The Most Desirable Value for the Newtonian Gravitational Constant

   If the numerical value of Newtonian gravitational constant [G] equals
that of the product of vacuum permeability and the square of the fine
structure constant and if the mismatch of units is recognized to be an
asset, not a liability, this implies that the [G] and fine structure
constant mysteries are inseparable and that gravity and magnetism are
linked.  Since that shared value differs by almost .3% from most [G]
values officially recommended, an unconsidered mechanism is indicated in
[G] value determination experiments.  One possible candidate for such a
mechanism can address both discrepancies among and variations in those
experiments as well as a number of local and cosmological mysteries: the
redistribution, by gravitational lensing, of inertia being delivered to
local masses according to Mach's principle.  The Allais effect suggests
that the best time to explore the possibility might be during eclipses.

          An Overmatch is a Mismatch of Units With Significance

   In the nineteenth century, as those seemingly unrelated phenomena of
electricity and magnetism were investigated, the separate, systematic
developments of CGS electrostatic (ESU) units and CGS electromagnetic
(EMU) units revealed that the experimentally determined EMU current, the
Biot, was 3 x 10^10 times as large as the experimentally founded ESU
current, the Franklin per second.  That overmatch, that coincidence of
numerical value with the experimentally determined value of the speed of
light and the pattern of overmatches it engendered helped to influence
Maxwell toward developing his theory of electromagnetism.  That pattern,
like electromagnetism, survived the purge of instantaneous action at a
distance and the theory of the ether, it survived both practicalization
and rationalization of systems of units, and that pattern can still be
discerned today in comparisons of and conversions between unit systems.

   The importance of that overmatch and the resulting overmatch pattern
is great enough that it would have become apparent even if history had
been different, even if speed of light [c] had been firmly established
after creation of the ESU and EMU systems of units.  The progression
would have been markedly different, however.  Since the advent of those
complementary systems significantly increases the number of possible
quantities and combinations of them, there is produced a much larger
collection of potential entries in any listing of quantities, a much
larger database wherein patterns and changes in them could have been
observed.  Investigations would have culminated in the same value for
[c], the importance of the overmatch pattern would have been recognized,
the EMU to ESU ratios and the [c] value would again have given validity
to each other, but early on, with poor enough results for the [c] value,
there would have been no overmatch pattern suspected.  A pattern would
have become seen as better values for [c] were attained, but only one of
near misses.  Later on, with better values for [c], the pattern would be
observed to be trending toward a pattern of exact numerical overmatches,
allowing a theoretical anticipation of the exact, correct value of [c].

   The above suggestion concerning an imaginary history can be made with
confidence because, using properly interconsistent Planck unit values,
just such a situation exists in the present in the context of attempts
to determine the value of [G].  For decades, a growing and numerically
sequenced archive of quantitative values supporting research into maps
of their interrelationships has shown an increasing number of clusters,
groups of entries with near miss, almost overmatching numerical values.

  In 2007 Fixler and his associates published their determination of the
[G] value using atomic interferometry, a value quite close to that of an
existing entry in the archive, the product of vacuum permeability and
the square of the fine structure constant: [d] x {[aa]^2}.  Since
atomic velocity over [light speed times fine structure constant] = 1,
permeability times permittivity times [light speed]^2            = 1,
light speed times permeability over vacuum impedance             = 1 and
light speed times permittivity times vacuum impedance            = 1,
the chance that the gravitational constant, magnetic constant and fine
structure constant could combine with no fudge factor to also be = 1
required thorough, objective investigation.

   Using the overmatch value for [G] with the required dependent Planck
mass, Planck length and Planck time values, revision of the quantity
calculus archive makes it clear that an extensive pattern does exist,
one requiring a simple choice.  Do the current definitions of units or
certain individual quantities generate a pervasive pattern of numerical
near misses, or, for only one [G] value, is it a pattern of overmatches,
is it a situation of significance like that known historical precedent?

