Abstract: We agree completely with the common critique that the framework of natural units is "just dimensional analysis and unit scaling." This paper demonstrates that this is not a critique but a correct, if incomplete, description of the nature of physical law itself. We will show that the "fundamental constants" of physics (c, h, G, k_B) are not properties of the universe, but are mathematically exact composite artifacts that arise solely from the relationship between our arbitrary SI coordinate system and the coherent, underlying Planck scale. Their existence and values are a feature of our chosen chart, not the territory. Note that these are the original planck units are are not reduced in any way. 1/2pi applied to unit scaling is a category error.
1. Introduction: A Statement of Agreement
The most potent and frequent criticism leveled against any attempt to unify physics through the lens of measurement is the assertion that the work is "just dimensional analysis and unit scaling." The intended purpose of this statement is to dismiss the work as trivial, a mere repackaging of undergraduate-level techniques that reveals nothing new about reality.
We agree with the premise of this criticism in its entirety.
The purpose of this paper is to demonstrate that what is perceived as a trivial technique is, in fact, the fundamental mechanism by which the apparent complexity of physical law is generated. The "fundamental constants" are not discovered; they are constructed. They are the precise mathematical artifacts of the scaling relationship between our human-centric SI unit system and the natural, unified Planck system.
2. The Two Coordinate Systems
To understand how the constants are constructed, one must first recognize the two distinct coordinate systems in play:
The Planck System (The Substrate Coordinates): This is the natural coordinate system of the universe. It is defined by four indivisible, fundamental units: the Planck Length (l_P), Planck Mass (m_P), Planck Time (t_P), and Planck Temperature (T_P). In this system, all of physics is simple because the base units are inherently coherent. The "value" of each base unit in its own system is simply 1.
The SI System (The Human Coordinates): This is our arbitrary, historical coordinate system. It is defined by the meter, the kilogram, the second, and the Kelvin. These units are based on human scales and historical accidents. They bear no intrinsic relationship to one another, and are thus a "misaligned" or "non-orthogonal" basis with which to view reality.
Physics as we know it is the process of describing the simple phenomena of the Planck system using the clumsy, misaligned language of the SI system. To do this, we require a dictionary of conversion factors. These conversion factors are what we call "fundamental constants."
3. The Construction of the Constants from Unit Scaling
A "fundamental constant" is the fixed numerical ratio that emerges when one measures a relationship between Planck units using SI units. The constants are not discovered within the Planck system; they are the result of projecting the Planck system onto the SI system.
Let us construct each constant explicitly.
c, The Speed of Light
In the Planck system, the natural unit of length is l_P and the natural unit of time is t_P. The natural speed—the rate at which causality propagates across one unit of length per one unit of time—is, by definition, l_P / t_P.
When we measure this natural speed using our SI units, we get a specific numerical value. We name this value c.
c = l_P / t_P
The constant c is not the speed of light. It is the scaling factor that dictates the relationship between our meter and our second, by measuring them against the universe's natural length and time.
h, The Planck Constant
In the Planck system, the natural unit of energy is E_P = m_P l_P² / t_P², and the natural unit of frequency is f_P = 1 / t_P. The relationship between them is direct. The constant of proportionality that relates energy to frequency is the Planck constant, h. We can define it as the natural unit of action, m_P l_P² / t_P.
h = m_P l_P² / t_P
The constant h is not the quantum of action. It is the scaling factor required to convert a measurement of mass, length, and time in our SI system into a coherent expression of frequency, as defined by the Planck scale.
G, The Gravitational Constant
In the Planck system, the force between two Planck masses separated by a Planck length over a Planck time is the Planck force. The constant that governs this interaction is G. Its structure is revealed when expressed in Planck units.
G = l_P³ / (m_P t_P²)
The constant G is not the intrinsic strength of gravity. It is the complex scaling factor required to make our independent, arbitrary units of meters, kilograms, and seconds correctly describe the interaction between two mass-concentrations in the substrate.
k_B, The Boltzmann Constant
In the Planck system, temperature is a measure of energy. The natural unit of temperature is T_P, and the natural unit of energy is E_P = m_P l_P² / t_P². The constant of proportionality between them is the Boltzmann constant, k_B.
k_B = m_P l_P² / (t_P² * T_P)
The constant k_B is not a law of thermodynamics. It is the purely historical scaling factor needed to convert our subjective, human-scale perception of "hotness" (Kelvin) into the actual physical quantity of energy, as defined by the Planck scale.
