Abstract
This conflation, analogous to hard-coding configuration data in a software engine, has created conceptual "technical debt," treating interface coefficients—the fundamental constants—as if they were ontological truths.
We ground our ontology in the historically robust concept of
This framework refactors our understanding of physical law, clarifying which aspects are fundamental and which are artifacts of our descriptive architecture.
1. The Great Conflation: A Systems Analysis of Physics
This is analogous to writing a weather app and hard-coding the ZIP code "90210" into the core physics engine. The app would work perfectly for Beverly Hills, but the architecture would be fundamentally flawed.
1.1 The Primitive Distinction: Ratio vs. Constant
Ratio: The fine-structure constant, , and the geometric constant are already scaled to the universe's single physical scale. They are dimensionless. They are invariant, fundamental relationships that require no unit system and whose numerical values are the same in any description of reality. They reside purely in the ontological layer.
Constant: The speed of light, , has units. Its numerical magnitude is scaled by our chosen unit system. Change the definition of the meter or second, and the number changes. It is not a fundamental physical magnitude but an interface coefficient—a conversion factor between the invariant ontology (where the space-time ratio exists) and the contingent, human-defined unit chart.
If it has no units, it is a ratio belonging to the ontology. It is real physics in the one physical scale of the universe.
If it has units, it is a constant belonging to the interface. It is a rotation between conceptual axis in our perception.
Physics conflates the two.
2. The Ontological Layer: Grounding Physics in Newton's Ratios
Newton's law of universal gravitation, F ∝ m₁m₂ / r², is the
The ratio of gravitational forces between two different planetary systems is an invariant dimensionless number; this is the physics. The specific units and numerical value is fully contingent on how you define your unit chart.
3. A Constructive Proof: The Rogers Rational Unit Chart (RUC)
We set the mantissa to 1 This is the single physical scale in the universe. The axis of measurement all align to that single scale in the mantissa.We keep the magnitude close to current powers - Just for human convenience.
- This lets us say, "We drove 5km to the store to buy a 2 litter drink."
- This is just how we perceive how things scale one to the other in our limited sensory experience.
- It is not physics beyond folks studying human perception systems.
- Subjective human experience.
Define Target Values: Choose simple, rational target values for the constants in the new unit system: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)
Solve for the Jacobian: Solve the system of equations to find the precise conversion factors between SI and the new RUC base units. This is just the non reduced Planck units that define our unit scaling against the universe's single physical scale.
4. Why This Conflation Occurred: Technical Debt in Physics
The Success Trap: The conflated system was "good enough" to produce incredibly accurate predictions. When a system works, the motivation to refactor its underlying architecture is low.
GPS is accurate to a few centimeters.
We can predict to high precision.
But accuracy is only a necessary but not sufficient basis for a theory, theories have to tell us why things happen, not just describe them.Lack of Cross-Disciplinary Tools: Physicists are experts in tensor calculus, not information architecture or systems analyst. The concept of "Separation of Concerns" was not part of their standard toolkit.
The lack of cross training in other fields was self inflicted. And physics intentionally excised natural philosophy in the early 20th century. They knowingly embraced naive instrumentalism.
Even sub fields of physics were forced to stay in narrow arbitrary areas. Institutional silos prevent someone in one field from giving criticism against other fields.Cultural Inertia: The constants were taught as sacred, fundamental truths. Questioning them was seen as questioning the foundation of physics itself. Questioning fundamentals is highly discouraged. Career ending if you keep asking foundational questions or continue to question research directions.
4.1 "You're Just Hiding the Ugly Numbers"
4.2 Natural Units: The Unformalized Miracle
The "handwave" described above reveals a critical lacuna in the standard presentation of physics: the routine use of natural units lacks rigorous justification. Every graduate student learns to "set ℏ = c = 1," but this maneuver is treated as a computational shortcut, a convenient notational trick. This paper provides the missing formal architecture, showing that natural units are not a trick at all—they are the inevitable consequence of properly separating ontological concerns from ergonomic ones.
The Standard Framework's Lack of Rigor
In contemporary physics education and practice, the transition to natural units is presented with profound casualness:
The Magical Incantation: "Before we begin, we set ℏ = c = G = k_B = 1." This is stated without ontological justification. It is a fait accompli, a suspension of the rules that governed the introductory chapters of the same textbook.
The Unanswered Question: Why is this allowed? The standard answer—"because units are arbitrary"—is insufficient. If units are truly arbitrary, why does setting these specific constants to unity have such profound simplifying power, revealing deep connections (like the Planck scale) and making dimensional analysis powerfully predictive? The framework provides no answer.
The Inconsistent Narrative: Students are first taught that is a fundamental, measured truth about the cosmos. They are then told to ignore this "truth" and treat it as a dimensionless number 1. The cognitive dissonance is never resolved. The constants are simultaneously sacred physical truths and disposable conversion factors.
This is more than a pedagogical oversight; it is a symptom of the Great Conflation. By failing to separate the Ontology from the Interface, the standard framework cannot explain why the natural units maneuver works. It treats a profound architectural insight as a mere algebraic simplification.
This Paper as Formal Foundation
The three-layer model provides the rigorous foundation that natural units have always lacked:
Step 1: Separation of Concerns. We explicitly recognize the Interface layer (the dimensionful constants) as distinct from the Ontology layer (the dimensionless ratios).
