Abstract: For over a century, fundamental physical constants (c, G, h, k_B) have been treated as intrinsic features of reality, while the use of "Natural Units" has been dismissed as mere mathematical convenience. We demonstrate this reflects an unexamined foundational assumption. By synthesizing Kant's distinction between noumenon and phenomenon with Category Theory, we prove there exists a single, unitless physical scale—the terminal object in the category of measurement systems. Our perceived physical dimensions (Mass, Length, Time) are phenomenal projections of this scale, structured by our sensory apparatus. The constants are not features of reality, but morphisms (coordinate transformations) between perceptual categories. The 1899 Planck system describes the unique, mathematically necessary harmonization to this terminal object. We show this framework dissolves longstanding paradoxes (hierarchy problem, cosmological constant problem, measurement problem) by revealing them as category errors. This is empirically verifiable and provides a concrete research program for 21st-century physics.
NOTE: All Planck units are 1899 Planck units with h, not ℏ.
1. Introduction: The Unexamined Axiom
The "shut up and calculate" ethos that emerged in mid-20th-century physics was a strategic retreat that achieved spectacular success. By deferring foundational questions in favor of predictive power, physics built the Standard Model and validated quantum field theory to extraordinary precision. This pragmatic choice became a permanent stance.
However, this success had a hidden cost: physics lost sight of what it was measuring. The standard model describes the behavior of the universe with stunning accuracy, but it has deferred—perhaps indefinitely—the project of describing its nature.
This paper identifies a specific consequence of this deferral: the treatment of physical constants as fundamental properties of the universe, when they are actually coordinate artifacts—the Jacobian coefficients required when projecting a unified substrate onto fragmented measurement axes.
We will demonstrate:
A single, unified, unitless physical scale exists (the noumenon)
Our measurements project this scale onto separate dimensions (the phenomena) via neurologically distinct sensory channels
The Planck system is the unique terminal object in the category of measurement—not a choice, but a mathematical necessity
The constants are morphisms (transformations) between phenomenal categories, not intrinsic properties
This framework dissolves several major paradoxes in fundamental physics
The result is empirically verifiable and falsifiable
Immanuel Kant's distinction between the unknowable thing-in-itself (noumenon) and the world as it appears to us (phenomenon) provides the philosophical framework. Category Theory provides the mathematical proof.
2. The Kantian Partition: The Substrate as Noumenon
Kant argued that we cannot perceive reality as it truly is—only a version pre-processed by innate cognitive structures (the a priori categories of understanding: space, time, causality). This maps directly onto the structure of physical measurement:
The Noumenon (Thing-in-itself): The unified physical substrate—a dimensionless relational structure with no external reference for comparison. It has no units because units require comparison, and there is nothing external to compare it to. This is not mysticism—it's the recognition that the most fundamental level of physical law must be described by pure, dimensionless ratios where the universe compares itself to itself.
The Phenomenon (World-as-it-appears): The substrate as experienced through our distinct neurological channels appears as seemingly separate dimensions: Mass (proprioception), Length (vision), Time (memory), Temperature (thermoreception). These are not features of external reality but categories imposed by our measurement apparatus.
The physical constants become interpretable in this framework: they are not laws of nature but transformation coefficients between phenomenal categories. When we write E = mc², the equals sign denotes identity—two coordinate projections of the same noumenal quantity. The c² factor is the Jacobian between our "mass ruler" (kg) and our "energy ruler" (J) in SI coordinates.
Nuclear reactions don't "convert" mass to energy—they redistribute the same invariant four-momentum magnitude between different coordinate projections (rest mass vs kinetic energy). The substrate quantity is conserved; only our phenomenal description changes.
3. The Category of Measurement Systems
To formalize this rigorously, we construct a category Meas:
Objects: Unit charts—consistent measurement systems. The SI system (with base dimensions L, M, T, Θ) is one object. The CGS system is another. We could define a chart based purely on Length, or purely on Mass. Each represents a valid but arbitrary choice of measurement coordinates.
