THE MAGNITUDE OF BEING

The Rydberg
constant, *R*_{∞ }= 1.0973731568527 × 10^{6} dm^{-1
}(CODATA 2006), is equal to the product *µ*_{0}^{2}*c*^{3}*e*^{4}*m*_{e}/8*h*^{3}.
*R*_{∞ }is the constant of magnitude of atomic and molecular
energy levels
(The
NIST Reference on Constants, Units, and Uncertainty). *R*_{∞
}assumes a hypothetical infinite nuclear mass (compare this to the infinite
density of the finite sphere in the Banach-Tarski paradox:
Banach-Tarski Paradox & the Real Trinary
Logic Matrix of the Fibonacci, Geometric, and Perfect Number Series).
Note that [ρ = *m*/*V*] → [ρ_{∞} ↔ *m*_{∞}]. The
*R*_{∞} infinite mass assumes V = 0, and, therefore, infinite
density. The infinite density of the Banach-Tarski sphere assumes V > 0, and,
therefore, infinitely infinite mass, infinite mass squared.

*Dimensional
Analysis*

Ignoring the
purely numerical multiples, *µ*_{0} and 8 (*µ*_{0} =
permeability of free space, exactly 4π × 10^{-7}),1
reduce the Rydberg constant to its pure physical dimensions, where L = length
(m), M = mass (kg), T = time (s), and Q = (ML)^{1/2} = charge:2

(L/T)^{3}Q^{4}M/(ML^{2}/T)^{3},

(L^{3}/T^{3})Q^{4}M/(M^{3}L^{6})/T^{3},

L^{3}Q^{4}M/(M^{3}L^{6}),

Q^{4}/(L^{3}M^{2}),

(M^{2}L^{2})/(M^{2}L^{3}),

L^{-1}.

Cf. D.G. Leahy, *
Foundation: Matter the Body Itself *(*Albany, 1996*)*,* pp. 132ff.,
“The absolute measure of the magnitude of being,” where *m* = mass, *c*
= velocity of light, *T* = time, and *L* = length:

*m*^{2}/*Tc*,

* *

*m*^{2}/*T*(*L/T*),

*m*^{2}/*L*,

Q^{4}/L^{3}.

^{ }

The absolute measure of the magnitude of being = the finite measure of the
magnitude of being (*m*^{2})3
identical with the infinite measure (*L*^{-1}):

*m*^{2}/*Tc*
*=* Q^{4}/L^{3}
= [Q^{4}/(L^{3}M^{2})]M^{2
}= *m*^{2}/*L*

Compare the
unique constitutionally perfect hypercubic structure of the absolute dead center
cube (Theorem
& Proof: The Uniqueness of the Absolute Dead center Cube), where,
when *x* = 10368^{1/2},

(*x*/6)^{4}/*x*^{3}
= 8*x*^{3}/*x*^{4}.

Employ the following analogy: let (*x*/6)^{4}/*x*^{3} = M and 8*x*^{3}/*x*^{4}
= L. Then for all cubes (*x*/6)^{4}/*x*^{3} × 8*x*^{3}/*x*^{4}
= ML. Unlike all other cubes, for the absolute dead center cube (*x*/6)^{4}/*x*^{3}
= 8*x*^{3}/*x*^{4 }= (ML)^{1/2}. This
analogue illustrates how the unique
hypercubic structure of the absolute dead center cube is the formal template for
understanding Q = (ML)^{1/2} = charge.

Ignoring the non-linear factors, 6 and 8, note also the analogy between the
linear ratio *x*^{4}/*x*^{3}, the *e*^{4}/*h*^{3}
dimensional ratio integral to the Rydberg constant, and “the absolute measure of
the magnitude of being,” the bridge ratio, Q^{4}/L^{3},
the volume density of hypercubic (infinitely flat) charge.

_____________________________

Beyond the formal analogy, note that where *x*^{4 }= the volume of
the absolute dead center hypercube, *i*(*x*^{4}) = the
integral product of that volume (Measure
Beyond Beyond Reach), and *R*_{∞ }= 1.0973731568527 × 10^{6}
dm^{-1},

*R*_{∞} dm/*x*^{2} ~ *x*^{4}/*i*(*x*^{4})
= 82944/784,

the ratio uniquely that of the absolute dead center hypercube volume to its
integral product (ibid.)—with a difference ~ (829.44/784)^{128-1} ~
(*e*^{4}/*h*^{3})^{(8∙128∙128)-1}
× ML^{3}/T^{3}.