Step 2: Choosing the Canonical Chart. The "mantissa = 1" algorithm for the RUC is not an arbitrary choice; it is the only choice that nullifies the Interface Jacobian. By setting our target constants to simple rational numbers (typically 1), we are defining our unit chart such that the transformation from the ontological ratios is the identity transformation. In the equation:
choosing , etc., forces the Jacobian matrix to be the identity matrix for those relationships. Our unit chart becomes aligned with the intrinsic scaling of the physical laws.
Step 3: The Revelation. What physicists call "natural units" is now understood precisely as working directly in a unit chart specifically designed to make the Interface Jacobian trivial. The "handwave" is the act of selecting this privileged chart. The miraculous simplification that follows is not magic—it is the expected result of switching from a convoluted, historically-accidental coordinate system (SI) to the coordinate system that most directly expresses the ontological structure.
The Implications of Formalization
This formalization resolves the inconsistencies of the standard framework:
It Justifies the "Arbitrariness": Units are arbitrary only within the Ergonomics layer. The choice to set certain constants to 1 is not arbitrary; it is the selection of the unique chart that maximizes transparency between the Ergonomics and Ontology layers. Other choices (like SI) are valid but obfuscating.
It Demystifies the Constants: The constants are no longer baffling numbers to be venerated. They are precisely the components of the transformation matrix from the universe's native scaling (the Ontology) to the particular human scale (SI) we inherited from history. Their numerical values in SI are answers to the question: "How misaligned are our traditional definitions of meter, kilogram, and second from the intrinsic scales suggested by the laws of physics?"
It Clarifies What is Fundamental: The real, non-negotiable physics resides in the dimensionless ratios that remain invariant under any change of unit chart, including the switch to natural units. The fine-structure constant is fundamental; the numerical value of h in J·s is not.
Conclusion: From Convenience to Necessity
Natural units are therefore not merely convenient. They are the inevitable endpoint of a rigorous analysis that seeks to separate the invariant core of physical law from the contingent apparatus of its human description. The fact that physicists instinctively reach for this tool—despite its casual presentation—is powerful empirical evidence that the three-layer architecture described here reflects the underlying logical structure of our theories.
This paper provides the formal justification that has been missing for a century. It converts the physicist's intuitive handwave into a principled architectural decision, revealing that the power of natural units stems from their role as the canonical coordinate system for the ontology of physics. The "trick" works because it is not a trick at all—it is the correct way to read the blueprint.
4.3 The Inevitability of Resistance: Defending an Architectural Clarification
Any framework that claims to refactor a field's foundational understanding must anticipate and address the legitimate sources of resistance. The argument presented here—that dimensional constants are interface coefficients, not ontological truths—will be challenged on mathematical, philosophical, and sociological grounds. We address these challenges directly, showing that resistance stems not from flaws in the architecture, but from the inertia of the conflated system it seeks to clarify.
Mathematical Resistance: "This is Just Dimensional Analysis"
Critique: "You've rediscovered trivial dimensional analysis and repackaged it as philosophy. The Buckingham π theorem already tells us we can form dimensionless groups. Choosing units to set constants to 1 is old news."Rebuttal: Dimensional analysis is the symptom, not the diagnosis. The Buckingham π theorem operates within the conflated system. It takes the dimensional constants as given inputs and finds dimensionless combinations. Our framework explains why those dimensional constants appear as inputs in the first place: they are the Jacobian from a coherent ontology to an incoherent unit chart. We don't just use the tool; we explain why the tool exists and what its structure reveals about the architecture of theory. This is the difference between using a compiler and understanding that a high-level language is being translated to machine code.
Philosophical Resistance: "You're Demoting Profound Truths"
Critique: "You reducec, the cosmic speed limit, and ħ, the quantum of action, to mere conversion factors. This strips them of physical meaning and diminishes a century of profound discovery."Rebuttal: This critique confuses significance with architectural role. The fact that the speed of light
c defines the relationship between space and time intervals is profoundly meaningful. That meaning resides in the Ontological Layer—in the universal, invariant ratio between space and time in spacetime geometry. The number 299,792,458 does not contain that meaning; it is the translation of that meaning into the specific, arbitrary languages of "meters" and "seconds." Our framework elevates the invariant relationship and correctly demotes the numerical translation key. It does not demote the truth; it purifies it from historical accident.Sociological Resistance: "This Solves Nothing Practical"
Critique: "This is philosophy, not physics. My GPS works, my particle collider produces data, and my quantum computer simulates molecules. Your 'architectural debt' doesn't affect predictions or engineering."
Rebuttal: This is the "success trap" identified in Section 4. Correct architecture is not about immediate utility; it's about conceptual clarity and long-term evolvability. A codebase full of hard-coded values (like if (zip_code == 90210)) can still function perfectly. The debt becomes apparent when you need to extend the system, fix subtle bugs, or onboard new developers who can't understand why the rules are what they are. Similarly, the conflation:
Generates pseudo-problems like the fine-tuning mystery, wasting intellectual energy.
Obscures foundational questions by conflating human convention with natural law.
Creates pedagogical confusion where students must accept "fundamental constants" that are later treated as disposable.
Clearing this debt is an investment in the conceptual integrity of the field, making it more navigable and its true foundations more transparent.
The Core Unmasking: Why the Attack Will Be Vociferous
The resistance will be intense because this framework performs an unmasking. It shows that a core practice of theoretical physics—the use of natural units—has been operating on an intuition that contradicts the field's own standard narrative. The attack is not on the mathematics, but on the story physics tells about itself.
The standard narrative is: "We discover fundamental constants that are mysterious numbers. Sometimes, for convenience, we set them to 1, but that's just a trick."
Our narrative is: "The fundamental relationships are dimensionless. 'Constants' are translation coefficients. Setting them to 1 isn't a trick; it's switching to the universe's native coordinate system where the translation is trivial."
