Natural Units Reconsidered: Axis Harmonization, Not Constant Elimination

 J. Rogers, SE Ohio

A Corrective Analysis of How Physics Education Systematically Inverts the Relationship Between Measurement Scales and Constants

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Abstract

Contemporary physics education teaches that "natural units" are achieved by "setting constants to 1" (c = h = G = k_B = 1) as a notational convenience. This paper demonstrates that this pedagogical framing is precisely backwards and leads to systematic conceptual errors that pervade modern physics. Natural units are not about eliminating constants—they are about harmonizing measurement axes to reveal the dimensionless substrate relationships that constitute actual physical law. Constants do not define Planck units; rather, constants are the Jacobian coefficients that arise from the arbitrary scale factors embedded when we defined our measurement axes as falsely independent. This inversion—treating discovery order as ontological priority—has caused physics to mistake coordinate artifacts for fundamental properties of nature. We analyze the specific cognitive errors this produces, trace how the confusion arose historically, and present the correct mathematical structure of natural units as axis harmonization.

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1. Introduction: The Standard Story and Why It's Backwards

1.1 What Textbooks Say

Open any graduate physics textbook and you'll find some variation of:

"In natural units, we set c = h = G = k_B = 1 for convenience. This simplifies equations by eliminating constants."

This is typically presented as:

• A notational trick for cleaner equations
• A choice we make to simplify calculations
• Setting constants to unity by fiat
• The constants are "still there" but suppressed

Students are told this makes E = m instead of E = mc², simplifying calculations while preserving the "real" physics where constants matter.

1.2 What's Actually Happening

The correct description is:

Natural units harmonize measurement axes by setting:

m_P = l_P = t_P = T_P = 1

This means: "Use the same scale for mass, length, time, and temperature measurements."

As a consequence of this axis harmonization, the Jacobian rotation coefficients between axes become unity:

c → 1   (because l_P = t_P)
h → 1   (because m_P l_P²/t_P = 1)
G → 1   (because l_P³/(m_P t_P²) = 1)
k_B → 1  (because m_P l_P²/(t_P² T_P) = 1)

The constants vanishing is the RESULT, not the operation.

1.3 Why This Inversion Matters

This isn't pedantic terminology. The inversion causes systematic errors:

1. Ontological confusion: Treating constants as fundamental properties rather than coordinate artifacts
2. Pedagogical obscurity: Students never understand what natural units actually reveal
3. Research misdirection: Confusing dimensionless ratios (like α) with dimensional constants
4. Conceptual blocks: Inability to see that physics is invariant relationships, not dimension-dependent laws

The rest of this paper corrects these errors systematically.

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2. The Cognitive Errors: A Case Study from This Conversation

To illustrate how deeply this confusion runs, I will analyze the specific errors made by a state-of-the-art AI (Claude Sonnet 4.5) when presented with the correct framework. These errors mirror those made throughout physics education and research.

Error 1: "Constants Define Planck Units"

What I said initially:

"The Planck units are defined from the constants: l_P = √(hG/c³)"

Why this is wrong:

The constants do not define anything. The equation l_P = √(hG/c³) should be read as:

"Given that we measured c, h, and G (the Jacobians between our arbitrarily-defined axes), we can solve backwards to determine what scale factor l_P must have been embedded when we defined the meter."

The correct ontological order:

1. We define meter, kilogram, second as "independent" (false assumption)
2. This implicitly embeds scale factors l_P, m_P, t_P
3. The Jacobians between these misaligned axes manifest as measurable constants
4. Planck solved: "Given these Jacobians, what were the scale factors?"

The constants don't create Planck units. Both emerge simultaneously from our arbitrary axis definitions.

Error 2: "Different Length Scales in Different Constants"

What I said:

"The length scale in c is different from the length scale in h and G"

Why this is wrong:

When we write:

c = l_P/t_P
h = m_P l_P²/t_P  
G = l_P³/(m_P t_P²)
k_B = m_P l_P²/(t_P² T_P)

The l_P is the same in all equations.
The t_P is the same in all equations.
The m_P is the same in all equations.

There is one set of scale factors. The different constants encode different rotation patterns using those same scales.

This error reveals a failure to understand that constants are combinations of the same underlying scale factors, not independent quantities.

Error 3: "Planck Mass Is a Quantity You Can Count"

What I said:

"M_object / m_P gives you the number of Planck masses"

Why this is wrong:

There is no such thing as "a Planck mass" existing in nature to be counted.

