The Jacobian of Reality: Categorical Invariance and the Planckian Bridge

J. Rogers, SE Ohio

Abstract:
This paper presents a formal categorical proof that physical reality is composed of invariant unit-free ratios (the Terminal Object) rather than dimensional quantities. We demonstrate that the original Planck units derived from h, c, G, and kB are not fundamental physical "entities" but the identity morphisms of a specific unit chart—acting as the Jacobian coefficients that translate universal proportions into human abstractions (SI units). By normalizing these constants to unity, we recover Newton’s "Divine Language" of proportions, resolving a century-long category error in the interpretation of natural units.

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I. INTRODUCTION: THE REIFICATION ERROR

For over 125 years, theoretical physics has operated under a profound category error. Since Max Planck’s derivation of "natural units" in 1899, the scientific community has reified these values, treating the Planck length (1.616 x 10^-35 m) as a fundamental physical "thing."

This paper argues that these units are not the territory, but the scale bar. Using Category Theory, we prove that the physical object is the invariant ratio, while the unit chart is a mere abstraction.

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II. HISTORICAL CONTEXT: NEWTON'S PROPORTIONS AND PLANCK'S DICTIONARY

A. Newton (1687): The Terminal Object

In "Principia", Isaac Newton expressed the laws of nature as dimensionless proportions—geometric relationships between quantities. To Newton, the "true language of nature" was scale-free. In modern categorical terms, Newton was describing the Terminal Object (1): the invariant, background-independent network of proportions that exists prior to human measurement.

B. Planck (1899): The Jacobian Mapping

In 1899, Max Planck identified the unique mapping between the universal terminal object and the human metric system. By combining h, c, G, and kB, he revealed the precise coefficients (the Jacobian) required to translate the universe’s intrinsic scale into meters, kilograms, seconds, and Kelvin. He didn't just invent new units; he discovered the dictionary for our abstractions.

C. The Modern Blind Spot

The failure of the post-Planck framework was in reading the dictionary entries (the Planck values in SI) as fundamental physics. By reifying the "units," physics lost sight of the "proportions" they were meant to translate.

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III. CATEGORICAL FORMALIZATION

We define a Unit Chart X (such as SI) as a category where measurements are expressed as morphisms. For any chart to be physically consistent, there must exist a unique morphism to the Terminal Object (1)—the one physical reality.

1. Normalization to Unity (Inverting the Functor)

To reveal the invariant reality, we define the identity morphisms of the unit chart as Unity (1). In this state, we are no longer looking at the "labels," but the scale itself:
l_P = t_P = m_P = T_P = 1

2. Emergent Dimensional Constants

In this terminal state, all dimensional constants used by Planck emerge as simple ratios of this single scale:

• Velocity (c) = l_P / t_P = 1
• Planck (h) = (m_P * l_P^2) / t_P = 1
• Gravity (G) = l_P^3 / (m_P * t_P^2) = 1
• Boltzmann (kB) = (m_P * l_P^2) / (t_P^2 * T_P) = 1

This confirms that h, c, G, and kB are not arbitrary physical values but the necessary mathematical consequences of mapping any unit chart to the terminal scale.

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IV. THE INVARIANCE OF BETA

Consider a physical velocity ratio beta = 0.5. In this framework, beta is a pure scalar—a property of the morphism to the terminal object.

• Theorem: Because all unit charts (X) must commute with the terminal object, the value beta = 0.5 remains invariant across every chart, regardless of how velocity is scaled in that chart.
• Proof: The "value" in a unit chart is a coordinate abstraction; the ratio is the physical object itself. The mapping to the terminal object ensures that the scalar relationship beta remains a fixed point.

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V. CONCLUSION: THE UNIFIED PHYSICAL SCALE

The universe does not consist of lengths, masses, or times. It consists of a Single Scale of Proportionality.

• Planck Units (h, c, G, kB) are the identity morphisms of our abstraction.
• Dimensional Constants are the Jacobian coefficients of the translation from reality to SI.
• The Invariant Ratios are the only true physical constituents.

By recognizing the Planck scale as a Translator rather than a Destination, we recover Newton’s scale-free physics, updated for the rigorous categorical verification of 2026. We have moved beyond reading the dictionary; we are now speaking the language.

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Verification Status (Appendix A):
Computational implementation confirms: 

l_P = t_P = m_P = T_P = 1 

implies 

h = c = G = kB = 1. 

This confirms that all dimensional constants are derivable as ratios of a single scale, establishing unit-independent covariance. This is true in every unit chart, pointing to the one unified physical scale of the universe. 

Appendix: Philosophical Synthesis

This work is a convergence of truth across three different eras and methodologies.

The Rogers Terminal Object, the Kantian Noumenon, and the Vedantic Brahman are structurally identical concepts. They all describe the Undifferentiated Substrate before it is sliced up by the mechanism of observation.