   In deciding, notice that this is a convergent/divergent phenomenon.
Any archive entry can be multiplied by positive or negative powers of
the overmatch ratio ([d]{aa]^2)/[G] as often as desired, generating a
cluster of numerically matching quantities.  Then, within each cluster,
an adjustment to the overmatch [G] value will cause all but one of the
entries to diverge, some increasing and some decreasing in value.  The
entries with higher powers of adjustable component factors will change
more in value for each increment of adjustment and unadjustment of the
[G] value will return every cluster to being numerically synchronized.

   Additional evidence for the desirability of this one [G] value comes
when quantitative values are not only listed with numerical sequencing,
but also arranged into scaled sequences, one dimensional maps of the
numerical interrelationships, and into multidimensional maps showing the
complete quantitative interrelationships among related entries.  Such a
more complete overview increases recognition of the plurality of and the
profundities of the quantity overmatches created.

   If T-33 = 10^-33, T+7 = 10^+7 and etc, [hB] as T-33, [c] as T+7 and
[G] as T-10 are grossly incorrect, but with [PM] = T-8, [PL] = T-32 and
[PT] = T-39, they not only supply the interconsistency lacking in the
2018 and 2022 [G], [PM], [PL] and [PT] subsets, they also allow a
simplistic introduction to some techniques that aid in understanding.

   They emphasize how numerical values expressed as logarithms reduce
multiplication and division to addition and subtraction of exponents:

     [G][PM]^2    = [hB][c]    and both precisely = T-26,              I
    T-10T-8T-8    = T-33T+7    and both precisely = T-26,

     {[PL]^2}/[G] = [hB]/[c^3] and both precisely = T-54,             II
     {[PT]^2}/[G] = [hB]/[c^5] and both precisely = T-68,            III
     [PM][PL]     = [hB]/[c]   and both precisely = T-40,             IV
     [PM][PT]     = [hB]/[c^2] and both precisely = T-47,              V
     [PL][PT]/[G] = [hB]/[c^4] and both precisely = T-61,             VI
     [G][PM]/[PL] = [c^2]      and both precisely = T+14,            VII
     [G][PM]/[PT] = [c^3]      and both precisely = T+21,           VIII
and  [PL]/[PT]    = [c]        and both precisely = T+7,              IX

with [hB]  = [c^2][PM][PT]  = [c][PM][PL]  = [PM]{[PL]^2}/[PT]
     T-33  = T+14T-8T-39    = T+7T-8T-32   = T-8T-32T-32T+39

and  [G]  = [hB][c]/{[PM]^2} = [c^2][PL]/[PM] = {[PL]^3}/([PM]{[PT]^2})
     T-10 = T-33T+7T+8T+8    = T+7T+7T-32T+8  = T-32T-32T-32T+8T+39T+39.

They enable a more germane, numerical sequencing in a quantity archive:

T+  7  [c] = [PL]/[PT] = [hB]/{[PM][PL]}
T-  8  [PM] = [hB]/{[c][PL]} = [c^2][PL]/[G]
T- 10  [G] = [c^2][PL]/[PM] = [hB][c]/{[PM]^2} = {[PL]^3}/([PM]{[PT]^2})
T+ 14  [c^2] = [G][PM]/[PL] = [hB]/{[PM][PT]}
T+ 21  [c^3] = [[G][PM]/[PT] = [hB][G]/{[PL]^2}
T- 26  [hB][c] = [G][PM]^2
T- 32  [PL] = [c][PT] = [hB]/{[c][PM]} 
T- 33  [hB] = [c^2][PM][PT] = [c][PM][PL] = [PM]{[PL]^2}/[PT]
T- 39  [PT] = [PL]/[c] = [hB]/{[PM][c^2]}
T- 40  [hB]/[c] = [PM][PL]
T- 47  [hB]/[c^2] = [PM][PT]
T- 54  [hB]/[c^3] = {[PL]^2}/[G]
T- 61  [hB]/[c^4] = [PL][PT]/[G]
T- 68  [hB]/[c^5] = {[PT]^2}/[G].

