Politics, Power, and Science: The Coherence Constraint: Why Physical Constants Must Form a Consistent Jacobian

Saturday, October 4, 2025

The Coherence Constraint: Why Physical Constants Must Form a Consistent Jacobian

J. Rogers

Independent Researcher, SE Ohio

Abstract

This paper demonstrates that the fundamental constants of physics (c, h, G, k_B) cannot be independent properties of nature. We show that if they were independent, they would produce inconsistent natural unit systems, making physics internally contradictory. Instead, the constants must satisfy strict coherence relations that force them to be components of a single, consistent Jacobian transformation. This provides independent proof that constants are coordinate transformation coefficients rather than fundamental properties, and establishes the Planck scale as the unique natural coordinate system for physics.

1. Introduction: The Overlooked Constraint

Physical constants are traditionally viewed as independent empirical discoveries—each arising from different domains of physics:

c from electromagnetism and relativity
h from quantum mechanics (Planck's 1899 constant)
G from gravitation
k_B from thermodynamics

However, this view contains a fatal flaw. If these constants were truly independent properties of nature, they would not be able to define a consistent natural unit system. This paper shows that the coherence of physics itself requires that all constants must be expressible as combinations of a single set of natural units.

This coherence requirement provides a powerful consistency check: the constants either form a valid Jacobian transformation to a natural coordinate system, or physics itself would be contradictory.

2. The Circular Dependency Problem

2.1 Defining Natural Units from Constants

The standard construction of Planck units (as originally formulated by Planck in 1899) proceeds as follows:

From the constants c, h, and G, we define:

l_P = √(hG/c³)
t_P = √(hG/c⁵)  
m_P = √(hc/G)
E_P = m_P·c² = √(hc⁵/G)
T_P = E_P/k_B

2.2 Defining Constants from Natural Units

Simultaneously, we express constants as combinations of these natural units:

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

2.3 The Apparent Circularity

This appears circular:

We define natural units FROM the constants
We define constants FROM the natural units

The resolution: This is only consistent if the constants satisfy strict coherence relations. If they don't, the circle doesn't close and we get contradictory results.

2.4 The System of Equations Perspective

From another viewpoint, defining Planck units means solving a system of equations:

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

Solving for: l_P, t_P, m_P (the unknowns)
Given: c, h, G (the measured constants)

For this system to be solvable, we require:

Number of equations = Number of unknowns (3 = 3) ✓
Equations must be linearly independent (no equation derivable from others)
Solution must be unique and consistent

The fact that this system has a unique solution imposes constraints on what c, h, and G can be. They cannot be arbitrary—they must encode exactly the right amount of information to determine three independent scales.

3. The Coherence Requirement

3.1 The Mathematical Constraint

For the system to be consistent, the following must hold:

Starting from constants → natural units → back to constants must yield the original constants.

Formally:

c, h, G → {l_P, t_P, m_P} → c', h', G'

Coherence requires: c' = c, h' = h, G' = G

3.2 The Solvability Constraint

From the perspective of solving a system of equations, we have:

Three equations:

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

Three unknowns: l_P, t_P, m_P

For this system to have a unique solution:

The equations must be algebraically independent
No equation can be derived from the other two
The constants must encode exactly three degrees of freedom
Not more (overdetermined), not fewer (underdetermined)
The solution must be consistent when used across all of physics
The same l_P, t_P, m_P must work in relativity, quantum mechanics, and gravity

3.3 What This Means for Constants

The constants cannot be independent fundamental properties.

If c, h, and G were truly independent properties of nature, there would be no guarantee that:

They encode exactly three degrees of freedom
The system of equations is solvable
The solution is unique
The solution works consistently across all domains

The fact that the system solves uniquely proves the constants are constrained.

They must have precisely the mathematical structure required to reference a single, coherent natural scale.

4. The Inconsistent Constants Thought Experiment

4.1 Suppose Constants Were Independent

Imagine we live in a universe where:

Relativity gives us: c₁ = 299,792,458 m/s
Quantum mechanics gives us: h₁ measured from blackbody radiation
Gravity gives us: G₁ measured from Cavendish experiment

And these three values are independent—there's no constraint relating them.