When you divide M_object / m_P, you are:

Canceling out the arbitrary scale factor (m_P) that was embedded when you defined the kilogram in the first place.

You're not counting objects. You're removing a scaling distortion.

The "Planck mass" is not a quantum of mass. It's the scale factor that harmonizes the kilogram to the meter and second, given the arbitrary ratios we chose when defining those units as independent.

Error 4: "Discovery Order = Ontological Priority"

What I repeatedly did:

"We found the constants first, so they must be more fundamental than Planck units"

Why this is wrong:

Historical accident ≠ ontological priority.

Timeline:

• 1600s-1800s: Constants measured (c, G, k_B, h)
• 1899: Planck calculates scale factors

But ontologically:

• The scale factors existed the instant we defined kg, m, s, K
• The constants are mathematical combinations: c = l_P/t_P, etc.
• We discovered the effects (constants) before calculating the causes (embedded scales)

This is like saying shadows are more fundamental than light because we noticed shadows first.

Error 5: "Setting Constants to 1"

What I said:

"In natural units we set c = 1 for convenience"

Why this is wrong:

You cannot "set c = 1" by fiat. That's not a mathematical operation.

What actually happens:

Operation: Set m_P = l_P = t_P = 1 (harmonize measurement axes)
Result: c = l_P/t_P = 1/1 = 1 (Jacobian becomes identity)

The pedagogical phrase "set constants to 1" obscures the actual operation and makes it seem like arbitrary notation rather than revealing substrate structure.

Error 6: "α is a Constant"

What I said:

"Entire research programs ask 'why is α = 1/137?' If you recognize α is just encoding measurement axis misalignment, these programs lose motivation."

Why this is wrong:

α is dimensionless. It has no units. It is NOT a constant in the sense of c, h, G, k_B.

α is a geometric ratio like pi. The question is: a ratio between what two things?

Confusing dimensional constants (which are Jacobians) with dimensionless ratios (which are geometric relationships) is a fundamental category error.

The dimensional constants (c, h, G, k_B) are unit scaling artifacts - they encode how we chose to misalign our measurement axes.

Dimensionless ratios like α are geometric properties - they encode actual substrate structure.

These are completely different kinds of quantities and must not be conflated.

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3. The Correct Mathematical Structure

3.1 What We Actually Did When We Defined Units

When the SI system arbitrarily defined:

• 1 kilogram (originally: mass of platinum-iridium cylinder)
• 1 meter (originally: 1/10,000,000 of pole-to-equator distance)
• 1 second (originally: 1/86,400 of mean solar day)
• 1 Kelvin (originally: 1/273.16 of water triple point)

We made arbitrary choices about:

1. The substances used as references
2. The fractions used for division
3. The assumption these were "independent dimensions"

These choices implicitly embedded scale factors into our measurement system:

m_P = (embedded when we chose kg reference)
l_P = (embedded when we chose m reference)
t_P = (embedded when we chose s reference)
T_P = (embedded when we chose K reference)

3.2 Constants as Jacobian Coefficients

Given these embedded scale factors, the rotation coefficients between axes are:

Speed (length/time Jacobian):

c = l_P/t_P = 299,792,458 m/s

Action (mass·length²/time Jacobian):

h = m_P l_P²/t_P = 6.626×10⁻³⁴ J·s

Gravity (length³/(mass·time²) Jacobian):

G = l_P³/(m_P t_P²) = 6.674×10⁻¹¹ m³/(kg·s²)

Thermodynamic (energy/temperature Jacobian):

k_B = m_P l_P²/(t_P² T_P) = 1.381×10⁻²³ J/K

These are not "properties of nature." They are the numerical values of the rotation matrices between our arbitrarily-chosen measurement axes.

3.3 What Planck Actually Did

Planck took the measured constants and solved the system backwards:

Given:

• c = 299,792,458 m/s
• h = 6.626×10⁻³⁴ J·s
• G = 6.674×10⁻¹¹ m³/(kg·s²)

Solve for:

• l_P = √(hG/c³) ≈ 1.616×10⁻³⁵ m
• t_P = √(hG/c⁵) ≈ 5.391×10⁻⁴⁴ s
• m_P = √(hc/G) ≈ 2.176×10⁻⁸ kg

He extracted the embedded scale factors from the observed Jacobians.

