The Architecture of Unity: Why Physics is the Language of Measurement

 J. Rogers, SE Ohio

1. The Ultimate Starting Point:

1               Unity

X/X = 1    Pick one point on unity 

X = X

In the study of theoretical foundations, we must acknowledge that physics does not begin with the observation of external phenomena, but with the imposition of a logical requirement: self-consistency. At the core of all physical inquiry lies the Ultimate Tautology. This is a content-free identity that demands nothing of reality other than it be identical to itself.

In this framework, we define a singular, dimensionless reality as X. This X represents Unity—the state of existence prior to the intervention of human measurement.

The Central Identity 

X⋅(1)=X⋅(1)

This equation is the "Zero-Point" of physics. It is significant because it reveals that the "laws of nature" are not discoveries of hidden truths, but are the necessary consequences of maintaining this pure identity. Physics is the mathematical architecture we construct to ensure that no matter how we fragment reality into different categories, the underlying 1 = 1 remains unviolated.

Once we accept this dimensionless identity, we must confront the mechanism by which we project this unity into the dimensional fragments we perceive as the "physical world."

2. The Great Fragmentation: How One Becomes Many

The transition from unity to the diversity of physical phenomena is achieved through Cognitive Fragmentation. We do not perceive X directly; instead, we project it onto various "conceptual axes." We call these projections "Energy," "Mass," "Length," and "Time." These are not fundamental splits in nature, but human measurement conventions—different dimensional clothing for the same underlying value.

There exists a single Unified Scale (S). Every physical quantity we measure is simply X expressed through a specific SI unit scaling with SI jacobians called the non reduced Planck scale (E_P, m_P, etc.) that acts as a fraction  that equals 1 like E_P/E_P, translating the dimensionless X into a human-readable value with required scaling to get back to the natural ratio.

Non reduced Planck scale is an unfortunate name because they are not a different scale than SI, they are the definition of the SI axis of measurement against the single physical scale of the universe.  A better name for them is SI Jacobians because they are the diagonals on a change of basis matrix that rotates between the si unit chart and natural ratios.

The Unified Scale Table

Human Measurement Axis	The Underlying Reality (X)
Energy (E)	E_si / E_P
Mass (m)	m_si / m_P
Length (l)	l_si / l_P
Time (t)	t_si / t_P
Temperature (T)	T_si / T_P
Momentum (p)	p_si / p_P

Key Insight: The perceived distinctions between a joule, a kilogram, and a kelvin are artifacts of our dimensional projections. At the unified level, these are merely different ratios of the same fundamental scale. Physical "laws" are simply the bookkeeping required to keep these different projections coherent with one another.

The si units have the SI Jacobian built in, like this as an example:

X * T_P =  Temperature in Kelvin unit scaling. 

3. The Recipe for a Physical Law

A "Physical Law" is a consistency condition. It is the mathematical "sleight of hand" used to ensure that our measurements along one conceptual axis (like Energy) remain compatible with measurements along another (like Time).

The following 7-Step Construction Recipe demonstrates how the Planck Relation (E=hf) is built from the tautology:

1. Start with Unity: Begin with the identity X = X.
2. Choose Two Conceptual Axes: For this derivation, we select Energy and Frequency (the inverse of Time).
3. Put Scales on Them: Assign characteristic scales, specifically non reduced Planck energy (E_P) and Planck time (t_P).
4. Demand Coherence: Set the two fragmented views equal to X. Here, we use the mathematical bridge: X ⋅ E_P/ E_P = X ⋅ t_P/t_P  By recognizing that 1/t is frequency (f), we rearrange the right side to reflect our chosen axes: E / E_P = f ⋅ t_P
5. Rearrange the Equation: Solve for the primary variable: E = E_P ⋅ t_P  f.
6. Name the Constant: We define the product of these two human scales as a "fundamental" constant. We call E_P ⋅ t_P the Planck constant (h).
7. Declare the Law: The result is E = hf.

