From Formless Reality to Formal Law: A Projection Theory of Physics

J. Rogers, SE Ohio, 15 Jun 2025, 1153

Before latitude and longitude, maps were disjointed and local. Only when we adopted a global coordinate system could cartography become a science. Likewise, the Physics Unit Coordinate System (PUCS) is a coordinate system for physical reality—treating units not as arbitrary labels, but as projections from a universal state, with physical constants as the structural distortions that arise between measurement axes.

Abstract
This paper presents a unified framework explaining the origin and nature of physical law. We resolve the "unreasonable effectiveness of mathematics" by positing that physical laws are not discovered properties of an objective universe, but are the inevitable, structured projections of a formless, unified reality through the cognitive and instrumental apparatus of the observer. We introduce a four-layer ontology—Reality, Perception, Measurement, and Theory—and provide its rigorous mathematical formalization using category theory. This reveals physical constants (G, c, h) not as fundamental properties of nature, but as the Jacobians of transformation required by our chosen unit systems. The framework is validated by a computational model that successfully derives the form of major physical laws—including gravitation, mass-energy equivalence, and quantum energy—from dimensionless postulates. The theory demonstrates that a physical law is a hybrid construct of an unknowable reality and a knowable measurement structure, thereby incorporating the observer into physics in a formal, non-trivial way and explaining how knowledge is possible even if ultimate reality is not.

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1. Introduction: The Puzzle of a Mathematical Universe

Since Eugene Wigner’s famous observation, the question of why mathematics so effectively describes the physical universe has haunted science and philosophy. Physical laws, expressed as elegant mathematical equations, possess a predictive power that seems almost magical. We propose a solution that inverts the traditional view: mathematics is not effective because it happens to describe reality; rather, what we call "physical law" is inevitably mathematical by its very construction.

This paper argues that a physical law is a projection. It is the shadow cast by a formless, unified reality (Layer 1) as it is filtered through the categorical axes of our perception (Layer 2) and scaled by our arbitrary systems of measurement (Layer 3). The resulting projection, which we call a physical law (Layer 4), is therefore a product of the structure of the measurement system itself. We will demonstrate that this is not merely a philosophical stance but a computationally valid and mathematically rigorous model.

2. The Four-Layer Ontological Framework

Our theory is built upon a four-layer model of how reality is translated into knowledge.

• Layer 1: The Formless Reality (The Brahman). The foundational layer is a single, unified, and indivisible reality. At this level, corresponding to the dimensionless Planck scale where all fundamental constants are unity, relationships exist as pure, formless ratios. This reality is, in its entirety, unknowable and unexplainable, as any act of explanation requires division and distinction, which are absent here.

• Layer 2: The Perceptual Axes. To interact with reality, a conscious observer must divide it into conceptual categories. This layer represents the cognitive act of creating distinct "axes" of perception—such as Length, Mass, Time, and Temperature. These axes are not fundamental to reality but are fundamental to our experience of it. They are the cognitive framework through which the formless becomes accessible.

• Layer 3: Measurement and Morphisms. Here, we formalize the perceptual axes by attaching arbitrary scales and units (meters, kilograms, seconds). Critically, because the underlying reality is unified, the axes we create remain interconnected. These connections manifest as "morphisms"—the conversion factors that relate one measured axis to another.

• Layer 4: Theory as a Formal Projection. A physical law is the final output of this process. It is a mathematical statement that describes the projection of a Layer 1 relationship, filtered through the perceptual axes of Layer 2, and expressed in the language of the scaled units and morphisms of Layer 3.

3. Mathematical Formalization: A Categorical Theory of Measurement

This ontological framework finds its perfect expression in the language of category theory.

• The Category of Universal State (𝒮_u): Formalizing Layer 1, this category consists of a single object, S_u*, representing the unified reality, and only the identity morphism. This elegantly captures a state that is whole and without internal structure or transformation.

• The Category of Measurement Axes (𝒜): Formalizing Layer 2, the objects of this category are the perceptual axes: Axis(Length), Axis(Mass), Axis(Time), etc. The morphisms are the crucial part: they represent the fundamental, pre-unit relationships between these concepts, such as rel_c: Axis(Length) ↔ Axis(Time). This category is a formal blueprint of the cognitive structure of the observer, showing how reality is parsed.