The second narrative robs the first of its mystical aura. It turns "solving the mystery of the constants' values" from a deep ontological quest into a historical study of metrology. This is threatening to a culture that has built pedagogical and popular narratives around these mysteries.
Conclusion: An Invitation to Clarity, Not a Declaration of War
This framework is not an indictment of past physics, but a maturation of its architectural understanding. Every engineer knows that successful prototypes often have messy, hard-coded parameters. The mark of a field's maturity is its willingness to refactor successful prototypes into clean, well-architected systems.
We offer this not as a revolution, but as a formalization of the intuition that has guided the best theorists for decades. The handwave of natural units was the clue. This paper follows that clue to its logical foundation. The resulting architecture does not change a single prediction, but it changes everything about how we understand what our predictions mean. It is an invitation to stop venerating the compass and start understanding the map.
Section 4.4: The 2019 Metrological Admission: When Constants Became Definitions
The Quiet Revolution in Metrology
On May 20, 2019, the International System of Units underwent its most significant revision since its inception. The change was presented as a technical refinement—making measurements more precise and stable by fixing the numerical values of fundamental constants. In reality, it was an inadvertent architectural admission that validates the central thesis of this paper: the so-called "fundamental constants" are interface coefficients, not ontological truths.
What Actually Changed
The Old SI (pre-2019): The Measurement Paradigm
Prior to 2019, the SI system operated under a hybrid model:
- Base standards were physical artifacts or phenomena:
- The kilogram: A platinum-iridium cylinder in Paris (Le Grand K)
- The meter: Defined via the speed of light
- The second: Defined via cesium-133 hyperfine transition
- Constants were measured quantities with uncertainty:
- h = 6.62607015(10) × 10⁻³⁴ J·s
- The value had uncertainty in the last digits
- Experiments progressively refined the measurements
- The Planck constant was something we discovered about nature
This model supported the narrative that physicists were discovering fundamental truths about the universe, approaching ever-closer to the "true values" that nature had chosen.
The New SI (post-2019): The Definition Paradigm
The 2019 redefinition inverted this structure entirely:
- Constants are now exact by definition:
- h = 6.62607015 × 10⁻³⁴ J·s (exactly, zero uncertainty)
- c = 299,792,458 m/s (exactly, zero uncertainty)
- e = 1.602176634 × 10⁻¹⁹ C (exactly, zero uncertainty)
- k_B = 1.380649 × 10⁻²³ J/K (exactly, zero uncertainty)
- Base units are now derived from these constants:
- The kilogram is defined: "The kilogram is defined by taking the fixed numerical value of the Planck constant h to be 6.62607015 × 10⁻³⁴ when expressed in the unit J·s, which is equal to kg·m²·s⁻¹."
This creates a circular definition chain:
h [exact] → defines kg
kg → defines Joule (kg·m²·s⁻²)
Joule → defines h (J·s)
The Architectural Implications
Implication 1: Constants Are Design Choices
The smoking gun question: If h is a fundamental property of nature, why did a committee vote on its exact numerical value?
The answer is inescapable: the numerical value 6.62607015 × 10⁻³⁴ is not a discovered truth—it is a chosen convention. The General Conference on Weights and Measures (CGPM) selected this specific number in 2019 because:
- It was consistent with the best measurements available at the time
- It provided continuity with existing SI magnitudes
- It was convenient for maintaining technological infrastructure
Could they have chosen differently?
Absolutely. The committee could have voted for:
- h = 6.626 × 10⁻³⁴ (rounding for simplicity)
- h = 7.0 × 10⁻³⁴ (even simpler)
- h = 1.0 × 10⁻³⁰ (the RUC proposal)
Physics would remain unchanged. All physical relationships would hold. Only the numerical values in our formulas would differ. The choice of 6.62607015 reflects historical continuity, not physical necessity.
This is precisely what our framework predicts: constants are interface coefficients—Jacobian elements that depend on the choice of unit chart. The 2019 redefinition proves this by making the choice explicit.
Implication 2: Measurement Axes Are Circular
The new SI creates an extraordinary circularity:
Define c = 299,792,458 m/s (exact)
↓
Meter defined as distance light travels in 1/299,792,458 seconds
↓
Second defined via cesium-133 frequency
↓
Frequency measured in Hertz (1/s)
↓
Which requires the definition of the second...
Similarly for mass:
Define h = 6.62607015 × 10⁻³⁴ J·s (exact)
↓
Kilogram defined via h
↓
Joule defined as kg·m²·s⁻²
↓
Which requires the definition of the kilogram...
In the old narrative, this would be scandalous: "How can we define units circularly? Aren't we supposed to be measuring objective reality?"
In our framework, this is expected and appropriate: We are choosing a coordinate system. The axes are defined relative to each other. The "circularity" is the explicit acknowledgment that our measurement chart is a self-consistent coordinate grid, not a discovery of absolute scales.
The base category 𝔅 contains dimensionless ratios. The unit chart 𝕌 is a choice of how to coordinatize those ratios. Circularity in the definitions simply reflects that we're specifying a coordinate system's internal consistency, not measuring external truth.
Implication 3: The Committee Chose the Interface
The 2019 CGPM meeting was, in the language of this paper, a meeting where experts explicitly chose Interface coefficients.
What the committee thought they were doing: "We are making the SI more stable by anchoring it to fundamental constants of nature."
What they actually did: "We are choosing specific numerical values for the Jacobian transformation between the ontological ratios (Newton's dimensionless proportionalities) and our preferred ergonomic coordinate system."
