G_SI ​as a Grothendieck Functor: Mapping Between Unit Spaces in the Physics Unit Coordinate System (PUCS) Abstract

The numerical value and dimensions of Newton's gravitational constant (

$G_{\mathrm{S}\mathrm{I}}$

) in the International System of Units (SI) are widely considered to be a fundamental and mysterious property of reality. However, within the framework of the Physics Unit Coordinate System (PUCS), we propose a radical re-interpretation: 

$G_{\mathrm{S}\mathrm{I}}$

 functions as a Grothendieck functor that maps physical calculations between two distinct "unit spaces"—the human-defined SI unit space and the universe's inherent Planck-Harmonized unit space. This paper demonstrates how the explicit structure of 

$G_{\mathrm{S}\mathrm{I}}$

 and the operation of Newton's Law of Gravitation encapsulate a dynamic process of converting inputs into dimensionless Planck-scale proportions, performing the core physical law, and then scaling the result back to SI units. This perspective reveals that the complexity of 

$G_{\mathrm{S}\mathrm{I}}$

 is an artifact of unit translation, not an inherent mystery of gravitational interaction, highlighting the underlying, almost "trivially easy" simplicity of fundamental physics at its natural scale.

---

1. Introduction: The Enigma of Physical Constants

For centuries, the fundamental physical constants—such as the speed of light (

$c$

), Planck's constant (

$h$

), the Boltzmann constant (

$k$

), and Newton's gravitational constant (

$G$

)—have been regarded as intrinsic, fixed values that define the very fabric of our universe. Their specific numerical values, particularly in the International System of Units (SI), often appear arbitrary and vastly disproportionate (e.g., 

$c\approx 3\times {10}^8\text{ m/s}$

, 

$G\approx 6.67\times {10}^{-11}{\text{ m}}^3{\text{ kg}}^{-1}{\text{ s}}^{-2}$

). This has led to deep philosophical questions about their origin and a prevailing belief that their unification requires a "Theory of Everything" that would fundamentally derive these values.

The Physics Unit Coordinate System (PUCS) framework offers an alternative perspective, rooted in the understanding that our system of measurement is a set of human-defined conventions. PUCS posits a layered model of scientific understanding:

• Layer 1: Physical Reality (

  $S_u$

  ): A continuous, dimensionless, conserved "Universal State" or "Stuff" that underpins all phenomena.

• Layer 2: Human Perception: Our innate cognitive filtering that segments continuous reality into distinct dimensions (mass, length, time, temperature, etc.).

• Layer 3: Measurement Systems: Human-defined base units (kg, m, s, K) and primary dimensionful constants (

  $c,h,k$

  ) which act as explicit numerical scaling factors between Layer 1/2 proportionalities and our arbitrary Layer 3 unit scales.

• Layer 4: Physical Laws & Theories: Mathematical models and conceptual frameworks built upon Layer 3 measurements.

This paper focuses on the role of 

$G_{\mathrm{S}\mathrm{I}}$

 as a Layer 3 constant within this framework, specifically applying the advanced mathematical analogy of a Grothendieck functor to describe its dynamic function in Newton's Law of Gravitation. We will demonstrate how 

$G_{\mathrm{S}\mathrm{I}}$

 acts as a mapping between SI unit space and a fundamental, Planck-Harmonized unit space, preserving the underlying physical law while managing the necessary unit transformations.

---

2. The Physics Unit Coordinate System (PUCS): A Unified View of Units and Constants

At the heart of PUCS is the realization that fundamental physical quantities (mass, energy, frequency, temperature, length, time) are not fundamentally distinct entities, but rather different manifestations or "projections" of the same underlying, dimensionless 

$S_u$

. Crucially, PUCS reveals that:

• Time is Central: All dimensions ultimately scale to time. Length is a time delay (

  $L\sim T$

  ), mass is a frequency (

  $M\sim f\sim T^{-1}$

  ), energy is a frequency (

  $E\sim f\sim T^{-1}$

  ), and temperature is a frequency (

  $T_{temp}\sim f\sim T^{-1}$

  ).

• Constants as Scaling Factors: Constants like 

  $c,h,k$

   are not mystical numbers but precise Jacobian coordinates in unit space. They are the numerical ratios required to convert values measured in one unit scale to an equivalent value in another, reflecting inherent Layer 1 equivalences.

  • $c$

    : Scales length to time (L/T ratio).