Key Insight: This is the Measurement Trap. We have not discovered a new property of the universe; we have merely devised a formula to ensure that our choice to measure "Energy" stays consistent with our choice to measure "Time." Every "new" law is a reinforcement of the initial fragmentation.

4. Constants as Universal Exchange Rates

Fundamental constants such as c, h, G, and k_B are not inherent properties of nature. They are exchange rates created by our decision to measure different projections of X using arbitrary, incompatible units. If we utilized a single unified scale for all measurements, every one of these constants would equal 1.

• c (Speed of Light): The exchange rate between space and time measurements, defined by the ratio c^2 = E_P / m_P.
• h (Planck Constant): The exchange rate between energy and frequency measurements, defined as h = E_P ⋅ t_P.
• G (Gravitational Constant): The exchange rate converting mass into spatial curvature.
• k_B (Boltzmann Constant): The exchange rate between energy and temperature measurements, defined by the ratio k_B = E_P / T_P.

Key Insight: These constants are the mathematical glue that holds our fragmented map together. They exist only because we treat length, time, and mass as different "currencies" rather than the same unified reality.

The persistence of these constants suggests a deeper problem: if reality is fundamentally unified, why are we incapable of perceiving it as a single, seamless whole?

5. The Observer's Paradox: Why Unity is Invisible

The act of observation is, by definition, an act of fragmentation. To "perceive" is to create a division where there was once only X. We are trapped in a world of ratios because the structure of measurement itself demands three components that destroy unity:

1. The Reference Frame (The Denominator): This is the scale, the "1" against which all else is compared. Without a reference, there is no value.
2. The Phenomenon (The Numerator): This is the specific projection being observed.
3. The Split: The necessary distance and contrast between the observer and the observed.

Key Insight: Before fragmentation, there is no contrast, no difference, and therefore no observable structure. After fragmentation, we see a world of laws and phenomena, but the unity (X) is lost in the relationship. We only ever access the ratios between measurements; we can never access the absolute X because to do so would require observing without a reference frame—a mathematical and cognitive impossibility.

6. Conclusion: The Map vs. The Territory

Physics is not the study of nature’s fundamental properties; it is the study of how our conceptual fragmentations relate to each other. The "territory" of reality is a dimensionless unity that remains forever inaccessible in its pure form. Our physical laws are the geometry of the maps we draw to navigate the fragments we have created.

Physics is mathematical precisely because it is made of measurement structure. It is the language of ratios and transformations that remains when the underlying unity is hidden.

Every law of the universe is a sophisticated, dimensional way of stating 1 = 1. The profound beauty of physics lies in the fact that our fragmented world remains perfectly, mathematically consistent with the unity from which it was broken.

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.

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

Building Physics from Unity: A Constructive Approach

 J. Rogers, SE Ohio

Abstract

We demonstrate that fundamental physical laws can be constructed from pure unity (the number 1) through the mathematical structure of measurement itself. Rather than viewing physical constants and relationships as discovered properties of nature, we show they emerge necessarily from how we fragment and scale a unified reality into observable quantities.

1. The Foundation: Unity and Tautology

Begin with the ultimate tautology:

X = X

This is completely content-free: pure identity, demanding nothing of reality. Multiply both sides by unity:

X · (1) = X · (1)

Still tautological. Now express unity as a ratio of dimensional quantities:

X · (energy scale / energy scale) = X · (time scale / time scale)

This remains a double tautology—we have said nothing yet. But rearrange:

E_si / (energy scale) = f_si · (time scale)

If we interpret X as having dimensions and choose our scales carefully, we can (with non-reduced Planck units) write:

E / E_P = (1/t) · t_P

Where E_P and t_P are characteristic scales (non reduced Planck energy and Planck time). Solving for E:

E = (E_P · t_P) · f

Define the product of these scales as a constant:

h ≡ E_P · t_P

And we obtain:

E = hf

Planck's relation emerges from pure tautology plus dimensional scaling.