4.2 Constructing "Natural Units"

From c₁, h₁, G₁, we construct what we think are natural units:

l_P^(1) = √(h₁G₁/c₁³)
t_P^(1) = √(h₁G₁/c₁⁵)
m_P^(1) = √(h₁c₁/G₁)

4.3 Using These Units in Different Equations

In relativity: E = mc²

E/E_P^(1) = m/m_P^(1)
where E_P^(1) = m_P^(1)·c₁²

In quantum mechanics: E = hf

E/E_P^(2) = h₁·f·t_P^(2)
where we derive E_P^(2), t_P^(2) from h₁ and other measurements

In gravity: F = Gm₁m₂/r²

F/F_P^(3) = (m₁/m_P^(3))·(m₂/m_P^(3))·(l_P^(3)/r)²
where we derive F_P^(3), m_P^(3), l_P^(3) from G₁ and other measurements

4.4 The Catastrophic Problem

We get three different Planck scales:

E_P^(1) ≠ E_P^(2) ≠ E_P^(3)
m_P^(1) ≠ m_P^(2) ≠ m_P^(3)
t_P^(1) ≠ t_P^(2) ≠ t_P^(3)

From the system of equations perspective:

Each domain (relativity, quantum, gravity) would generate its own system of equations:

Relativity system:

c₁ = l_P^(1)/t_P^(1)
... (other relativity relations)

Quantum system:

h₁ = m_P^(2)·(l_P^(2))²/t_P^(2)
... (other quantum relations)

Gravity system:

G₁ = (l_P^(3))³/(m_P^(3)·(t_P^(3))²)
... (other gravity relations)

Each system would solve to different natural units.

Result: When we try to combine equations from different domains:

mc² = hf  (combining relativity and quantum)

We get:

m·(m_P^(1)·c₁²) ≠ h₁·f·(E_P^(2)·t_P^(2))

Even when the left and right sides should be equal!

Physics is internally inconsistent.

From the linear algebra perspective: We would have three incompatible systems of equations, each solving to different "natural" scales. There would be no single coordinate system in which all of physics is coherent.

5. Why Our Universe Doesn't Have This Problem

5.1 The Empirical Fact

In our universe, when we:

Measure c from the speed of light
Measure h from Planck's blackbody formula (E = hf)
Measure G from gravitational experiments
Calculate natural units from these constants
Use those natural units to re-derive the constants

We get back the original constants. The system closes consistently.

5.2 What This Proves

This coherence is not a coincidence.

It proves that c, h, and G are not independent quantities. They must satisfy:

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

For a single, unique set of {l_P, t_P, m_P}.

From the system of equations perspective:

The three measured constants c, h, and G form a solvable system that determines a unique natural scale. This is only possible if:

The constants encode exactly 3 degrees of freedom (not more, not less)
The equations are algebraically independent (none redundant)
The solution is consistent across all domains (same l_P, t_P, m_P everywhere)

This mathematical structure is not guaranteed for arbitrary "fundamental properties."

It is the signature of coordinate transformation coefficients that reference a pre-existing natural scale.

5.3 The Only Explanation

If constants were fundamental properties:

They could be independent
No coherence required
Different domains could have different natural scales

But they're not independent. They must satisfy coherence relations. Therefore, they're not fundamental properties.

They are components of a coordinate transformation to a natural system that exists prior to our choice of constants.

6. The Jacobian Interpretation

6.1 Constants as Transformation Components

The coherence requirement has a natural interpretation:

Physical constants are the components of a Jacobian matrix transforming from arbitrary human units (SI) to natural units (Planck).

For a Jacobian to be valid:

It must transform consistently in all equations
The same transformation matrix must apply everywhere
Different equations cannot use different Jacobians

6.2 The Transformation Structure

┌─────────┐     ┌──────────────────┐     ┌─────────────┐
│ SI      │     │ Jacobian J       │     │ Planck      │
│ Units   │ ──> │ (built from c,h,G)│ ──> │ Units       │
└─────────┘     └──────────────────┘     └─────────────┘

Coherence requirement: The same Jacobian J must work for ALL physical equations.