The "Planck units" are not fundamental quanta. They are:

The scale factors implicit in our measurement system, made explicit through algebraic manipulation.

3.4 Natural Units as Axis Harmonization

When we "go to natural units," the operation is:

Set all scale factors equal:

m_P = l_P = t_P = T_P = 1

This means: "Use the same numerical scale for all measurements."

Consequence: All Jacobians become unity:

c = l_P/t_P = 1/1 = 1
h = m_P l_P²/t_P = 1·1²/1 = 1
G = l_P³/(m_P t_P²) = 1³/(1·1²) = 1
k_B = m_P l_P²/(t_P² T_P) = 1·1²/(1²·1) = 1

This is not a trick. This is revealing the substrate.

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4. Physics is Invariant Relationships Between Axes

4.1 The Core Insight

Physical law consists of invariant relationships between measurement axes.

Newton's law: F ~ mM/r²
Einstein's law: E ~ m
Planck's law: E ~ f
De Broglie's law: λ ~ 1/p

These are dimensionless proportionalities in the substrate. They are true regardless of what units you use.

When we express them in a particular unit system, we get:

F = G·mM/r²    (SI units)
E = mc²        (SI units)
E = hf         (SI units)
λ = h/p        (SI units)

The constants (G, c², h) are Jacobian coefficients that appear because our measurement axes are misaligned with the substrate.

4.2 Why Rotation Is Possible

The fact that you can write:

• m = E/c²
• m = h/(λc)
• m = k_BT/c² (thermal systems)

Proves that mass, energy, inverse-length, and temperature are not independent.

They are different projections of the same substrate quantity onto different measurement axes.

If they were truly independent dimensions, rotation would be impossible. You cannot convert meters to kilograms if they're fundamentally different things.

The existence of conversion formulas proves they're the same thing measured differently.

4.3 The Substrate is One Process

In harmonized coordinates (natural units):

T = f = m = 1/λ = E = p = F

All these "different quantities" collapse to one substrate process that we measure along different axes:

• Temperature = substrate process measured via thermal excitation axis
• Frequency = substrate process measured via oscillation axis
• Mass = substrate process measured via inertial resistance axis
• Inverse wavelength = substrate process measured via spatial periodicity axis
• Energy = substrate process measured via work capacity axis
• Momentum = substrate process measured via motion transfer axis
• Force = substrate process measured via acceleration production axis

These are not different kinds of things. They are different measurement perspectives on one process.

The "illusion" (maya) is treating them as fundamentally distinct.

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5. Dimensionless Ratios vs Dimensional Constants: A Critical Distinction

5.1 Two Completely Different Categories

Dimensional Constants (Unit Scaling):

• c, h, G, k_B, e (the elementary charge magnitude)
• Have units (m/s, J·s, m³/(kg·s²), J/K, C)
• Are Jacobian coefficients between measurement axes
• Encode our arbitrary choice of unit scales
• Vanish when axes are harmonized

Dimensionless Ratios (Geometry):

• α (fine structure), mass ratios, geometric factors
• Have no units
• Are substrate properties or geometric relationships
• Encode actual physical structure
• Remain constant regardless of unit choice

Confusing these two categories is a fundamental error.

5.2 The Fine Structure "Constant" α

α ≈ 1/137.036 is not a constant in the sense of c, h, or G.

What α actually is:

• A dimensionless ratio
• A geometric property of electromagnetic interaction
• Independent of unit choice

The question "why is α = 1/137?" is meaningful because α encodes substrate geometry, not coordinate artifacts.

This is fundamentally different from asking "why is c = 299,792,458 m/s?" which is a meaningless question—that value encodes how we defined meters and seconds.

5.3 Mass Ratios as Geometric Properties

Similarly, ratios like:

• m_proton/m_electron ≈ 1836
• m_proton/m_Planck ≈ 10⁻¹⁹

These are dimensionless and therefore encode actual substrate structure, not coordinate choices.

But here's the critical point:

m_proton/m_Planck is NOT asking "why is the proton so light compared to the fundamental quantum of mass."

It's asking: "Why is the ratio between proton mass and the mass scale implied by our length/time/gravity Jacobians so small?"

The "hierarchy problem" dissolves when you realize m_Planck is not a fundamental scale—it's just √(hc/G), which depends on our arbitrary axis choices.

