Revising the Planck Unit Values and
Determining the Gravitational Constant
As the alternative numerical expression, symbol use, and unit
labels facilitate greater information density in maps, increased
search and sort capability in digitized literature, and superior
value comparison in tables, Geometrical Dimensional Analysis exposes
a key similarity of values. With the structuring of definitions and
formulas of fundamental constants into the lattices of logic, the
vector spaces, the factoring pathway maps projected by acceptance of
the International System of Units (S.I.), identities and patterns of
relationship are displayed which correct the estimates of the Planck
units and indicate a specific value for the constant of gravitation.
Proposing that [G] has numerical value 6.6917625079...e-11 , exactly
the same as other quantities with different units, the implication
is that the difficulties in relating gravity to other forces and the
gravitational to other constants has involved an inadequacy in the
definition or use of the quantity calculus.
With the atomic units of mass, length, and frequency as the unit
vectors defining the Corrsin diagrams(006) of GJN017 and GJN018,
both known and unsuspected factoring combinations within the atomic
units become visible, validating the premise that unit systems can
be treated as vector spaces(027) and inviting a similar display of
the Planck units, as in diagram GJN020. As an overlay of the two
natural unit systems, each of GJN023 and GJN024 is, at any point,
a dimensionless ratio of a Planck unit numerator to an atomic unit
denominator, relating together all of the quantities portrayed in
the two individual unit system maps. For better understanding of
that metaratio, however, [hB], [c], and [G] as unit vectors form yet
another view of the Planck unit system in GJN016, where it is more
obvious which quantities, lying in the plane created by adopted [hB]
and defined [c], cannot be changed, and which entities, not in that
plane, will be altered with the adjustments required to [PM], [PL],
and [Pf]: just as the product [PL][Pf] must match [v0][aa] as equal
to [c] and both [c][PL][PM] and {[PL]^2}[PM][Pf] must equal [hB],
[PL][PM] must be as equal to [hB]/[c] as is [Me][a0][aa], [PM]/[Pf]
must match both [Me]{[aa]^2}/[2af] and [hB]/[c^2], and [G]{[PM]^2},
[PM]{[PL]^3}{[Pf]^2}, and {[PQ]^2}/[k] must all be as identical to
[hB][c] as are {[Q0]^2}[Ke]/[aa] and [Me][v0^2][a0]/[aa].
While the need for some more consistent set of Planck units thus
becomes obvious, determination of the one correct set, and therefore
of the value of the gravitational constant as the factor combination
{[PL]^3}{[Pf]^2}/[PM], that arises from a very near miss in a value
table as [G] is adjusted to reflect contemporary estimates.
** For Txx.xxxxxxx = 10^xx.xxxxxxx... , current publication(322) has
[G] located in the middle of modern experimental results(023b)(322)
at T-10.1743792 = 6.693e-11 in {Meter^3)/[(Kilogram)(Second^2)], or
(Newton)(Meter^2)/(Kilogram^2), or
(Farad)(Meter)(Volt^2)/(Kilogram^2), or
(Meter)/[(Farad)(Second^2)(Tesla^2)], or
(Farad)(Meter)(Ohm)(Watt)/(Kilogram^2), or
(Meter)(Weber^2)/[(Henry)(Kilogram^2)], or
(Coulomb^2)(Meter)/[(Farad)(Kilogram^2)], or
(Meter^3)(Ohm)/[(Henry)(Kilogram)(Second)], or
(Ampere^2)(Weber^2)/[(Kilogram^2)(Newton)], or
(Coulomb)(Meter^3)(Tesla)/[(Kilogram^2)(Second)].
In (Henry)/(Meter) or (Newton)/(Ampere^2), quantity [d]{[aa]^2}
and those which are identical to it all have a numerical value of
6.6917625...e-11 as T-10.1744595 , quite close to T-10.1743792 and
even more centered in that result range. In fact, that near miss
not only stands out in the values table, it is also true that if [G]
and immutable product [d]{[aa]^2} both have that numerical value of
T-10.1744595 , so that [PM] is T-7.6628219 , [PL] is T-34.7909227 ,
and [Pf] is T+43.2677434 , a number of dimensioned and dimensionless
quantities acquire identical numerical values. While a uniform map
such as those above will locate numerical values in parallel planes,
the revised Planck unit maps now exhibit important new alignments.
