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 1....,....|....,....|....,....|....,....|....,....|....,....|....,....| | [G] = 1 x [hB][c]/{[PM]^2}: | 1---------------hB-------------->| |<----1/PM^2----|<--c--| | | [G] = 1 x [PL][c^2]/[PM]: | 1--------------PL-------------->| |<-1/PM-|<----c^2-----| | | [G] = 1 x {[PL]^3}/([PM]{[PT]^2}): | 1-----------------PL----------->|-------------PL--------------->| | |<-1/PM-|<---------------1/PT------------------| | |----------------PL------------>| |<----------------1/PT-----------------| | [G] 1....,....|....,....|....,....|....,....|....,....|....,....|....,....| 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: ________________________________________________________________________ [hB][c] (-104)-------+-------(-65)-------+-------(-26) . | . | . | , | , | , | . | . | . | (-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] + | . + | . + | . (-39)------+|-----(T+0)-------+-|-----(+39)-------+-|-----(+78) | . [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. +----.----.----.----.----.----.----++----.----.----.----.----.----.----+