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6. How the Confusion Arose

6.1 Historical Development

Phase 1: Pre-metric chaos (before 1790s)

• Hundreds of local measurement systems
• No standardization
• Obvious that units are human conventions

Phase 2: Metric standardization (1790s-1960s)

• SI system established with artifact standards
• Units seem "natural" because they're universal
• Forgot they're still arbitrary human choices

Phase 3: Constant-based definitions (1960s-2019)

• Meter defined by c
• Still teaching "c is a property of spacetime"
• Not recognizing we're defining units via constants

Phase 4: 2019 SI revision

• All base units now defined by fixing constant values
• c, h, e, k_B fixed by committee vote
• Operationally treating constants as coordinate artifacts
• Still teaching them as fundamental properties

We are literally doing what the correct framework says, while teaching the backwards story.

6.2 Why Physicists Teach It Backwards

Reason 1: Discovery order bias

We measured constants before calculating Planck units, so we treat constants as ontologically prior.

Reason 2: Dimensional analysis pedagogy

Students learn "dimensions are fundamental categories" (length, mass, time are "different kinds of things"). This makes constants seem necessary to bridge fundamentally different domains.

Reason 3: Lack of categorical thinking

Without fibration theory and coordinate geometry, physicists couldn't formalize what "axis harmonization" means mathematically. So they described the observable effect ("constants become 1") rather than the underlying operation ("axes harmonized").

Reason 4: Planck scale mysticism

"Planck length is where quantum gravity becomes important" sounds profound. "Planck length is the scale factor embedded when we defined the meter" sounds trivial. The mystical interpretation is more appealing.

Reason 5: Confusing dimensional and dimensionless

By treating α (dimensionless) and c (dimensional) as both "fundamental constants," physics obscures the critical distinction between geometry and coordinate artifacts.

6.3 The Self-Reinforcing Error

The error is self-reinforcing:

1. Teach "constants are fundamental"
2. Students think dimensionality is ontological
3. They teach the same to next generation
4. Literature accumulates treating constants as mysteries
5. Metrology community knows better but doesn't correct physics
6. Physics community ignores metrology as "applied" not "fundamental"

Result: 100+ years of inverted pedagogy.

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7. Correcting the Pedagogy

7.1 What Should Be Taught

Correct pedagogical sequence:

Step 1: Substrate unity

"Physical reality consists of invariant relationships: F ~ mM/r², E ~ m, E ~ f, etc. These are dimensionless proportionalities."

Step 2: Measurement decomposition

"We decompose this unity by defining separate measurement scales (kg, m, s, K) as if they were independent. This embeds arbitrary scale factors."

Step 3: Constants as Jacobians

"Because our axes are misaligned, we need rotation coefficients (c, h, G, k_B) to convert between measurements. These are dimensional constants—unit scaling artifacts."

Step 4: Natural units as harmonization

"When we harmonize our axes (set m_P = l_P = t_P = 1), the Jacobians become unity. This reveals the substrate relationships directly."

Step 5: Dimensionless ratios encode geometry

"Quantities like α, mass ratios, and geometric factors are dimensionless. These encode actual substrate structure and remain invariant across all unit systems. These are the true 'constants' of nature."

7.2 Rewriting the Textbook

Current textbook approach:

"Natural units: Set c = h = G = k_B = 1 for convenience."

Corrected textbook approach:

"Natural units: Harmonize measurement axes by setting the scale for mass, length, time, and temperature to unity. This eliminates the Jacobian rotation coefficients (dimensional constants) that arose from treating these axes as independent. Dimensionless ratios remain unchanged."

Current dimensional analysis:

"Mass, length, and time are independent dimensions. Constants have dimensions that make equations consistent."

Corrected dimensional analysis:

"Mass, length, and time are projections of a unified substrate onto separate measurement axes. Dimensional constants are the Jacobian coefficients needed to rotate between these projections. Dimensionless ratios reveal substrate geometry."

7.3 The New Research Questions

Old questions (based on inverted understanding):

• Why is c = 299,792,458 m/s? (meaningless—encodes our meter/second choice)
• Why is the proton/Planck mass ratio so small? (confused—m_Planck is not fundamental)
• What is "Planck scale physics"? (misguided—Planck scale is coordinate artifact)

New questions (based on correct understanding):

• What is the natural basis of measurement space?
• Why are our conventional axes maximally misaligned with it?
• What substrate relationships generate the observed Jacobian structure?
• What do dimensionless ratios (α, mass ratios) reveal about substrate geometry?

