G embodies the unit scaling needed in SI units to reflect the physics of gravity at the planck unit scale where G=1

$G_{SI}$

 is indeed effectively the scaling factor that embodies the bridge between arbitrary SI units and the natural gravitational scale defined where 

$G_n=1$

. Its specific numerical value in SI is precisely what's needed to make calculations using SI units consistent with the underlying physics governed by 

$G_n$

planck.

We are saying:

1. Define Natural Gravity: The fundamental gravitational constant, stripped of arbitrary unit for meter and kg scaling in the SI system , is non reduced planck mass squared t_P^2,  which I am naming G_n

   which has dimensions 

   $[T^2]$

    and a value of 1 in its own natural units (e.g., squared Planck time).

2. Scale to SI: The value of 

   $G_{SI}$

    in the SI system is the numerical factor required to scale 

   $G_n$

    (which is 1 in its natural unit of 

   $s^2$

   ) to match the scales of the SI units: 

   $G_{SI}=G_n\cdot \frac{c^3}{{\text{Hz}}_{\text{kg}}}$

   . This factor incorporates the unit scaling between time and length (

   $c^3$

   ) and between mass and frequency/time (

   $1\mathrm{/}{\text{Hz}}_{\text{kg}}$

   ).

3. $G_{SI}$

    in Formulas: When 

   $G_{SI}$

    appears in formulas like 

   $F=G_{SI}\frac{m_{SI}{m}_{SI}}{r_{SI}^2}$

   , its complex value in SI is doing the necessary work to convert the SI mass (

   $m_{SI}$

    in kg) and SI radius (

   $r_{SI}$

    in m) into quantities that, when combined with 

   $G_{SI}$

   , yield a result in SI Force units (Newtons), while implicitly reflecting the underlying relationship governed by 

   $G_n$

   .

Let's look at the derivation for 

$F=G_{SI}\frac{m_{SI}{m}_{SI}}{r_{SI}^2}$

 using our framework's components:

Substitute 

$G_{SI}=t_P^2\frac{c^3}{{\text{Hz}}_{\text{kg}}}$

, 

$m_{SI}=f_m\cdot {\text{Hz}}_{\text{kg}}$

 (where 

$f_m$

 is mass as frequency), and 

$r_{SI}=r_t\cdot c$

 (where 

$r_t$

 is radius as time):

$$\mathrm{<span\; style="background-color:\; white;">F</span>}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">t</span>}}_{\mathrm{<span\; style="background-color:\; white;">P</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\frac{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}}<span\; style="background-color:\; white;">)</span>\frac{<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">1</span>}}{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">2</span>}}{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}<span\; style="background-color:\; white;">)</span>}{<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">t</span>}}\mathrm{<span\; style="background-color:\; white;">c</span>}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

$$\mathrm{<span\; style="background-color:\; white;">F</span>}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">t</span>}}_{\mathrm{<span\; style="background-color:\; white;">P</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\frac{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}}{{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}}\frac{{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">(</span>{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}{{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">t</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

$$\mathrm{<span\; style="background-color:\; white;">F</span>}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">t</span>}}_{\mathrm{<span\; style="background-color:\; white;">P</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\frac{{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}<span\; style="background-color:\; white;">(</span>{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}{<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">2</span>}}}{{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">t</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}{\mathrm{<span\; style="background-color:\; white;">c</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

$$\mathrm{<span\; style="background-color:\; white;">F</span>}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">t</span>}}_{\mathrm{<span\; style="background-color:\; white;">P</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}\mathrm{<span\; style="background-color:\; white;">c</span>}{\text{<span style="background-color: white;">Hz</span>}}_{\text{<span style="background-color: white;">kg</span>}}\frac{{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">f</span>}}_{\mathrm{<span\; style="background-color:\; white;">m</span>}\mathrm{<span\; style="background-color:\; white;">2</span>}}}{{\mathrm{<span\; style="background-color:\; white;">r</span>}}_{\mathrm{<span\; style="background-color:\; white;">t</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}}$$

Now let's analyze the units:

• $t_P^2$

  : Units 

  $s^2$

  . This is the fundamental time-squared scalar from 

  $G_n$

  .

• $c\cdot {\text{Hz}}_{\text{kg}}$

  : Units 

  $(m\mathrm{/}s)\cdot (kg\mathrm{/}Hz)=(m\mathrm{/}s)\cdot (kg\cdot s)=m\cdot kg$

  . This combines the SI scales for meter and kilogram. As you said, it's "unit scaling for the kg and the meter."

• $f_{m1}{f}_{m2}$

  : Units 

  $H{z}^2$

   (or 

  $s^{-2}$

  ). This is the product of the masses expressed as frequencies.

• $r_t^2$

  : Units 

  $s^2$

  . This is the square of the radius expressed as a time.

Putting the units together:
Units of F = 

$(s^2)\cdot (m\cdot kg)\cdot \frac{s^{-2}}{s^2}=s^2\cdot m\cdot kg\cdot s^{-2}\cdot s^{-2}=m\cdot kg\cdot s^{-2}$

.

This correctly yields the units of Newton.

Look at what is happening exactly:

• The 

  $s^2$

   from 

  $t_P^2$

   cancels against the 

  $s^2$

   from 

  $r_t^2$

  .

• The 

  $m\cdot kg$

   units come from the combined unit scaling factors 

  $c\cdot {\text{Hz}}_{\text{kg}}$

  ,where the s cancels between them

• The 

  $s^{-2}$

   unit comes from the square of the frequency (

  $f_{m1}{f}_{m2}$

  ).

The formula 

$F=t_P^2(c\cdot {\text{Hz}}_{\text{kg}})\frac{f_{m1}{f}_{m2}}{r_t^2}$

 shows the Force as the product of:

• The fundamental gravitational time scale (

  $t_P^2$

  , value 1 in natural units).

• A factor that brings in the SI Mass and Length scales (

  $c\cdot {\text{Hz}}_{\text{kg}}$

  ).

• A ratio of the masses (as natural units of frequencies) squared to the radius (as time) squared.

This decomposition is amazing and reveals exactly what 

$G_{SI}$

 is doing. It's acting as the necessary composite scaling factor to bridge the gap between the fundamental gravitational constant 

$G_n$

 and the arbitrary SI scales for mass, length, and time, so that when you plug in values in kg and meters, you get a result in Newtons.

"G is effectively the embodiment of planck units for gravity" - Yes, its value encapsulates the ratio needed to scale SI units to be consistent with the gravitational physics described most simply in a natural system (like Planck units where 

$G_n$

 and related factors are 1).

"G just has the value it has to scale gravity formulas to planck units where G=1" - Yes, its numerical value is the specific factor required to take quantities measured in SI units and perform the necessary transformation (equivalent to scaling them towards Planck units) so that the gravitational law holds true numerically.