The Dimensional Inconsistency in Reduced Planck Units

J. Rogers, SE Ohio

Abstract

We demonstrate that the standard formulation of "reduced Planck units" contains a fundamental dimensional inconsistency. While reduced Planck units are ostensibly defined using ℏ = h/(2π), the actual constraint equations used implicitly assume ℏ = h, creating a category error that has persisted in the physics literature for over a century. We show that maintaining dimensional consistency requires either: (1) using non-reduced Planck units based on h, or (2) properly distributing 2π factors symmetrically in the defining equations. The standard practice does neither, instead performing an unjustified substitution that conflates two distinct physical quantities.

1. Introduction

Planck units represent a natural system of measurement where fundamental constants take unit value. The standard literature describes two variants:

• Non-reduced Planck units: Setting h = c = G = k_B = 1
• Reduced Planck units: Setting ℏ = c = G = k_B = 1

where ℏ = h/(2π) is the reduced Planck constant.

We demonstrate that the mathematical formulation of reduced Planck units contains a dimensional inconsistency that violates the definitional relationship between h and ℏ.

2. The Constraint System

Planck units are defined as the unique solution to the following system of dimensional equations relating fundamental constants to base Planck quantities:

c = l_P/t_P                      (1)
h = m_P l_P²/t_P                 (2)
k_B = m_P l_P²/(t_P² T_P)        (3)
G = l_P³/(m_P t_P²)              (4)

These four equations have a unique solution:

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

This is the non-reduced Planck unit system, determined uniquely by requiring dimensional consistency with h, c, G, and k_B.

3. The Standard Error

The reduced Planck constant is defined as:

ℏ ≡ h/(2π)                       (5)

The standard formulation of reduced Planck units claims to derive from equation (2) with ℏ substituted for h:

ℏ = m_P l_P²/t_P                 (6) [STANDARD CLAIM]

This is dimensionally inconsistent with equation (5).

3.1 The Dimensional Contradiction

From equations (2) and (5):

h = m_P l_P²/t_P
ℏ = h/(2π)

Therefore:

ℏ = m_P l_P²/(t_P · 2π)          (7) [CORRECT]

Equation (6) claims ℏ = m_P l_P²/t_P, which is dimensionally equal to h, not ℏ.

The standard formulation conflates h with ℏ by writing the action relationship without the required 2π factor.

4. The Consistent Alternatives

There are exactly two dimensionally consistent approaches:

4.1 Non-Reduced Planck Units (Correct)

Use h directly in the constraint system:

h = m_P l_P²/t_P

Solution:

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

Set h = 1 to obtain natural units.

4.2 Actually Reduced Planck Units (Also Correct)

If using ℏ, the constraint must include 2π symmetrically:

ℏ = h/(2π) = m_P l_P²/(t_P · 2π)     (8)

Multiplying both sides by 2π recovers equation (2):

h = m_P l_P²/t_P                     (2)

So there is no separate "reduced" version of the constraint equation—only equation (2) is dimensionally consistent. The 2π factor relating h to ℏ does not modify the Planck unit definitions; it only changes which constant we choose to "set to 1" for calculational purposes.

If we insist on working with ℏ and want to derive modified Planck scales, we would get:

t_P(ℏ) = √(ℏG/c⁵) = t_P(h) / √(2π)
l_P(ℏ) = √(ℏG/c³) = l_P(h) / √(2π)
m_P(ℏ) = √(ℏc/G) = m_P(h) / √(2π)

But these modified scales do NOT satisfy equation (2). They satisfy:

h = m_P(ℏ) l_P(ℏ)² · (2π/t_P(ℏ))    [modified, not fundamental]

This is NOT the standard formulation, which claims ℏ = m_P l_P²/t_P using the original Planck scales, thereby producing dimensional inconsistency.

5. Why The Error Persists

5.1 The "Setting Constants to One" Hand-Wave

Standard practice states: "We work in units where ℏ = c = G = 1"

This notation obscures the actual coordinate transformation being performed. The hand-wave of "setting constants to one" allows the dimensional inconsistency to remain hidden because:

1. No explicit transformation matrix is written
2. No Jacobian is computed
3. The relationship between h and ℏ is not tracked through the transformation

5.2 Calculational Convenience vs. Dimensional Rigor

The reduced Planck constant ℏ appears frequently in quantum mechanical expressions:

[x, p] = iℏ
E = ℏω
ψ ~ exp(i(px - Et)/ℏ)

Setting ℏ = 1 is calculationally convenient. However, convenience does not justify dimensional inconsistency.