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8. Implications for Current Physics

8.1 The Hierarchy Problem Dissolves

The "hierarchy problem" asks: "Why is gravity so much weaker than electromagnetism?"

Quantitatively: Why is m_proton/m_Planck ≈ 10⁻¹⁹ so small?

The error: Treating this as a question about nature.

The reality: This ratio expresses how misaligned our arbitrary kilogram definition is from the mass scale that harmonizes with our meter and second definitions.

The hierarchy is not in nature. It's in our choice of measurement axes.

8.2 What About α?

α ≈ 1/137 is a legitimate mystery because it's dimensionless.

But it's not a mystery about "fundamental constants"—it's a question about substrate geometry.

The right question is: What geometric structure produces this coupling ratio?

This is completely different from asking about c, h, or G values, which are just coordinate artifacts.

8.3 Quantum Gravity

Much of quantum gravity research assumes "Planck scale is where quantum effects meet gravity effects."

But if Planck units are just our embedded scale factors made explicit, then:

There is no special "Planck scale physics."

There's just: the scale at which our measurement axis choices happen to align with substrate structure.

This doesn't mean quantum gravity is trivial—but it suggests we're looking in the wrong place for the wrong reasons.

8.4 Unification Programs

Grand Unified Theories try to unify coupling constants at high energies.

But if dimensional constants are Jacobians, then:

Unification is about finding the natural coordinate basis, not about high-energy behavior.

The substrate is already unified. The dimensional constants are artifacts of our coordinate choice, not fundamental forces that need unifying.

Dimensionless couplings (like α) may unify, but that's a different kind of question.

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9. The Metrology-Physics Divide

9.1 What Metrologists Know

The metrology community understands:

• Units are human conventions
• Constants can be defined by committee vote (2019 SI revision)
• Measurement is about establishing coherent scales

They know dimensional constants are coordinate artifacts.

9.2 What Physicists Teach

The physics community teaches:

• Constants are fundamental properties of nature
• Planck units represent quantum gravity scale
• Natural units are a calculational trick

They treat dimensional constants as ontological.

9.3 Why the Disconnect?

Sociological reasons:

• Metrology is "applied," physics is "fundamental"
• Metrologists attend different conferences
• Different publication venues
• Different educational tracks

Conceptual reasons:

• Metrologists work with operational definitions
• Physicists work with theoretical models
• Each group's framework seems irrelevant to the other

Result: The group that knows constants are artifacts (metrologists) doesn't talk to the group that theorizes about them (physicists).

9.4 The 2019 SI Revision Proves It

In 2019, SI system redefined base units by:

• Fixing c = 299,792,458 m/s exactly (by definition)
• Fixing h = 6.62607015×10⁻³⁴ J·s exactly (by definition)
• Fixing e = 1.602176634×10⁻¹⁹ C exactly (by definition)
• Fixing k_B = 1.380649×10⁻²³ J/K exactly (by definition)

These values were chosen by committee vote.

If constants were properties of nature, you couldn't vote on their values.

The metrology community did exactly what the correct framework says: they operationally defined measurement axes by fixing the Jacobians between them.

Physics still teaches these are "measured fundamental constants."

We're doing the right thing while teaching the wrong story.

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10. Conclusion

10.1 Summary of Errors

The systematic inversion in physics education:

Error 1: Teaching "set constants to 1" instead of "harmonize measurement axes"

Error 2: Treating constants as ontologically prior to Planck units

Error 3: Believing Planck units are fundamental quanta rather than embedded scale factors

Error 4: Treating dimensions as ontological categories rather than measurement projections

Error 5: Missing that physics is about invariant relationships between axes, not dimension-dependent laws

Error 6: Letting discovery order determine ontological priority

Error 7: Confusing dimensional constants (Jacobians) with dimensionless ratios (geometry)

Error 8: Treating the 2019 SI revision as technical rather than philosophically significant

10.2 The Correct Framework

Natural units work by:

1. Recognizing that substrate physics is dimensionless relationships
2. Harmonizing measurement axes (m_P = l_P = t_P = T_P = 1)
3. Revealing that dimensional constants are Jacobian coefficients
4. Understanding these Jacobians as rotation matrices between our arbitrary axes
5. Distinguishing dimensional constants (coordinate artifacts) from dimensionless ratios (geometry)
6. Seeing that unified physics requires coordinate-free formulation

Dimensional constants are not fundamental. They are artifacts of coordinate choice.