• The Category of Unit Systems (𝒰): Formalizing the arbitrary aspect of Layer 3, the objects are specific unit systems, such as U_SI or U_Planck. This category represents the choice of scale.

• The Measurement Functors (ℳ): The engine of projection from reality to law is a two-step functorial process:

  1. Projection Functor (P_X): This functor maps the single object in 𝒮_u to an object in 𝒜. It is the formal act of "looking at" reality through a specific perceptual lens (e.g., P_Mass).

  2. Scaling Functor (Scale_U): This functor maps the conceptual object from 𝒜 to a concrete measurement in 𝒰. For example, it takes the conceptual Axis(Length) and maps it to a value like (10, "meters").

The climax of this formalization is the origin of physical constants. When the Scale_U functor translates a relationship (a morphism in 𝒜, like rel_c) into a specific unit system (an object in 𝒰, like U_SI), the numerical scaling factor it must use is the Jacobian of the transformation. We find that physical constants are these Jacobians. The constant c is not a property of the universe, but the Jacobian of the Scale_SI functor acting on the rel_c morphism. In the Planck system, the functor is designed so this Jacobian is 1, and the constant "vanishes."

4. Computational Validation

We validated this theory with a computational system (pyym) that automates the projection process. The system takes a dimensionless postulate from Layer 1 and derives its corresponding Layer 4 physical law.

Methodology:

1. Postulate: A dimensionless relationship is proposed, e.g., that the dimensionless force (F/F_P) is proportional to the dimensionless gravitational interaction term (M₁M₂/m_P²) × (l_P²/r²).

2. Projection: The system algebraically solves for the dimensional quantity (F) in a target unit system (e.g., SI).

3. Result: The system outputs the dimensionally correct form of the physical law.

Validation Results:
The system successfully re-derives the form of numerous fundamental laws:

• Newton's Gravitation: From a postulate relating dimensionless force and mass/distance, it derives F = G*M1*M2/r**2.

• Mass-Energy Equivalence: From a postulate relating dimensionless energy and mass, it derives E = M*c**2.

• Planck's Relation: From a postulate relating dimensionless energy and frequency, it derives E = f*h.

• Stefan-Boltzmann Law (Pressure): From a postulate relating dimensionless pressure and temperature to the fourth power, it derives P ∝ T⁴*k_B⁴/(c³*h³).

The Case of the Unity Morphism: F=ma
A powerful confirmation of the theory is its handling of laws seemingly without constants. By postulating F/F_P = (m/m_P) * (a/A_P) and solving for F, the scaling morphism Π = [F_P / (m_P * A_P)] resolves, by definition of the Planck units, to exactly 1. This demonstrates that F=ma is not an exception but a perfect example of the projection framework where the Jacobian of the transformation is unity.

Note: This method derives the dimensional form of laws. It does not account for dimensionless geometric ratios (e.g., the 1/2 in kinetic energy), which likely arise from a deeper geometric structure within the morphisms of Category 𝒜.

5. Philosophical Implications

The success of this framework provides profound answers to foundational questions.

• The Nature of Law and Constants: Physical constants are artifacts of our measurement systems. Physical laws are necessary constructions, not discovered truths of an objective reality. They are mathematically constrained by the structure of our perception and measurement.

• Realism vs. Constructivism: Our theory offers a middle path. A real, independent reality exists (Layer 1), but our knowledge of it is a structured construction (Layer 4). The construction is not arbitrary, but is bound by mathematical necessity, explaining the universality of science.

• The Paradox of Knowledge: The framework resolves how we can have precise, formal knowledge of a universe that is itself formless and unexplainable. It is because our knowledge is not of the universe-in-itself (𝒮_u), but is a description of the consistent, stable output of our projection system. We understand the projection, not the source.

• The Formal Role of the Observer: This theory incorporates the observer (consciousness and instrumentation) into the formation of physical law in a formal, non-trivial way. The observer’s cognitive structure is modeled by the Category of Measurement Axes (𝒜), an essential component without which no law could be formed.

6. Conclusion

We have moved beyond asking why mathematics is so effective in physics. The answer provided by the Projection Theory is that physics could be nothing else. Physical laws are mathematical because they are the output of a system whose constituent parts—perception as categorical division and measurement as scaled mapping—are inherently structural and mathematical.