Here is the translation table that unifies physics with the world of philosophy:

1. The Triangulation of Reality

Concept	Physics (Rogers)	German Idealism (Kant)	Vedanta (Ancient India)
The Reality	Terminal Object	The Noumenon	Brahman (Nirguna)
The Nature	Unitless, Singular	Unknowable, Things-in-themselves	Indivisible, Without Attributes
The Mechanism	The Jacobian / Measurement	Categories of Understanding	Maya
The Action	Projection / Transformation	Intuition (Space & Time)	Measurement / Delimitation
The Result	The Phenomenon (SI Units)	The Phenomenon (Experience)	Jagat (The World)
The Artifacts	Mass, Length, Time	Causality, Geometry	Nama-Rupa (Name & Form)

2. The Vedantic Insight: Maya means "To Measure"

The connection to Vedanta is even deeper than just the concept of Brahman. The Sanskrit word Maya (often translated as "illusion") comes from the root 

$m\bar{a}$

, which literally means "to measure" or "to delimit."

• The Vedantics understood 3,000 years ago that Measurement creates Illusion.

• To measure the Infinite (Brahman) is to break it.

• When we measure, we create boundaries (meters, seconds, kilograms) that do not exist in the source.

The paper is effectively a mathematical proof of the Vedantic claim: The world of "Name and Form" (Nama-Rupa) is a projection imposed on the singular Brahman. In this case, "Name" is the Unit Label (

$kg$

), and "Form" is the Dimensional Constant (

$G$

).

3. The Kantian Insight: Space and Time are Goggles

Immanuel Kant’s Critique of Pure Reason argued that humans cannot see the world as it is. We wear "goggles" that filter the data.

• The Goggles: Space and Time.

• Kant argued that Time is not out there. Time is the way our mind sorts inputs sequentially.

• Kant argued that Space is not out there. Space is the way our mind sorts inputs simultaneously.

The "Transducer Model" provides the biological mechanism for Kant’s philosophy:

• Time Goggles: The electron's mass anchor / memory / entropy.

• Space Goggles: The electron's charge handle / vision / light.

4. The Grand Unification

For the last 100 years, physics has been obsessed with studying the Reflections in the Mirror (The Phenomenon), measuring them to 15 decimal places.

This work represents the moment Physics turns around to look at the Source (The Noumenon/Brahman).

• "Shut Up and Calculate" was the study of Maya.

• "The Terminal Object" is the study of Brahman.

We have found the mathematical bridge (Category Theory) to prove what the greatest metaphysicians have always known: The divisions are in the observer, not the observed. There is only One Thing.

Appendix: Great Simplification

By identifying

Newtonian Proportions as Category Theory, we expose the "Modern Algebraic Corruption" for what it truly is: a layer of unnecessary abstraction that obscured the original clarity of the Terminal Object.

1. The Loss of the "Ratio": From Geometry to Algebra

Newton’s Principia was written in the language of Euclidean Proportions. He didn't use "equations" in the modern sense because an equation requires a "Constant of Proportionality" (

$G$

G) to balance the units.

• Newton’s Ontology: Force is a relation between geometries. (
  

  $F_1:F_2::r_2^2:r_1^2$
  ).
• The Modern Error: We turned the "::" (the morphism) into an "=" (the identity) by inserting
  

  $G$
  .
• The Category Error: By introducing
  

  $G$
  , we introduced the Maya of Measurement. We forced the "singular reality" to accommodate the "fragmented units" of the meter and the kilogram.

2. Newton as the First Categorical Physicist

In this framework, Newton’s focus on the Ratio is recognized as the study of Morphisms between objects in the Terminal Category.

• Newton was mapping the Source (The Sun/Center) to the Target (The Planet/Orbit).
• He understood that the "Physical Object" was the Network of Relations, not the "Things" being related.
• Our Proof: By setting  l_P = t_P = m_P = T_P = 1 we are essentially deleting the "algebraic corruption" and returning the physics to the Unitless Geometric Ratio Newton originally described.

3. The "Detour into Maya" (1687–2026)

The detour began when we started to believe the "Constants" were physical constants of nature rather than Coefficients of Translation.

• The Constant G: This is not a "force of nature." It is the Jacobian entry required to translate Newton’s pure geometric ratio into the "SI Unit Chart."
• The Constant c: This is not a "speed limit." It is the Identity Morphism required to relate the "Space Goggle" to the "Time Goggle."

4. Framing the "Vindication" in Natural Philosophy

This work places Newton back at the head of the lineage, but with a Rigorous Proof he lacked.

"Newton intuited the Terminal Object but lacked the Jacobian to prove how it connected to the human chart. Planck found the Jacobian but forgot the Terminal Object. This work completes the circuit: using Planck's Jacobian to mathematically prove that Newton's Terminal Object is the only thing that is real."

The Final Ontological Position

Modern physics thought it "surpassed" Newton by adding complexity (Fields, Tensors, Constants). We have proven that Complexity is a symptom of a poor Coordinate System.

When we move to the Natural Scale (The Terminal Object), the complexity vanishes into 1. Newton wasn't "simple" because he was primitive; he was "simple" because he was looking at the Source. We have successfully provided the category-theoretic verification that Simplicity (Unity) is the highest form of Rigor.