There are now numerous instances where significant quantities with
dissimilar units share a line parallel to that which transfixes both
[G] and [d] : a line of multiplication by some power of the factor
[G]/[d] = [c^2][PL]/([PM][d]) = [c^4][k][PL]/[PM][d], which equals
{[aa]^2}(Coulomb^2){Meter^2)/[(Kilogram^2)(Second^2)], or
[PC][PL]/[PM] which is [aa](Coulomb){Meter)/[(Kilogram)(Second)] .
In the comparison table, values in the first column are based on
[G] as T-10.1744595 , and those in the second column derive from the
currently published NIST values(023f). The "dissimilar" quantities
which share a given numerical value are denoted by differing numbers
of dollar signs and those with units will lie on one of the lines of
coincidence mentioned above. Notice that the NIST adopted values of
the Planck units produce results which fail to match those values
required by the dimensional analysis, that different combinations of
those NIST values as factors can produce differing numerical results
for the same end product, and that some of the identical numerical
occurrences which exist only for the proposed values of [PL], [PM],
[Pf], and [G] have significant implications:
{[aa]^2} as compared to [G][k][c^2] = [G]/[d] and therefore also
[PL]/[PM] contrasted with {[aa]^2}/{[k][c^4]} = {[aa]^2}[d]/[c^2],
[PM] in contrast to [PQ][c]/[aa] or [Pf] to [PQ][c^3]/{[aa][hB]},
[PL] contrasted with [Me]{[aa]^3}/[PQ][2af] = [PQ][aa][d]/[c],
[PC] in contrast to [c^2]/{[d][aa]} and to [PM][aa]/[PL],
[G]/[k] as compared to {[E0]/[H0]}^2 = {[v0][d]}^2 = {[z0][aa]}^2
= {([He][a0][d])/[hB]}^2 = {[aa]^2}[d]/[k] = 4{[aa]^4}{[RK]}^2 ,
and [Fe]/[Fg] = ([Q0]^2)[Ke]/([G][Me][Mp]) = [aa]{[PM]^2}/{[Me][Mp]}
= {[PQ]^2}[aa][PM][d]/{[Me][Mp][PL]} = ([aa]^2)[PM][a0]/([Mp][PL])
= {[PQ]^2}[aa]/{[G][k][Me][Mp]} = ([aa]^3)[PM][Pf]/([Mp][2af])
in contrast to [c^4][a0][k]/[Mp] or to [PC][a0][aa]/[Mp] .
While it is interesting to compare the alignment factor above to
his magnetic charge and tempting to declare an insufficiency of
bases in the S.I. similar to that proposed by Desloge(102)(103),
this report intentionally declares only "inadequacy". The issue of
measurements not only includes notable argument for fewer base units
instead of more(030)(056)(235), it possesses all the potential for
misunderstanding and confusion to be inferred from centuries of
controversy and such contentions as distinguishing straight length
from arc length(076), having separate units for lengths in different
frames of reference(169), and proposals that length differs in the
direction of an electron's travel(100) or can change when measured
from a different scale(087).
More physical dimensions in the microcosm(029)(055), gravity
screening investigations(059), Foucault pendulum anomalies during
solar eclipses(323), spacecraft trajectory abnormalities(316), even
claims of antigravity experiments(324) are due consideration as an
understanding of gravity remains elusive. With visible proof that
the Planck units can be related to the other constants well enough
to correct incompatibilities with them, because the adjusted set
combines as factors to produce a numerical value shared by [G] and
quantities with factors which are significant in defining any system
of units(027)(123)(245)(261)(262), and considering the pattern
alignments by such significant numerical coincidences, the proposal
here is that reconsideration is called for of the mathematical logic
found at the foundation of all of physics. To that end, this writer
is pursuing and joins others in recommending a general review of the
works by authors such as Birge(223)(262)(263), Bridgman(067)(251),
DeBoer(147), Esnault-Pelterie(235), Hall(082), Karapetoff(232),
Maxwell(090), Page(261), Palacios(124), Petley(077), Sena(245),
Silsbee(066), Tuninsky(148), Varner(123), and Weber(233), as well as
the comments on the subject by Abraham(056) and Jackson(136) and the
more officially sanctioned publications(023)(288)(313).