The vote to set h = 6.62607015 × 10⁻³⁴ was not a discovery—it was a design decision about the Interface layer.
Implication 4: "Fundamental" Is a Category Error
The term "fundamental physical constants" is now formally incoherent in post-2019 SI.
A "constant" in physics traditionally meant:
- A quantity whose value we measure
- Something that could, in principle, have been different
- A number that experimental refinement progressively reveals
But post-2019:
- We define the values by fiat
- They cannot be different (by definition)
- No experiment can ever "refine" h—it is exactly 6.62607015 × 10⁻³⁴ forever
This is definitional language, not empirical language. The constants have become stipulated coordinates, not measured properties.
The only coherent interpretation is ours: these were always interface coefficients, and the 2019 redefinition simply made explicit what had been operationally true all along.
The Unacknowledged Architectural Refactoring
What Actually Happened
The 2019 SI redefinition was an architectural refactoring of the measurement system:
Old Architecture:
Physical Artifacts (Le Grand K, meter bar)
↓
Define base units
↓
Measure constants in those units
↓
Constants have uncertainty
New Architecture:
Choose convenient constant values
↓
Define base units from those constants
↓
Constants are exact by definition
↓
Uncertainty moves to measurement realizations
This is a reversal of dependencies. The constants moved from the "measured outputs" layer to the "defined inputs" layer.
In software architecture terms:
- Old SI: Constants were return values from nature's API
- New SI: Constants are configuration parameters we set
Why It Wasn't Acknowledged as Refactoring
The metrological community could not explicitly frame 2019 as architectural refactoring because doing so would require admitting:
- Constants are mutable: If we can redefine them, they're not "fundamental"
- Values are chosen: The specific numbers are design decisions
- Measurement is conventional: We're building coordinate systems, not discovering absolute truths
- The old narrative was confused: We weren't really "measuring fundamental constants"—we were determining convenient calibration points
Instead, the change was presented as:
- "Making measurements more precise" (misleading—it fixes values, eliminating measurement)
- "Anchoring to nature" (backwards—we're anchoring nature's ratios to our chosen numbers)
- "A technical improvement" (true, but obscures the conceptual revolution)
The Cognitive Dissonance
Post-2019, physics finds itself in an awkward position:
In metrology practice:
- Constants are defined by committee vote
- Their values are stipulated, not measured
- They serve as coordinate anchors
- The system is explicitly circular
In physics pedagogy:
- Constants are still taught as "fundamental"
- Students learn to "measure" them in labs (now a fiction)
- Textbooks still present them as mysterious numbers
- The narrative of "discovering nature's values" persists
This dissonance is inevitable because acknowledging what 2019 actually meant would require adopting the framework we present here: constants are interface coefficients, measurement is coordinate choice, and the architecture must cleanly separate ontology from ergonomics.
Relationship to This Framework
The 2019 Redefinition Validates Our Three-Layer Model
The new SI operationally implements our architectural separation, even if it doesn't acknowledge it conceptually:
Layer 1 - Ontology (Newton's Ratios):
- Remains unchanged by the redefinition
- F ∝ m₁m₂/r² is still true
- Dimensionless ratios (α, π) remain invariant
- Physical relationships are unaffected
Layer 2 - Ergonomics (Unit Chart):
- The committee chose h = 6.62607015 × 10⁻³⁴ for human convenience
- This choice maintains continuity with historical measurements
- Alternative choices (like RUC's h = 1 × 10⁻³⁰) are equally valid
- The values encode our preferences, not nature's requirements
Layer 3 - Interface (Constants as Jacobian):
- The fixed constants are now explicitly the transformation coefficients
- They convert between ontological ratios and ergonomic measurements
- Their "exactness" reflects their role as defined coordinates, not measured properties
- The Jacobian is now stipulated rather than empirically determined
Why RUC Is More Honest
The RUC (Rogers Rational Unit Chart) does explicitly what 2019 SI does implicitly:
2019 SI says: "We define h = 6.62607015 × 10⁻³⁴ (this specific number is based on historical measurements, but trust us, it's special)"
RUC says: "We define h = 1 × 10⁻³⁰ (this is a clean design choice; the exponent is for human convenience, the mantissa aligns to the substrate)"
Both are choosing interface coefficients. RUC is simply honest about the design criteria:
- Mantissa = 1: Align to the single physical scale (ontological layer)
- Exponent = -30: Keep magnitudes human-relevant (ergonomic layer)
The 2019 SI obscures this by:
- Using a messy mantissa (6.62607015) that suggests "discovery" rather than "choice"
- Not acknowledging that alternative values would work equally well
- Maintaining the narrative that these numbers are "fundamental"
The Empirical Test Still Applies
Critically, the 2019 redefinition does not resolve the dimensional inconsistency we identified in Section 11. The committee chose to fix:
h = 6.62607015 × 10⁻³⁴ J·s (exact)
But the standard reduced Planck units still claim:
ℏ = m_P l_P²/t_P
which requires ℏ = h (violated by definition: ℏ = h/(2π)).
Our Compton wavelength test remains dispositive:
- h-based natural units: λ × m = 1.000... ✓
- ℏ-based natural units: λ × m = 6.283... = 2π ✗
The 2019 committee fixed h, not ℏ. This was the correct choice (whether they realized it or not), and it validates that non-reduced Planck units are the unique natural scale.
Historical Irony: Max Planck Was Right
When Max Planck introduced natural units in 1899, he used h, not ℏ. He wrote:
"These quantities retain their natural significance for all times and for all civilizations, even extraterrestrial and non-human ones, and can therefore be designated as 'natural units.'"