Where the Planck charge *q _{P}*

[*q _{P}*

and, since (*x*/6)^{4}/*x*^{3} = 8*x*^{3}/*x*^{4
}= (ML)^{1/2 }(the formal analogue of Q = charge, as above), where
8*x*^{3}/*x*^{4}, the surface/volume ratio of the
absolute dead center hypercube, is substituted for (*x*/6)^{4}/*x*^{3},

[*q _{P}*

with a difference in both cases ~

1 + [(*q _{P}*

_____________________________

Since [(*x*/6)^{4}/*x*^{3}]^{2} = [8*x*^{3}/*x*^{4}]^{2},
by analogy (as above) M^{2} = L^{2},5
at once ML = ML, the unique hypercubic structure of the absolute dead center cube shows it to be
not only the absolute balance point in the infinite array of all cubes (Theorem
& Proof: The Uniqueness of the Absolute Dead Center Cube;
illustrated in 2 dimensions)

but as the center for all cubes at once the fundamental template for the
formula determining the center of mass where* m* = mass, and **R**,** r**,
= the position vector:6

simplifying, transposing, and substituting M for *m* = mass and L for **R**,**
r**, = the position vector,

ML = ML.

The physical,
non-tautological, nature of this otherwise purely formal equation is a function
of the fact that, by analogy (as above), M = L = (ML)^{1/2}. In
turn, the unique rational constitution of the absolute dead center cube is
founded in the operations of real trinary logic, where, since 00 =
= 0
= 1, 0 = =
0.
For all cubes
other than the absolute dead center cube, although ML = ML, M ≠ (ML)^{1/2
}and L ≠ (ML)^{1/2}. The rationally unbalanced constitution of these cubes
inadequate to the real balance of physical forces is
consistent with dyadic trinary logic, where, although 0U = 1, since 00 ≠ UU ≠
0U, 0 ≠ U ≠ 0U.7 See, on this web,
Quantum Gravitational vs. Quantum Logic:
Virtually Left-handed Real Trinary Logic.

_____________________________

Cf. Leahy,
*Foundation*, Section IV.2, pp. 519-29, where the *L* series
is the arithmetic transcription of the sequential summing of the real
trinary logic geometric series, 1, 0,
, 1, 0,
, 1, 0,
, . . ., that is, 1,
1.5, 2, 3, 3.5, 4, 5, 5.5, . . .. The inverse alternating *L* series,

= .6444762592780 . . ..

Recall that
the fourth-power volume/surface ratio of the absolute dead center cube
divided by the cube volume, (*x*/6)^{4}/*x*^{3}, =
the surface/volume ratio of the absolute dead center hypercube, 8*x*^{3}/*x*^{4}.

Where
gravitational force = centripetal force, *G* is the gravitational
constant, (*x*/6)^{4}/*x*^{3} and 8*x*^{3}/*x*^{4}
the uniquely equal elements of the absolute dead center cube (the balance
point of the infinite array of cubes), and ~*L*^{-1} the
inverse alternating *L* series,8

[(*x*/6)^{4}/*x*^{3}
× ~*L*^{-1}]^{-1} + [8*x*^{3}/*x*^{4}
× ~*L*^{-1}]^{-1} ~ 4π^{2} = *G*(M
+ M)T^{2}/L^{3},

which is
Newton’s refinement of Kepler’s 3^{rd} law, viz., that the cube of
the radius of the orbit is proportional to the square of the period for *
all* pairs of objects orbiting their
center of mass.

Since

*G*(M)T^{2}/L^{3}
= (L^{3}/MT^{2})[(M)T^{2}/L^{3}],

the gravitational/centripetal force reduces by dimensional analysis to a function of the unique purely geometric dimensions of the absolute dead center cube:

[(*x*/6)^{4}/*x*^{3}
× ~*L*^{-1}]^{-1} + [8*x*^{3}/*x*^{4}
× ~*L*^{-1}]^{-1} ~ 4π^{2} = 2π^{2}
+ 2π^{2}.9

_____________________________

Here is a wide-angle
partial view of the Plot for (*x*/6)^{4}/*x*^{3} = 8*x*^{3}/*x*^{4},
showing the semi-axis lengths [4 | 4] and the focal parameter [2 √2 =
2.828427125 = 8^{1/2}]:

When Einstein’s
field equations are expressed in
geometrized units, the circumference of the circle intersecting the (*x*/6)^{4}/*x*^{3} = 8*x*^{3}/*x*^{4}
hyperbola = the ratio of the curvature of spacetime to its matter/energy
content:

8π = *G*_{μν}/*T*_{μν}.^{10}

**Notes**

1
µ_{0} = 4π × 10^{−7} N·A^{−2}. Cf.
http://en.wikipedia.org/wiki/Magnetic_constant,
http://en.wikipedia.org/wiki/Newton_(unit), and
http://en.wikipedia.org/wiki/Ampere. In keeping with the dimensional
analysis here employed, 1 N = 1 kg m s^{-2} = 1 ML/T^{2} and
1 A = 1 C s^{-1} = 1 (ML)^{1/2}/T, N·A^{−2 }= (ML/T^{2})/(ML/T^{2})
= 1.

2
Cf., on this web,
Common Biophysical Variables as
Combinations of Planck Length (*l*_{p}), Planck Time (*t*_{p})
and Planck Mass (*m*_{p}) & the Equivalent Golden Section
Factors, (prepared by S.B. Hoath). For the
choice of the ampere as the basic SI unit of electric current and its
historical relation to the dimensional notion of charge, cf. H.A. Klein, *
The World of Measurements* (New York, 1974), pp. 427ff.

3
For *m*^{2}, cf. N.
Redington and M.A.K. Lodhi, “Discrete Character of Meson Masses.” Online at:
http://web.mit.edu/redingtn/www/twolines.pdf. March 2010.

^{4} *q _{P}*

5
Compare the ratio, M^{2}/L^{2} = 1, to the
fundamental logical ratio *i*^{2}/*j*^{2} = 1, the
proof form of the "unity of the minimum order of the elements of the
foundation" (cf. Leahy, *Foundation*, pp. 276ff.; also, pp. 387ff., pp.
399ff., and pp. 491ff.). See also, on this web,
The Deep Epidermal
Surface: The Cornerstone Construction Order, Minimum Order Tetrahedron
Hypercube, & Absolute Dead Center Hypercube.

6 Cf. http://en.wikipedia.org/wiki/Center_of_mass.

^{7}
For a
different approach to M = L, see, on
this web, **
Theorem for a Flat Universe: The ρ _{crit} = 3H_{0}^{2}/8πG
Building Block 1000^{.125}**.

^{8} ~*L*^{-1
}= .6444762592780 . . .. The rational product of the Hebrew
נקדת העיון, ‘point of balance’, = 33.6^{-1}
and the rational product of יחוס,
‘relation’, = 8^{-1}. Consonant with its function in what
immediately follows, ~*L*^{-1 }= .6444762592780 . . . ~ (33.6^{-1})^{8-1} =
^{יחוס}נקדת
העיון, ‘point of balance’ raised to the ‘relation’ power. Cf. Leahy,
*Foundation*, pp. 521ff. For the new, essentially rational, reading of
the mathematics of the ancient languages, see, on this web,
Transdecimal Calculation of Number Identity: A Note on *Integral Product* & Related Terms.
Where the electric constant *ε*_{0} (see
the Planck charge, above, n. 4) = 8.8541878176 × 10^{-12} F/m, 144π
= the circumference inscribing the face of the absolute dead center cube = the area of each of the circles inscribing each of the 36 equal parts
of that face (an identity unique to the absolute dead center cube [cf.
Leahy, *Foundation*, p. 524, n. 98]), and ~*L*^{-1} =
.6444762592780, then (~*L*^{-1})^{16}/(144π)^{(162∙82)-1} ×
10^{-8 }F/m = *ε*_{0}, accurate
to 1 part in 10 million. Further for ~*L*^{-1}, see, on
this web,
The Diagonal Logic of the Triple Absolute Dead Center Cube & Certain
Fundamental
Mathematical
& Physical Constants.

^{9} 4π^{2 }=
2π^{2} + 2π^{2} = the total
surface area of two 4D unit hyperspheres.

^{10} Cf.
http://en.wikipedia.org/wiki/Einstein_field_equations. See also,
on this web, **
Sorted Diagonals,
**Addenda 2000-2006, 3.

.