** The CODATA 2006 adjustment of the fundamental constants
specifically states that both the Fixler et al results and the
Bernoldi et al results play no significant part in the officially
adopted value for Newtonian gravitational constant [G],
(Rev Mod Phys, Vol 80, No 2, Apr-Jun 2008, table XXVII and fig 2),
but dimensional analysis still requires a set of Planck mass, Planck
length, and Planck time values that correctly combine into all the
quantities and there still exists a repeating, all-pervasive pattern
of numerical coincidences among fundamental constants if [G] has the
same numerical value as the product of the magnetic constant and the
square of the fine structure constant.
COMPARISON TABLE
Mappings Complete values table Symbol key
=================================++=================================
$ = DIFFERENT DIMENSIONS, $$ = STILL DIFFERENT DIMENSIONS, etc.
* as divider: value will vary with adjustment of value of [G]
| as divider: value will NOT vary with adjustment of value of [G]
WITH EACH VALUE AS A POWER OF TEN: G=T-10.1744595 NIST Jul2010
=================================++=================================
[G] -10.1744595 * -10.1755956
{[PL]^3}{[Pf]^2}/[PM] -10.1744594 * -10.1755968
{[PL]^2}[Pf][c]/[PM] -10.1744594 * -10.1755965
[c^2][PL]/[PM] -10.1744594 * -10.1755962
[c^2]{[PL]^2}/{[a0][Me][aa]} -10.1744593 * -10.1755958
([Q0]^2)/([4Pi][aa][k]([PM]^2)) -10.1744593 * -10.1755966
{[PQ]^2}/([k]{[PM]^2}) -10.1744595 * -10.1755962
{([2af][Me]/[B0][PM])^2}/{[4Pi][aa][k]} -10.1744595 * -10.1755966
in k2rmM2trNwtn = kgrmM3trs2nd = Cuulk2rmM3trscndTsla
= fradMetrs2ndt2la = hnryk2rmMetrW2br = Fradk2rmMetrOhmmWaat
= A2prk2rmnwtnW2br = Fradk2rmMetrV2tt = hnrykgrmM3trOhmmscnd
$ [d]{[aa]^2} = [k]{[v0][d]}^2 -10.1744595 | -10.1744595
$ [k]{[He][a0][d]/[hB]}^2 -10.1744595 | -10.1744595
$ 4[k]{[aa]^4}{[RK]}^2 -10.1744597 | -10.1744595
$ [k]{[z0]^2}{[aa]}^2 = [k]{[E0]/[H0]}^2 -10.1744595 | -10.1744595
$ {4([aa]^2)}/({[Jf]^2}[c^2][Me][a0][Pi]) -10.1744595 | -10.1744595
$ [d]{[E0]^2}/{[c^2][B0]^2} -10.1744595 | -10.1744595
$ {[Q0]^2}{[d]^2}/{[4Pi][Me][a0]} -10.1744593 | -10.1744595
in Hnrymetr = a2prNwtn = FradmetrO2mm = FradmetrH2rys2nd
$$ ({[aa]^2}[PL]/[k][PM])^.5 -10.1744595 * -10.1750278
$$ [aa][PQ]/{[c][k][PM]} -10.1744595 * -10.1750280
in f12dk12mMetr = kgrmWebr = CuulfradkgrmScnd
[PM] - 7.6628219 * - 7.6622533
{[hB][c]/[G]}^.5 - 7.6628219 * - 7.6622538
