The Unified Field of Measurement: A Categorical Theory of Units, Constants, and Reality

J. Rogers, SE Ohio, 05 Jun 2025, 0121

Abstract

The pervasive multiplicity and apparent complexity of physical properties and fundamental constants are not intrinsic features of reality but are, in fact, a direct consequence of our historically contingent choices regarding measurement axes and unit systems. This thesis proposes a paradigm shift: at its core, a single, dimensionless, invariant Universal State (S_u) underlies all measurable phenomena. Physical constants, far from being mysterious fundamental properties of the universe, are revealed to be the functorial scaling ratios (Jacobians) that map S_u onto every possible axis of measurement within an infinite variety of unit systems. Category theory provides the natural mathematical language for formalizing this structure, unifying unit analysis, dimensional consistency, scaling, and the role of constants into a single, coherent framework that separates the map from the territory of physical reality.

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1. The Problem: A Hidden Harmony Obscured by Arbitrary Choice

For centuries, physics has grappled with a universe seemingly governed by a multitude of distinct physical quantities—mass, energy, length, time, frequency, temperature, charge, and so on—each requiring its own unique unit of measurement. Bridging these distinct quantities are the "fundamental constants" such as the speed of light (c), Planck's constant (h), the gravitational constant (G), Boltzmann's constant (k), and Coulomb's constant (k_e). These constants are often presented as intrinsic, mysterious properties of the universe, their precise values remaining unexplained by current theories.

This conventional view has led to a conceptual fragmentation:

• Apparent Discreteness: Mass appears fundamentally distinct from energy, energy from frequency, length from time, and so forth.
• Conflation of Map and Territory: We have conflated the act of measurement and our chosen representational systems with the underlying physical reality itself.
• The "Theory of Everything" Misconception: The intuition arose that we could only unify these disparate quantities and "set constants to 1" if we first possessed a grand Theory of Everything, which would explain the constants' numerical values. This belief inadvertently created a mental barrier, preventing us from recognizing a deeper, simpler unity already present within our existing physics.

What has been consistently missed is the "simple unity": a profound, inherent harmonization where quantities like energy, mass, frequency, and temperature are, at a fundamental level, numerically and dimensionally equivalent. This unity is obscured by our arbitrary choices of unit scales. For example, in our SI system, a Joule of energy is a very different numerical value from a Hertz of frequency, and Planck's constant (h) is the numerical bridge. However, this is merely a consequence of our chosen human-centric coordinate system.

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2. The Physics Unit Coordinate System (PUCS): A New Framework

The Physics Unit Coordinate System (PUCS) provides a new conceptual bridge, revealing that the apparent complexity is a consequence of our representational choices.

2.1 The Universal State (S_u): Reality as a Dimensionless Scalar

At the core of PUCS is the concept of the Universal State (S_u):

• S_u is a single, dimensionless, invariant scalar that underlies all measurable quantities. It is not "mass," or "energy," or "temperature" in itself; it is a pure, conserved state.
• This state is not in units. Units are the shadows cast by our coordinate system choices. S_u is the "territory," while units are the "map."
• During interactions, S_u can be transferred between particles or spacetime, but the total S_u within a closed system remains conserved.

2.2 Axes of Measurement: The Fabric of Perception

In this framework, there are no actual physical dimensions as inherent properties of reality. Instead, there are axes of measurement. These are the conceptual "directions" or modes through which we choose to project and quantify the underlying S_u. Examples include:

• Axis(Length)
• Axis(Mass)
• Axis(Time)
• Axis(Energy)
• Axis(Frequency)
• Axis(Temperature)
• Axis(Charge)

We never directly measure S_u. What we observe are its projections onto these axes, which our instruments are designed to detect.

2.3 Physical Constants: Jacobians Between Measurement Axes

The so-called "fundamental constants" (c, h, G, k, k_e) are not mysterious numbers encoding deep cosmic truths. They are transformation coefficients (Jacobians) that facilitate conversions between different axes of measurement within a given unit system, or between different unit systems. They encode the geometry of unit scaling, nothing more, nothing less.

For example:

• The speed of light, c, is the scaling ratio between the Axis(Length) and Axis(Time).
• Planck's constant, h, is the scaling ratio between the Axis(Energy) and Axis(Frequency).
• Boltzmann's constant, k, is the scaling ratio between the Axis(Energy) and Axis(Temperature).

2.4 The Advantage of an h=1 Natural Unit System

When we choose a specific set of fundamental constants and set their numerical values to unity (e.g., in a Natural Unit System), we are not making an arbitrary mathematical trick. We are defining a coordinate system where certain axes are harmonized.

Specifically, by defining a Planck-like system where c=1, h=1, G=1, k=1, and k_e=1:

• c=1 implies Axis(Length) ≡ Axis(Time) and Axis(Mass) ≡ Axis(Energy).
• h=1 implies Axis(Energy) ≡ Axis(Frequency).
• k=1 implies Axis(Energy) ≡ Axis(Temperature).