This work provides a cohesive philosophy, a rigorous mathematical formalism, and a computational validation for a new understanding of science. It suggests that mathematics does not describe reality. Rather, reality is the formless input, and mathematics is the language that describes the structured projection of that reality through the lens of a conscious observer. The laws of nature are the laws of our own perception.

Appendix A. The categorical description.

3. Mathematical Formalization: A Categorical Theory of Measurement (Revised)

To rigorously formalize the Physics Unit Coordinate System (PUCS), we adopt the framework of category theory. This provides a language for distinguishing between the invariant structure of physical reality, the algebraic rules of dimensional analysis, and the scaling behavior imposed by unit systems. The goal is to clarify the representational role of measurement and physical constants, not to define a physical theory per se. This is a meta-theory concerning the structure of units and their relationship to physical law.

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3.1 Category of Universal States (𝒮ᵤ)

• Objects: A single object $S_u^*$, representing the underlying, dimensionless Universal State.

• Morphisms: Identity only; $\text{id}_{S_u^*}$.

• Interpretation: $\mathcal{S}_u$ encodes the ontological substrate—reality itself before conceptual projection or scaling.

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3.2 Category of Conceptual Axes (𝒜)

• Objects: Conceptual measurement axes: Length, Time, Mass, Energy, Frequency, Temperature, Charge, etc.

• Structure: A symmetric monoidal category, where the monoidal product ⊗ encodes dimensional composition (e.g., Energy = Mass ⊗ Length² ⊗ Time⁻²).

• Morphisms: Structural relationships between axes that become isomorphisms when constants are set to unity:

  • $\text{rel}_c: \text{Length} \leftrightarrow \text{Time}$

  • $\text{rel}_h: \text{Mass} \leftrightarrow \text{Frequency}$

  • $\text{rel}_k: \text{Temperature} \leftrightarrow \text{Frequency}$

  • $\text{rel}_G: \text{Mass} \leftrightarrow \text{Length}$

• Interpretation: This category formalizes the intrinsic relationships between measurement types as discovered through physical law.

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3.3 Category of Unit Systems (𝒰)

• Objects: Unit systems $U$ (e.g., SI, Planck_h, CGS).

• Morphisms: Unit conversion functors $\text{conv}_{U \to U'}$ that rescale base units.

• Interpretation: Captures the conventional (human-imposed) coordinate charts on the conceptual space 𝒜.

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3.4 Measurement as a Grothendieck Bifibration

Measurement is modeled as a bifibration:

$\mathcal{M} : \mathcal{A}^{op} \times \mathcal{U} \to \textbf{Cat}$

This structure defines how a measurement depends simultaneously on a conceptual axis (𝒜) and a unit system (𝒰), and how these dependencies cohere under changes in axis or unit system.

• Fibers over $(X, U)$ give categories of measured values $\mathcal{M}_{U,X}$ with unit-tagged scalars like (value, "kg") or (value, "m").

• Sections define consistent mappings of universal states $S_u^*$ into all axes and units.

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3.5 The Hierarchy of Constants: Base Morphisms and Composite Invariants

PUCS reveals a key insight: physical constants are not homogeneous. They separate into two tiers:

3.5.1 Base Jacobians (with respect to Time)

Each base SI unit axis (Mass, Length, Temperature) scales fundamentally relative to Time. These are the "base morphisms" or Jacobian constants, and they serve as the foundation of all scaling. These Jacobians are:

• Hz_kg: the mass-frequency conversion factor, defined as the Jacobian from Mass to Frequency:
  $\text{Hz\_kg} := \frac{1}{\text{Planck Mass in Hz}} = \frac{1}{f_{\text{mass}}}$

• m_s: the length-time conversion factor, i.e. the speed of light in the unit system:
  $\text{m\_s} := \text{Length} / \text{Time}$

• K_Hz: the temperature-frequency Jacobian:
  $\text{K\_Hz} := \frac{1}{\text{Planck Temperature in Hz}} = \frac{1}{f_{\text{temp}}}$

Each of these defines a projection from an axis to Time through a definable scaling ratio. These base Jacobians are what the constants "are" in the categorical structure—they are the primitive structure-preserving maps from axes back to time units.

Time is primary: All other axes define their fundamental scale through these base Jacobians to Time. That is, Mass, Length, and Temperature are not scaled independently—they are scaled via Time.