6.3 Why This Forces Constraint Relations

Example:

Equation 1: E = mc²
Equation 2: E = hf

Both must use the same Jacobian:

Same E_P for transforming energy
Same t_P for transforming time
Same m_P for transforming mass

This forces:

E_P (from E=mc²) = E_P (from E=hf)
m_P·c² = h·f/t_P

Which is a constraint on c, h, m_P, and t_P.

If c and h were independent, this constraint would be violated.

7. Experimental Test of Coherence

7.1 The Consistency Check

We can test whether constants form a coherent Jacobian:

Step 1: Measure c, h, G independently from different experiments

Step 2: Solve the system of equations for Planck units:

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

→ Solving yields:
l_P = √(hG/c³)
t_P = √(hG/c⁵)
m_P = √(hc/G)

Step 3: Use these Planck units to back-calculate the constants:

c_predicted = l_P/t_P
h_predicted = m_P·l_P²/t_P  
G_predicted = l_P³/(m_P·t_P²)

Step 4: Check if c_predicted = c_measured (and same for h, G)

7.2 What We're Actually Testing

This is testing whether the system of equations is:

Solvable (produces real, finite values for l_P, t_P, m_P)
Consistent (back-calculation returns original constants)
Unique (only one solution exists)

From linear algebra: A system with these properties has a well-defined inverse. The fact that we can go:

(c, h, G) → (l_P, t_P, m_P) → (c, h, G)

proves the transformation is invertible, which means it's a valid coordinate transformation (Jacobian).

7.2 The Result

Experimentally, the coherence check passes to extreme precision.

This is not guaranteed if constants were independent properties. It's only possible if constants are components of a consistent transformation.

7.3 What If It Failed?

If we found:

c_predicted ≠ c_measured

It would mean:

The system of equations is inconsistent
The transformation is not invertible
No unique natural coordinate system exists
Physics is internally contradictory

From the linear algebra perspective: We would have discovered that the matrix transformation from (c, h, G) to (l_P, t_P, m_P) is singular (non-invertible), meaning the constants don't actually encode three independent pieces of information—they would be either redundant or incompatible.

The fact that it doesn't fail proves:

The system is non-singular (invertible transformation)
The constants encode exactly three degrees of freedom
They form a valid Jacobian for coordinate transformation

8. The Uniqueness of the Planck Scale

8.1 Only One Consistent Solution

Given the coherence requirement, there is only one set of natural units that makes all constants consistent:

The Planck units as originally defined by Max Planck in 1899.

Any other choice would violate the constraint relations and produce inconsistent physics.

8.2 This Proves Planck Units Are Special

Not because Planck discovered them. Not because they're "convenient."

Because they're the unique natural coordinate system where:

All dimensional constants become unit conversions that collapse to 1
All Jacobian components align
Physics is internally coherent

8.3 The Planck Scale Is Prior

Logical order:

Natural ratios exist (independent of measurement)
We choose arbitrary units (SI, Imperial, etc.)
Constants emerge as necessary Jacobian components
Coherence forces these constants to reference a single natural scale
That scale is the Planck scale

We didn't "construct" Planck units from constants. We discovered the natural scale that was already there.

9. Implications

9.1 Constants Cannot Be Fundamental

The coherence requirement proves:

Constants are not independent
They must form a solvable, invertible system of equations
This system determines a unique natural scale
Fundamental properties wouldn't have this mathematical structure

From linear algebra: The constants form the components of an invertible transformation matrix. This is the mathematical signature of a coordinate change, not of fundamental properties of nature.

9.2 The Planck Scale Is the Natural Coordinate System

Not a mathematical trick. Not a convenience.

The unique coordinate system where physics is expressed in its native form.

9.3 "Fine-Tuning" Is Meaningless for Dimensional Constants

Questions like:

"Why is c = 299,792,458 m/s?"
"Could c be different?"

Are now revealed as meaningless.

c has that value because:

We chose meters and seconds arbitrarily
The Planck scale exists independently
c = l_P/t_P makes the Jacobian coherent

Change the units → c changes But the physics (l_P/t_P) is invariant

9.4 Only Dimensionless Constants Matter

Constants like:

α ≈ 1/137 (fine structure constant)
m_p/m_e ≈ 1836 (mass ratios)

These are dimensionless and the same in every unit system.