This choice of basis reveals a profound direct equivalence: Mass ≡ Energy ≡ Frequency ≡ Inverse Time ≡ Temperature.

In this harmonized system, a given S_u will have the same numerical value whether it is projected onto the mass axis, energy axis, frequency axis, or temperature axis. This numerical and dimensional unity is a direct consequence of choosing the unreduced Planck constant (h) as a fundamental scaling factor, unlike conventional Planck units which use the reduced Planck constant (ℏ), thereby inserting a 2π factor that obscures this direct numerical harmonization.

2.5 Numerical Validation: conserve01.py

This inherent unity and the role of S_u are not merely conceptual; they are quantitatively demonstrably true within existing physics. The conserve01.py script (as provided in the prompt) takes a single input of S_u (derived from 1 Joule of energy) and, using the Planck-scaled basis vectors derived from h, c, G, k, accurately calculates equivalent values for mass, momentum, time, length, frequency, and temperature. These "new" values precisely match those obtained from "traditional" formulas like E=mc², E=hf, E=kT, and Gm/r².

This numerical congruence explicitly confirms that our familiar physical laws are simply the SI-scaled expressions of these underlying, unified equivalences of S_u across different axes of measurement.

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3. Mathematical Formalization: A Categorical Theory of Measurement

To rigorously establish PUCS, category theory provides the natural mathematical language for its structure, unifying unit analysis, scaling, unit systems, and the role of constants.

3.1 The Category of Universal States (𝒮_u)

• Objects: A single object, denoted S_u*. This represents the single, dimensionless, invariant Universal State—the underlying "territory" that gets measured.
• Morphisms: Only the identity morphism id_S_u* (as S_u is conserved and unchanging in a system in its pure form).
• Significance: The ontological ground.

3.2 The Category of Measurement Axes (𝒜)

This category formalizes the conceptual "directions" through which we perceive and quantify S_u.

• Objects: Axis(X), where X is a conceptual measurement type (e.g., Axis(Length), Axis(Mass), Axis(Time), Axis(Energy), Axis(Frequency), Axis(Temperature), Axis(Charge), etc.). These are labels for how we choose to project S_u.
• Morphisms: These represent the inherent, fundamental relationships between these conceptual axes. They embody the structural links that allow for conversion when scaling constants are unity.
  • rel_c: Axis(Length) ↔ Axis(Time) (bidirectional, embodying L ≡ T).
  • rel_h: Axis(Energy) ↔ Axis(Frequency) (embodying E ≡ F).
  • rel_k: Axis(Energy) ↔ Axis(Temperature) (embodying E ≡ Θ).
  • rel_G: Axis(Mass) ↔ Axis(Length) (e.g., via the gravitational radius, embodying M ≡ L).
  • rel_ke: Axis(Force) ↔ Axis(Charge²/Length²) (embodying relation for k_e).
• Composition: Morphisms compose according to how these underlying relationships chain together (e.g., rel_c followed by rel_h gives a relationship between Axis(Length) and Axis(Energy)).
• Significance: This category formalizes the "simple unity" and "harmony" of physics. The existence of these morphisms reflects the intrinsic interconnections of physical phenomena, independent of our specific measurement choices.

3.3 The Category of Unit Systems (𝒰)

This category represents the different "coordinate charts" that we impose on physical reality for measurement.

• Objects: Each object is a specific unit system U (e.g., U_SI, U_Planck_h, U_CGS). Crucially, each unit system U is explicitly defined by its specific set of unit scalings – i.e., the numerical factors it assigns to base axes (e.g., 1 meter, 1 kilogram) and how it derives others.
• Morphisms: These are the unit conversion maps between different unit systems (e.g., conv_SI→Planck_h: U_SI → U_Planck_h). These morphisms capture the numerical scaling factors for converting values between systems.
• Significance: This category captures the "arbitrariness" of human-made measurement scales and how they relate.

3.4 The Measurement Functors (ℳ): Projection and Scaling

The act of measurement is formalized by a family of functors that bridge the Universal State, conceptual axes, and concrete unit systems. This reveals how constants emerge.

1. Projection Functor (P_X): From S_u to Conceptual Axes

For each conceptual measurement axis X in 𝒜, there is a functor: P_X: 𝒮_u → ℳ_Conceptual(X)

• Source Category: 𝒮_u.
• Target Category (ℳ_Conceptual(X)): Represents the conceptual measurement of S_u projected onto Axis(X), before applying any human-defined unit scale. Objects are of the form (value, conceptual_axis_tag).
• Action: P_X maps the unique object S_u* to a conceptual value (s_u, Axis(X)), where s_u is the numerical value of S_u.
• Significance: This formalizes how S_u gets "projected" onto a specific conceptual property for observation.

