Politics, Power, and Science: The Categorical Distinction Between Jacobian Constants and Geometric Invariants

Tuesday, December 16, 2025

The Categorical Distinction Between Jacobian Constants and Geometric Invariants

Why You Can Set c=h=G=1 But Not α=π=m_p/m_e=1

J. Rogers, SE Ohio
December 2025

Abstract

We provide a rigorous categorical framework that distinguishes two classes of "constants" in physics: (1) dimensional Jacobian coefficients that encode coordinate transformations between measurement charts, and (2) dimensionless geometric invariants that represent substrate relationships independent of any coordinate choice. This distinction, while intuitively recognized by physicists who "set c=h=G=1 for convenience," has never been formally articulated in the literature. We show that the practice of natural units is not merely computational convenience but represents a specific coordinate chart selection in a fibered category structure. The framework resolves the apparent paradox of why ℏ cannot be independently set to 1 when h=1: ℏ=h/(2π) makes it dependent on both a Jacobian constant (h) and a geometric invariant (π), revealing that the "reduced" formulation obscures the natural coordinate structure.

1. The Unarticulated Distinction

1.1 The Standard Practice

In modern theoretical physics, it is common practice to "set constants to 1":

High-energy physics: ℏ = c = 1
General relativity: G = c = 1
Planck units: h = c = G = k_B = 1

This is justified as "choosing units for convenience" or "working in natural units." Yet physicists never attempt to set:

α (fine structure constant) = 1
π = 1
m_p/m_e (proton-to-electron mass ratio) = 1

When asked why, the typical response is: "Those are different. Those are real physics."

1.2 The Problem

No formal framework exists in standard physics to articulate this distinction.

Physicists know intuitively that c, h, G are "different" from α, π, mass ratios—but:

They cannot formally define what makes them different
They cannot explain what operation "setting to 1" actually performs
They cannot rigorously justify why one class can be modified and the other cannot
They have no mathematical structure that separates these two categories

This paper provides that structure.

2. The Categorical Framework

2.1 Measurement as Fibered Projection

We model physical measurement as a Grothendieck fibration:

π : 𝓔 → 𝓑

Where:

𝓑 = Base category of dimensionless conceptual axes (Time, Mass, Length, Temperature, etc.)
𝓔 = Total category of measured quantities (numbers with units)
π = Fibration functor mapping each measurement to its conceptual type

Each fiber π⁻¹(X) over an object X ∈ 𝓑 contains all possible unit-chart realizations of that measurement type.

Example: The fiber over "Mass" contains:

(1, kg) in SI coordinates
(1000, g) in CGS coordinates
(1, m_P) in Planck coordinates
(0.001, tonne) in metric tons
etc.

2.2 Physical Law as Cartesian Lifting

A physical law is a Cartesian lifting of a morphism in 𝓑.

Given a substrate relationship φ : X → Y in 𝓑 (example: Mass → Energy), a physical law is:

f : (x, U₁) → (y, U₂) in 𝓔

such that π(f) = φ and f is universal over φ.

Example: The substrate says E ∝ m (Energy is proportional to Mass).

In different fibers (unit charts):

Planck chart: E = m (direct proportionality, axes aligned)
SI chart: E = m·c² (needs Jacobian coefficient c² for axis misalignment)

The constant c² is the cocycle data ensuring coherence of the lifting across different unit charts.

3. The Two Classes of Constants

3.1 Class 1: Jacobian Coefficients (Can Be Set to 1)

Definition: Dimensional constants that appear as cocycle data in the fibration π : 𝓔 → 𝓑. These encode the transformation coefficients between different coordinate charts (unit systems).

Properties:

Have physical dimensions (transform under unit changes)
Are artifacts of coordinate choice
Encode axis misalignment between measurement charts
Can be "set to 1" by choosing the Planck coordinate chart

Examples:

Constant	Dimensions	Geometric Meaning
c	[LT⁻¹]	Space/Time axis exchange rate
h	[ML²T⁻¹]	Action/Frequency axis exchange rate
G	[L³M⁻¹T⁻²]	Mass/Curvature axis exchange rate
k_B	[ML²T⁻²Θ⁻¹]	Temperature/Energy axis exchange rate

What "Setting to 1" Means:

When we "set c=1," we are choosing coordinates where:

1 unit of Space = 1 unit of Time (no axis rotation needed)
The Space and Time axes maintain their natural 1:1 ratio

This is a coordinate chart selection, not a change in physics.

3.2 Class 2: Geometric Invariants (Cannot Be Set to 1)

Definition: Dimensionless ratios that represent morphisms in the base category 𝓑 itself. These are coordinate-free geometric truths about the substrate.

Properties:

Are dimensionless (don't transform under unit changes)
Are properties of substrate geometry
Are the same in every fiber (every unit chart)
Cannot be changed by coordinate choice

Examples:

Constant	Value	Geometric Meaning
α	~1/137.036	Electromagnetic coupling strength
π	~3.14159	Circle circumference/diameter ratio
m_p/m_e	~1836.15	Proton/electron mass ratio
m_p/m_μ	~8.88	Proton/muon mass ratio

Why You Cannot Set These to 1:

These are substrate relationships—geometric truths that hold in all coordinate charts. Setting α=1 would not be "choosing coordinates," it would be changing the substrate geometry itself (making electromagnetism infinitely strong).