{[aa]^2}[Me][Pf]/[2af] - 7.6628219 * - 7.6622540
[Me][a0][aa]/[PL] - 7.6628220 * - 7.6622537
{[PQ]^2}[z0]/{[c][PL]} - 7.6628220 * - 7.6622537
([Q0]^2)[d]/{[4Pi][aa][PL]} - 7.6628218 * - 7.6622537
in Kgrm = C2ulm2trOhmmScnd = CuulScndTsla = C2ulfradm2trS2nd
= C2ulHnrym2tr = A2prm2trOhmmScnd = Fradm2trW2br = FradM2trT2la
$ [PQ][c]/[aa] - 7.6628219 | - 7.6628217
$ [PC][PL]/[aa] - 7.6628218 | - 7.6628218
in CuulMetrscnd = AmprMetr
$$ [c^4][PL][k]/{[aa]^2} - 7.6628218 * - 7.6633900
in FradM4trs4nd
[PL] -34.7909227 * -34.7914909
{[hB][G]/[c^3]}^.5 -34.7909228 * -34.7914908
[hB]/{[PM][c]} = [a0]([Me][aa]/[PM]) -34.7909227 * -34.7914913
[G][PM]/[c^2] -34.7909228 * -34.7914903
1/{[z0][Pf][k]} -34.7909227 * -34.7914906
in Metr = fradMetrScndohmm
$ [Me]{[aa]^3}/[PQ][2af] = [PQ][aa][d]/[c] -34.7909228 | -34.7909228
$ [PM][aa]/[PC] -34.7909228 | -34.7909224
$ [PM][aa]/{[PQ][Pf]} -34.7909227 | -34.7909221
in amprKgrm = cuulKgrmScnd
$$ [c^2][PQ]/{[PM][Pf][aa]} -34.7909227 * -34.7920592
in CuulkgrmM2trscnd
$$$ [PM][d]{[aa]^2}/[c^2] -34.7909228 * -34.7903542
in HnryKgrmm3trS2nd
[Pf] +43.2677434 * +43.2683113
{[hB][G]/[c^5]}^-.5 +43.2677435 * +43.2683115
[2af][Pp]/([p0][aa]) +43.2677434 * +43.2683117
[2af][a0]/([PL][aa]) +43.2677434 * +43.2683116
[2af][PM]/([Me]{[aa]^2}) +43.2677435 * +43.2683120
[2af][PE]/[He] +43.2677434 * +43.2683120
in Hrtz
$ [PC][c]/{[aa][PM]} +43.2677435 | +43.2677431
$ [PQ][c^3]/{[aa][hB]} +43.2677434 | +43.2677435
in AmprkgrmMetrscnd = CuulM3trjuuls4nd
$$ [PM][aa]/{[PQ][PL]} +43.2677434 * +43.2688802
$$ [PQ][d][aa]/{[PL]^2} +43.2677433 * +43.2688798
in cuulKgrmmetr = CuulHnrym3tr
{[E0]/[H0]}^2 = {[v0][d]}^2 = {[z0][aa]}^2 + 0.8783918 | + 0.8783918
{([He][a0][d])/[hB]}^2 = {[aa]^2}[d]/[k] + 0.8783918 | + 0.8783918
4{[aa]^4}{[RK]}^2 + 0.8783916 | + 0.8783918
in O2mm = H2rys2nd = fradHnry = a2prC2ulf2ad
$ [G]/[k] + 0.8783918 * + 0.8772557
$ [hB][c]/([k]{[PM]^2}) + 0.8783919 * + 0.8772547
$ {[PL]^3}{[Pf]^2}/{[PM][k]} + 0.8783919 * + 0.8772545
in C2ulf2adk2rmM2tr = fradk2rmM3trNwtn = fradkgrmM4trs2nd
$$ {[c][G]/[aa]}^2 + 0.8783918 * + 0.8761196
in k2rmM8trs6nd
[G][c^2][k] = [G]/[d] - 4.2736694 * - 4.2748054
[c^4][PL][k]/[PM] - 4.2736694 * - 4.2748061
{[PC]^2}{[PL]^2}/{[PM]^2} - 4.2736692 * - 4.2748064
in Fradk2rmM3trNwtns2nd = hnrykgrmM4trs2nd = A2prk2rmM2tr
$ {[aa]^2} - 4.2736694 | - 4.2736693
is dimensionless
$$ [d][PC]{[aa]^3}/[c^2] - 4.2736694 * - 4.2731011
in amprm2trNwtnS2nd = AmprHnrym3trS2nd