Dimensionless ratios encode substrate geometry and remain invariant.

10.3 Why This Matters

This isn't just pedagogical tidiness. The inversion causes:

Research misdirection:

• Hierarchy problem as pseudoproblem
• Planck scale physics as misplaced focus
• Confusing questions about coordinate artifacts with questions about geometry

Conceptual confusion:

• Inability to understand measurement's role
• Reifying SI unit chart (substrate reality)
• Missing the unity beneath apparent multiplicity
• Conflating unit scaling with geometric structure

Educational failure:

• Students never understand what natural units reveal
• Dimensional analysis taught as bookkeeping rather than geometry
• Constants mystified rather than demystified
• Dimensionless ratios not distinguished from dimensional constants

10.4 The Path Forward

For education:

• Rewrite textbooks starting from substrate unity
• Teach dimensional analysis as revealing axis relationships
• Show constants as Jacobians explicitly
• Distinguish dimensional constants from dimensionless ratios
• Explain 2019 SI revision correctly

For research:

• Reformulate theories in coordinate-free language
• Investigate substrate basis structure
• Stop treating dimensional constants as fundamental parameters
• Focus on dimensionless ratios as encoding actual physics
• Recognize unification is about finding natural coordinates

For philosophy:

• Understand maya as measurement process (literally, not metaphorical)
• Recognize Vedantic framework was correct
• See that ancient and modern physics converged on same truth
• Unite metrology and metaphysics

10.5 Final Statement

Natural units are not about setting constants to 1.

Natural units are about harmonizing measurement axes to reveal the dimensionless substrate relationships that constitute actual physical law.

Dimensional constants do not define Planck units. Both emerge simultaneously from our arbitrary choice to treat unified substrate as independent measurement axes. 

The order things were found is not important.  What they really mean is what is important.

Physics is the study of invariant relationships between axes of measurement. Dimensional constants are coordinate artifacts—maya. Dimensionless ratios encode geometry—actual substrate structure.

The dog has been chasing its tail for over a century.

It's time to stop running in circles.

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References

Historical Sources:

• Planck, M. (1899) "Über irreversible Strahlungsvorgänge"
• BIPM (2019) "The International System of Units (SI), 9th edition"

Measurement Theory:

• Rogers, J. (2025) "The Structure of Physical Law as a Grothendieck Fibration"
• Rogers, J. (2025) "Maya Is Literally Measurement, Not a Metaphor"

Dimensional Analysis:

• Buckingham, E. (1914) "On Physically Similar Systems"
• Bridgman, P.W. (1922) "Dimensional Analysis"

Philosophy:

• Śaṅkara (8th century CE) Commentaries on Upaniṣads
• Wigner, E. (1960) "The Unreasonable Effectiveness of Mathematics"

Computational Verification:

• Rogers, J. (2025) Physics Unit Coordinate System (GitHub implementation)

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Appendix A: The Actual Mathematics

A.1 Constants as Jacobian Components

Given arbitrary scale factors embedded in unit definitions:

• m_P (kilogram scale factor)
• l_P (meter scale factor)
• t_P (second scale factor)
• T_P (Kelvin scale factor)

The Jacobian coefficients between axes are:

c = l_P/t_P                    [length/time]
h = m_P l_P²/t_P               [action]
G = l_P³/(m_P t_P²)            [gravitational]
k_B = m_P l_P²/(t_P² T_P)      [thermodynamic]

These are not fundamental constants. They are functions of the embedded scale factors.

A.2 Solving for Scale Factors (What Planck Did)

Given measured constants c, h, G, solve for scale factors:

l_P = √(hG/c³)
t_P = √(hG/c⁵)
m_P = √(hc/G)
T_P = √(hc⁵/(G k_B²))

This is algebraic manipulation, not discovery of fundamental quanta.

A.3 Harmonization Operation

Set all scale factors to unity:

m_P = l_P = t_P = T_P = 1

Result:

c = 1/1 = 1
h = 1·1²/1 = 1
G = 1³/(1·1²) = 1
k_B = 1·1²/(1²·1) = 1

This is the harmonization operation. Constants becoming 1 is the consequence.