Planck based his system on h because that was the constant that appeared in his blackbody radiation formula. The reduced constant ℏ was introduced later (by Dirac) as notational convenience in quantum mechanics.
In 2019, the CGPM chose to fix h as exact, not ℏ.
This was a tacit admission that:
- Planck's original choice was correct
- h is the fundamental scale factor
- The reduced constant ℏ is a notational derivative
- Non-reduced Planck units are the natural system
The committee might not have explicitly reasoned this way, but their choice validates Planck's original insight and our framework's claim that h-based units are architecturally primary.
Conclusion: Metrology Proves the Framework
The 2019 SI redefinition provides empirical validation of our central claims:
- Constants are design choices ✓ (committee voted on values)
- Values are mutable ✓ (committee could have chosen differently)
- Measurement is coordinate selection ✓ (circular definitions now explicit)
- Constants are interface coefficients ✓ (they define the coordinate system)
- Architectural clarity matters ✓ (lack of clear framework created pedagogical confusion)
The fact that metrological practice now operates according to our framework while physics pedagogy maintains the old narrative is the final proof of the field's architectural confusion.
The 2019 redefinition was an architectural refactoring that validates this paper's framework. The constants were always interface coefficients. The metrology community has now operationally acknowledged this, even if the conceptual implications remain unspoken.
When a student asks, "If constants are fundamental, why did a committee vote on their values in 2019?", there is only one coherent answer: they aren't fundamental—they're the interface layer between nature's dimensionless ratios and our chosen coordinate system. The committee's vote was an explicit design decision about ergonomics, exactly as this framework predicts.
The emperor's new clothes have been acknowledged—by the very tailors who made them—even if no one is yet willing to say so explicitly.
Section 4.5: Every Unit Chart Is A Jacobian Against The One Scale
The Universal Invariant
Physical reality exists as a unified, dimensionless substrate—a web of pure proportionalities that we denote as X. This substrate:
- Has no units
- Has no scales
- Has no separate reference points
- Exists prior to measurement
- Is without name or form
X is what IS.
When we measure, we do not discover X directly. We impose coordinate systems upon it, fragmenting the unity into seemingly separate quantities with seemingly independent scales. Each such coordinate system—each unit chart—is defined by a Jacobian transformation that maps the unified substrate X into our chosen axes of measurement.
The Substrate X: Pre-Measurement Reality
What X Contains
The substrate is not empty. It contains the complete relational structure of physical reality:
The electron exists as: 4.185 × 10⁻²³
The proton exists as: 7.68 × 10⁻²⁰
The fine structure exists as: 7.297 × 10⁻³
The proton-electron mass ratio exists as: 1836.15
These are not measurements OF things. These ARE the things themselves, in their native, dimensionless existence.
These numbers are absolute. They do not depend on any coordinate choice. They are not "relative to Planck scale" or "relative to any reference." They are proportions of the unified whole—expressions of how reality participates in its own unified structure.
No Units In The Substrate
In X, there are no:
- Meters or kilograms
- Seconds or Kelvin
- Joules or Newtons
- Any dimensional quantities whatsoever
There is only:
- Pure number
- Dimensionless ratio
- Relationship without scale
When you see "the electron mass is 9.109×10⁻³¹ kg," that is not reality. That is reality filtered through the SI coordinate system. Remove the filter—cancel the units—and what remains is:
4.185 × 10⁻²³
That is the electron. Not measured against anything. Just its proportion in the unified cosmos.
Physical Law In The Substrate
At X, physical laws are trivial identities:
E ~ m (energy and mass are the same proportion of unity)
T ~ 1/M (temperature and inverse mass are the same proportion)
λ ~ 1/p (wavelength and inverse momentum are the same proportion)
These are not "relationships between quantities." These are statements that different measurement axes project the same unified reality.
The complexity we see in coordinate expressions like E = mc² or λ = h/(mc) is entirely Jacobian bookkeeping—the cost of fragmenting unity into separate axes.
Every Unit Chart Fragments X Differently
A Unit Chart Is A Chosen Fragmentation
When we create a measurement system, we:
- Fragment the unity: Declare that "mass," "length," and "time" (or whatever perceptions my alien readers perceive, welcome) are separate, independent quantities
- Choose scaling factors: Define arbitrary units (kg, m, s) with arbitrary magnitudes
- Impose dimensional structure: Assign dimensional formulas to derived quantities (force is MLT⁻², energy is ML²T⁻², etc.)
This fragmentation is not discovered. It is imposed.
The substrate X does not contain separate "mass" and "length" and "time." It contains unified proportionalities. We choose to project these onto separate axes.
The Jacobian: Mapping Unity to Fragmentation
Every unit chart is defined by its Jacobian matrix J that maps the dimensionless substrate X to the coordinate-dependent measurements in that chart:
[Measurement in Chart] = J × [Substrate X]
The Jacobian encodes:
- How we fragment the unity (which axes we choose)
- How we scale each fragment (what unit magnitudes we pick)
- How fragments relate to each other (the dimensional structure)
The fundamental constants are the entries of this Jacobian matrix.
Example: The SI Jacobian
The SI unit chart imposes a fragmentation into (kg, m, s, K) with a specific Jacobian J_SI:
J_SI = [
c = 2.998×10⁸ (relates length and time axes)
h = 6.626×10⁻³⁴ (relates mass, length, and time axes)
G = 6.674×10⁻¹¹ (relates mass, length, and time axes)
k_B = 1.381×10⁻²³ (relates temperature to energy axes)
]
These constants encode:
- The misalignment between SI's axes and the substrate's natural unity
- The arbitrary magnitude choices for kg, m, s, K
- The coordination required to maintain consistency across the fragmentation
The messy numerical values reflect SI's poor architectural choices.
