Dimensional Analysis as Unit Chart Scaling

J. Rogers, SE Ohio

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Abstract

Standard dimensional analysis assumes base units—Length (L), Mass (M), and Time (T)—are independent abstract categories. This paper demonstrates this framework is ontologically inverted. Physical reality consists of dimensionless natural ratios that are identical in every possible unit chart. The SI unit chart and Planck scale are the same coordinate system. The Planck units are just the scaling of that one scale to qll our differ perceptual channels. Constants (c, G, h) encode how our arbitrary measurement axes relate to each other. Dimensional analysis verifies that coordinate-dependent expressions preserve the coordinate-independent natural ratios.

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1. The Simple Truth

1.1 There Are Only Two Things

Physical Reality:

• Dimensionless natural ratios

• Single physical scale

• Example: v/c = 0.5

• This ratio is identical in SI, Planck, Rogers, and every possible unit chart

• This invariance proves it's the actual physics

Measurement of Perception (The SI Unit Chart):

• We project natural ratios onto arbitrary scales

• v_SI = 0.5 × c_SI

• v_Planck = 0.5 × c_Planck

• The 0.5 natural ratio never changes

• The c values change (because they're coordinate-dependent)

1.2 The Ontological Inversion

Standard framework claims:

• Mass, length, time are fundamentally independent

• Constants relate these independent dimensions

• Dimensional analysis balances abstract categories

Reality:

• There is one unified physical scale

• "Mass," "length," "time" are the same thing viewed from different perceptual channels

• These angles are illusions of our senses (we evolved them)

• Constants rotate between our evolved perceptual axes

• Dimensional analysis verifies coordinate transformations preserve invariants

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2. Natural Ratios: The Invariant Reality

2.1 What a Natural Ratio Is

Example: Velocity

Physical reality:

    v/c = 0.5

This ratio is the same in:

• SI units: (1.499×10⁸ m/s) / (2.998×10⁸ m/s) = 0.5

• Planck units: (0.5) / (1) = 0.5

• Rogers units: (5×10⁹ m_r/s_r) / (1×10¹⁰ m_r/s_r) = 0.5

The invariance proves this ratio is the physics.

2.2 The Sameness Is The Point

Not: "Some dimensionless ratio that varies by context"

Yes: "THE invariant ratio that is identical across all possible unit charts"

This sameness is what makes it real:

• Coordinate-independent = physical

• Coordinate-dependent = measurement artifact

2.3 All Physics Is Natural Ratios

Kepler's Third Law:

    t/t_ref = (r/r_ref)^1.5 × (m_ref/m)^0.5

This ratio relation is identical in every unit chart.

Einstein's Mass-Energy:

    E/(mc²) = 1

This ratio is identical in every unit chart.

The physics is the invariant ratio structure.

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3. The SI Unit Chart = Planck Scale

3.1 One Chart, Not Two

WRONG: "SI is human scale, Planck is nature's scale"

RIGHT: "SI chart and Planck scale are the exact same coordinate system"
The Planck units are inside every unit in our SI chart

3.2 Planck "Units" Are SI Values

1899 definition with c, h, G

Planck "units" are specific numerical values on the SI chart:

• l_P = 4.051 × 10⁻³⁵ meters (SI unit)

• m_P = 5.455 × 10⁻⁸ kilograms (SI unit)

• t_P = 1.351 × 10⁻⁴³ seconds (SI unit)

Derived from SI constants:

m_P = √(hc/G)
t_P = h/(c^2 m_P)
l_P = c t_P

Where h, G, c are SI-defined constants.

3.3 What This Means

The SI unit chart IS the Planck scale.

When you write "1 kilogram," you're already on the Planck scale chart. The base SI units are defined by their relationship to Planck values through the constants. The following is the defintion of the constants by the SI unit scales:

c   =     l_P   /  t_P
h   = m_P l_P^2 /  t_P
k_B = m_P l_P^2 / (t_P^2 T_P)
G   =     l_P^3 / (t_P^2 m_P)

And yes, this is a tautology, that is the point.

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4. Constants: Rotations Between perceptual Axes

4.1 What Constants Actually Encode

The speed of light:

c = 299,792,458 m/s (in SI)
c = 1 (in Planck units)
c = 1×10¹⁰ m_r/s_r (in Rogers units)

c is coordinate-dependent. It changes with unit chart.

But the natural ratio v/c is coordinate-independent. It never changes.