4. Conclusion: The Triviality is the Point
The critics are correct. This is just dimensional analysis and unit scaling.
They fail to realize that this is all that the "fundamental constants" are. They are not deep properties of the universe waiting to be discovered. They are the Jacobian coefficients of the coordinate transformation between the human world of measurement and the real world. They have no existence or meaning outside of our chosen chart. Every unit chart has its own version of the constants that perform the same task, to scale those axis of measurement to the natural unified ratios of the universe.
The entire edifice of physical law, with its beautiful and complex formulas, is a grand and intricate tapestry woven from these simple scaling factors. The complexity is not in the universe; it is in our description. The laws are the emergent patterns that appear when one simple reality is viewed through our complex, misaligned lens.
Therefore, we do not reject the critique. We embrace it as the central truth. The goal of fundamental physics is not to measure these scaling factors with ever-greater precision, but to perform the coordinate transformation—to recognize that the Planck system is the true origin of all unit charts—and in so doing, to that the constants are only present because we have scaled unity differently along every axis of measurement.
Appendix A: Numerical Verification
Using the CODATA 2018 values for the SI-expressed Planck units:
l_P = 1.616255 x 10⁻³⁵ m
t_P = 5.391247 x 10⁻⁴⁴ s
m_P = 2.176434 x 10⁻⁸ kg
T_P = 1.416784 x 10³² K
We can construct the constants:
c_calculated = l_P / t_P
h_calculated = m_P * l_P**2 / t_P
G_calculated = l_P**3 / (m_P * t_P**2)
k_B_calculated = m_P * l_P**2 / (t_P**2 * T_P)
=====================================================================================
CONSTRUCTION OF FUNDAMENTAL CONSTANTS FROM PLANCK UNIT SCALING
=====================================================================================
Constant | CODATA 2018 Value | Calculated from Planck Units | Ratio
-------------------------------------------------------------------------------------
c | 299792458.0 m/s | 299792458.0 | 1.00000000
h | 6.62607015e-34 J·s | 6.62607015e-34 | 1.00000000
G | 6.674300e-11 N·m²/kg² | 6.674300e-11 | 1.00000000
k_B | 1.380649e-23 J/K | 1.380649e-23 | 1.00000000
-------------------------------------------------------------------------------------
Conclusion: The 'fundamental constants' are not fundamental.
They are the precise, composite scaling factors that emerge from
expressing the coherent Planck scale using the arbitrary SI system.
import math
from math import sqrt
# --- CODATA 2018 Recommended Values ---
# Official SI values for the "Fundamental Constants" for comparison
c = 299792458.0
h = 6.62607015e-34
G = 6.67430e-11
k_B = 1.380649e-23
l_P = sqrt(h * G / c**3) # Planck Length (m)
t_P = sqrt(h * G / c**5) # Planck Time (s)
m_P = sqrt(h * c / G) # Planck Mass (kg)
T_P = sqrt(h * c**5 / G) / k_B # Planck Temperature (K)
# --- Constructing the Constants from Planck Unit Scaling ---
print()
# c is the scaling factor between the Planck length and Planck time.
print (" c_calculated = l_P / t_P")
c_calculated = l_P / t_P
# h is the scaling factor for action (mass * length^2 / time).
print (" h_calculated = m_P * l_P**2 / t_P")
h_calculated = m_P * l_P**2 / t_P
# G is the scaling factor for gravitational interaction (length^3 / (mass * time^2)).
print (" G_calculated = l_P**3 / (m_P * t_P**2)")
G_calculated = l_P**3 / (m_P * t_P**2)
# k_B is the scaling factor between energy (mass * length^2 / time^2) and temperature.
print (" k_B_calculated = m_P * l_P**2 / (t_P**2 * T_P)")
k_B_calculated = m_P * l_P**2 / (t_P**2 * T_P)
print()