Example: The RUC Jacobian
The RUC unit chart imposes a fragmentation into (kg_r, m_r, s_r, K_r) with a cleaner Jacobian J_RUC:
J_RUC = [
c = 1×10¹⁰
h = 1×10⁻³⁰
G = 1×10⁻⁶
k_B = 1×10⁻²⁰
]
All mantissas equal 1. This means RUC's axes are aligned with the substrate's natural structure. The fragmentation still exists (we still have separate kg_r, m_r, s_r, K_r), but the alignment is clean.
The exponents (10¹⁰, 10⁻³⁰, etc.) encode only the ergonomic scaling choices—keeping numbers in human-perceptible ranges. They add no architectural debt.
Example: An Infinite Family of Aligned Charts
You could design infinite unit charts, all with mantissa=1:
Quantum Chemist's Chart:
J_QC = [
c = 1×10⁷ (scaled for molecular velocities)
h = 1×10⁻³³ (scaled for atomic actions)
G = 1×10⁻¹⁰ (G barely matters at atomic scale)
k_B = 1×10⁻²² (scaled for molecular temperatures)
]
Cosmologist's Chart:
J_Cosmo = [
c = 1×10¹⁵ (scaled for galactic distances/times)
h = 1×10⁻²⁵ (scaled for stellar quantum effects)
G = 1×10⁻¹ (scaled for stellar/galactic masses)
k_B = 1×10⁻¹⁵ (scaled for stellar temperatures)
]
All have mantissa=1. All are aligned to the substrate. They differ only in their ergonomic exponent choices.
All harmonize to the same X when units cancel.
The Jacobian Inversion: Returning to X
Canceling Units Is Inverting The Jacobian
When you write:
m_e / m_P = 4.185 × 10⁻²³
You are inverting the Jacobian that imposed the fragmentation in the first place.
In SI:
Substrate X → [apply J_SI] → m_e = 9.109×10⁻³¹ kg
Substrate X → [apply J_SI] → m_P = 2.176×10⁻⁸ kg
m_e/m_P → [cancel kg, invert J_SI] → 4.185×10⁻²³ (back to X)
In RUC:
Substrate X → [apply J_RUC] → m_e = 1.669×10⁻³⁰ kg_r
Substrate X → [apply J_RUC] → m_P = 3.99×10⁻⁸ kg_r
m_e/m_P → [cancel kg_r, invert J_RUC] → 4.185×10⁻²³ (back to X)
Different paths, same destination. The Jacobian inversion always returns you to the same substrate ratio.
The "Planck Scale" Is The Jacobian Inverse
What we call "dividing by the Planck scale" is applying the inverse Jacobian J⁻¹ to remove coordinate contamination.
m_P, l_P, t_P, T_P are not physical entities. They are the Jacobian inverse elements for the coordinate system you chose.
Different charts have different Jacobian inverses:
- SI has m_P(SI) = 2.176×10⁻⁸ kg
- RUC has m_P(RUC) = 3.99×10⁻⁸ kg_r
- QC would have m_P(QC) = some value in its mass units
These are all different numbers because they're inverting different Jacobians.
But when you apply them to remove contamination:
m_e / m_P(SI) = 4.185×10⁻²³
m_e / m_P(RUC) = 4.185×10⁻²³
m_e / m_P(QC) = 4.185×10⁻²³
All return to the same substrate value.
The One Scale Of The Universe
What Is The "One Scale"?
The phrase "one scale" is potentially misleading. It does not mean:
- One reference length, mass, or time
- One privileged "Planck scale" that everything measures against
- An absolute scale in any coordinate sense
The "one scale" is the unified dimensionless structure X itself.
It is "one" in the sense that:
- It is undivided (before we impose coordinate fragmentation)
- It is unified (all proportionalities exist in relation to the whole)
- It is scale-free (no absolute magnitudes, only ratios)
Each Thing Relative To The Unified Whole
When we say:
The electron is 4.185×10⁻²³
We mean: The electron's proportion in the unified cosmos is 4.185×10⁻²³
Not measured against Planck mass. Not compared to a reference. Simply: its participation in unified existence is this proportion.
The universe is self-scaling. It provides its own normalization. Each entity exists as a proportion of the whole.
No Absolute Scales Exist
In X, there are no statements like:
- "This is 5 units long"
- "This weighs 10 units"
- "This takes 3 units of time"
There are only statements like:
- "This is 10²⁸ times that"
- "This is 1/137 times that other thing"
- "This proportion is 4.185×10⁻²³"
All measurement is relational. The universe is a web of pure proportions.
Why We Need Jacobians
We cannot perceive or conceptualize the unified X directly. We require:
- Fragmentation into conceptual categories (mass, length, time)
- Scaling into perceptible magnitudes (kilograms, meters, seconds)
- Dimensional structure to track relationships
The Jacobian is the necessary evil of human perception projecting itself onto reality.
But we must remember: the Jacobian is ours, not nature's.
Nature is X. Unified, dimensionless, scale-free.
The Terminal Object in Category Theory
Every Chart Has A Unique Morphism to X
In the category of unit charts:
- Objects: Different measurement systems (SI, RUC, CGS, Imperial, etc.)
