The Structure of Physical Law as a Grothendieck Fibration

J. Rogers, SE Ohio, 18 Jun 2025, 1757

Abstract

We present a categorical framework for understanding the emergence of physical law from measurement. By modeling physical quantities as objects in a total category 𝓔 fibered over a base category 𝓑 of dimensionless conceptual axes, we show that all empirical laws arise as Cartesian liftings of morphisms in 𝓑. The so-called "fundamental constants"—ℏ, c, G, k_B—are revealed to be cocycle data, or connection coefficients, ensuring coherence under changes of unit systems. Our framework elevates metrology to its proper foundational role: not just supporting physics, but structuring its very possibility.

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1. Ontological Cascade: The Four-Layer Architecture

This theory rests on a layered ontology, where each stratum reflects a deepening act of projection:

Layer 1 – The Coherent Substrate (𝒮ᵤ):
A unified, pre-conceptual substrate of dimensionless relations. This is the raw coherence of reality, undivided and unmeasured.

Layer 2 – The Conceptual Axes (𝒜):
The decomposition of coherence into distinct measurement categories—Mass, Time, Length, etc. This structure is formalized as a symmetric monoidal category, with morphisms reflecting natural, proportional equivalences like relₖ: Temp ↔ Freq.

Layer 3 – Coordinate Charts of Units (𝒰):
Human-imposed unit systems (SI, Planck, CGS), forming a category where morphisms are unit scaling maps or “gauge transformations.”

Layer 4 – The Measurement World (𝓔):
A total category whose objects are concrete numerical quantities, e.g., (9.8, m/s²). Physical laws live as arrows in this world—morphisms that relate quantities within and across fibers.

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2. Formal Structure: π : 𝓔 → 𝓑 as a Grothendieck Fibration

We now define:

• 𝓑: The category of dimensionless measurement types. Objects include Time, Mass, etc. Morphisms are conceptual: e.g., φ: Mass → Energy denotes an invariant, unit-free relationship.

• 𝓔: The category of measured quantities. Objects are pairs like (value, unit); morphisms include physical laws and conversion rules.

• π: A fibration functor π : 𝓔 → 𝓑, mapping each measurable quantity to its conceptual base. The fiber π⁻¹(X) over each object X ∈ 𝓑 contains all possible unit-anchored realizations of X.

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3. Physical Law as Cartesian Lifting

Given a morphism φ : X → Y in 𝓑 (e.g. Mass → Energy), a physical law is defined as a Cartesian lifting:

$$f : (x, U₁) → (y, U₂) \quad \text{in} \quad 𝓔$$

such that:

$$π(f) = φ, \quad \text{and } f \text{ is universal over } φ.$$

This construction ensures that unit-dependent expressions like E = m·c² are liftings of simpler proportional laws, with constants like $c^2$ encoding the fiber-wise transition data.

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4. Constants as Cocycles and Connection Coefficients

The coherence of physics across unit systems imposes commutativity conditions on the liftings. When transitioning between unit charts U₁ → U₂, the law's form must remain valid. Constants like c, h, k_B, G are not fundamental quantities but cocycle data—transition maps ensuring these liftings agree.

In this view:

• ℏ encodes the Mass ↔ Frequency scaling transition (via Hzₖg).

• k_B encodes the Temp ↔ Energy connection (via K_Hz).

• G arises from a composite morphism through Time, Mass, and Length projections.

These constants are artifacts of the measurement bundle—not properties of nature, but glue for projection coherence.

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5. The Law Compiler Λ: Functorial Derivation

We define a compiler functor:

$$Λ : \text{Hom}(𝓑) \to \text{Sect}(π)$$

which assigns to each conceptual relationship in 𝓑 a section of the fibration π. These sections correspond to all consistent laws in 𝓔 realizing that relationship under various unit conditions.

Λ is your Formula Forge in functorial form. It automates law synthesis from dimensional primitives.

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6. Observers as Section Selectors

Each observer defines a fibered view of 𝓔 by selecting:

• A conceptual decomposition (choice of axes)

• A unit scheme (choice of chart in 𝒰)

• A coordinate basis (point within each fiber)

Thus, physical laws become observer-indexed liftings through π. Constants encode the observer's scaling imprint upon the substrate. This recasts the observer not as a nuisance, but as the constructor of law’s visible form.