4.2 Constants Are Unit Chart Structure

Constants show how our arbitrary measurement axes relate to each other:

c: How the meter-axis relates to the second-axis
G: How the meter³-axis relates to kilogram-axis and second²-axis
h: How the joule-axis relates to the second-axis

They're not properties of nature. They're properties of our coordinate chart.

4.3 Proof: Change the Chart

SI chart: c = 2.998×10⁸ m/s
Planck chart: c = 1
Rogers chart: c = 1×10¹⁰ m_r/s_r

Same physical reality. Different coordinate expressions.

The constant changes → it's coordinate-dependent → it's not fundamental.

The natural ratio stays same → it's coordinate-independent → it's the physics.

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5. The Illusion of Independent Dimensions

5.1 Same Thing, Different Viewing Angles

Physical reality: One unified thing

We measure it from different perceptual axis:

• View through "mass axis" → appears as mass

• View through "length axis" → appears as length

• View through "time axis" → appears as time

Same thing. Different projections.

5.2 The Projection Formula

To express the natural ratio in SI measurements:

Mass_in_kg       = (natural_ratio) × m_P
Length_in_meters = (natural_ratio) × l_P
Time_in_seconds  = (natural_ratio) × t_P

SAME natural ratio.
Different projection axes.

5.3 Where The Illusion Comes From

We invented "mass," "length," and "time" as if they're fundamentally different things.

They're not different.

They're the same unified thing, viewed through different measurement projections we created.

The independence of dimensions is our perceptual illusion.

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6. Dimensional Analysis: Verifying Invariant Preservation

6.1 What Dimensional Analysis Actually Does

NOT: "Balancing abstract dimension categories"

YES: "Verifying that coordinate transformations preserve the coordinate-independent natural ratios"

6.2 Example: Velocity Formula

Start with invariant natural ratio:

    v/c = 0.5

Project to SI:

v_SI = 0.5 × c_SI
v_SI = 0.5 × (2.998×10⁸ m/s)
v_SI = 1.499×10⁸ m/s

Dimensional check:

    [v] = [m/s] ✓

6.3 What Just Happened

1. Started with coordinate-independent ratio: v/c = 0.5

2. Multiplied by coordinate-dependent constant: c_SI

3. Got coordinate-dependent measurement: v_SI

Dimensional analysis verified: Does v have dimensions [m/s]? Yes.

What we actually checked: Is the natural ratio preserved through the coordinate transformation? Yes.

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7. The Tautology: kg = √(hc/G)

7.1 Expanding the Dimensions

    kg = √(hc/G)

Substitute dimensional values:

[kg] = √([J·s][m/s] / [m³/(kg·s²)])
[kg] = √([kg·m²/s][m/s] / [m³/(kg·s²)])
[kg] = √(kg²)
[kg] = kg

7.2 What This Reveals

This is a tautology: kg = kg

It proves: The constants encode nothing but the internal consistency of our unit chart.

If constants were "fundamental properties of nature," this would be a profound equation.

Since it's just kg = kg, constants are coordinate artifacts.

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8. Examples of Invariant Natural Ratios

8.1 Energy-Mass Equivalence

Natural ratio (coordinate-independent):

    E/(mc²) = 1

This ratio is identical in:

• SI: (9×10¹⁶ J) / ((1 kg)(3×10⁸ m/s)²) = 1

• Planck: (1 E_P) / ((1 m_P)(1)²) = 1

• Any chart: always 1

Coordinate-dependent SI form:

    E = mc²

c² appears because we're rotating from mass-projection to energy-projection on our SI chart.

8.2 Newton's Gravitational Law

Natural ratio (coordinate-independent):

    F × r² / (M₁M₂) = constant_ratio

Coordinate-dependent SI form:

    F = G × M₁M₂/r²

G appears because we're converting between force-projection, mass-projection, and distance-projection on our SI chart.

8.3 The Pattern

Every "law of physics" is:

1. A coordinate-independent natural ratio (the actual physics)

2. Dressed in coordinate-dependent constants (the measurement apparatus)

Constants appear when we project natural ratios onto our arbitrary unit chart.