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3.5.2 Composite Constants as Ratios Between Axes

What we commonly call physical constants (like $h, G, c, k_B$) are composite constants, each constructed as a ratio or product of base Jacobians:

• Planck Constant (h): arises from combining $\text{Hz\_kg}$ and $\text{m\_s}^2$.

• Boltzmann Constant (k_B): constructed from $\text{K\_Hz}^{-1}$ and $\text{Hz\_kg}$.

• Gravitational Constant (G): constructed as a ratio involving $t_P^2$, $\text{m\_s}^3$, and $\text{Hz\_kg}$.

These constants are not stand-alone axioms. They emerge from the relationships between scaling morphisms in a particular unit system and define how multiple axes relate numerically when projected and scaled.

In this view:

• $h$ is not an independent physical "thing" but a composite morphism expressing a structural identity across multiple axes.

• Constants are not intrinsic to physics, but to the way we scale physical relations in our chosen coordinate chart.

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3.6 Open Questions and Next Steps

• Categorical treatment of dimensionless numbers (like $\pi$, $\frac{1}{16\pi^2}$, etc.) and naturally arising scalar factors remains to be integrated into the formalism.

• Full functorial proofs of bifibration coherence and natural transformations across 𝒜 and 𝒰 are open work.

• The possibility of modeling constants as 2-morphisms in a bicategory of measurements remains to be explored.

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This framework provides a categorical, interpretable basis for understanding physical constants and dimensional analysis. Constants emerge not as brute facts, but as structural invariants of how we measure the world, and Time sits at the center of that web.

Appendix B, ouput from the postulate to law algebraic conversion script.

 These are all calculated by dimensional analysis of Π groups,

     so this does not account for geometric ratios.

--- Deriving formula for: T ---

1. Postulate: T/T_P, m_P/M

   Symbolic Form: Eq(sqrt(G)*T*k_B/(c**(5/2)*sqrt(h)), sqrt(c)*sqrt(h)/(sqrt(G)*M))

2. Solved for T: Eq(T, c**3*h/(G*M*k_B))

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(T, c**3*h/(G*M*k_B))

--- Deriving formula for: F ---

1. Postulate: F/F_P, ((M1*M2/m_P**2) * (l_P**2/r**2))

   Symbolic Form: Eq(F*G/c**4, G**2*M1*M2/(c**4*r**2))

2. Solved for F: Eq(F, G*M1*M2/r**2)

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(F, G*M1*M2/r**2)

--- Deriving formula for: lambda ---

1. Postulate: wavelength/l_P, 1/sqrt(M*T / (m_P*T_P))

   Symbolic Form: Eq(c**(3/2)*lambda/(sqrt(G)*sqrt(h)), c**(3/2)*sqrt(h)/(sqrt(G)*sqrt(k_B)*sqrt(M*T)))

2. Solved for lambda: Eq(lambda, h/(sqrt(k_B)*sqrt(M*T)))

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(lambda, h/(sqrt(k_B)*sqrt(M*T)))

--- Deriving formula for: E ---

1. Postulate: E/E_P, M/m_P

   Symbolic Form: Eq(E*sqrt(G)/(c**(5/2)*sqrt(h)), sqrt(G)*M/(sqrt(c)*sqrt(h)))

2. Solved for E: Eq(E, M*c**2)

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(E, M*c**2)

--- Deriving formula for: E ---

1. Postulate: E/E_P, f*t_P

   Symbolic Form: Eq(E*sqrt(G)/(c**(5/2)*sqrt(h)), sqrt(G)*f*sqrt(h)/c**(5/2))

2. Solved for E: Eq(E, f*h)

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(E, f*h)

--- Deriving formula for: r_s ---

1. Postulate: r_s/l_P, M/m_P

   Symbolic Form: Eq(c**(3/2)*r_s/(sqrt(G)*sqrt(h)), sqrt(G)*M/(sqrt(c)*sqrt(h)))

2. Solved for r_s: Eq(r_s, G*M/c**2)

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(r_s, G*M/c**2)

--- Deriving formula for: P ---

1. Postulate: P/P_P, (T/T_P)**4

   Symbolic Form: Eq(G**2*P*h/c**7, G**2*T**4*k_B**4/(c**10*h**2))

2. Solved for P: Eq(P, T**4*k_B**4/(c**3*h**3))

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(P, T**4*k_B**4/(c**3*h**3))