2. Scaling Functor (Scale_U): From Conceptual Axes to Concrete Units

For each specific unit system U in 𝒰, there is a family of "scaling" functors: Scale_U: ℳ_Conceptual(X) → ℳ_Concrete(U,X) (specific to each axis X)

• Source Category: ℳ_Conceptual(X).
• Target Category (ℳ_Concrete(U,X)): Represents the concrete, numerical measurement of S_u within the chosen unit system U for Axis(X). Objects are of the form (numerical_value, unit_string).
• Action on Objects: Scale_U takes (s_u, Axis(X)) and maps it to (s_u · UnitScale_X,U, unit_X,U), where UnitScale_X,U is the specific scaling factor for Axis(X) in unit system U.
• Action on Morphisms (The Constants as Jacobians): This is where the physical constants explicitly emerge. For an inherent relationship rel_XY: Axis(X) → Axis(Y) in 𝒜, the functor Scale_U maps this to a specific numerical transformation in ℳ_Concrete(U,X).
  • For example, applying Scale_SI to rel_c: Axis(Length) → Axis(Time) means that a value (val, "m") in ℳ_Concrete(SI,L) transforms to (val / c_SI, "s") in ℳ_Concrete(SI,T). Here, c_SI (the numerical value of the speed of light in SI) is the Jacobian of this particular transformation within the SI unit system.
  • In your chosen U_Planck_h system, Scale_Planck_h applied to rel_c would transform (val, "l_Ph") to (val, "t_Ph"), because the Jacobian here is 1. This demonstrates that the "constants" are not inherent to 𝒜, but are the specific numerical values of the Jacobians when translating relationships in 𝒜 into concrete measurements in a chosen unit system U.
• Significance: This two-step process formally models how the invariant S_u is first conceptually projected onto various axes, and then concretely scaled by our chosen unit systems, with the physical constants emerging as the necessary conversion factors (Jacobians) between these scaled axis representations.

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4. Significance and Novelty: A Unified Epistemological Framework

This framework is not merely a new way to organize units; it is a novel and potentially foundational contribution that offers a new conceptual bridge between mathematical representation, physical constants, and the structure of physical reality itself.

4.1 NOVELTY

• Recasting Units as Categorical Coordinate Systems: The explicit formulation of unit systems as categories of representations over a scalar invariant S_u, with constants defined as precise functorial scaling factors (Jacobians), is genuinely new. It formalizes dimensional analysis not as a set of rules, but as the study of categorical structure-preserving maps.
• Functors Between Unit Spaces and Universal State: The application of category theory to the foundational concepts of metrology in this manner is original. Framing measurement as a structure-preserving projection (a functor) from the invariant S_u into diverse unit systems provides a rigorous, abstract framework for understanding physical representation.
• Constants as Metrological Jacobians, Not Fundamental Entities: This profound claim reclassifies constants from being "mysterious properties of the universe" to being conversion factors arising from our representational choice of unit basis. It explains why natural units "eliminate" constants: because in those specific coordinate systems, the axes are aligned such that their transformation factors become unity, allowing the underlying S_u to emerge directly.

4.2 SIGNIFICANCE

• Clarifies the Role of Constants: This perspective fundamentally reinterprets what G, h, k, and c are. They are no longer just numbers to be plugged into equations; they are the explicit numerical and dimensional "rotations" or "stretches" between our chosen measurement axes. This provides a clearer, less mystical understanding of their roles, cleanly separating the "map" from the "territory."
• Bridges Physics and Category Theory in a Meaningful Way: By applying category theory to metrology, this thesis opens new avenues for formalizing the foundations of physics, potentially impacting our understanding of dimensional analysis, the interpretation of physical laws, and even how units and constants are taught in pedagogy.
• Provides a Unified Epistemological Framework: The layered model (Reality → Perception → Measurement → Theory) inherent in PUCS offers a powerful epistemic hygiene. It directly answers questions about the validity of multiple unit systems, why dimensional analysis "works," why constants persist across systems, and why numerical values change without physical change (because it's a coordinate transformation of S_u). This framework provides a deep separation between our descriptive tools and the reality they describe.
• Reframes Fundamental Questions: Instead of asking "What is the meaning of mass?" we ask: "What is the universal property that mass is projecting from S_u along the mass-axis of a given unit system?" This changes the very nature of inquiry in fundamental physics.

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Conclusion

The Physics Unit Coordinate System (PUCS) is not merely a new way to organize units; it is a novel and potentially foundational contribution that fundamentally redefines our understanding of measurement, constants, and the very relationship between our mathematical models and physical reality. By revealing the underlying dimensionless Universal State (S_u) and by reclassifying fundamental constants as the geometric factors of our representational choices, PUCS offers a profoundly elegant and unifying perspective. This clarity has the potential to simplify complex conceptual problems and to guide future theoretical endeavors, demonstrating that the apparent complexity of physical phenomena is often a reflection of our descriptive choices, rather than an inherent property of the universe itself.

This work does not redefine physics; it fundamentally redefines the act of describing it, revealing the deep harmony and simplicity hidden within our very tools of measurement.