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7. Extensions: Higher-Category Generalization

To go further:

• Model constants as 2-morphisms: Natural transformations mediating lifts across overlapping charts.

• Consider bicategories of measurement: where morphisms are liftings and 2-morphisms are consistent rewritings under unit or basis changes.

• Express measurement theory as a stack: capturing local law forms over the base 𝓑, glued by constants.

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8. The Projection Calculus: Physical Laws as Basis Rotations

8.1 Why the Calculus Works: Categorical Projection Theory

The computational derivation system demonstrated above operates through a precise categorical mechanism. What appears to be "formula derivation" is actually basis rotation through fibered projection. Here we explain why this calculus necessarily works and what it reveals about the nature of physical law.

The Three-Stage Projection Process

Every physical law emerges through a canonical three-stage process:

• Stage 1 – Substrate Conception:
  A dimensionless relationship like T ~ 1/M exists as a morphism in the conceptual category 𝒜. This is pure proportionality—a basis rotation between measurement axes with no coordinate dependence.

• Stage 2 – Planck Normalization:
  The conceptual relationship is scaled to natural units, expressing both sides as ratios to their respective Planck quantities: T/T_P ~ (m_P/M). This anchors the rotation in the coherent substrate 𝒮_u.

• Stage 3 – Coordinate Projection:
  The normalized relationship is lifted through the fibration π : 𝓔 → 𝓑 into our chosen unit coordinate system, generating the coordinate-dependent formula via Jacobian transformation.

Why This Process is Universal

The calculus works because every physical law is fundamentally a basis rotation. What we interpret as complex relationships between different physical quantities are actually simple rotations between measurement axes, obscured by the Jacobian factors required to express these rotations in non-orthogonal coordinate systems.

Example – Temperature-Mass Relationship

• Substrate reality: T ~ 1/M

• Coordinate expression: T = c³ℏ/(GM k_B)

The complexity of the coordinate expression reflects only a basis misalignment. The constants act as the Jacobian matrix element for rotating from the mass axis to the temperature axis.

The Categorical Mechanism

Formally, the calculus implements:

• Morphism Identification: Base morphism φ : X → Y in 𝓑

• Planck Lifting: Canonical lift φ̃ : (X, U_P) → (Y, U_P)

• Coordinate Transport: Fibered transport along u : U_P → U_SI yields φ̂ : (X, U_SI) → (Y, U_SI)

"Fundamental constants" are just coordinate transformation coefficients from step 3.

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8.2 The Basis Rotation Interpretation

All physical complexity emerges from basis choice. The substrate contains only simple proportionalities between measurement axes.

Examples of Simple Rotations

• E ~ m: Einstein relation

• T ~ 1/M: Thermodynamic relation

• P ~ T⁴: Stefan-Boltzmann law

• F ~ ma: Newton's second law

Each is a trivial rotation in measurement space. The constants in their expressions (c², ℏ, etc.) reflect coordinate system misalignment—not physical complexity.

The Illusion of Fundamental Complexity

We mistake coordinate artifacts for deep laws. A formula like T = c³ℏ/(GM k_B) seems profound—but the actual relation T ~ 1/M is trivial in the substrate. The constants are pure Jacobian bookkeeping.

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8.3 Implications for Physical Theory

Laws as Coordinate Artifacts

Physical laws are coordinate-dependent expressions of coordinate-free proportionalities.

Constants as Measurement Geometry

"Fundamental constants" encode our measurement geometry, not reality itself.

Complexity as Basis Misalignment

Traditional axes are misaligned with the substrate basis. That misalignment creates the illusion of complexity.

Theory as Coordinate Translation

Physical theory is translation between coordinate views of the same substrate truths. The goal is to discover the coordinate-free relationships underlying all forms.

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Conclusion: A Meta-Theory of Lawful Perception

This Grothendieckian model shows that law is not imposed on reality—it is extracted from our interaction with a coherent substrate, through structured projections and consistent scaling. Constants fall out as transition data. Laws are liftings. Complexity is an artifact of chart choice. And measurement is not peripheral to theory—it is the act that generates theory.