They inspire logarithmic scales where calculations are represented by
changes of position on the scale, thus enabling a graphical addition
and subtraction of powers to accomplish multiplication and division
on any meter stick, any ten meter tape or in a diagram such as this,

     [1/c]  1/[c^2]   [hB][c]  [hB]   [hB]/[c] <<<<< unadjustable values
      -7     -14         -26    -33    -40           above
       |      |           |      |      |

        T-10      T-20      T-30      T-40      T-50      T-60      T-70
         [G] = 1 x [hB][c]/{[PM]^2}:
         [G] = 1 x [PL][c^2]/[PM]:
         [G] = 1 x {[PL]^3}/([PM]{[PT]^2}):
          |      |<-1/PM-|<---------------1/PT------------------|
          |      |----------------PL------------>|
        T-10      T-20      T-30      T-40      T-50      T-60      T-70

        | |                     |      |
      -8  -10                  -32    -39        <<<<< adjustable values
    [PM]   [G]                 [PL]    [PT]            below

and, if [OR] is the origin, [PE] = Planck energy, [Pp] = Planck momentum
and [Pf] = Planck frequency, then [PM], [PL] and [Pf] as unit vectors
form a multidimensional map displaying the interrelationships among both
numerical values and units of measure in this quantity calculus network:


                          . |                 . |                 . |
                        ,   |               ,   |               ,   |
                      .     |             .     |             .     |
                  (-96)-------+-------(-57)-------+-------(-18)     |
                  . |[PL]^3 +         . |       +         . |       +
                ,   |       |       ,   |       |       ,   |       |
              .     |       |     .     |       |  [G].     |       |
          (-88)-------+-------(-49)-------+-------(-10)     |       |
            |       +       |   |       +   [hB]|   |       +   [PE]|
            |       |     (-72)-|-----+-|-----(-33)-|-----+-|-----( +6)
            |       |     . |   |       |     . |   |       |     . |
            |       |   ,   |   |       |   ,   |   |       |   ,   |
            +       | .     |   +       | .     |   +       | .     |
            |     (-64)-------+-|-----(-25)-------+-|-----(+14)     |
            |     . |[PL]^2 +   |     . |       +   |     . |[c^2]  +
            |   ,   |       |   |   ,   |       |   |   ,   |       |
            | .     |       |   | .     |       |   | .     |       |
          (-56)-------+-------(-17)-------+-------(+22)     |       |
            |       +       |   |       +   [Pp]|   |       +       |
            |       |     (-40)-|-----+-|-----( -1)-|-----+-|-----(+38)
            |       |     . |   |       |     . |   |       |     . |
            |       |   ,   |   |       |   ,   |   |       |   ,   |
            +       | .     |   +       | .     |   +       | .     |
            |     (-32)-------+-|-----( +7)-------+-|-----(+46)     |
            |     . |[PL]   +   |     . |[c]    +   |     . |       +
            |   ,   |       |   |   ,   |       |   |   ,   |       |
            | .     |       |   | .     |       |   | .     |       |
          (-24)-------+-------(+15)-------+-------(+54)     |       |
            |       +       |   |       +       |   |       +       |
            |       |     ( -8)-|-----+-|-----(+31)-|-----+-|-----(+70)
            |       |     .[PM] |       |     .     |       |     .
            |       |   ,       |       |   ,       |       |   ,
  [PT]      +       | .         +       | .         +       | .
            |     . [OR]        |     . [Pf]        |     . [Pf]^2
            |   ,               |   ,               |   ,
            | .                 | .                 | .
          ( +8)-------+-------(+47)-------+-------(+86)

     Accurate Enough Values in the Real Present Convey the Elegance

   Using the methods exemplified above and CODATA 2018 or CODATA 2022
recommended values of the fundamental constants expressed as common logs
rounded off to seven decimal places, a unified context quantity calculus
map of Atomic units is self validating because completely populated by
known values and thus provides a trustworthy template for mapping Planck
units.  Seeing there the quantity calculation 1 x [G] = [G] represented
among the others in that network, inspection provides a better vector in
the map to examine, a better question to ask: is the value of the vector
from the plot point for vacuum permeability [d] to that of [G] a match
to a combination of powers of established constants, does the value of
[G]/[d] match that of [aa]^2 , or is it just some close near miss?