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9. What Dimensional Analysis Really Checks

9.1 Not Balancing Categories

Standard view: "Check that [L], [M], [T] balance on both sides"

Actual function: "Verify that the coordinate transformation preserves the natural ratio"

9.2 The Verification Process

Step 1: Start with natural ratio (coordinate-independent)

Step 2: Apply coordinate transformation (multiply by constants/Planck values)

Step 3: Verify dimensions match in target coordinate system

Step 4: If dimensions match, natural ratio was preserved ✓

9.3 Why It Works

Dimensional analysis works because:

• Natural ratios are invariant (same in all charts)

• Constants correctly encode coordinate transformations

• If dimensions balance, the transformation preserved the invariant

Not because "dimensions must balance."

Because coordinate transformations must preserve coordinate-independent reality.

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10. The Corrected Framework

10.1 Two Layers

Layer 1: Physical Reality

• Dimensionless natural ratios

• v/c, E/(mc²), F×r²/(M₁M₂), etc.

• Identical in every possible unit chart

• Coordinate-independent

• This is what exists

Layer 2: Measurement (SI Unit Chart)

• Projection onto arbitrary axes (mass, length, time)

• Constants encode rotations between axes

• Planck values are specific SI values where constants = 1

• Coordinate-dependent

• This is how we measure what exists

10.2 The Relationship

Natural ratio → multiply by constants → SI measurement

Natural: v/c = 0.5

↓ (multiply by c_SI)
SI: v = 1.499×10⁸ m/s

The 0.5 is real (invariant across all charts)
The 1.499×10⁸ m/s is our SI projection (changes with chart choice)

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11. Why This Matters

11.1 Constants Are Not Mysterious

Question: "Why does c = 299,792,458 m/s?"

Wrong answer: "That's the fundamental speed limit"

Right answer: "Because we defined the meter and second with that ratio in 2019"

Evidence: Change the unit chart → c changes value → not fundamental

11.2 Dimensions Are Not Independent

Question: "Why are mass, length, time independent?"

Wrong answer: "They're fundamentally different categories"

Right answer: "They're not independent. They're the same thing viewed from different angles we invented"

Evidence: Same natural ratio projects onto all "dimensions"

11.3 Dimensional Analysis Is Geometry

Question: "Why does dimensional analysis work?"

Wrong answer: "Because dimensions must balance"

Right answer: "Because our coordinate transformations (constants) correctly preserve coordinate-independent natural ratios"

Evidence: The math checks that invariants are preserved

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12. The Ultimate Simplification

12.1 The Standard Framework

Claims:

• Multiple independent dimensions exist

• Constants are fundamental properties

• Dimensional analysis balances categories

• Planck scale is special

• Complex mathematical structure

12.2 The Actual Reality

Truth:

• One unified thing exists

• One physical scale

• Natural ratios describe it

• "Dimensions" are viewing angles we invented

• Constants rotate between our angles

• Dimensional analysis preserves invariants

• Dead simple

12.3 Why Standard Framework Is Complicated

Because it's wrong.

When you build a framework on false ontology (independent dimensions), you need elaborate structures to make it work.

When you build on correct ontology (unified reality + measurement projections), it's trivial.

Complexity is the bug, not the feature.

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13. Conclusion

13.1 The Framework

Physical reality:

• Dimensionless natural ratios

• v/c = 0.5 is identical in every possible unit chart

• This invariance proves it's the physics

Measurement:

• SI unit chart = Planck scale (same thing)

• Constants encode coordinate transformations

• "Dimensions" are imaginary viewing angles

Dimensional analysis:

• Verifies coordinate transformations preserve invariants

• Not balancing abstract categories

• Checking geometry of coordinate rotations

13.2 The Key Insight

The natural ratio v/c = 0.5 is the same in:

• SI units

• Planck units

• Rogers units

• Every possible unit chart ever conceived

• All of them. Without exception.

This sameness IS the physics.

Everything else is measurement apparatus.

13.3 The Simplicity

There's one thing.

We measure it from different angles.

Constants rotate between the angles.

That's all dimensional analysis is.

Dead simple.

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Appendix: The Invariance Proof

The Critical Demonstration

Take any natural ratio, express it in different unit charts:

SI:

    v/c = (1.5×10⁸ m/s) / (3×10⁸ m/s) = 0.5

Planck:

    v/c = (0.5) / (1) = 0.5

Rogers:

    v/c = (5×10⁹ m_r/s_r) / (10¹⁰ m_r/s_r) = 0.5

The ratio is IDENTICAL.

The numerical values change.

The ratio doesn't.

The ratio is the reality.

The numerical values are the coordinate projection.

QED.

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End of Paper