  • $h=(m\mathrm{/}f)c^2$

    : Scales mass/frequency to energy, also acting as the fundamental mass-frequency-light speed ratio.

  • $k=(K\mathrm{/}f)h$

    : Scales temperature/frequency to energy.

• Planck Scale as Unit Harmony: The Planck units (

  $t_P,l_P,m_P,T_P$

  ) are not physical boundaries where physics breaks down. Instead, they represent the specific unit definitions that cause all fundamental constants to become unity (or simple dimensionless integers) when expressed in terms of these "natural" units. This is the scale where the universe's inherent proportionalities are most clearly and elegantly revealed, making the physics "almost trivially easy."

In this framework, any physical quantity 

$X_{\mathrm{S}\mathrm{I}}$

 in SI units can be expressed as its "natural" Planck-harmonized counterpart 

$X_{Ph}$

 multiplied by a specific Planck-harmonized scaling factor for that unit:

$X_{\mathrm{S}\mathrm{I}}=X_{Ph}\cdot (\text{scaling factor for }X)$

---

3. Defining the "Unit Categories" for the Functor Analogy

To model 

$G_{\mathrm{S}\mathrm{I}}$

 as a Grothendieck functor, we define two distinct "unit categories":

• Category 

  $\mathcal{S}$

   (SI Unit Space):

  • Objects: Physical quantities (mass, length, force) measured and represented using SI base units (kilogram, meter, second) or derived SI units (Newton). An object representing the inputs to Newton's law would be 

    $(m_{1,\mathrm{S}\mathrm{I}},m_{2,\mathrm{S}\mathrm{I}},r_{\mathrm{S}\mathrm{I}})$

    .

  • Morphisms: Physical operations or laws defined within this SI unit space. The morphism representing Newton's Law of Gravitation is 

    $F=G_{\mathrm{S}\mathrm{I}}\frac{m_{1,\mathrm{S}\mathrm{I}}{m}_{2,\mathrm{S}\mathrm{I}}}{r_{\mathrm{S}\mathrm{I}}^2}$

    .

• Category 

  $\mathcal{P}$

   (Planck-Harmonized Unit Space):

  • Objects: Physical quantities expressed as dimensionless ratios relative to their Planck-harmonized counterparts. An object representing inputs would be 

    $(m_{1,Ph},m_{2,Ph},r_{Ph})$

    , where these are pure numbers (e.g., 

    $m_{1,Ph}=m_{1,\mathrm{S}\mathrm{I}}\mathrm{/}{m}_P$

    ).

  • Morphisms: The fundamental, dimensionless physical laws that operate at the most harmonious scale. The core gravitational interaction is a simple proportionality, 

    $F_{Ph}=\frac{m_{1,Ph}{m}_{2,Ph}}{r_{Ph}^2}$

    , where the natural gravitational constant 

    $G_{Ph}$

     is unity (or a dimensionless 1).

The Grothendieck functor, which we will call 

$F_{G_{\mathrm{S}\mathrm{I}}}$

, will map from Category 

$\mathcal{S}$

 to Category 

$\mathcal{P}$

 and back again, preserving the fundamental physical law.

---

4. 

$G_{\mathrm{S}\mathrm{I}}$

 as the Grothendieck Functor: A Dynamic Mapping

Let's examine how Newton's Law of Gravitation, featuring 

$G_{\mathrm{S}\mathrm{I}}$

, implicitly performs this mapping:

The standard formula in SI units is:

$${\mathrm{<span\; style="background-color:\; white;">F</span>}}_{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">I</span>}}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">G</span>}}_{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">I</span>}}\frac{{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">I</span>}}{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">I</span>}}}{{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">I</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

Within the PUCS framework, we have the following Layer 3 (SI) to Layer 1/2 (natural) scaling definitions:

1. Mass in SI (

   $m_{\mathrm{S}\mathrm{I}}$

   ): Expressed as an equivalent frequency (

   $m_n$

   ) scaled by 

   $H{z}_{\mathrm{k}\mathrm{g}}$

    (the mass-per-frequency scaling factor):
   

   $m_{\mathrm{S}\mathrm{I}}=m_n\cdot H{z}_{\mathrm{k}\mathrm{g}}[\text{kg}]$

2. Radius in SI (

   $r_{\mathrm{S}\mathrm{I}}$

   ): Expressed as an equivalent time (

   $r_n$

   ) scaled by 

   $c$

    (the length-per-time scaling factor):
   

   $r_{\mathrm{S}\mathrm{I}}=r_n\cdot c[\text{m}]$

3. Gravitational Constant in SI (

   $G_{\mathrm{S}\mathrm{I}}$

   ): This is the key. We derived 

   $G_{\mathrm{S}\mathrm{I}}$

    as a composite of fundamental factors that align SI units with the Planck scale. 