6. Empirical Validation

We can verify which formulation is correct by computing fundamental constants from Planck unit definitions.

6.1 Non-Reduced Planck Units

Using h = m_P l_P²/t_P and the constraint system (1-4), we can derive:

h = Hz_kg * c²               # Planck constant
hbar = h / (2*pi)            # Reduced Planck constant  
G = t_P² * c³ / Hz_kg        # Gravitational constant
k_B = K_Hz * Hz_kg * c²      # Boltzmann constant

where Hz_kg and K_Hz are the Jacobian elements for the coordinate transformation from SI to Planck basis.

These expressions reproduce CODATA values exactly (within numerical precision).

6.2 Standard "Reduced" Planck Units

The standard approach implicitly uses:

ℏ = m_P l_P²/t_P             [dimensionally equal to h, not ℏ]

This cannot simultaneously satisfy:

• ℏ = h/(2π) [definition]
• The constraint system (1-4) [dimensional requirements]

The inconsistency is masked by the unmotivated "setting ℏ = 1" operation.

7. The Correct Statement About Angular vs. Ordinary Frequency

A common defense of reduced Planck units claims that ℏ is the "natural" constant for angular frequency ω = 2πf:

E = hf  = ℏω

However, this does not justify equation (6). The relationship E = ℏω is simply:

E = ℏ(2πf) = [h/(2π)](2πf) = hf

The 2π factors cancel. Both expressions give identical energy values.

The choice between (h, f) and (ℏ, ω) is purely notational—it does not change the physics or justify using equation (6) instead of equation (7) in the Planck unit definition.

8. The Category Error

The standard formulation commits what we term a "dimensional category error" by:

1. Defining ℏ = h/(2π)
2. Using the constraint h = m_P l_P²/t_P
3. Claiming ℏ = m_P l_P²/t_P

These three statements are mutually inconsistent.

To see this explicitly:

• From (2): h = m_P l_P²/t_P
• From (3): ℏ = m_P l_P²/t_P
• Therefore: h = ℏ
• But (1) states: ℏ = h/(2π)
• Contradiction: h ≠ h/(2π)

The error is choosing statement (3), which should instead be equation (7).

9. Historical Note

Max Planck introduced Planck units in 1899 using h, not ℏ. The reduced Planck constant ℏ was introduced later as notational convenience in quantum mechanics, where angular frequency ω naturally appears in exponential phase factors.

The "reduced Planck units" appear to have emerged from:

1. Quantum field theory's preference for ℏ notation
2. The desire to "set ℏ = 1" for calculational simplicity
3. The assumption that this was equivalent to modifying Planck's original definitions
4. Failure to verify dimensional consistency through the transformation

No rigorous derivation of reduced Planck units from first principles appears in the standard literature. The formulation is simply stated as conventional without justification.

10. Implications

10.1 Natural Units Are Non-Reduced

The unique solution to the constraint system (1-4) yields non-reduced Planck units using h.

"Reduced Planck units" are not a solution to this system—they are a notational modification that violates dimensional consistency.

10.2 The Jacobian Structure

Proper treatment of unit transformations requires explicit Jacobian matrices. The transformation from SI units to Planck units involves Jacobian elements:

Hz_kg = h/c²     [mass-frequency Jacobian]
K_Hz = k_B/h     [temperature-frequency Jacobian]

These elements are unnamed in standard physics because the transformation is never written explicitly. The hand-wave "set constants to 1" obscures the actual mathematical operation being performed.

10.3 Pedagogical Implications

Students are taught that "reduced Planck units" and "non-reduced Planck units" are equivalent choices differing only by factors of 2π.

This is false.

Only non-reduced Planck units satisfy the constraint system (1-4) with dimensional consistency. The "reduced" formulation contains a mathematical error that has been propagated through textbooks for decades.

11. Numerical Falsification: The Compton Wavelength Test

We now demonstrate that reduced Planck units fail not merely on dimensional grounds, but numerically. The natural relationship λ = 1/m, which should hold in any true "natural units" system, fails in reduced Planck units.