   The self validating Atomic unit map portrays a set of entities which
are interconsistent.  Every pathway to a plot point from another depicts
an equivalent calculation, all equivalent calculations achieve the same
result, however computed, by whatever path.  That set is also a coherent
set.  With [2Pi] and [4Pi] as constants, every resultant quantity can be
seen as a combination of constants of established value multiplied by 1.

   While many sets of Planck unit values map out to be interconsistent,
only one [G] value, one subset of [G], [PM], [PL], [PT] values, one set
of Planck unit values makes [G]/[d] numerically match [aa]^2, bringing
the complete coherence of the Atomic units network and map to the Planck
units network and map.  It is the same value supplying the pervasive
overmatch pattern seen as the multitude of numerically synchronized
clusters of entries in the sequenced archive.  It is the same value
providing the maximum synchronization of locations on the logarithmic
scale of numerical values of combinations of constants.  It is the same
one value replicating the undeniably significant historical precedent.

   How desirable should a [G] value be?


             Redistribution of Inertia: A Fibrous Universe?

   Gravitational lenses, altering space time geodesics, redistribute
what travels them, concentrating it here, diluting it there.  If Mach's
principle is valid in that inertia of local matter is the gravitational
influence arriving from the rest of the mass in the universe, it must
travel those same geodesics, no faster than the speed of light, and it
must also be redistributed.  While electromagnetic radiation is lensed
around a mass or blocked, gravitational screening does not exist, and,
finding mass transparent, some redistributed arriving inertia (RAI) will
have travelled the more radically altered geodesics within lensing mass.
Having energy, and hence mass, and possibly in transit for billions of
years, some RAI might concentrate by self interation as well.  If any
factors segregate volumes of RAI into streams, fibers of more amid the
accompanying volumes of less, the consequences can be greater than those
of concentrated light, possibly explaining a number of mysteries.

   On a cosmlogical scale, RAI may explain the organizing of structures
such as the sheets and strings of galaxies seen at great distance or the
disorganizing of structures to yield such things as quasars.  Encounters
or cessation of encounters of galaxies with RAI, at different angles,
might explain the existence of or anomalous directions of galactic jets,
barred spiral galaxies, such spectacles as Hoag's object, the existence
of globular clusters or those observations leading to theories of the
unverified dark matter.  As the Earth and the solar system move through
space, passage through concentrations or dilutions of RAI might even
explain such dire occurrences as the natural cataclysms and/or mass 
extinctions that fossil, geological and historical records indicate.

   Less energetic RAI might explain not only the anomalies in paths of
satellites and space probes and the gap between the Kuiper belt and the
Oort cloud, the Kuiper cliff, it might explain both the presence of and
variations in the Allais effect and both less desirable values for and
variations in determinations of the Newtonian gravitational constant.

   On October 2, 2024, during the eclipse of the Sun due to occur just
nine days after the autumnal equinox, the path of totality will pass
over a location close to 133 degrees west longitude and only two degrees
below the equator, just as it did on April 8, 2024, eighteen days after
the vernal equinox, and just as it will again on November 14, 2031,
twenty two days after the autumnal equinox when maximum totality will be
seen at that location.  In each case, there is arriving, after more than
eight minutes of travel time, what was emmitted by the Sun, what was
gravitationally lensed around the Sun, and, traversing the more strongly
altered geodesics, what finds the Sun and the Moon to be transparent.