4. The Formal Distinction

4.1 Location in the Category Structure

Jacobian Constants (Class 1):

Live as cocycle data in the fibration structure
Mediate transitions between fibers (unit chart transformations)
Are properties of the projection π, not of 𝓑 or individual fibers

Geometric Invariants (Class 2):

Live as morphisms in 𝓑 (the base category)
Are fiber-independent (same in all unit charts)
Are properties of the substrate geometry 𝒮ᵤ

4.2 Transformation Behavior

Under change of unit chart U₁ → U₂:

Class 1 (Jacobian) constants:

Transform: c_SI ≠ c_CGS ≠ c_Planck (different numerical values)
In Planck chart: All equal 1 (axes aligned with natural ratios)

Class 2 (Geometric) constants:

Invariant: α_SI = α_CGS = α_Planck = 1/137.036
Cannot be made equal to 1 by any coordinate choice

4.3 Dimensional Analysis

Class 1: Have dimensions → Are chart-dependent → Are Jacobian coefficients

Class 2: Dimensionless → Are chart-independent → Are substrate geometry

This is the formal criterion:

If it has dimensions, it's a Jacobian. If it's dimensionless, it's substrate geometry.

5. The ℏ Problem: Why Reduced Planck Constant Cannot Be Set to 1

5.1 The Composition Structure

The reduced Planck constant is defined as:

ℏ = h/(2π)

This is a composition of:

h: A Jacobian coefficient (Class 1)
2π: Contains π, a geometric invariant (Class 2)

5.2 The Contradiction

Claim: "Set ℏ=1 for convenience"

Analysis: If ℏ = h/(2π) = 1, then h = 2π.

But:

Setting h=1 means choosing Planck coordinates (valid coordinate chart selection)
Setting π=1 means changing substrate geometry (impossible)

You cannot have both h=1 and ℏ=1 simultaneously because:

h=1 and ℏ=1 → 1 = 1/(2π) → π = 1/2 (contradiction!)

5.3 The Natural Units Are h-Based, Not ℏ-Based

The actual Planck coordinate chart is defined by:

l_P = √(hG/c³)
t_P = √(hG/c⁵)
m_P = √(hc/G)
T_P = √(hc⁵/Gk_B²)

Note: These use h, not ℏ.

In true Planck coordinates:

h = 1 (Jacobian set by coordinate choice)
ℏ = 1/(2π) ≈ 0.159 (contains geometric invariant π)

The "reduced" formulation obscures the natural coordinate structure.

5.4 Why Physicists Use ℏ Anyway

The factor 2π appears frequently in quantum mechanics:

Energy eigenvalues: E_n = nℏω (not n(h/2π)ω)
Commutation relations: [x,p] = iℏ (not ih/2π)

Using ℏ absorbs these factors for notational convenience. But this is computational convenience, not fundamental structure.

The price: You lose the clean separation between Jacobian coefficients and geometric invariants.

6. Why This Distinction Matters

6.1 Conceptual Clarity

Without this framework:

Students are told "set constants to 1" with no explanation of what this means
The difference between c and α is intuitive but not rigorous
Natural units appear as magic rather than coordinate geometry

With this framework:

"Setting constants to 1" = Choosing Planck coordinate chart
Class 1 vs Class 2 = Jacobian coefficients vs substrate geometry
Natural units = The coordinate chart where measurement axes preserve their natural 1:1 ratios

6.2 Pedagogical Implications

Current Teaching:

"These constants can be set to 1 for convenience"
"These other constants are fundamental"
No formal distinction provided

Correct Teaching:

"Dimensional constants are Jacobian coefficients encoding axis misalignment"
"Dimensionless ratios are substrate geometry"
"Natural units align axes with substrate, eliminating Jacobian factors"

6.3 Research Implications

Understanding this distinction reveals:

All "fundamental constants with dimensions" are measurement artifacts
The search for "why c has this value" is meaningless (it's a Jacobian)
The search for "why α ≈ 1/137" is meaningful (it's substrate geometry)
Dimensional analysis reveals coordinate structure, not physical content

7. The 2019 SI Redefinition as Implicit Recognition

7.1 What Actually Happened

In 2019, the SI system was redefined by setting exact numerical values for:

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

This was setting Jacobian coefficients by committee vote.

7.2 The Unspoken Recognition

The metrology community implicitly recognized:

These "constants" are not being measured
They are conversion factors between unit charts
We can define them because they encode coordinate choice, not physics

But they never stated this explicitly. The official justification was "reproducibility" and "quantum standards."

7.3 What They Didn't Do

Notice they did not define:

α = 1/137 (exact)
m_p/m_e = 1836 (exact)
Any dimensionless ratio

Because those are actual physics—substrate geometry that must be measured, not defined.

The 2019 redefinition was humanity admitting (without saying) that dimensional constants are coordinate artifacts.