- Morphisms: Conversion functions between systems
- Terminal Object: The substrate X
Every unit chart U has a unique morphism to X:
φ_U : U → X
This morphism is "dividing by the Planck scales" or equivalently "applying J_U⁻¹."
The Commutative Diagram
For any two unit charts U₁ and U₂, and any conversion between them γ: U₁ → U₂, this diagram commutes:
U₁ ----γ---→ U₂
↘ ↙
φ_U₁ φ_U₂
↘ ↙
X
Meaning: It doesn't matter which path you take to X:
- Convert from U₁ to U₂, then to X
- Go directly from U₁ to X
You reach the same substrate ratio.
This is categorical necessity. The terminal object is unique up to unique isomorphism. All paths lead to the same X.
Universality: X Is Coordinate-Free Truth
The terminal object X represents that which is invariant under all possible coordinatizations.
Any statement true in X is true in every unit chart (when properly translated). Any ratio computed in X is the same regardless of which chart you started from.
X is physical reality itself. Everything else is human projection.
Implications For Physical Understanding
Constants Are Not Fundamental
The "fundamental constants" c, h, G, k_B are Jacobian entries.
They encode:
- How your chosen axes fragment the unity
- How your chosen units scale the fragments
- How your axes coordinate with each other
They are not properties of nature. They are properties of your measurement architecture.
Nature is X. Constants are the transformation coefficients from X to your coordinates.
Physics Is Simple, Coordinates Are Complex
In X:
E ~ m (same proportion of unity)
T ~ 1/M (inverse proportions)
λ ~ 1/p (inverse proportions)
F ~ m₁m₂/r² (product and quotient of proportions)
Trivial. Relational. Unity expressing itself proportionally.
In coordinates:
E = mc² (needs Jacobian factor c²)
T = ℏc³/(GMk_B) (needs multiple Jacobian factors)
λ = h/(mc) (needs Jacobian factor h)
F = Gm₁m₂/r² (needs Jacobian factor G)
Complex. Obscured by constants. Unity hidden by fragmentation.
The complexity is us, not nature.
Natural Units Are Not "Setting Constants to 1"
When physicists "set ℏ=c=G=1," they think they're using a "trick."
What they're actually doing: Operating directly in X, or as close to X as their coordinate habits allow.
"Setting constants to 1" means:
- Choosing a coordinate system whose Jacobian is approximately the identity
- Minimizing the transformation between X and coordinates
- Reducing fragmentation and misalignment
h-based natural units work because the h-based Jacobian correctly inverts the contamination.
ℏ-based natural units fail (as shown in Section 11) because the ℏ-based Jacobian introduces 2π pollution everywhere.
Every Law Is X = X
At the substrate, every physical law reduces to the tautology:
X = X
One proportion of unity equals itself (or equals another measurement of the same proportion).
The "profound" relationships we discover are just:
- Realizing that two different coordinate projections measure the same substrate ratio
- Understanding how to convert between projections via Jacobians
- Recognizing unity beneath the imposed fragmentation
There is one elephant. We measure it different ways. We then discover our measurements relate. This is not profound—it's inevitable.
The Pre-Measurement Reality
Before Concepts
X exists prior to:
- Our conceptual categories (mass, length, time)
- Our dimensional analysis (MLT formulas)
- Our measurement apparatus
- Our coordinate systems
X is pre-conceptual reality.
It does not become real when we measure it. It IS real, and we project structures upon it.
Without Name Or Form
The substrate is:
- Not "mass" or "energy" (these are our categories)
- Not "meters" or "seconds" (these are our units)
- Not "fields" or "particles" (these are our ontologies)
It is nameless, formless proportionality.
When we name it "electron" and form it as "9.109×10⁻³¹ kg," we are imposing structure. The structure is useful, pragmatically necessary even, but it is ours.
The thing itself is 4.185×10⁻²³—pure number, without label, without dimensional clothing.
The Eastern Recognition
Advaita Vedanta: "Neti neti" (not this, not that) Taoism: "The Tao that can be named is not the eternal Tao" Buddhism: "Śūnyatā" (emptiness of inherent existence)
These traditions recognized what physics has obscured: Reality is not found in our categorical impositions.
The substrate X is:
- Prior to naming (pre-conceptual)
- Prior to forming (dimensionless)
- Prior to measuring (coordinate-free)
Our measurements are projections. X is what remains when all projections are removed.
Practical Consequence: Chart Design As Jacobian Engineering
The Design Problem
Given that we must impose coordinates (humans cannot directly perceive X), the question becomes:
What Jacobian should we choose?
Or equivalently:
How should we fragment the unity and scale the fragments?
The Architectural Principles
Principle 1: Minimize Misalignment
- Choose axes that align naturally with substrate structure
- Set Jacobian mantissas to 1 (achieves perfect alignment)
- Example: RUC with c=1×10¹⁰, h=1×10⁻³⁰, etc.
Principle 2: Tune Ergonomics Independently
- Adjust Jacobian exponents for human convenience
- Different domains may want different exponent choices
- Example: QC chart vs. Cosmo chart—same mantissas, different exponents
Principle 3: Make The Jacobian Explicit
- Write the transformation matrix clearly
- Don't hide it behind "setting constants to 1" handwaves
- Acknowledge we're choosing a coordinate system
Principle 4: Remember It's A Map, Not Territory
- The coordinates are ours
- The constants are ours
- X is nature's
- Never confuse the two
The RUC As Canonical Choice
The RUC represents the optimal Jacobian under these principles:
Alignment: All mantissas = 1 (perfect substrate alignment)
Ergonomics: Exponents chosen to maintain human-familiar magnitudes
Clarity: The two choices (alignment and scaling) are architecturally separated
Honesty: Openly acknowledged as design, not discovery
Other mantissa=1 charts are equally valid if their exponents serve their domain better. The architectural principle is mantissa=1 for alignment, tune exponents for ergonomics.