   $G_{\mathrm{S}\mathrm{I}}$

    implicitly contains the fundamental time-squared constant 

   $t_P^2$

    (where 

   $t_P$

    is the non-reduced Planck time).
   

   $G_{\mathrm{S}\mathrm{I}}=t_P^2\cdot \frac{c^3}{H{z}_{\mathrm{k}\mathrm{g}}}[{\text{m}}^3{\text{ kg}}^{-1}{\text{ s}}^{-2}]$

Now, we substitute these PUCS definitions into Newton's Law (

$F_{\mathrm{S}\mathrm{I}}$

):

$${\mathrm{<span\; style="background-color:\; white;">F</span>}}_{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">I</span>}}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">t</span>}}_{\mathrm{<span\; style="background-color:\; white;">P</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\frac{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{\mathrm{<span\; style="background-color:\; white;">H</span>}{\mathrm{<span\; style="background-color:\; white;">z</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}}<span\; style="background-color:\; white;">)</span>\frac{<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">n</span>}}\mathrm{<span\; style="background-color:\; white;">H</span>}{\mathrm{<span\; style="background-color:\; white;">z</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">n</span>}}\mathrm{<span\; style="background-color:\; white;">H</span>}{\mathrm{<span\; style="background-color:\; white;">z</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}<span\; style="background-color:\; white;">)</span>}{<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}\mathrm{<span\; style="background-color:\; white;">c</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

Simplify the terms algebraically:

$${\mathrm{<span\; style="background-color:\; white;">F</span>}}_{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">I</span>}}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">t</span>}}_{\mathrm{<span\; style="background-color:\; white;">P</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\frac{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">n</span>}}{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">\cdot </span><span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">H</span>}{\mathrm{<span\; style="background-color:\; white;">z</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{\mathrm{<span\; style="background-color:\; white;">H</span>}{\mathrm{<span\; style="background-color:\; white;">z</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}<span\; style="background-color:\; white;">\cdot </span>{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

Cancel common factors (

$c^3\mathrm{/}{c}^2=c$

; 

$(H{z}_{\mathrm{k}\mathrm{g}}{)}^2\mathrm{/}H{z}_{\mathrm{k}\mathrm{g}}=H{z}_{\mathrm{k}\mathrm{g}}$

):

$${\mathrm{<span\; style="background-color:\; white;">F</span>}}_{\mathrm{<span\; style="background-color:\; white;">S</span>}\mathrm{<span\; style="background-color:\; white;">I</span>}}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">t</span>}}_{\mathrm{<span\; style="background-color:\; white;">P</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">c</span>}<span\; style="background-color:\; white;">\cdot </span>\mathrm{<span\; style="background-color:\; white;">H</span>}{\mathrm{<span\; style="background-color:\; white;">z</span>}}_{\mathrm{<span\; style="background-color:\; white;">k</span>}\mathrm{<span\; style="background-color:\; white;">g</span>}}<span\; style="background-color:\; white;">\cdot </span>\frac{{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">n</span>}}{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">n</span>}}}{{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

This formula still contains SI-derived units (implicitly in 

$c$

, 

$H{z}_{kg}$

, and 

$t_P$

 which is scaled to SI seconds). To express it in the pure dimensionless Planck-Harmonized space (

$\mathcal{P}$

), we define the dimensionless Planck-harmonized quantities:

$m_{Ph}=m_n\cdot t_P(\text{dimensionless})$

$r_{Ph}=r_n\mathrm{/}{t}_P(\text{dimensionless})$

Then, the core dimensionless gravitational relationship is:

$${\mathrm{<span\; style="background-color:\; white;">F</span>}}_{\mathrm{<span\; style="background-color:\; white;">P</span>}\mathrm{<span\; style="background-color:\; white;">h</span>}}<span\; style="background-color:\; white;">=</span>\frac{{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">1</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">P</span>}\mathrm{<span\; style="background-color:\; white;">h</span>}}{\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\mathrm{<span\; style="background-color:\; white;">2</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">P</span>}\mathrm{<span\; style="background-color:\; white;">h</span>}}}{{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">P</span>}\mathrm{<span\; style="background-color:\; white;">h</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