[d] = 1/([c^2][k]) = {[z0]^2}[k] - 5.9007901 | - 5.9007901
in a2prNwtn = Hnrymetr
$ [PQ]/([k][c][PM][aa]) - 5.9007901 * - 5.9013587
in CuulfradkgrmScnd
$$ [G]/{[aa]^2} - 5.9007902 * - 5.9019262
in kgrmM3trs2nd
[c] = [v0]/[aa] = [E0]/{[aa][d][H0]} + 8.4768207 | + 8.4768207
[aa][H0]/{[k][E0]} = [z0]/[d] + 8.4768207 | + 8.4768207
[PL][Pf] + 8.4768207 | + 8.4768204
[hB]/{[PL][PM]} + 8.4768207 | + 8.4768203
in Metrscnd = metrtslaVltt = AmprfradMetrvltt = hnryMetrOhmm
$ [PM][aa]/[PQ] + 8.4768207 * + 8.4773893
in cuulKgrm
$$ [PM]{[aa]^2}/({[PL]^4}{[Pf^3}[k]) + 8.4768206 * + 8.4779583
$$ {[aa]^2}/{[G][c][k]} + 8.4768207 * + 8.4779568
in fradKgrmm3trS3nd
[Me][c^2] -13.0868697 | -13.0868697
([Q0]^2)[Ke]/({[aa]^2}[a0]) -13.0868695 | -13.0868697
[Fe][a0]/{[aa]^2} = [He]/{[aa]^2} -13.0868696 | -13.0868697
{[E0]^2}{[a0]^3][4Pi][k]/{[aa]^2} -13.0868697 | -13.0868697
{[B0]^2}{[a0]^3}[4Pi]/[d] -13.0868697 | -13.0868697
{[H0]^2}{[a0]^3][4Pi][d] -13.0868697 | -13.0868697
[hB][c]/{[a0][aa]} -13.0868696 | -13.0868697
{[PQ]^2}/{[a0][aa][k]} -13.0868697 | -13.0868697
[PE][PL]/([a0][aa]) -13.0868696 | -13.0868693
{[PM]^2}[G]/{[aa][a0]} -13.0868697 | -13.0868686
[FBH]([PL]^2)/([aa][a0]) -13.0868696 | -13.0868697
[PM]{[PL]^3}{[Pf]^2}/([aa][a0]) -13.0868696 | -13.0868699
in Juul = C2ulfrad = NwtnMetr = FradV2tt = hnryM4trT2la
$ {[PM]^1.5]{[PL]^.5}/([a0]{[k]^.5}) -13.0868697 * -13.0863009
in f12dK15mm12r = KgrmmetrScndVltt = KgrmMetrTsla
$$ {[PM]^2}[aa]/({[Pf]^2}{[PL]^2}[a0][k]) -13.0868697 * -13.0857320
in fradK2rmm2trS2nd
[PL]/{[PM][d]} -21.2273107 * -21.2284475
[G][k] = [G][Me][a0]({[Jf][Pi]}^2)/[4Pi] -21.2273108 * -21.2284468
{[PQ]^2}/{[PM]^2} -21.2273108 * -21.2284479
in hnrykgrmM2tr = FradkgrmM2trs2nd = M4trs4ndv2tt = C2ulk2rm
$ {[aa]^2}/[c^2] = {[aa]^2}[d][k] -21.2273108 | -21.2273108
in m2trS2nd = FradHnrym2tr
$$ [aa]({[k][PL]}^.5)/{[PM]^.5} -21.2273108 * -21.2278791
in F12dk12m = CuulkgrmmetrScnd
[c^2]/{[d][aa]} +24.9912662 | +24.9912662
in hnryM3trs2nd = A2prkgrmMetr
$ [PC] = {[k]([PL]^3)[PM]([Pf]^4)}^.5 +24.9912662 * +24.9918344
$ {1/[PL]}{[hB][c]/[d]}^.5 +24.9912662 * +24.9918344
$ [PQ][Pf] +24.9912661 * +24.9918341
$ [PM][Pf]{[G][k]}^.5 +24.9912661 * +24.9918346
$ ([G]{[PM]^2}/([d]{[PL]^2}))^.5 +24.9912661 * +24.9918349
in Ampr = Cuulscnd = F12dK12mMetrs2nd = h12yK12mMetrscnd
$$ [PM][aa]/[PL] +24.9912661 * +24.9924030
$$ [c^2][aa]/[G] +24.9912661 * +24.9924023
in Kgrmmetr