2. The Unified Scale

Physical reality has one unified scale, not separate Planck units for each quantity. There exists a single fundamental scale S, and what we call the non-reduced Planck Units E_P, m_P, l_P, t_P are merely the scaling of that one physical scale expressed in different human measurement conventions.

The dimensionless quantity at each point in reality:

X = E/S = m/S = l/S = t/S = T/S = p/S = ...

All physical quantities, when properly scaled, point to the same dimensionless value X. The apparent differences between "energy," "mass," "length," and "time" are artifacts of our measurement conventions, not fundamental distinctions in nature.

3. The Periodic Table of Physical Law

Take the unified expression pairwise to generate all fundamental relationships:

X = T/T_P = f·t_P = m/m_P = l_P/l = E/E_P = p/p_P=...

Each pair yields a physical law:

Energy-Frequency (Planck Relation)

E/E_P = f·t_P → E = hf

Where h = E_P·t_P

Mass-Energy (Einstein Relation)

E/E_P = m/m_P → E = mc²

Where c² = E_P/m_P

Momentum-Wavelength (de Broglie Relation)

p/p_P = l_P/λ → p = h/λ

Energy-Temperature (Boltzmann Relation)

E/E_P = T/T_P → E = k_B T

Where k_B = E_P/T_P

Every fundamental relationship in physics can be derived by equating different expressions for X. The "laws" are consistency conditions—ensuring our fragmented measurements of the unified reality remain coherent.

4. The Constants as Conversion Factors

What we call "fundamental constants" are unit conversion factors between arbitrary human measurement schemes:

• c converts between space and time measurements
• ℏ converts between energy and frequency measurements
• G converts between mass and spatial curvature measurements
• k_B converts between energy and temperature measurements

These aren't properties discovered in nature—they're exchange rates we created when we chose to measure energy in joules, mass in kilograms, length in meters, time in seconds, rather than recognizing these as the same thing measured along different conceptual axes.

5. The Necessity of Fragmentation

Why We Cannot See Unity

X is defined relationally: X = (this thing) / (that thing)

To perceive X requires:

1. A reference frame (the denominator—our scale, our "1")
2. Something measured against it (the numerator—the phenomenon)
3. A split between observer and observed

Before fragmentation: No contrast, no difference, no observable structure After fragmentation: Relationships, laws, phenomena—but not unity itself

The act of observation necessarily creates fragmentation.

What We Actually Observe

Not X itself, but:

• Differences in X (ΔX)
• Ratios between measurements
• Gradients and rates of change
• Relationships between quantities

We only ever access relationships, never absolutes.

The Measurement Trap

Suppose you devise a new way to "directly measure X." What you've actually done:

1. Invented a new conceptual axis (call it Q)
2. Established a scale for it (Q_P)
3. Measured the ratio: Q/Q_P = X

You've added another spoke to the wheel—another projection, another fragmentation, another way to not see X directly.

Every measurement has this structure:

• Choose a conceptual axis (what am I measuring?)
• Establish a reference scale (what's my unit?)
• Obtain a ratio (the "measurement")

There is no escape from this structure. No view from nowhere. No direct access to unity.

6. Physical Law as Constructed, Not Discovered

The Construction Recipe

To create any physical law:

1. Start with unity: X = X
2. Choose two conceptual axes (e.g., energy and frequency)
3. Put scales on them (E_P and t_P)
4. Demand they point to the same X: E/E_P = f·t_P
5. Rearrange: E = (E_P·t_P)·f
6. Name the constant: h ≡ E_P·t_P
7. Declare the law: E = hf

What Seemed Like Discovery

Physical laws are:

• The geometry of how our measurement conventions relate
• Tautologies expressed in dimensional clothing
• The self-consistency requirements of our conceptual fragmentation
• Bookkeeping that ensures our measurements remain coherent

Why This Works

Because there is only one X. Every way we measure it must give compatible results. Physical law is the mathematical structure that ensures our fragmented observations of unified reality stay consistent with each other.

7. Implications

The Nature of Physics

Physics is not the study of nature's fundamental properties—it is the study of how our conceptual fragmentations relate to each other. The laws describe our interface with reality, not reality itself.