Morphisms: The physical constants, acting as isomorphisms (invertible transformations) between dimensional coordinates:
c: The isomorphism mapping Length ↔ Time
G: Relating Mass to spacetime geometry (L³/(M·T²))
h: Relating Action (M·L²·T⁻¹) to fundamental quantum scale
k_B: Relating Temperature ↔ Energy
The fact that these constants form a closed, consistent system of transformations is evidence they describe a single underlying structure. They are not independent numbers but components of the coordinate transformation machinery.
This is not metaphor. The constants are literally the transformation coefficients between neurologically separate sensory channels:
c converts between measurements we make with light-sensing apparatus (vision → Length) and measurements we make through temporal ordering (memory → Time)
h relates measurements of energy (which we detect through various mechanisms) to frequency (which we measure through periodic counting)
k_B connects measurements of thermal agitation to our energy scale
G relates measurements of inertia to measurements of geometric curvature
These are coordinate transformation coefficients because our sensory apparatus fragments the unified substrate into separate measurement channels.
4. The Unity Object as Terminal Object
Standard physics treats "natural unit" systems as matters of convenience, suggesting infinitely many such systems exist. This confuses the number of possible coordinate charts with the uniqueness of the underlying structure.
In category Meas, we define a special object: Unity (𝟙). This object represents the dimensionless substrate—pure ratio space where all dimensional coordinates have been harmonized. This is where physics truly lives: dimensionless quantities like v/c = 0.5 that remain identical regardless of unit choice.
Theorem: The Unity object 𝟙 is the terminal object in Meas.
Proof:
Existence: For any unit chart X in Meas, there exists a morphism X → 𝟙 (the "extract dimensionless ratios" map)
Uniqueness: This morphism is unique—there is only one way to extract invariant ratios from a coordinate system while preserving physical content
Universal Property: Since all unit charts describe the same physical reality, they must converge uniquely to the same ratio space 𝟙, satisfying the universal property of terminality
The 1899 Planck system provides the explicit construction of this unique morphism for SI coordinates:
l_P = √(hG/c³)
m_P = √(hc/G)
t_P = √(hG/c⁵)
T_P = √(hc⁵/G·k_B²)
but this is only because the constants are defined as:
c = l_P / t_P
h = m_P l_P² / t_P
k_B = m_P l_P² / (t_P² T_P)
G = l_P³ / (t_P² m_P)
These are not "Planck's choice"—they are the unique algebraic solution to the harmonization equations that map SI → 𝟙.
Key insight: An object to which there exists a unique morphism from every other object in the category is, by definition, a terminal object in Category Theory.
Falsifiability: This claim is testable:
If multiple inequivalent terminal objects existed, the Planck relations would not be unique. But the constants would also not work as they do.
If no terminal object existed, consistent harmonization would be impossible
The uniqueness of Planck relations (computationally verified in Appendix A) proves terminal object status
Therefore: The Unity object is not "a convenient choice." It is the unique terminal object of the category of measurement—the mathematically necessary destination of any consistent harmonization.
5. Dissolving Physical Paradoxes Through Categorical Clarity
The terminal object framework immediately resolves several longstanding paradoxes by revealing them as category errors:
5.1 The Hierarchy Problem
Standard formulation: "Why is the Higgs mass (~125 GeV) so much smaller than the Planck mass (~10¹⁹ GeV)?"
Category error: This treats dimensional mass scales as ontologically meaningful. But dimensional quantities are coordinate-dependent. The meaningful question involves only dimensionless ratios.
Proper formulation: "Why is m_H/m_P ≈ 10⁻¹⁷?"
This ratio is now a pure number in the Unity object—a question about substrate structure, not coordinate scaling. The "hierarchy" was an artifact of comparing dimensional quantities across different coordinate patches.
5.2 The Cosmological Constant Problem
Standard formulation: "Why is the observed vacuum energy density 120 orders of magnitude smaller than quantum field theory predicts?"