The overall process encapsulated by 

$G_{\mathrm{S}\mathrm{I}}$

 in the original SI formula is a dynamic, multi-step mapping (a functor's action):

1. Forward Map (

   $\mathcal{S}\to \mathcal{P}$

   ): 

   $G_{\mathrm{S}\mathrm{I}}$

    implicitly translates the SI inputs (

   $m_{1,\mathrm{S}\mathrm{I}},m_{2,\mathrm{S}\mathrm{I}},r_{\mathrm{S}\mathrm{I}}$

   ) into their dimensionless Planck-Harmonized equivalents (

   $m_{1,Ph},m_{2,Ph},r_{Ph}$

   ). This translation relies on 

   $G_{\mathrm{S}\mathrm{I}}$

   's internal structure that involves 

   $t_P$

   , 

   $c$

   , and 

   $H{z}_{kg}$

   .

2. Core Physics Operation in 

   $\mathcal{P}$

   : The truly fundamental physical law (

   $F_{Ph}=\frac{m_{1,Ph}{m}_{2,Ph}}{r_{Ph}^2}$

   ) operates in this pure, dimensionless space, where the gravitational constant (

   $G_{Ph}$

   ) is unity. This is the "almost trivially easy" calculation.

3. Backward Map (

   $\mathcal{P}\to \mathcal{S}$

   ): The result from the Planck-Harmonized space (

   $F_{Ph}$

   ) is then implicitly scaled back by the remaining factors (also embedded within 

   $G_{\mathrm{S}\mathrm{I}}$

    and the implicit unit definitions) to yield the force in SI Newtons (

   $F_{\mathrm{S}\mathrm{I}}$

   ).

This entire sequence, from SI input to Planck-scale physics to SI output, is seamlessly mediated by 

$G_{\mathrm{S}\mathrm{I}}$

. 

$G_{\mathrm{S}\mathrm{I}}$

 thus functions as the Grothendieck functor that maps across these unit spaces, preserving the fundamental physical relationship while managing all the complex unit transformations necessary for calculations in our human-defined SI system.

---

5. Implications and Conclusion

The interpretation of 

$G_{\mathrm{S}\mathrm{I}}$

 as a Grothendieck functor has several profound implications:

• Demystification of 

  $G_{\mathrm{S}\mathrm{I}}$

  : The complex numerical value of 

  $G_{\mathrm{S}\mathrm{I}}$

   is no longer a Layer 1 mystery but a precisely defined composite scaling factor within Layer 3. Its role is to translate between arbitrary SI base units and the universe's inherent, harmonious dimensional proportionalities.

• Planck Scale is Not a Breakdown: This framework strongly refutes the dogma that physics breaks down at the Planck scale. Instead, it shows that the Planck scale is precisely where the fundamental laws operate in their simplest, dimensionless form. The "math of a black hole" (e.g., 

  $r_{Ph}=2m_{Ph}$

  ) is the same dimensionless math as for two interacting feathers; it works universally.

• Physics is "Trivially Easy": The underlying physical laws are almost trivially simple when expressed in their native, dimensionless, Planck-harmonized units. The perceived complexity arises from the arbitrary unit systems we impose.

• Unification of Concepts: This analogy unifies seemingly disparate concepts: unit analysis, unit scaling, natural units, and the nature of constants are all aspects of the same underlying system. 

  $G_{\mathrm{S}\mathrm{I}}$

   encapsulates the Jacobain coordinates necessary for this unit space "rotation."

• Redefining "Fundamental": True fundamentality resides in the Layer 1 dimensionless 

  $S_u$

   and its inherent proportionalities, not in the specific numerical values of Layer 3 constants.

In conclusion, viewing 

$G_{\mathrm{S}\mathrm{I}}$

 as a Grothendieck functor between SI and Planck-Harmonized unit spaces provides an elegant and powerful re-interpretation of fundamental physical laws. It reveals that the constants are not merely static numbers but dynamic operators that implicitly perform complex unit transformations, preserving the simple, dimensionless core of physical reality. This framework offers a radical simplification of physics, bringing clarity to long-standing mysteries by properly distinguishing between the territory of objective reality and the map of our measurement systems.