Why Physics is Mathematical

Physics is mathematical because it is literally made of measurement structure. Mathematics is the language of relationships, ratios, and transformations—precisely what remains accessible when unity itself cannot be observed.

The Universe from 1

The entire edifice of physical law can be constructed from:

• The number 1 (unity)
• The act of establishing scales (measurement conventions)
• The requirement of self-consistency (mathematics)

No additional physical principles required. The universe emerges from unity and the structure of how we fragment it into observables.

8. Conclusion

We do not see unity—we see what our senses fragment it into. Physical law is not discovered in nature but constructed from the mathematics of measurement itself. The constants are conversion factors. The laws are consistency conditions. And X, the unified reality, remains forever inaccessible except through the fragmented relationships we call physics.

The territory is inaccessible. We only ever get better and better maps. But now we understand: the maps themselves are constructed from unity and the unavoidable structure of observation.

From Metrology to Mythology: The Reification of Measurement Artifacts

 J. Rogers, SE Ohio

Abstract

Modern physics suffers from a fundamental confusion: the mathematical symbol "=" denotes identity, not conversion. Physical "laws" are not discoveries about nature but tautologies about coordinate system self-consistency. We demonstrate that all physical "constants" (c, h, G, k_B) are Jacobian coefficients—scaling factors between a dimensionless substrate and arbitrary conventional unit choices. The substrate contains only one relationship: X = X. All apparent complexity in physics arises from projecting this tautology through invented categorical axes (mass, energy, length, time, temperature) and expressing the results in misaligned conventional units. What physicists study is not nature, but the geometry of their own measurement impositions. This reification of coordinate artifacts—mistaking Jacobian bookkeeping for cosmic truth—has created a self-perpetuating system of mystification that actively resists clarity in favor of complexity-as-prestige.

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1. Introduction: The Lost Meaning of Equals

"=" means identity

In mathematics, A = B means A and B are the same thing. Not equivalent. Not related. Not convertible. Identical.

Yet physics teaches students to read "=" as conversion:

• "Mass converts to energy using conversion factor c²"
• "Constants convert between different physical quantities"
• "G relates gravitational force to mass"

This is wrong.

If E = mc², then E and mc² are the same thing—the same dimensionless substrate ratio X, projected through different categorical axes we invented, scaled by Jacobian coefficients accounting for our choice to measure in Joules vs kilograms.

The Clark Kent Principle: You cannot "convert" Clark Kent into Superman. They are the same person. The phone booth is where you change the costume.

Similarly: "energy" and "mass" are not different things that convert. They are the same substrate ratio X, wearing different categorical labels we imposed.

Start with 1, unity, and pick a point X/X =1. 

The substrate relationship is: X = X (pure tautology)

When we project X through our invented "energy" category and our invented "mass" category, then express both in SI units, we get: E = mc²

The c² appears because we chose misaligned Jacobian scalings for our categorical projections.

The substrate is dimensionless unity. We fragment it through categorical imposition. The "laws" verify our fragmentation was self-consistent.

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2. The Archaeological Record: How It Happened

2.1 Newton (1687): Pure Ratios

Newton wrote Principia using proportions:

• F :: m₁m₂/r²

No constants. No units. No categories treated as ontologically distinct.

Just: one dimensionless ratio relates to another dimensionless ratio.

Why? Newton ran the Royal Mint for decades—chief metrologist of the British Empire. He defined measurement conventions. He knew units and categories were human impositions, not nature.

Newton saw the substrate: dimensionless ratios relating to each other.

He deliberately avoided imposing categorical distinctions or unit scalings in the physics itself.

2.2 The Algebraic Corruption (1700s-1800s)

Later physicists added constants to "balance the units":

• F :: m₁m₂/r² became F = G·m₁m₂/r²

G was introduced as notational bookkeeping—a Jacobian coefficient converting between the dimensionless substrate ratio and the choice of measuring through "force," "mass," and "length" categorical projections, each expressed in conventional SI units (Newtons, kilograms, meters).