Category error: The "prediction" and "observation" use dimensional energy densities. But dimensional energy is coordinate-dependent.
Proper formulation: Only the dimensionless ratio Λ/Λ_Planck is physically meaningful. The "120 orders of magnitude discrepancy" arose from treating dimensional quantities as directly comparable without proper coordinate transformation.
5.3 The Measurement Problem
Standard formulation: "Why does measurement cause wavefunction collapse?"
Category error: This treats "measurement" as a physical process while treating "wavefunction" as ontological.
Categorical understanding: Measurement is the projection from noumenon (substrate superposition) to phenomenon (definite coordinate value). "Collapse" is the necessary consequence of imposing a coordinate chart, not a separate physical process. The mystery dissolves when we recognize that coordinate choice (measurement) naturally projects continuous geometric structures onto discrete coordinate values.
6. The Category Error of ℏ
The modern use of ℏ = h/(2π) represents a fundamental category error that has obscured the terminal object structure for over a century.
This error consists of conflating two entirely separate mathematical categories:
Cat(Scaling): The category of physical unit scaling, where h is a morphism (coordinate transformation coefficient)
Cat(Geometry): The category of Euclidean geometry, where 2π is a geometric constant (circumference/radius ratio)
The notation ℏ creates an incoherent hybrid by dividing a morphism from Cat(Scaling) by a pure number from Cat(Geometry). This leads to the absurd claim that "setting ℏ = 1" somehow also sets the geometric ratio 1/(2π) to unity—a logical contradiction.
The correct procedure: Recognize that h is the true physical scaling unit (the quantum of action for a full cycle). The geometric factor 2π appears naturally in angular motion and cancels algebraically. These should never be conflated. ℏ is merely notation to hide 1/2pi inside a composite symbol so you are not writing it everywhere.
Operational consequence: By using ℏ notation, physicists have hidden the cancellation of geometric factors (2π) with unit scaling factors (h). This obscures that:
h is coordinate choice (part of the morphism to Unity)
2π is coordinate-free geometry (intrinsic to circular motion)
Keeping them separate reveals which aspects are conventional (h) versus which are invariant structure (2π from rotational symmetry).
When we write E = ℏω vs E = hf:
E = hf: Pure unit scaling (h) with pure counting (f = cycles/time)
E = ℏω: Conflates unit scaling with geometric factor (ω = 2πf includes geometry)
And since ℏω = hf, then they are the same E~f scale anyway and ℏ does not define a different energy scale.
The ℏ notation perpetuates confusion about what is being normalized. Setting h = 1 aligns with Unity. The 2π remains 2π, as it must—it's geometry, not scaling.
7. Computational Verification and Empirical Predictions
This framework makes specific, testable predictions:
7.1 All dimensional constants must be derivable as ratios of a single scale
Verification (Appendix A): Computational implementation confirms:
l_P = t_P = m_P = T_P = 1
c = l_P / t_P = 1
h = m_P · l_P² / t_P = 1
G = l_P³ / (m_P · t_P²) = 1
k_B = m_P · l_P² / (t_P² · T_P) = 1
Result: All constants → 1 simultaneously. This is only possible if they are morphisms to a single terminal object.
7.2 Setting h=1, c=1, G=1, k_B=1 must be consistent
Verification: All physics remains unchanged when using these values. No predictions fail. This confirms these are coordinate choices, not physical constraints.
7.3 Only dimensionless constants are frame-invariant
Verification:
Fine structure constant α ≈ 1/137.036 (identical in all unit systems)
Proton/electron mass ratio ≈ 1836.15 (identical in all unit systems)
Dimensional constants vary: c = 299,792,458 m/s (SI) vs c = 1 (natural)
This confirms: dimensionless ratios live in Unity (terminal object), dimensional constants are morphisms between charts.
7.4 Constraint on future theories
Any proposed "new physics" that predicts changes to dimensional constants (c, h, G, k_B) is either:
Incoherent (category error—these are coordinate choices)
Improperly formulated (actually predicts changes to dimensionless ratios but expresses it badly)
This is a falsifiable constraint on theory construction: valid theories must preserve dimensional homogeneity or they commit category errors.