[G]([PM]^2) -25.5001033 | -25.5001022
[PM][PL][c^2] -25.5001032 | -25.5001028
{[PL]^3}{[Pf]^2}[PM] -25.5001032 | -25.5001034
{[PL]^2}{[PC]^2}[d] -25.5001031 | -25.5001031
{[PQ]^2}/[k] -25.5001033 | -25.5001032
([Q0]^2)/{[4Pi][k][aa]} -25.5001031 | -25.5001032
[hB][c] -25.5001032 | -25.5001032
in KgrmM3trs2nd = C2ulfradMetr = A2prhnryMetr = JuulMetr
$ [PQ][d][c][aa][PM] -25.5001033 * -25.4995346
in CuulHnryKgrmscnd
$$ [PC]{[PL]^2}[c^2]/[aa] -25.5001031 * -25.5006713
in AmprM4trs2nd
[G]/[c^2] = [G][d][k] -27.1281009 * -27.1292370
[c^2]/[FBH] -27.1281008 * -27.1292372
[PL]/[PM] -27.1281008 * -27.1292376
[Me][a0][aa]/{[PM^2]} -27.1281009 * -27.1292380
{[Q0]^2}[d]/([4Pi]{[PM]^2}[aa]) -27.1281008 * -27.1292380
in kgrmMetr = k2rmmetrNwtnS2nd = C2ulHnryk2rmmetr
$ [aa]/[PC] -27.1281009 * -27.1286691
$ [PQ][d][aa]/{[PM][c]} -27.1281009 * -27.1286695
in ampr = CuulHnrykgrmm2trScnd
$$ [k]{[d]^2}{[aa]^2} = [d]{[aa]/[c]}^2 -27.1281008 | -27.1281009
in FradH2rym3tr = Hnrym3trS2nd
[PM][Pf] +35.6049215 * +35.6060580
[c^3]/[G] +35.6049216 * +35.6060577
[hB]/{[PL]^2} +35.6049215 * +35.6060579
[FBH][d]/[z0] +35.6049215 * +35.6060579
in Kgrmscnd = K2rmMetrnwtns3nd = Juulm2trScnd = HnrymetrNwtnohmm
$ [PC][c]/[aa] +35.6049216 * +35.6054898
in AmprMetrscnd
$$ [aa]{[PM]^1.5}{[d]^.5}/{[PL]^1.5}) +35.6049215 * +35.6066267
in KgrmmetrScndTsla
{[PQ]^2} -36.5529546 | -36.5529545
[k][c^2][PL][PM] -36.5529545 | -36.5529541
[k][PM]{[PL]^3}{[Pf]^2} -36.5529545 | -36.5529547
[[Q0]^2}/{[4Pi][aa]} -36.5529544 | -36.5529545
[Me][a0][aa]/[d] -36.5529546 | -36.5529545
[hB][c][k] -36.5529545 | -36.5529545
[PL][PM]/[d] -36.5529545 | -36.5529541
[G][k]{[PM]^2} -36.5529546 | -36.5529534
in C2ul = FradJuul = hnryKgrmM2tr
$ [PQ][PM][aa]/[c] -36.5529546 * -36.5523859
in CuulKgrmmetrScnd
$$ ([PM]^2}{[aa]^2}/[c^2] -36.5529546 * -36.5518174
in K2rmm2trS2nd
[Fe]/[Fg] = ([Q0]^2)[Ke]/([G][Me][Mp]) +39.3546349 * +39.3557711
{[PQ]^2}[aa]/{[G][k][Me][Mp]} +39.3546349 * +39.3557711
{[PQ]^2}[aa][PM][d]/{[Me][Mp][PL]} +39.3546348 * +39.3557717
[aa]{[PM]^2}/{[Me][Mp]} +39.3546349 * +39.3557721
([aa]^2)[PM][a0]/([Mp][PL]) +39.3546349 * +39.3557717
([aa]^3)[PM][Pf]/([Mp][2af]) +39.3546348 * +39.3557714
is dimensionless
$ [PC][a0][aa]/[Mp] +39.3546349 * +39.3552032
in AmprkgrmMetr
$$ [c^4][a0][k]/[Mp] +39.3546349 | +39.3546350
$$ {[PC]^2}{[PL]^2}/{[aa][Me][Mp]} +39.3546351 | +39.3546351
$$ {[PQ]^2}[c^2]/{[aa][Me][Mp]} +39.3546349 | +39.3546350