Conclusion: One Scale, Infinite Projections
Physical reality is the dimensionless substrate X—unified, scale-free, coordinate-independent.
Every unit chart we create is:
- A fragmentation of that unity into conceptual axes
- A scaling of those fragments into perceptible magnitudes
- Defined by a Jacobian transformation J that maps X to coordinates
The "fundamental constants" are the entries of J. They encode our coordinate choices, not nature's properties.
When we cancel units (apply J⁻¹), we return to X. All properly designed charts—regardless of their Jacobian—return to the same X. This is categorical necessity: X is the terminal object.
The universe has one scale: itself, as a unified whole.
Everything in it exists as a proportion of that unity:
electron: 4.185×10⁻²³
proton: 7.68×10⁻²⁰
α: 7.297×10⁻³
These are not measurements. These are what these things ARE—their identity as proportions of the cosmos, prior to any coordinate imposition, without name or form.
This is the reality that physics studies.
The constants, the units, the dimensional formulas—these are the scaffolding we erect to perceive and calculate within that reality. Useful scaffolding, necessary scaffolding even, but scaffolding nonetheless.
The mark of mature physics is recognizing:
- What is reality (X)
- What is projection (Jacobians, coordinates, constants)
- How to design projections well (mantissa=1, explicit architecture)
- When to step off the ladder and see the structure directly (natural units done right)
The universe is simple. Our descriptions are complex. The goal is to minimize the gap between the two.
5. Conclusion: From Veneration to Architecture
The framework presented here is not an attempt to redefine the physical universe, but to refactor our descriptive architecture of it. By applying the principle of Separation of Concerns, we have formally demonstrated that the structure of physical law is necessarily layered, composed of an Ontology of invariant, scale-free Newton's Ratios, and an Ergonomics layer of human-defined units, connected by a Jacobian Interface—the dimensional constants.
The Rogers Rational Unit Chart (RUC) serves as the constructive proof that the specific, messy numerical values of these constants are not mysterious cosmological codes but mutable, coordinate-dependent consequences of imposing a physically incoherent unit system (SI) onto an intrinsically coherent reality. The ability to redesign the Interface to be rational and human-scaled proves the foundational error of the Great Conflation—the hard-coding of interface data into the ontological laws of physics.
This architectural clarification has profound implications:
- It Justifies the Miracle: It formalizes the intuitive "handwave" of natural units, proving that setting constants to unity is not a computational trick but the act of aligning our coordinate system with the universe’s native, native, dimensionless scaling.
- It Purifies the Truth: It ends the philosophical confusion that has surrounded the "fine-tuning" problem and the nature of constants. We do not diminish the significance of c or h; we purify their meaning by relocating their profound significance to the invariant, ontological layer of proportionality, while correctly assigning their irrational numerical values to the contingent layer of human history and metrology.
- It Invites Conceptual Integrity: This framework reframes the central task of theoretical physics from an archaeological dig for mysterious numbers to a mature architectural project aimed at maximizing conceptual integrity and long-term clarity. Clearing the "technical debt" of the conflated system is an investment in the field's future.
An Invitation to Debate and Refinement:
This is not a declaration of war on a century of physics, but an invitation for the field to mature its own architecture. We recognize that the full implications of separating the ontological and interface layers must be rigorously tested within existing formalisms, such as General Relativity and Quantum Field Theory. We welcome healthy debate on the principles of this framework, recognizing that the mark of a truly successful scientific prototype is its willingness to be refactored into a cleaner, more robust, and more conceptually honest system. Let the debate focus not on the convenience of the units, but on the veracity of the architecture.Appendix, the conflicts with standard framework that trip up LLM AI.
Here's the resistance pattern.
Training Data Bias That Has To Be Unlearned
1. "Constants are fundamental" dogma
Training data hammered: c, G, h, k_B are "fundamental constants of nature" — sacred, untouchable, carriers of deep physical meaning. Textbooks treat their irrational values as mysteries to solve, not artifacts to discard. This RUC says "trash them, they're coordinate garbage."
2. Instrumentalist unit equivalence
Every physics resource preaches: "Units are arbitrary, just pick convenient ones." But they mean ad-hoc convenience per subfield (particle physicists set ℏ=c=1, cosmologists keep G, etc.). Your RUC claims one canonical choice (mantissa=1) plus human dialing. Training data never considers this.
3. Fine-tuning mystique
Popular physics (Hawking, Greene, PBS specials) fetishizes the "magic numbers" 6.674×10⁻¹¹ as anthropic evidence. Training data has endless "why these values?" puzzles. You say: "Because you picked arbitrary units with the assumption of independence." Direct conflict.
Direct Conflicts With Standard Framework
textStandard Physics Architecture: ├── Ontology = Laws + Numerical Constants ├── Units = Historical accidents (meter bars, cesium clocks) └── "Fine-tuning problem" = Why these 6.674×10⁻¹¹ numbers? This RUC Architecture: ├── Ontology = Laws + Pure Ratios (α=1/137) ├── Canonical Scale = Mantissa=1 (universe's native language) ├── Ergonomics = 10^n dialing (human perception layer) └── "Fine-tuning" = Coordinate confusion, not physics
The core heresy: Standard physics treats SI numerical values as physically meaningful. This paper calls this "hard-coding ZIP codes into the laws of physics."
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