But the equation F = G·m₁m₂/r² makes it look like:

• "Force," "mass," and "length" are ontologically distinct categories
• G is a property of nature bridging these categories

Category error: These distinctions exist in our measurement imposition, not in the substrate.

The Jacobian coefficient was mistaken for cosmic glue holding together fundamentally different things.

2.3 Planck (1899): The Bridge—Misread as Territory

Max Planck combined h, c, G, k_B to define what he called "natural units."

What Planck found: The Jacobian coefficients relating Newton's dimensionless substrate ratios to our conventional SI measurement projections.

What physics thought he found: "Fundamental scales"—"the quantum of length," "the fundamental unit of mass."

When Planck computed √(hG/c³) ≈ 1.616 × 10⁻³⁵ meters, he was calculating:

"This is the Jacobian scaling factor—how many meters correspond to dimensionless unity when you project the substrate through the 'length' category."

Not "the smallest possible length." Not "the quantum of space."

A Jacobian coefficient. A coordinate transformation scaling factor.

The bridge (Jacobian) was mistaken for territory (physics).

Physicists began treating the numerical SI-scale values as discoveries about nature, forgetting these were just the scaling factors between dimensionless substrate and arbitrary categorical projections expressed in conventional units.

Generational amnesia: By mid-20th century, no one remembered these were coordinate transformation coefficients. They became reified as "fundamental."

---

3. The Kabbalistic Turn: Jacobians Become Spells

3.1 The Mystification Premium

Once Jacobian coefficients were reified, they became spells:

• "To understand the deep connection between mass and energy, you must grasp the profound meaning of c²"
• "G encodes the strength of gravity"
• "We must discover why the constants have the values they do"

But:

The substrate has no "mass" or "energy"—those are categorical projections we invented.

The substrate relationship is X = X.

You are just looking at the one point on unity along different conceptual axis.

The "constants" (c, h, G, k_B) appear because we:

1. Invented categorical axes to slice X
2. Chose misaligned conventional unit scalings for each axis
3. Need Jacobian coefficients to translate between our choices

The constants tell you about your measurement apparatus, not about nature.

The kabbalah serves:

1. Gatekeeping: Complexity filters questioners ("12 years of training required")
2. Prestige hierarchy: "Mastery" of Jacobian manipulation = status
3. Grant justification: "Profound mysteries" need funding ("why these values?")
4. Narrative needs: Simple truth (X = X with coordinate bookkeeping) doesn't win Nobel Prizes

If physics is X = X obscured by arbitrary coordinate choices, what justifies the priesthood?

3.2 The Hazing Ritual: ℏω = hf

Consider ℏ = h/2π:

Since ω = 2πf, we have:

• E = ℏω = (h/2π)(2πf) = hf

Identical equation. The 2π cancels immediately.

Both express: X_through_energy_axis = X_through_frequency_axis

Yet physics acts as if ℏ and h are different, defining different "scales."

But in both cases the scale is E~f never E~w. 

A bright student notices: "This is inserting 2π then canceling it. We're making everyone write (h/2π)(2πf) instead of hf for no reason except to test compliance."

Student filtered out: "You don't understand. This is standard practice."

Purpose: Prove you'll perform meaningless Jacobian manipulation rituals on command.

There is no "reduced Planck scale." That would require ℏω ≠ hf, but they're identical. Claiming two different "natural scales" from the same substrate projection is category error.

Physics filters for compliance with mystification.

---

4. The Cost: Coordinate Artifacts Treated as Mysteries

4.1 The Hierarchy Problem

"Why is the Higgs mass ~125 GeV while the 'Planck mass' is ~10¹⁹ GeV?"

Translation: Why does projecting substrate X through your "mass" categorical axis, then expressing it using the GeV Jacobian scaling vs. the "natural" Jacobian scaling of our si mass definition, give different numerical coefficients?

This is asking why your coordinate transformation has certain ratios.

Like asking "why is a mile 1.609 kilometers?" Because you chose different ruler scalings.