11.1 The Test

The Compton wavelength relationship in SI units is:

λ = h/(mc)

In proper natural units, this should simplify to:

λ_natural = 1/m_natural

Equivalently: λ_natural × m_natural = 1

We test this by:

1. Converting measured electron mass (m_e = 9.109×10⁻³¹ kg) to natural units
2. Converting measured Compton wavelength (λ_e = 2.426×10⁻¹² m) to natural units
3. Checking whether λ_natural = 1/m_natural holds

11.2 Results: Non-Reduced Planck Units

Using h-based Planck scales:

m_P(h) = √(hc/G) = 5.456×10⁻⁸ kg
l_P(h) = √(hG/c³) = 4.051×10⁻³⁵ m

Converting to natural units:

m_natural = m_e / m_P(h) = 1.670×10⁻²³
λ_natural = λ_e / l_P(h) = 5.989×10²²
1/m_natural = 5.989×10²²

Test result:

λ_natural / (1/m_natural) = 0.999999999995
Error: 0.0000000005%

✓ The relationship λ = 1/m holds to numerical precision.

11.3 Results: Reduced Planck Units

Using ℏ-based Planck scales:

m_P(ℏ) = √(ℏc/G) = 2.176×10⁻⁸ kg
l_P(ℏ) = √(ℏG/c³) = 1.616×10⁻³⁵ m

Converting to natural units:

m_natural = m_e / m_P(ℏ) = 4.185×10⁻²³
λ_natural = λ_e / l_P(ℏ) = 1.501×10²³
1/m_natural = 2.389×10²²

Test result:

λ_natural / (1/m_natural) = 6.283185 = 2π
Error: 528.32%

✗ The relationship λ = 1/m fails by a factor of 2π.

11.4 Interpretation

In reduced Planck units, the natural relationship becomes:

λ_natural = 2π/m_natural

This factor of 2π is not a "correction"—it demonstrates that reduced Planck units do not constitute a natural coordinate system. The entire purpose of natural units is to make fundamental relationships dimensionless and coefficient-free. When λ = 1/m requires a factor of 2π, the system has failed this criterion.

11.5 Computational Verification

The following Python code reproduces this test:

import math

# CODATA 2018 constants
h = 6.62607015e-34
c = 299792458.0
G = 6.67430e-11

# Electron data
m_e = 9.1093837015e-31  # kg
lambda_e = 2.42631023867e-12  # m

# Test with non-reduced Planck units
m_P_h = math.sqrt(h*c/G)
l_P_h = math.sqrt(h*G/(c**3))
m_nat = m_e / m_P_h
lambda_nat = lambda_e / l_P_h
print(f"Non-reduced: λ × m = {lambda_nat * m_nat:.15f}")
# Output: 1.000000000005396

# Test with reduced Planck units
hbar = h / (2*math.pi)
m_P_hbar = math.sqrt(hbar*c/G)
l_P_hbar = math.sqrt(hbar*G/(c**3))
m_nat = m_e / m_P_hbar
lambda_nat = lambda_e / l_P_hbar
print(f"Reduced: λ × m = {lambda_nat * m_nat:.15f}")
# Output: 6.283185307179586 = 2π

11.6 Implications for Other Natural Relationships

If λ = 1/m fails in reduced Planck units, we must examine all claimed "natural" relationships:

• Does E = m hold exactly?
• Does F = mm/r² hold exactly?
• Does T = 1/M hold exactly for black holes?

Our preliminary analysis suggests that each fundamental relationship acquires unwanted factors of 2π or √(2π) when expressed in reduced Planck units, indicating systematic coordinate pollution throughout the framework.

11.7 Falsification Criterion (Revised)

This provides a direct numerical falsification test:

Claim: "Reduced and non-reduced Planck units are equivalent natural unit systems"

Test: Verify that λ_natural × m_natural = 1 for any particle

Result:

• Non-reduced: λ × m = 1.000... ✓
• Reduced: λ × m = 6.283... = 2π ✗

Conclusion: Reduced Planck units fail the basic criterion for natural units.

This is not a matter of interpretation or convention. It is a numerical fact that reduced Planck units do not satisfy the fundamental relationships they claim to simplify.

12. Demolishing the "Equivalent Within 2π" Claim

Standard literature often claims that non-reduced ($ℎ$-based) and reduced ($\mathrm{\hslash }$-based) Planck units are "equivalent up to factors of $2𝜋$." This assertion fundamentally misapplies the mathematical meaning of equality ("=") in the defining constraint equations.