   As the Allais effect suggests, these may be the best times to test
for local changes in inertia as rotating, moving, transparently lensing
bodies, some possessing internal differential rotation, align while they
redistribute what is arriving from all directions.  Before, during and
after each eclipse, at the mentioned location in the Pacific ocean, at
the antipodal location off the coast of Somalia and at locations off the
Earth along that line, conclusive empirical evidence might be obtained.


    Proper Interconsistency Among Planck Unit Values is a Necessity

   Sizing the wheels of an automobile so that each could be part of some
matched set is inadequate.  They must form that matched set on that car.

   For gravitational constant [G], Planck mass [PM], Planck length [PL],
Planck time [PT], speed of light [c] and unit of action (h-bar) = [hB],

[PM]^2 = [hB][c]/[G], [PL]^2 = [hB][G]/[c^3] and [PT]^2 = [hB][G]/[c^5].

Defining [hB] and [c] to be exact by convention therefore requires that

      {[PM]^2}[G]                        = unadjustable [hB][c],       I
      {[PL]^2}/[G]                       = unadjustable [hB]/[c^3]    II
and   {[PT]^2}/[G]                       = unadjustable [hB]/[c^5].  III

   The method of simultaneous equations allows broader consideration.

Since [I x II]^.5   is [PM][PL]          = unadjustable [hB]/[c],     IV
      [I x III]^.5  is [PM][PT]          = unadjustable [hB]/[c^2],    V
      [II x III]^.5 is [PL][PT]/[G]      = unadjustable [hB]/[c^4],   VI
      [I / II]^.5   is [G][PM]/[PL]      = unadjustable [c^2],       VII
      [I / III]^.5  is [G][PM]/[PT]      = unadjustable [c^3]       VIII
and   [II / III]^.5 is [PL]/[PT]         = unadjustable [c],          IX

it is clear that in the network of interrelationships among Planck unit
quantities a subset of [G], [PM], [PL] and [PT] values must be precisely
interdependent and that adjustment of any one of the four must accompany
precise, appropriate changes in the numerical values of the other three.

   Using [c] = 299792458, [hB] = 1.054571817e-34, [G] = 6.67430e-11,
[PM] = 2.176434e-8, [PL] = 1.616255e-35 and [PT] = 5.391247e-44 from
CODATA2018 and CODATA2022 in rules I - IX, resultant values do approach
unadjustables, but in the whole network of Planck interrelationships,

while     [G] as {[PL]^3}/([PM]{[PT]^2}) = 6.674 2993824...e-11,
          [G] as [hB][c]/{[PM]^2}        = 6.674 302097883...e-11,
and       [G] as [c^2][PL]/[PM]          = 6.674 30095012...e-11
do match CODATA2018 and CODATA2022 [G]   = 6.674 30e-11,

          [c] as [hB]/{[PM][PL]}         = 299792 509.554... ,
          [c] as [G]{[PM]^2}/[hB]        = 299792 363.768... ,
and       [c] as {[PM][G]/[PL]}^.5       = 299792 436.661...
though unadjustable [c]                  = 299792 458,

and       [hB] as [c][PM][PL]            = 1.054571 6356493...e-34,
          [hB] as [c^2][PM][PT]          = 1.054571 75949960...e-34,
          [hB] as [PM]{[PL]^2}/[PT]      = 1.054571 5117990...e-34
and       [hB] as [G]{[PM]^2)}/[c]       = 1.054571 485524...e-34
though unadjustable [hB]                 = 1.054571 817e-34.