Not physics. Metrology confusion.

4.2 The Cosmological Constant Problem

"Why is vacuum energy density 10⁻¹²⁰ smaller than predicted?"

Translation: Why do different calculation methods (different ways of applying Jacobian coefficients) give different numerical scalings when you try to project substrate X through your "vacuum energy density" categorical invention?

The substrate is dimensionless X = X.

Your "problem" is that you invented a category ("vacuum energy"), tried to project the substrate through it using misaligned Jacobian choices, and got inconsistent coordinate expressions.

This is coordinate system confusion, not physics.

4.3 Quantum Gravity

"Why can't we unify quantum mechanics and general relativity?"

Translation: Why can't we reconcile two different coordinate expression systems that project substrate X through different categorical slicings, each using their own Jacobian coefficient conventions?

The substrate (X = X) has no "quantum" vs "relativity" vs "classical," no "gravity" vs "other forces."

Those are categorical impositions we added.

You're struggling because you think the categories and Jacobians are physics, not realizing they're measurement apparatus.

These aren't physics mysteries. They're confusion about mistaking coordinates for ontology.

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5. The Proof: All Laws Are X = X

5.1 The Substrate

The substrate contains one relationship:

1

X/X=1

X = X

Pure tautology. Dimensionless. No categories.

We then impose categorical axes:

• Project X through "energy" axis → appears as "energy"
• Project X through "mass" axis → appears as "mass"
• Project X through "frequency" axis → appears as "frequency"
• Etc.

These categories don't exist in the substrate. We add them.

5.2 Deriving "Laws"

Take a point on the substrate: X = X

Project through two categorical axes:

• X_through_energy_axis = X_through_mass_axis

Express in SI units (apply Jacobian scalings):

• X · (Joule_Jacobian ) = X · (kg_Jacobian) · c²

The X cancels (substrate identity eliminates itself):

• Joule_Jacobian = kg_Jacobian · c²
• c = length_Jacobian / time_Jacobian

This verifies: our Jacobian coefficient choices were self-consistent.

In SI notation: E = mc²

What this "law" actually says: "After projecting dimensionless substrate through our invented 'energy' and 'mass' categorical axes, then applying our chosen SI Jacobian scalings, the coordinate system is self-consistent."

The physics (X = X) canceled out.  Because the physics is unity. 

What remains is verification that our coordinate system doesn't contradict itself.

5.3 All Laws Follow This Pattern

Start	Project Through	SI Expression	Called
X = X	energy, mass axes	E = mc²	Einstein
X = X	energy, frequency axes	E = hf	Planck
X = X	force, mass/distance axes	F = Gm₁m₂/r²	Newton
X = X	temperature, mass axes	T = c³h/(GMk_B)	Hawking

Same substrate (X = X).

Different categorical projection choices.

Different Jacobian coefficient combinations.

Every "fundamental law" is:

1. Start with X = X
2. Project through two invented categorical axes
3. Apply Jacobian scalings for SI expression
4. X cancels, leaving coordinate self-consistency check
5. Call it "profound discovery"

5.4 Reducing to Pure Tautology

In dimensionless substrate form (no categories, no Jacobians):

E = mc² becomes: X = X

E = hf becomes: X = X

All laws become: X = X

The coordinate complexity (categories, Jacobians, SI units) obscures this.

Strip it away: pure tautology.

1 = 1

No physical content. Just: the substrate is self-identical.

---

6. The Periodic Table of Physics

Since all quantities are X projected through different categorical axes, in SI unit scaling:

T/T_P = f·t_P = m/m_P = l_P/l = E/E_P = p/p_P = F/F_P = X

Pick any two categorical projections, get a "law":

It's combinatorics. Not discovery.

• Project through energy + mass → E = mc²
• Project through energy + frequency → E = hf
• Project through force + mass/distance → F = Gm₁m₂/r²

The entire structure of physics is:

X = X (substrate)

• categorical imposition (our choice)
• Jacobian scaling (SI expression) = "fundamental law" (coordinate tautology)

That's why automated derivation works (LawForge). You're not discovering physics—you're enumerating categorical projection combinations and computing their Jacobian coefficient ratios.