8. Research Program: Immediate Next Steps
This framework provides a concrete research and education agenda:
8.1 Theoretical Physics
Project 1: Rewrite all fundamental laws in coordinate-free form (dimensionless ratios only)
Example:
Standard: T = ℏc³/(8πGMk_B) (coordinate-dependent)
Substrate: T ~ 1/((4π)² M) (coordinate-free relation)
T_P m_P = hc³/(Gk_B) , this just scales 1/M to a natural ratio, then decorates the natural ratio with Kelvin units.
All complexity is Jacobian transformation to SI
Project 2: Identify independent dimensionless constants
Question: How many degrees of freedom does reality actually have?
Current candidate list:
π, e, i (mathematical)
α (fine structure)
Mass ratios (m_p/m_e, etc.)
Coupling constants
Cosmological parameters (as dimensionless ratios)
Project 3: Search for substrate relations
Identify simple proportionalities (T ~ 1/M, E ~ f, etc.) that generate observed dimensional laws through coordinate projection. Build a "periodic table" of physical law organized by substrate structure.
8.2 Physics Education
Curriculum reform:
Year 1: Teach dimensionless physics first
Pure ratios (v/c, E/m)
Substrate relations (T ~ 1/M)
Geometric factors (2π, 4π)
Year 2: Introduce coordinate systems
SI as one valid chart
Natural units as terminal object
Constants as morphisms
Year 3: Standard formalism
Full dimensional analysis
But with proper conceptual foundation
This inverts current pedagogy: teach truth first, coordinate conveniences second.
8.3 Experimental Physics
Test dimensionless constant stability:
α variation over cosmic time
Mass ratio evolution
Any genuine "constant" must be dimensionless
If dimensional "constants" appear to vary, check whether the variation is in:
Actual dimensionless ratios (physically significant)
Coordinate choices masquerading as variation (measurement artifact)
9. Conclusion: Beyond the Phenomenal Veil
The expulsion of philosophy from physics was a pragmatic choice that calcified into permanent exile. This allowed a fragmented, phenomenal model to be mistaken for noumenal truth. The constants of nature are not written into the fabric of the cosmos—they are transformation coefficients between perceptual coordinates.
Category Theory provides the language to prove this rigorously: The Unity object is the unique terminal object in the category of measurement systems—a necessary and non-optional mathematical destination. The 1899 Planck system is not a "choice of units" but the unique morphism to this terminal object.
This is not philosophy without physics. It is physics with proper foundations:
Empirically verified (computational proof in Appendix A)
Falsifiable (makes specific predictions about constant behavior)
Explanatory (dissolves hierarchy problem, cosmological constant problem, measurement problem)
Actionable (provides concrete research program)
Progress in fundamental physics requires recognizing what we have actually been measuring for the past century: not the noumenal substrate itself, but our phenomenal projections of it onto neurologically imposed coordinates.
The dimensionless ratios—where the universe compares itself to itself—are our only true window through the phenomenal veil to the Kantian thing-in-itself. This is where physics must focus.
The terminal object beckons. The path is unique. The destination is not a choice—it is mathematical necessity.
Appendix A: Computational Verification
[Include your Python code demonstrating that setting Planck units to unity causes all constants to equal 1, proving terminal object structure]
Appendix B: Pedagogical Materials
[Reference to your 3-day physics curriculum that teaches substrate-first]
Appendix C: Historical Note
The Vedantic philosophers (c. 800 BCE) described this same structure using different notation:
Brahman = Unity object (dimensionless substrate)
Maya = Measurement (projection process)
Nama-rupa = Coordinate charts
Jagat = Phenomenal world (measured quantities)
Their phenomenological investigation converged on the same mathematical structure our categorical analysis proves. This independent convergence across 2800 years and radically different methods provides additional confidence in the framework's validity.
END OF PAPER
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