# --- Calculate Ratios for Comparison ---
# The ratio will be 1.0 if the construction is identical to the official value.
c_ratio = c_calculated / c
h_ratio = h_calculated / h
G_ratio = G_calculated / G
k_B_ratio = k_B_calculated / k_B
# --- Print Results in a Formatted Table ---
print(" ", "="*85)
print(" CONSTRUCTION OF FUNDAMENTAL CONSTANTS FROM PLANCK UNIT SCALING")
print(" ", "="*85)
print(f" {'Constant':<10} | {'CODATA 2018 Value':<24} | {'Calculated from Planck Units':<30} | {'Ratio':<10}")
print(" ", "-"*85)
print(f" {'c':<10} | {c:<9.1f} m/s | {c_calculated:<30.1f} | {c_ratio:<10.8f}")
print(f" {'h':<10} | {h:<9.8e} J·s | {h_calculated:<30.8e} | {h_ratio:<10.8f}")
print(f" {'G':<10} | {G:<9.6e} N·m²/kg² | {G_calculated:<30.6e} | {G_ratio:<10.8f}")
print(f" {'k_B':<10} | {k_B:<9.6e} J/K | {k_B_calculated:<30.6e} | {k_B_ratio:<10.8f}")
print(" ", "-"*85)
print(" \nConclusion: The 'fundamental constants' are not fundamental.")
print(" They are the precise, composite scaling factors that emerge from")
print(" expressing the coherent Planck scale using the arbitrary SI system.")
The identities are not just conceptual; they are numerically exact.
Appendix C: On the Charge of Circular Reasoning - The Logic of Factoring
A common and understandable objection to the methodology presented in this paper is the charge of circular reasoning. The critique can be summarized as follows: "You have used the constants (G, h, c) to define the Planck units (l_P, m_P, t_P), and then used the Planck units to reconstruct the constants. This is a self-referential loop that proves nothing."
This objection, while appearing valid, misinterprets the purpose of the exercise. The goal is not to prove a physical proposition via a linear, deductive argument. The goal is to demonstrate the structural identity of a definitional system through the process of factoring and reconstruction.
1. The Distinction: A Fallacious Proof vs. A Valid Definition
A fallacious circular proof attempts to prove a proposition about the world is true by assuming its own conclusion. For example: "My theory is correct because the data supports it, and the data is valid because my theory says it is." This is a fallacy because it adds no new knowledge and proves nothing about external reality.
What is presented in this paper is a valid definitional analysis. We are not proving that G must have its specific value. We are demonstrating what that value represents by breaking it down into its constituent parts. This is analogous to factoring a composite number.
2. An Analogy: Factoring a Composite Number
Consider the number 30. This number is a composite entity. We can analyze it by finding its prime factors:
30 = 2 × 3 × 5
This is the process of analysis, or factoring. We have taken a complex, composite value (30) and broken it down into simpler, more fundamental components (2, 3, 5).
Now, we can perform the inverse operation of synthesis, or reconstruction:
2 × 3 × 5 = 30
This is not a circular proof that 30 is 30. It is the verification that our factorization was correct and complete. The fact that the loop closes perfectly is the proof that we have correctly identified the fundamental components of the original number.
3. Applying the Analogy to Physical Constants
The "fundamental constants" are the composite numbers of physics. They are the complex values we encounter first. The Planck units are their "prime factors."
Analysis (Factoring): When we derive the Planck units from the constants (e.g., l_P = sqrt(hG/c³)), we are performing a factorization. We are asking, "What are the simpler, underlying units of Mass, Length, and Time that compose this complex, multi-dimensional constant G?" We are isolating the simple factors.
Synthesis (Reconstruction): When we then use these isolated factors to reconstruct the constants (e.g., G = l_P³ / (m_P t_P²)), we are performing the verification step.
The perfect 1.00000000 ratios in the verification table are not a fallacious proof of a physical law. They are the mathematical confirmation that our factorization of the SI unit system was successful.
4. Conclusion: A Structural, Not Physical, Proof
The circularity is not a flaw in the argument; it is the discovered property of the system being analyzed. We have not proven a new fact about the universe. We have proven a profound fact about the structure of our description of the universe.
The analysis demonstrates that the set of constants (G, h, c, k_B) and the set of Planck units (l_P, m_P, t_P, T_P) are definitionally equivalent. They are two different bases for the same underlying system of relationships. The perfect, circular closure of the logic is the ultimate proof that the constants are not fundamental entities, but are composite artifacts of our chosen coordinate system—epicycles constructed from the simpler, cleaner factors of the Planck scale.
We are not proving a proposition. We are revealing a structure. The argument is not circular; it is a successful demonstration of a closed, self-consistent, and entirely human-constructed definitional system.
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