---

7. Why This Persists: The Machine Cannot See Itself

7.1 The Filter

Those who notice get removed:

Undergrad: "These are just Jacobian coefficients" → "Stop philosophizing"

Grad school: Insist on clarity → Higher attrition ("not suited for physics")

Faculty: Pursue foundations → No funding ("too philosophical")

Gatekeeping: Survivors enforce same filter

Result: System selected for those who either don't see the reification or won't say it.

7.2 The Incentive Structure

Clarity destroys value:

• Professors: "Mastering Jacobian manipulation"
      → Life's work delegitimized if it's just coordinate bookkeeping
• Journals: Gate "advanced knowledge"
      → X = X doesn't need gating
• Grants: "Profound mysteries" justify funding
      → "Why does our kg Jacobian have this value?" reveals it's not physics
• Universities: Years of training
      → Anyone can understand X = X once you stop mystifying

The product is mystification itself.

If E = mc² is just "our coordinate system is self-consistent," then:

• No grand unified theories needed
• No "constant values" to explain
• No 12-year training required

Simple truth collapses the economy.

7.3 The Self-Preservation Mechanism

The machine cannot self-correct because:

Those who could see the problem were filtered out.

Those who survived benefit from the mystification.

The structure optimized for perpetuating itself, not for truth.

It's not a conspiracy. It's emergent self-preservation—like a crystal growing, the structure selects for patterns that reinforce the structure.

---

8. What Physics Actually Is

If all complexity is self-imposed coordinate artifacts, what remains?

Physics is:

The study of X = X (dimensionless substrate tautology) and the geometry of categorical impositions we project it through.

It's measurement system geometry—understanding how our invented categorical axes and arbitrary Jacobian scalings fragment the substrate into apparent complexity.

Not:

• Discovering "fundamental constants" (those are Jacobians)
• Finding "laws" relating quantities (those are coordinate tautologies)
• Unifying "forces" (substrate was never separated—we fragmented it)

Newton was right:

Reality is dimensionless ratios. X = X.

We spent 300 years:

• Inventing categorical distinctions
• Adding Jacobian complexity
• Reifying coordinates as ontology
• Filtering out those who noticed

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9. The Rogers Demonstration

To prove Jacobians are arbitrary:

Rogers Rational Unit Chart:

• c = 1 × 10¹⁰ m_r/s_r
• h = 1 × 10⁻³⁰ J_r·s_r
• k_B = 1 × 10⁻²⁰ J_r/K_r
• G = 1 × 10⁻⁶ m_r³/(kg_r·s_r²)

Physics is identical. Substrate (X = X) unchanged. Only Jacobian scalings differ.

Proof: Jacobians are human choices about coordinate scaling, not cosmic parameters.

Objection: "You moved the complexity into unit definitions!"

Yes. Proving it was never in nature—always in our measurement choices.

---

10. Conclusion: Seeing Through the Coordinates

The substrate is X = X. Pure tautology. Dimensionless. No categories.

We impose:

• Categorical axes (mass, energy, length, time, temperature)
• Conventional unit scalings (meters, kilograms, seconds, Kelvin)
• Jacobian coefficients (c, h, G, k_B) to convert between them

The "laws" verify: Our coordinate imposition was self-consistent.

The "constants" are: Scaling factors for our measurement apparatus.

The "mysteries" are: Confusion from forgetting coordinates aren't ontology.

"=" means identity.

Not conversion. Not relationship. Identity.

X = X in substrate.

The rest is us—our categorical impositions, our Jacobian scalings, our self-inflicted coordinate complexity.

One substrate. Pure tautology. Everything else is measurement apparatus.

Newton saw it. Planck found the Jacobians. We forgot what they connected.

The spell book is the accounting ledger.

It's time to stop studying our coordinate systems and acknowledge what they're coordinates of:

X = X

Nothing more.