Appendix A - Jacobian Rotation in Unit Space.

This is an implementation that proves that the constants encode the Jacobian coordinates by rotating the basis of the SI unit system to the Planck unit system of measurement. These Jacobians are the base unit scaled to time. This is what they actually really mean with the magical hand wave of setting constants to 1, but they don't understand that this is how to actually really formally mathematically perform the operation. They don't even name the Hz_kg or K_Hz Jacobian coordinates in the standard framework.

cat modular/unit_scaling/planck_h_scaling.py  https://github.com/BuckRogers1965/Physics-Unit-Coordinate-System/blob/main/examples/physics_unit_coordinate_scaling.py

import math

# Dynamically loads reusable modules from specified file paths to keep the program modular and extensible. from load_mods import load_module

# Define standard CODATA 2018 values for necessary constants in SI G   = 6.67430e-11 e   = 1.602176634e-19 c   = 299792458.0 k_B = 1.380649e-23 h   = 6.62607015e-34  k   = 1.3806490000e-23

Hz_kg = h / c2 K_Hz  = k / h G_n   = G * Hz_kg / c3 t_P   = G_n(1/2) e_scaling = (  1e7 * Hz_kg * c  )(1/2) #e_scaling = (  1e7 * Hz_kg * c )(1/2) * 3.4079462030e-02 (1/4) #e_scaling = (  1e7 * Hz_kg * c / 0.26397533357678554*(2))**(1/2) 

def calculate_scaling_factors(constants):

    rescale_factors = [         {"symbol": "s",  "factor": t_P,             "swap_with": "t_Ph"},         {"symbol": "m",  "factor": t_P * c,         "swap_with": "l_Ph"},         {"symbol": "kg", "factor": Hz_kg/t_P,       "swap_with": "m_Ph"},         {"symbol": "K",  "factor": 1/(t_P *  K_Hz), "swap_with": "T_Ph"},         #{"symbol": "C",  "factor": e_scaling,       "swap_with": "C_Ph"},         {"symbol": "A",  "factor": e_scaling / t_P, "swap_with": "A_Ph"},         {"symbol": "mol","factor": Hz_kg/t_P,       "swap_with": "mol_Ph"},         {"symbol": "pi", "factor": 1.0,             "swap_with": "pi"},         {"symbol": "amu","factor": 1.0,             "swap_with": "amu"},     ]

    composite_unit_module = load_module("./modular/composite_units.py", "composite_units")

    return composite_unit_module.rescale_composite_units(rescale_factors, "_Ph")

The output: python physics_unit_coordinate_scaling.py 

   Symbol   Constant Name                    Original Value   Rescaled Value   Units Applied                 Ratio    -----------------------------------------------------------------------------------------------------------------------------

   --- Core Scaling Constants ---    c        speed_of_light_c                 2.9979245800e+08 1.0000000000e+00 l_Ph t_Ph⁻¹                   2.99792458e+08    h        planck_constant_h                6.6260701500e-34 1.0000000000e+00 J_Ph t_Ph                     6.62607015e-34    Hz_kg    Hz_kg                            7.3724973238e-51 1.0000000000e+00 m_Ph Hz_Ph⁻¹                  7.37249732e-51    k        boltzmann_constant_k             1.3806490000e-23 1.0000000000e+00 m_Ph l_Ph² t_Ph⁻² T_Ph⁻¹      1.38064900e-23    K_Hz     K_Hz                             2.0836619123e+10 1.0000000000e+00 Hz_Ph T_Ph⁻¹                  2.08366191e+10    G        gravitational_constant_G         6.6743000000e-11 1.0000000000e+00 l_Ph³ m_Ph⁻¹ t_Ph⁻²           6.67430000e-11    k_e      coulombs_constant_k_e            8.9875517870e+09 9.9999999996e-01 m_Ph l_Ph³ t_Ph⁻² C_Ph⁻²      8.98755179e+09

Appendix B - Calculus of Physical Law

https://github.com/BuckRogers1965/Physics-Unit-Coordinate-System/blob/main/examples/buckingham_pi_group.py

python buckingham_pi_group.py 

 These formulas are derived by dimensional analysis of Π groups,

     showing the deep unity underlying all of physics.