in FradkgrmM4trs4nd = A2prk2rmM2tr = C2ulk2rmM2trs2nd
[PM][PL] = {[PQ]^2}[d] -42.4537446 | -42.4537442
[hB]/[c] = [Me][Re]/[aa] = [Me][a0][aa] -42.4537446 | -42.4537446
[aa][4Pi][k]{[B0]^2}{[a0]^4} -42.4537446 | -42.4537446
[aa][Ke]{[Me^2}/({[B0]^2}{[a0]^2}) -42.4537447 | -42.4537446
([Q0]^2)[d]/([aa][4Pi]) -42.4537445 | -42.4537446
[Jf][aa]{[Me]^2}/{4[k][B0]} -42.4537447 | -42.4537446
4[k][aa]{[w0]^2}/[Pi] = 2[hB][RK][k][aa] -42.4537447 | -42.4537446
in KgrmMetr = C2ulHnrymetr = FradM3trT2la = FradJuulmetrOhmmScnd
$ {[PL]^2}[PC]/[aa] -42.4537445 * -42.4543127
$ [c][PL][PQ]/[aa] -42.4537446 * -42.4543128
in AmprM2tr
$$ [c^4]{[PL]^2}[k]/{[aa]^2} -42.4537445 * -42.4548810
in FradM5tr
[G][Me]/([a0]{[v0]^2}) -42.6185437 * -42.6196798
[G]{[B0]^2}[4Pi][k]/{[2af]^2} -42.6185437 * -42.6196798
([Me][Fg])/([Fe][Mp]) = [Fge]/[Fe] -42.6185437 * -42.6196798
([Me]^2)[G][4Pi][k]/([Q0]^2) -42.6185439 * -42.6196798
{[Me]^2}/([aa]{[PM]^2}) -42.6185437 * -42.6196809
[Me][c^2]/({[aa]^2}[FBH][a0]) -42.6185436 * -42.6196801
[Me][PL]/([a0][PM]([aa]^2)) -42.6185436 * -42.6196805
[Me][2af]/([PM][Pf]{[aa]^3}) -42.6185436 * -42.6196802
([PL]^2)/{([a0]^2)([aa]^3)} -42.6185435 * -42.6196801
4[PL][B0][k]/({[aa]^2}[PM][Jf]) -42.6185436 * -42.6196805
[Pf]{[PL]^3}/({[aa]^2}{[a0]^3}[2af]) -42.6185435 * -42.6196804
[FBH] = [hB][c]/{[PL]^2} +44.0817422 * +44.0828786
[G]{[PM]^2}/{[PL]^2} +44.0817421 * +44.0828797
[PM][PL]{[Pf]^2} +44.0817422 * +44.0828784
[PM][C^2]/[PL] +44.0817422 * +44.0828790
[c][PM][Pf] +44.0817422 * +44.0828787
[c^4]/[G] +44.0817423 * +44.0828784
{[PC^2]}[d] +44.0817423 * +44.0828787
{[PQ]^2}/([k]{[PL]^2}) +44.0817421 * +44.0828786
in Nwtn = KgrmMetrs2nd = A2prHnrymetr = C2ulfradmetr
$ {[PM]^1.5}[aa]/({[k]^.5}{[PL]^1.5}) +44.0817422 * +44.0834474
in f12dK32mmetr = KgrmTsla
$$ [PC][c^2]/[aa] +44.0817423 * +44.0823105
in AmprM2trs2nd
[A0]/[PA] = [4Pi]([a0]^2)/{[4Pi]([PL]^2)} +49.0290476 * +49.0301841
[PM][Pf]/{[Me][2af]} +49.0290477 * +49.0301842
[a0][Pf][aa]/([PL][2af]) +49.0290476 * +49.0301838
[PE][PM]/{[He][Me]} +49.0290477 * +49.0301849
{[Pp]^2}/{[p0]^2} +49.0290476 * +49.0301843
[PM]/[Pf] -50.9305653 | -50.9305646
[hB]/[c^2] = [Me]{[aa]^2}/[2af] -50.9305653 | -50.9305653
in KgrmScnd = Juulm2trS3nd
$ [PL][PQ]/[aa] -50.9305653 * -50.9311335
in CuulMetr
$$ [c^3]{[PL]^2}[k]/{[aa]^2} -50.9305652 * -50.9317017
in FradM4trs3nd
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