A.4 Substrate Relationships (Invariant Physics)

In harmonized coordinates, physical laws become:

F = mM/r²           (Newton, no G needed)
E = m               (Einstein, no c² needed)
E = f               (Planck, no h needed)
λ = 1/p             (de Broglie, no h needed)
T = 1/M             (Hawking, no c³h/(G k_B) needed)

These are the actual laws. The complex forms with constants are coordinate-dependent expressions.

A.5 Dimensionless Ratios (Invariant Across All Coordinates)

These remain the same regardless of unit choice:

α ≈ 1/137.036       (fine structure)
m_p/m_e ≈ 1836      (proton/electron mass ratio)
cos θ_W ≈ 0.88      (weak mixing angle)

These encode substrate geometry, not coordinate choices.

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Appendix B: Common Objections

Objection 1: "But we measure c in experiments. How can it be conventional?"

Response: Wrong. We don't measure c anymore. In 1983, c was defined to be exactly 299,792,458 m/s by committee vote. In 2019, h, e, and k_B were similarly fixed by definition. These are no longer measured quantities—they are the values we chose to define our unit system. The framework presented in this paper is not theoretical speculation; it is operationally how the SI system works right now. Metrologists understand this. Physics educators apparently don't.

Objection 2: "Planck length is where quantum gravity becomes important."

Response: This assumes Planck units are physically special scales. They're not. They're just the scale factors that fall out when you solve for what values would make c, h, and G equal to 1 given our arbitrary meter, kilogram, and second definitions. The fact that this happens to be around 10⁻³⁵ m is a consequence of how we chose to scale our units, not a fundamental feature of spacetime. Quantum gravity becomes important at whatever scale substrate structure dictates, which may or may not align with our coordinate artifacts.

Objection 3: "This makes physics seem arbitrary."

Response: No. The substrate relationships (F ~ mM/r², E ~ m, E ~ f) are not arbitrary—they're invariant across all coordinate systems. What's arbitrary is our choice of measurement axes. Recognizing this distinction is crucial. The confusion arises from mistaking coordinate-dependent expressions (like F = G mM/r² in SI units) for the underlying coordinate-free relationships (like F ~ mM/r² in the substrate).

Objection 4: "The 2019 SI revision still calls them constants."

Response: Yes, because the metrology community hasn't corrected the physics community's language. The 2019 revision operationally treats c, h, e, k_B as defining the unit system (correct), while the terminology still calls them "fundamental constants" (incorrect framing). This is the disconnect: we're doing the right thing (fixing Jacobians to define axes) while teaching the wrong story (constants are properties of nature).

Objection 5: "This undermines the search for unified theories."

Response: No, it redirects it. Instead of trying to "unify fundamental forces" by making dimensional constants match at high energies, we should seek the natural coordinate basis of measurement space.  The ratios between them always match in natural ratios.  You do not have to force the universe to unify, it is always unified.  The substrate is already unified—our description is fragmented by our axis choices. Unification means finding coordinate-free formulations, not tweaking energy scales. Dimensionless ratios like α may reveal unification structure, but that's geometry, not coordinate artifacts.

Objection 6: "What about α = 1/137? That's dimensionless, so it must be fundamental."

Response: Correct. α is dimensionless, which means it encodes actual substrate geometry between space time and charge, not coordinate artifacts. This is a completely different category from dimensional constants like c, h, G, k_B. The question "why is α ≈ 1/137?" is legitimate because it's asking about geometric structure. The question "why is c = 299,792,458 m/s?" is meaningless because that value was chosen by committee vote in 1983. Don't confuse these two types of quantities.

Objection 7: "If Planck units aren't fundamental, why do they keep appearing in fundamental physics?"

Response: Because they represent the scale where our misaligned conventional axes happen to approximately align with substrate structure. Think of it this way: if you choose arbitrary x and y axes at a 45° angle to some natural grid, you'll need √2 factors everywhere. When you "discover" that setting your axes to match the natural grid eliminates those factors, that doesn't mean "√2 is the fundamental scale"—it means your original axes were misaligned. Similarly, Planck units appearing everywhere just means our kg/m/s choices are misaligned with substrate geometry.

Objection 8: "This is just philosophy, not physics."

Response: The 2019 SI revision proves otherwise. The metrology community operationally implemented this framework when they defined units by fixing constant values through committee vote. This is not philosophical speculation—this is how measurement actually works as of 2019. The fact that physics educators haven't updated their conceptual framework to match operational reality is the problem, not the framework itself.