 # GRAVITATIONAL AND RELATIVISTIC PHYSICS 

--- Deriving formula for: T in Hawking radiation temperature of a black hole ---

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 in Newton's law of universal gravitation ---

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: r_s in Schwarzschild radius of a black hole ---

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: E in Einstein's mass-energy equivalence ---

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)

 # QUANTUM MECHANICS 

--- Deriving formula for: E in Planck-Einstein energy-frequency relation ---

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: lambda in Thermal de Broglie wavelength ---

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: p in de Broglie momentum-wavelength relation ---

1. Postulate: p/p_P, l_P/wavelength

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

2. Solved for p: Eq(p, h/lambda)

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(p, h/lambda)

--- Deriving formula for: x in Heisenberg uncertainty principle (order of magnitude) ---

1. Postulate: (x/l_P) * (p/p_P), 1

   Symbolic Form: Eq(p*x/h, 1)

2. Solved for x: Eq(x, h/p)

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(x, h/p)

 # THERMODYNAMICS AND STATISTICAL MECHANICS 

--- Deriving formula for: P in Stefan-Boltzmann radiation pressure ---

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))

--- Deriving formula for: rho in Blackbody energy density ---

1. Postulate: rho/rho_P, (T/T_P)**4

   Symbolic Form: Eq(G**2*h*rho/c**5, G**2*T**4*k_B**4/(c**10*h**2))

2. Solved for rho: Eq(rho, T**4*k_B**4/(c**5*h**3))

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(rho, T**4*k_B**4/(c**5*h**3))

--- Deriving formula for: E in Blackbody energy ---

1. Postulate: E/E_P, (T/T_P)**4

   Symbolic Form: Eq(E*sqrt(G)/(c**(5/2)*sqrt(h)), G**2*T**4*k_B**4/(c**10*h**2))

2. Solved for E: Eq(E, G**(3/2)*T**4*k_B**4/(c**(15/2)*h**(3/2)))

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(E, G**(3/2)*T**4*k_B**4/(c**(15/2)*h**(3/2)))

--- Deriving formula for: lambda in Wien's displacement law ---

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

   Symbolic Form: Eq(T*k_B*lambda/(c*h), 1)

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

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(lambda, c*h/(T*k_B))

--- Deriving formula for: P in Ideal gas pressure ---

1. Postulate: P/P_P, (T/T_P) * (l_P**3/l**3)

   Symbolic Form: Eq(G**2*P*h/c**7, G**2*T*h*k_B/(c**7*l**3))

2. Solved for P: Eq(P, T*k_B/l**3)

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(P, T*k_B/l**3)

 # CLASSICAL MECHANICS 

These classic formulas the input units match the ouput unit so the ratio of the constants is 1.

--- Deriving formula for: F in Newton's second law of motion ---

1. Postulate: F/F_P, (m/m_P) * (a/a_P)

   Symbolic Form: Eq(F*G/c**4, G*a*m/c**4)

2. Solved for F: Eq(F, a*m)

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(F, a*m)

--- Deriving formula for: E in Classical kinetic energy ---

1. Postulate: E/E_P, (m/m_P) * (v/v_P)**2

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

2. Solved for E: Eq(E, m*v**2)

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(E, m*v**2)

--- Deriving formula for: p in Classical momentum ---

1. Postulate: p/p_P, (m/m_P) * (v/v_P)

   Symbolic Form: Eq(sqrt(G)*p/(c**(3/2)*sqrt(h)), sqrt(G)*m*v/(c**(3/2)*sqrt(h)))

2. Solved for p: Eq(p, m*v)

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(p, m*v)

--- Deriving formula for: E in Gravitational potential energy ---

1. Postulate: E/E_P, (M1*M2/m_P**2) * (l_P/r)

   Symbolic Form: Eq(E*sqrt(G)/(c**(5/2)*sqrt(h)), G**(3/2)*M1*M2/(c**(5/2)*sqrt(h)*r))

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

3. Substituted Planck definitions...

4. Simplified Result:

   >>> Eq(E, G*M1*M2/r)