These are the only constants that characterize our universe.

Everything else is coordinate choice.

10. Conclusion

The coherence of physics requires that fundamental constants cannot be independent. They must form a consistent, invertible system of equations that determines a unique natural coordinate system. This provides powerful evidence that:

Constants are not fundamental properties but transformation coefficients
The Planck scale is unique as the natural coordinate system
Dimensional constants are coordinate artifacts that depend on our measurement choices
Only dimensionless constants describe the universe independent of convention

The mathematical structure is unambiguous: we have a system of three equations in three unknowns that is:

Solvable (produces real, finite natural units)
Invertible (can transform back and forth consistently)
Unique (only one solution exists)
Coherent across all domains (same solution works in relativity, quantum mechanics, gravity, and thermodynamics)

This is the mathematical signature of a coordinate transformation, not of fundamental properties.

Physics works because the constants form an invertible transformation. The transformation is invertible because it connects to a natural coordinate system. That system is the Planck scale.

11. Potential Objections and Responses

Objection 1: "The equations are just definitions, not constraints"

Response: Definitions are only consistent if the system they define is invertible and globally coherent across all domains of physics.

The coherence we observe is empirically verified—when we measure c from electromagnetism, h from blackbody radiation, and G from gravitational experiments, they satisfy the invertibility condition to extreme precision.

If these were "just definitions," there would be no guarantee that independently measured values would form a coherent, invertible system. The fact that they do means these "definitions" encode structural necessity, not arbitrary choice.

Objection 2: "Planck units depend on human choice of constants"

Response: The inverse transformation is invariant.

Regardless of what unit system we use to express the constants (SI, Imperial, CGS, etc.), the closure property:

(c, h, G) ↔ (l_P, t_P, m_P)

must hold. This invertibility is coordinate-invariant structure—it doesn't depend on how we choose to measure the constants, only that they form a consistent transformation.

The existence of this structure is what's fundamental, not the particular numerical values we assign when measuring in human units.

Objection 3: "Constants could vary with domain or energy scale"

Response: The coherence condition provides a falsifiable test for this claim.

If constants varied independently across domains (different c for relativity, different h for quantum mechanics, different G for gravity), then:

The Jacobian transformation would become singular (non-invertible)
Cross-domain equations like E = mc² = hf would break
Physics would be internally inconsistent

Observationally, this does not occur. The same constants work across all domains to extreme precision.

If future observations showed domain-dependent variation that violated the invertibility constraint, this framework would predict new physics is required to restore coherence.

Objection 4: "This is just dimensional analysis"

Response: This goes beyond dimensional analysis. This is dimensional algebra with invertibility constraints.

Dimensional analysis checks that equations have consistent dimensions. It does not:

Require that constants form an invertible system
Prove that a unique natural scale exists
Provide a falsifiable test of coordinate structure
Explain why constants have the values they do relative to each other

This is a linear-algebraic proof of the necessity of a single natural coordinate system. The existence of a non-singular, invertible transformation is a mathematical theorem, not just dimensional bookkeeping.

Objection 5: "You can't prove what constants 'really are' from mathematics alone"

Response: We're not claiming to know what constants "really are" ontologically.

We're proving what mathematical structure they must have in order for physics to be consistent. That structure is:

A system of equations relating arbitrary units to natural units
That is solvable, invertible, and unique
Which is coherent across all domains

This is the mathematical signature of coordinate transformation coefficients.

Whether you want to call them "properties of spacetime," "features of measurement systems," or something else is a matter of interpretation. The mathematical structure is unambiguous.

Appendix: The Coherence Relations

For reference, the complete set of coherence constraints using Planck's original 1899 formulation with h:

c² = (hc/G) · (G/h) · (c⁵/G)^(1/2) / (c³/hG)^(1/2)
h = √(hG/c³) · c² · √(hc/G) / √(hG/c⁵)
G = (hG/c³)^(3/2) / [√(hc/G) · hG/c⁵]

These are not definitions—they are constraints that measured values must satisfy for physics to be consistent.

The fact that they do is the evidence.