple proportion: Energy = Mass. In this chart, the ratio Energy / Mass is unity.

Choose a Human Abstraction (The SI Chart): We wish to express this reality using our abstract units of Joules (for energy) and kilograms (for mass).

Apply the Bridge (The Jacobians): To translate our SI measurements back to the invariant reality of the Planck Chart, we must divide by the appropriate conversion factors. Let E_si be a value in Joules and m_si be a value in kilograms.

Energy_natural = E_si / E_P

Mass_natural = m_si / m_P

Equate and Solve: In the Planck Chart, the natural quantities are equal:
Energy_natural = Mass_natural
E_si / E_P = m_si / m_P

Solving for our SI energy value, E_si:
E_si = m_si * (E_P / m_P)

Calculate the Composite Jacobian: The term (E_P / m_P) is a composite conversion factor built from the fundamental Jacobians. Using their definitions in terms of h, c, and G:
E_P / m_P = √(h c⁵ / G) / √(h c / G) = √(c⁴) = c²

The Result: The final formula is E_si = m_si * c².

This is a proof by construction. The constant c² is not a part of the original physical reality. It is the composite Jacobian required to project the simple, invariant reality (E=m) into our specific, abstract measurement system (SI).

5. Consequences of the Clarification

This architectural correction is not a semantic game; it resolves the deepest conceptual problems in physics.

The "Fine-Tuning Problem" is Dissolved: The question "Why do the constants have their specific values?" is revealed to be a non-question. It is equivalent to asking, "Why is the conversion factor between the inch and the centimeter 2.54?" The answer lies in the history of human convention, not in the physics of the universe.

The 2019 SI Redefinition is Re-framed: The act of fixing the values of h, c, and k_B was the act of fixing the Planck-SI Jacobians by international treaty. It transformed the SI system from an artifact-based abstraction into a projection-based abstraction, permanently locking its relationship to the invariant Planck Chart.

Physics is Simplified: The apparent complexity of physics is revealed to be an artifact of a convoluted measurement system. The reality itself—the Planck Chart—is simple and elegant.

6. Conclusion

What we call "Planck units" are a misnomer for a deeper concept: the Planck-SI Jacobians. They are not the fundamental scales of reality, nor are they a feature of the true physics. They are the conversion factors that form a bridge between the invariant, dimensionless reality of natural law (the Planck Chart) and the arbitrary, abstract systems of human measurement (like the SI Chart).

By making this clarification, we demystify the constants of nature, simplify the structure of physical law, and correct a foundational category error that has confused physics for over a century. The constants are not the physics; they are the exchange rate we must pay to translate the pure physics into our chosen language.

Note: Separation of concerns

The Three Separate Concerns

1. Reality (The Planck Chart)

Invariant, coordinate-free ratios
E = m, p = 1/l, F = m²/l²
Pure physics
No numbers, no units
Cannot be changed by human convention

2. Abstraction (The SI Chart, or any measurement system)

Arbitrary, independently-scaled axes
Meters, kilograms, seconds
Human convention
Defined by agreement
Can be changed by committee vote

3. The Bridge (The Planck-SI Jacobians)

Neither reality nor abstraction
The transformation components connecting them
l_P ≈ 1.616 × 10⁻³⁵ m, m_P ≈ 2.176 × 10⁻⁸ kg
Changes when measurement changes
Not fundamental, not arbitrary - relational

Why This Matters

The Jacobians don't belong to either system.

They're not part of:

The Planck Chart (which has no units)
The SI Chart ( which is arbitrary independently scaled axes of measurement )

They exist in the transformation space between the two systems. They convert from measurements to natural ratios.

The Category Error Physics Makes

Current physics conflates all three:

❌ "Planck units are fundamental scales where quantum gravity happens"

Confusing Jacobians (#3) with reality (#1)

❌ "Constants are fundamental properties we measure"

Confusing Jacobians (#3) with abstraction (#2)

❌ "The Planck length is 10⁻³⁵ meters"

Putting a number from the bridge onto the reality side

The Correct Architecture

┌─────────────────────┐
│   PLANCK CHART      │  ← Reality
│   (invariant)       │     Pure ratios
│   E = m             │     No units
│   p = 1/l           │
└──────────┬──────────┘
           │
           │  ┌──────────────────────┐
           │  │  JACOBIAN BRIDGE     │  ← Neither/Bridge
           ├──┤  l_P, m_P, t_P       │     Transformation
           │  │  (relational)        │     Changes with measurement
           │  └──────────────────────┘
           │
┌──────────┴──────────┐
│   SI CHART          │  ← Abstraction  
│   (arbitrary)       │     Human convention
│   meters, kg, sec   │     Changeable
└─────────────────────┘

The Jacobians Are Relational

The Jacobians don't "live" anywhere - they're the relationship itself.

Like:

The exchange rate between dollars and euros isn't part of "the dollar system" or "the euro system"
It's the relationship between the two systems
Change either currency → exchange rate changes
But the underlying economic value (reality) stays the same

What Changes When

Change the SI Chart (redefine the kilogram):

Planck Chart: unchanged (still E = m)
Jacobians: changed (m_P gets new numerical value in new kg)
SI Chart: changed (new definition)

Change the Planck Chart (impossible - it's reality):

Can't be done

Change the Jacobians directly ( forces measurement to change):

They're determined by the relationship between the two charts
Not independent entities

Why Physics Is Confused

Physics treats the Jacobians as if they belong to one of the two systems:

Error 1: "Planck units are fundamental"

Treats Jacobians as part of reality (#1)

Error 2: "Constants are properties we measure like mass or length"

Treats Jacobians as part of abstraction (#2)
We didn't measure the constant, we calibrated an error bar.

Truth: Jacobians are neither - they're the transformation function, which is a third category.

The Three Categories Are Ontologically Distinct

Category Nature Can Change? Contains Units?
Planck Chart Reality No No (pure ratios)
Jacobians Relation Yes (if measurement changes) Yes (bridge values)
SI Chart Convention Yes (by agreement) Yes (defined scales)

Category	Nature	Can Change?	Contains Units?
Planck Chart	Reality	No	No (pure ratios)
Jacobians	Relation	Yes (if measurement changes)	Yes (bridge values)
SI Chart	Convention	Yes (by agreement)	Yes (defined scales)

Physics has been trying to cram Jacobians into one of the first two categories, when they're fundamentally a third kind of thing.

The Paper Should Make This Explicit

The paper needs a section (maybe expanded from current Section 2) that says:

"There are three ontologically distinct categories that physics has failed to properly separate:

1. The Invariant Reality (Planck Chart - pure ratios, no numbers)

2. The Conventional Abstraction (SI Chart - arbitrary scales)

3. The Relational Bridge (Jacobians - transformation components)

The Jacobians belong to neither system. They exist in the transformation space between them. This is why they have properties of both (units like SI, but referenced to natural structure like Planck) yet belong to neither.

Failure to recognize this third category has caused physics to misidentify constants as 'fundamental properties' when they are actually 'relational transformations.'"

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