8. Common Misconceptions

8.1 "Planck Units Are God's Units"

Wrong. Planck units are not privileged by the universe. They are the coordinate chart where human measurement axes happen to align with their natural geometric ratios (1:1 exchange rates).

An alien civilization with different measurement conventions would have:

Different numerical values for c, h, G in their units
The same Planck coordinate chart (the one with axes aligned)
The same dimensionless ratios (α, π, mass ratios)

8.2 "Setting ℏ=1 Is Just As Valid As h=1"

Wrong. Setting h=1 is a valid coordinate choice (Planck chart). Setting ℏ=1 requires either:

h ≠ 1 (not in Planck coordinates), or
π ≠ π (changing substrate geometry—impossible)

The ℏ-based formulation is computationally convenient but conceptually confused.

8.3 "Constants Are Arbitrary Because We Defined Them"

Partially wrong. Dimensional constants (Class 1) are arbitrary coordinate choices—we can define them. Dimensionless ratios (Class 2) are substrate geometry—we must measure them.

The 2019 SI redefinition set Jacobian coefficients, not geometric invariants.

9. Implications for Physical Theory

9.1 The Nature of Physical Law

Physical laws are not "discovered equations with mysterious constants." They are:

Coordinate projections of substrate proportionalities, with constants encoding the projection geometry.

Example:

Substrate: E ∝ m (dimensionless proportionality)
SI projection: E = mc² (with Jacobian c²)
Planck projection: E = m (no Jacobian needed)

9.2 The Role of Dimensional Analysis

Dimensional analysis doesn't reveal "the structure of physical law." It reveals the structure of our coordinate choice.

When you see dimensions [L³M⁻¹T⁻²] in G, you're seeing:

The Jacobian structure for transforming gravitational force
From Planck coordinates (where axes are aligned)
To SI coordinates (where axes are rotated)

9.3 What Questions Are Meaningful

Meaningless questions:

"Why does c = 299,792,458 m/s?" (It's a Jacobian—we defined it)
"Why is h so small?" (Small compared to what? It's a unit conversion factor)
"Could physics work with different c or h?" (Yes—different coordinate charts)

Meaningful questions:

"Why is α ≈ 1/137?" (Substrate geometry—must be explained)
"Why is m_p/m_e ≈ 1836?" (Substrate geometry—must be explained)
"Why these substrate proportionalities?" (The actual physics)

10. Conclusion

We have provided a rigorous categorical framework distinguishing:

Class 1: Jacobian Constants (c, h, G, k_B)

Have dimensions
Are cocycle data in fibration π : 𝓔 → 𝓑
Encode coordinate transformations between unit charts
Can be "set to 1" by choosing Planck coordinates
Are measurement artifacts, not physics

Class 2: Geometric Invariants (α, π, mass ratios)

Are dimensionless
Are morphisms in base category 𝓑
Represent substrate geometry independent of coordinates
Cannot be changed by coordinate choice
Are the actual physics

This distinction, while intuitively recognized by practicing physicists, has never been formally articulated. The framework reveals that:

"Natural units" means choosing the Planck coordinate chart
"Setting constants to 1" means aligning axes with natural ratios
ℏ cannot independently equal 1 because ℏ = h/(2π) mixes a Jacobian with a geometric invariant
The 2019 SI redefinition was implicitly recognizing dimensional constants as coordinate artifacts

The proper formulation uses h (non-reduced), not ℏ (reduced), because this preserves the clean categorical separation between Jacobian coefficients and geometric invariants.

Physics expelled the philosophy needed to ask these questions, then lost the mathematics needed to answer them. This paper recovers both, showing that the distinction between "constants you can set to 1" and "constants you cannot" is not merely intuitive but rigorously definable through fibered category theory.

The framework transforms metrology from a practical necessity into a foundational structure: measurement is not peripheral to physics—it is the fibered projection that generates the appearance of physical law.

References

Newton, I. (1687). Philosophiæ Naturalis Principia Mathematica
Planck, M. (1899). "Über irreversible Strahlungsvorgänge"
Grothendieck, A. (1971). Revêtements Étales et Groupe Fondamental (SGA 1)
BIPM (2019). "The International System of Units (SI)", 9th edition
Rogers, J. (2025). "The Structure of Physical Law as a Grothendieck Fibration"
Rogers, J. (2025). "Newton's Natural Ratios and the Planck Coordinate Chart"

Appendix: Quick Reference Table

Property	Jacobian Constants	Geometric Invariants
Examples	c, h, G, k_B	α, π, m_p/m_e
Dimensions	Yes	No
Category Location	Cocycle data in π	Morphisms in 𝓑
Coordinate Dependence	Transform between charts	Invariant across charts
Can Set to 1?	Yes (choose Planck chart)	No (substrate geometry)
Physical Meaning	Axis exchange rates	Substrate relationships
Must Be Measured?	No (can be defined)	Yes (cannot be defined)
Changed in 2019 SI?	Yes (set exact values)	No (still measured)

The simple rule: If it has dimensions, it's a Jacobian. If it's dimensionless, it's geometry.