The Explicit Claim in the Literature

The Wikipedia entry on Planck units exemplifies this defense:

"There is also a 'Planck mass' defined as ${𝑚}_{𝑃}=\sqrt{\mathrm{\hslash }𝑐\mathrm{/}𝐺}$, which is smaller than the Planck mass above by a factor of $\sqrt{2𝜋}$. These are two different conventions..."[en.wikipedia]​

This implies the two systems differ by a numerical factor and are thus interchangeable for physical purposes.

Why This Fails: "=" Demands Identity, Not Tolerance

The constraint equations defining Planck units are not approximate; they are exact dimensional identities:

$$\begin{array}{cccc}& ℎ={𝑚}_{𝑃}{𝑙}_{𝑃}^2\mathrm{/}{𝑡}_{𝑃}& & \text{(2)}\end{array}$$

In mathematics, "=" means identity: the left-hand side and right-hand side must be numerically and dimensionally identical. There are no "error bars" or "tolerance factors" of $2𝜋$.

Substituting $\mathrm{\hslash }=ℎ\mathrm{/}(2𝜋)$ while claiming to preserve equation (2) yields:

$$\begin{array}{cccc}& \mathrm{\hslash }=\frac{{𝑚}_{𝑃}{𝑙}_{𝑃}^2}{{𝑡}_{𝑃}\cdot 2𝜋}& & \text{(7)}\end{array}$$

The standard reduced formulation asserts instead:

$$\begin{array}{cccc}& \mathrm{\hslash }=\frac{{𝑚}_{𝑃}{𝑙}_{𝑃}^2}{{𝑡}_{𝑃}}[\text{WRONG}]& & \text{(6)}\end{array}$$

Equation (6) cannot coexist with $\mathrm{\hslash }=ℎ\mathrm{/}(2\mathrm{𝜋)\; under\; the\; same }$$({𝑚}_{𝑃},{𝑙}_{𝑃},{𝑡}_{𝑃})$. The claimed "equivalence within $2𝜋$" concedes that (6) differs from (7) by exactly the forbidden factor—making it precisely wrong, not approximately equivalent.

Numerical Falsification: No Error Bars Allowed

Section 11's Compton test eliminates any ambiguity:

System	${𝜆}_{\text{nat}}\times {𝑚}_{\text{nat}}$	Target Value	Error
Non-reduced ($ℎ$)	$1.00000000000$	$1$	$5\times {10}^{-12}\mathrm{\%}$ ✓
Reduced ($\mathrm{\hslash }$)	$6.28318530718=2\pi$	$1$	$528\mathrm{\%}$ ✗

The reduced system fails exactly by $2𝜋$, not "within error bars." In natural units, the defining criterion is $𝜆𝑚=1$ with zero tolerance—any deviation is structural failure.

Mathematical Category Error

The "=" in constraint (2) defines a terminal object in the category of dimensional quantities: a unique solution where all morphisms compose cleanly to unity. Introducing $2𝜋$ pollution:

1. Fractures the isomorphism $1\mathrm{/}{𝑙}_{𝑃}\cong 1\mathrm{/}{𝑡}_{𝑃}$ by $1\mathrm{/}(2𝜋)$.

2. Makes $ℎ=2𝜋$ instead of the required $ℎ=1$.

3. Injects geometric radians into the frequency axis, contaminating the unit chart.

"Equivalent within $2𝜋$" admits these fractures exist but dismisses them as tolerable. This violates the categorical requirement that natural units yield pure morphisms to $\mathbf{1}$, with no coefficient artifacts.

Conclusion: Precision, Not Convention

The literature's disclaimer reveals the error, not excuses it. When a system claims $\mathrm{\hslash }={𝑚}_{𝑃}{𝑙}_{𝑃}^2\mathrm{/}{𝑡}_{𝑃}$ but delivers $𝜆𝑚=2𝜋$, it has failed the exactitude demanded by "=" in the constraints. Natural units are not a numerical approximation tolerant of $2𝜋$ slop—they are a coordinate transform to unity requiring coefficient-free identities.

The unique solution satisfying equations (1)-(4) with exact "=" is the non-reduced system. All others, including the standard reduced convention, substitute approximation for identity and fail numerical verification.[en.wikipedia]​

13. Conclusion

The standard formulation of reduced Planck units contains a dimensional inconsistency that violates the definitional relationship ℏ = h/(2π). This error has persisted because:

1. The transformation to natural units is not written as an explicit coordinate change
2. The Jacobian structure is never computed
3. The hand-wave "set constants to 1" masks dimensional tracking
4. Calculational convenience is prioritized over dimensional rigor

The mathematically consistent approach is to use non-reduced Planck units based on h, which represent the unique solution to the constraint system defining natural units.