Compare 2018 atomic interconsistency with AUN (atomic unit of) charge
[Q0] = 1.602176634e-19, AUN velocity [v0] = 2.18769126364e+6, AUN mass
[Me] = 9.1093837015e-31, AUN length [a0] = 5.29177210903e-11, AUN time
[AT] = 2.4188843265857e-17, Hartree energy [He] = 4.3597447222071e-18
and fine structure constant [aa] = 7.2973525693e-3 so that

       [c] as [a0]/{[AT][aa]}            = 29979245 7.99973... ,
       [c] as [hB]/{[Me][a0][aa]}        = 29979245 7.8171... ,
       [c] as ({[He]/[Me]}^.5)/[aa]      = 29979245 8.000...
and    [c] as {10^7}[hB][aa]/{[Q0]^2}    = 29979245 7.9804...
while  unadjustable [c]                  = 29979245 8,

and    [hB] as [v0][a0][Me]              = 1.054571817 641...e-34,
       [hB] as {[v0]^2}[Me][AT]          = 1.054571817 641...e-34,
       [hB] as {[a0]^2}[Me]/[AT]         = 1.054571817 642...e-34
and    [hB] as [He][AT]                  = 1.054571817 6461...e-34
while  unadjustable [hB]                 = 1.054571817 e-34.

Then compare as well the 2022 atomic interconsistency where AUN charge
[Q0] = 1.602176634e-19, AUN velocity [v0] = 2.18769126216e6, AUN mass
[Me] = 9.1093837139e-31, AUN length [a0] = 5.29177210544e-11, AUN time
[AT] = 2.4188843265864e-17, Hartree energy [He] = 4.3597447222060e-18
and fine structure constant [aa] = 7.2973525643e-3 so that

       [c] as [a0]/{[AT][aa]}            = 29979245 8.001677883... ,
       [c] as [hB]/{[Me][a0][aa]}        = 29979245 7.817863791... ,
       [c] as ({[He]/[Me]}^.5)/[aa]      = 29979245 8.001619510...
and    [c] as {10^7}[hB][aa]/{[Q0]^2}    = 29979245 7.775014808...
while  unadjustable [c]                  = 29979245 8,

and    [hB] as [v0][a0][Me]              = 1.0545718176 48360149...e-34,
       [hB] as {[v0]^2}[Me][AT]          = 1.0545718176 50122440...e-34,
       [hB] as {[a0]^2}[Me]/[AT]         = 1.0545718176 46597858...e-34
and    [hB] as [He][AT]                  = 1.0545718176 461871...e-34
while  unadjustable [hB]                 = 1.054571817 e-34.

Only one [G] value (= T-10.1744595) provides both interconsistency and
coherence to the network and map of Planck unit interrelationships:

In CODATA2018, [aa] = 7.2973525693e-3 and [d] = 1.25663706212e-6 so that
[d]{[aa]^2} has [G] = T-10.1744594 7...  = 6.6917625 698...e-11
requiring that [PM] = T- 7.6628218 2...  = 2.1735927 167...e-8,
requiring that [PL] = T-34.7909227 1...  = 1.6183680 192...e-35 and
requiring that [PT] = T-43.2677434 1...  = 5.3982946 403...e-44 with

[c], [hB] and [G] precisely accurate, however calculated, since they and
rules I - IX were used to generate the [PM], [PL] and [PT] values.

In CODATA2022, [aa] = 7.2973525643e-3 and [d] = 1.25663706127e-6 so that
[d]{[aa]^2} has [G] = T-10.1744594 7...  = 6.6917625 561...e-11
requiring that [PM] = T- 7.6628218 2...  = 2.1735927 189...e-8,
requiring that [PL] = T-34.7909227 1...  = 1.6183680 175...e-35 and
requiring that [PT] = T-43.2677434 1...  = 5.3982946 348...e-44 with

[c], [hB] and [G] again precisely accurate.

If [d] is [4Pi][T-7], [aa] = 7.2973525693e-3 and [d] = 1.25663706...e-6,
[d]{[aa]^2} has [G] = T-10.1744594 7...  = 6.6917625 662...e-11
requiring that [PM] = T- 7.6628218 2...  = 2.1735927 173...e-8,
requiring that [PL] = T-34.7909227 1...  = 1.6183680 187...e-35 and
requiring that [PT] = T-43.2677434 1...  = 5.3982946 388...e-44, and,

once again, [c], [hB] and [G] calculate back out as precisely accurate.