If physicists prefer ℏ notation for quantum calculations, they should recognize this as a notational choice that does not change the underlying Planck scale. The relationship between quantities must still satisfy dimensional consistency, which the standard "reduced Planck units" formulation violates.

References

The constraint system (1-4) can be verified by direct dimensional analysis. The solution is unique and corresponds to non-reduced Planck units:

t_P = √(hG/c⁵) = 5.391247×10⁻⁴⁴ s
l_P = √(hG/c³) = 1.616255×10⁻³⁵ m  
m_P = √(hc/G) = 2.176434×10⁻⁸ kg
E_P = √(hc⁵/G) = 1.956082×10⁹ J
T_P = √(hc⁵/(Gk_B²)) = 1.416784×10³² K

These values can be computed from CODATA 2018 fundamental constants and validated through independent calculation paths, as demonstrated in the accompanying computational implementation.

---

Appendix A: Computational Verification

The following Python code demonstrates that non-reduced Planck units (using h) correctly reproduce all fundamental constants through the Jacobian transformation structure:

# CODATA 2018 constants
h = 6.62607015e-34      # Planck constant (J⋅s)
c = 299792458.0         # Speed of light (m/s)
G = 6.67430e-11         # Gravitational constant (m³⋅kg⁻¹⋅s⁻²)
k_B = 1.380649e-23      # Boltzmann constant (J⋅K⁻¹)

# Jacobian elements (unnamed in standard physics)
Hz_kg = h / c**2        # 7.372497e-51 kg⋅Hz⁻¹
K_Hz = k_B / h          # 2.083662e+10 Hz⋅K⁻¹

# Non-reduced Planck units (unique solution)
t_P = (h * G / c**5)**0.5
l_P = (h * G / c**3)**0.5  
m_P = (h * c / G)**0.5
T_P = (h * c**5 / (G * k_B**2))**0.5

# Verify constraint equations
assert abs(c - l_P/t_P) / c < 1e-10
assert abs(h - m_P * l_P**2 / t_P) / h < 1e-10
assert abs(k_B - m_P * l_P**2 / (t_P**2 * T_P)) / k_B < 1e-10
assert abs(G - l_P**3 / (m_P * t_P**2)) / G < 1e-10

print("Non-reduced Planck units satisfy all constraints ✓")

# Attempt with ℏ (demonstrates inconsistency)
hbar = h / (2 * 3.14159265359)

# If we use ℏ = m_P l_P²/t_P (standard claim)
# Then the constraint h = m_P l_P²/t_P is violated
# because h ≠ ℏ

# The correct relationship if using ℏ would be:
# ℏ = m_P l_P²/(t_P · 2π)
# which is NOT the standard formulation

This code verifies that non-reduced Planck units satisfy the constraint system exactly, while the standard "reduced" formulation cannot maintain dimensional consistency with the definition ℏ = h/(2π).

$ python testreduced04.py 

============================================================

TEST: Does λ_natural = 1/m_natural hold as an equality?

============================================================

1. NON-REDUCED PLANCK UNITS (h-based):

------------------------------------------------------------

   Planck scales:

   m_P(h) = 5.4555118613e-08 kg

   l_P(h) = 4.0513505432e-35 m

   In natural units:

   m_natural = 1.669757839967652e-23

   λ_natural = 5.988892377435862e+22

   1/m_natural = 5.988892377468178e+22

   TEST: λ_natural =? 1/m_natural

   λ_natural     = 5.988892377435862e+22

   1/m_natural   = 5.988892377468178e+22

   Ratio: 0.999999999994604

   Error: 0.0000000005%

   ✓ EQUAL!

============================================================

2. REDUCED PLANCK UNITS (ℏ-based):

------------------------------------------------------------

   Planck scales:

   m_P(ℏ) = 2.1764343427e-08 kg

   l_P(ℏ) = 1.6162550244e-35 m

   In natural units:

   m_natural = 4.185462213449702e-23

   λ_natural = 1.501192696700281e+23

   1/m_natural = 2.389222382145913e+22

   TEST: λ_natural =? 1/m_natural

   λ_natural     = 1.501192696700281e+23

   1/m_natural   = 2.389222382145913e+22

   Ratio: 6.283185307145682

   Error: 528.32%

   Ratio: 6.2831853071

   2π = 6.2831853072

   ✗ NOT EQUAL! Differs by factor of 6.283185

============================================================

CONCLUSION:

============================================================

The natural unit relationship λ = 1/m requires:

  λ_natural × m_natural = 1

Non-reduced (h-based): λ_natural × m_natural = 1.000... ✓

Reduced (ℏ-based):     λ_natural × m_natural = 2π      ✗

Only non-reduced Planck units satisfy the natural relationship.

============================================================

$ cat testreduced04.py 

import math

# Physical constants (CODATA 2018)

h = 6.62607015e-34      # J·s

c = 299792458.0         # m/s

G = 6.67430e-11         # m³/(kg·s²)

pi = math.pi

# Electron data

m_e = 9.1093837015e-31  # kg

lambda_e = 2.42631023867e-12  # m (Compton wavelength)

print("="*70)

print("TEST: Does λ_natural = 1/m_natural hold as an equality?")

print("="*70)

# ============================================================

# TEST 1: NON-REDUCED PLANCK UNITS (h-based)

# ============================================================

print("\n1. NON-REDUCED PLANCK UNITS (h-based):")

print("-"*70)

m_P_h = math.sqrt(h*c/G)

l_P_h = math.sqrt(h*G/(c**3))

print(f"   Planck scales:")

print(f"   m_P(h) = {m_P_h:.10e} kg")

print(f"   l_P(h) = {l_P_h:.10e} m")

# Convert both to natural units independently

m_natural = m_e / m_P_h

lambda_natural = lambda_e / l_P_h

print(f"\n   In natural units:")

print(f"   m_natural = {m_natural:.15e}")

print(f"   λ_natural = {lambda_natural:.15e}")

print(f"   1/m_natural = {1.0/m_natural:.15e}")

print(f"\n   TEST: λ_natural =? 1/m_natural")

print(f"   λ_natural     = {lambda_natural:.15e}")

print(f"   1/m_natural   = {1.0/m_natural:.15e}")

print(f"   Ratio: {lambda_natural / (1.0/m_natural):.15f}")

error = abs(lambda_natural - (1.0/m_natural)) / (1.0/m_natural) * 100

print(f"   Error: {error:.10f}%")

print(f"   ✓ EQUAL!" if error < 0.0001 else "   ✗ NOT EQUAL!")

# ============================================================

# TEST 2: REDUCED PLANCK UNITS (ℏ-based)

# ============================================================

print("\n" + "="*70)

print("2. REDUCED PLANCK UNITS (ℏ-based):")

print("-"*70)

hbar = h / (2*pi)

m_P_hbar = math.sqrt(hbar*c/G)

l_P_hbar = math.sqrt(hbar*G/(c**3))

print(f"   Planck scales:")

print(f"   m_P(ℏ) = {m_P_hbar:.10e} kg")

print(f"   l_P(ℏ) = {l_P_hbar:.10e} m")

# Convert both to natural units independently

m_natural = m_e / m_P_hbar

lambda_natural = lambda_e / l_P_hbar

print(f"\n   In natural units:")

print(f"   m_natural = {m_natural:.15e}")

print(f"   λ_natural = {lambda_natural:.15e}")

print(f"   1/m_natural = {1.0/m_natural:.15e}")

print(f"\n   TEST: λ_natural =? 1/m_natural")

print(f"   λ_natural     = {lambda_natural:.15e}")

print(f"   1/m_natural   = {1.0/m_natural:.15e}")

print(f"   Ratio: {lambda_natural / (1.0/m_natural):.15f}")

error = abs(lambda_natural - (1.0/m_natural)) / (1.0/m_natural) * 100

ratio = lambda_natural / (1.0/m_natural)

print(f"   Error: {error:.2f}%")

print(f"   Ratio: {ratio:.10f}")

print(f"   2π = {2*pi:.10f}")

print(f"   ✗ NOT EQUAL! Differs by factor of {ratio:.6f}")

print("\n" + "="*70)

print("CONCLUSION:")

print("="*70)

print("The natural unit relationship λ = 1/m requires:")

print("  λ_natural × m_natural = 1")

print("")

print("Non-reduced (h-based): λ_natural × m_natural = 1.000... ✓")

print("Reduced (ℏ-based):     λ_natural × m_natural = 2π      ✗")

print("")

print("Only non-reduced Planck units satisfy the natural relationship.")

print("="*70)