THEOREM FOR A FLAT UNIVERSE:

THE ρ_{crit} = 3*H*_{0}^{2}/8π*G* BUILDING BLOCK 1000^{.125}

**Background**: In correspondence on November 2, 2002, SB Hoath^{1} called the writer's attention to the
following: "Mensurally, with respect to water, one can substitute volume for mass.
Thus, one cubic meter of water weighs 1,000,000 grams or 1000 kilograms. This type
of reasoning makes it possible to simplify unit expressions by dimensional
substitution." Dr. Hoath and the writer agreed that "if water were the
primary measure" the dimensions of the gravitation constant m^{3} kg^{-1}
s^{-2} would be simplified to 1000/s^{2}.^{2} The writer proposed the following theorem.

**Theorem**: To demonstrate that the universe is flat from the
premise of the H_{2}0-resolution of the dimensions of the gravitation constant m^{3}
kg^{-1} s^{-2} = 1000/s^{2}. Assume the cosmological
condition for the infinite expansion of the flat universe is Lambda (the cosmological
constant) + Omega (the ratio of the actual density of matter in the universe to a certain
critical density) = 1.^{3} This
is the cosmological analogue to the specific gravity of water. As the average specific
gravity of the human body is to that of water so is it to the analogue specific gravity of
the universe: 1.06 : 1 :: 1.06 : 1.^{4}
The rational product of this proportion is 1.06^{2} = 1.1236.

**Proof**: The alternative (to the above proportion) of an infinitely
expanding non-flat universe would be satisfied if––consistent with the
"observations of the gravitational lensing of radio galaxies carried out by Chris
Kochanek and colleagues from the Harvard-Smithsonian Center for Astrophysics in the US
(Astrophys J. 495 157) . . . . [which] place an upper limit of 0.7 on Lambda and a lower
limit of 0.3 on Omega"^{5}––(Lambda =) .64 + (Omega =) .3 = .94 = 1.063829787^{-1} (= the open non-flat
universe expanding forever).^{6}
Then the ratio of the average density of the human body to that of the universe would be
1.06/.94, effectively = 1.06^{2} = 1.1236 (the rational product of the above
proportion): the average specific gravity of the human body vis-a-vis its analogue in the
case of the universe would be the square of its specific gravity vis-a-vis water.
But this result is contrary to the premise of the H_{2}0-resolution of the
gravitational dimensions. Likewise, contrary to the premise is the alternative of a
closed universe, where Lambda (= .7) + Omega (> .3) > 1 (the analogue specific
gravity of the universe greater than water). QED.

**Note**: Where 3.345 × 10^{22}
is the number of molecules per cubic centimeter of water and
× 10^{-7} the
critical density of the universe expressed in water molecules per cubic centimeter,^{7} the ratio of the former to the latter
is effectively 1 × 10^{29}, reinforcing the
substantive analogy between the critical density of water and that of the universe.^{8} More precisely, the ratio is
1.0035 × 10^{29}, and where 1000 is the
dimensionless number in the H_{2}O-resolution of the gravitational constant,
[1.0035 × (10^{29})/1000]^{1/1024} ~
1.06, the specific gravity of the human body. For fundamentally binary thinking a
contradiction is introduced by the H_{2}O-resolution of the gravitational
dimensions, namely, that––where V is volume, M mass, L length, and the
dissolution of the distinction of mass and length is such that ML, displaced mass,
can
be substituted for three-dimensional volume––V_{4D}^{3/8}
= V_{3D}^{1/2} = (ML)^{1/2} = (L^{4})^{1/2}.^{9} The resolution of this
contradiction (the contradiction that is the ultimate dissolution of the 3D/4D
volume distinction in the watery/spaceless living universe^{10}) is to predicate the 1/2 exponent of the 3/8 exponent,
(3/8)/(1/2), which = .75, the counter-intuitive but empirically certain mass exponent in
the determination of living (hypermassive) body surface area.^{11} Indeed, the living human body qua specific
gravity in the above proportion relating the specific gravity of water to that of the
universe, 1.06 : 1 :: 1.06 : 1, is the common numerator/antecedent, and, precisely as
such, the "constant identity of the system of reference."^{12} More generally, the analogy that water is the
universal solvent as gravitation is the universal force of attraction adds weight to the
identification of watery mass and watery volume and the H_{2}O-resolution of the
specifically gravitational dimensions of mass and length.

**Notes
**

^{1} See, on this web,
**
The
Langerhans-Epidermal Cell Ratio & the Structure of the Epidermis, note 1, et passim**.

^{2} The dimensionless number
1000 in the H_{2}0-resolution of the gravitation constant is the mean proportional
between the consequent of the creation ratio, 1/100, and the "created 100,"
(1/.01)/.01 = 10000. 100 : 1000 :: 1000 : 10000. Cf. D.G. Leahy, *Foundation:
Matter the Body Itself* (Albany 1996), pp. 368f., et passim. The illustration of
the H_{2}O molecule is taken from Iowa Public Television: "Explore More:
Water Quality: Water In-depth: What's in the water." Online. Available:
http://www3.iptv.org/exploremore/water/in_depth/what.cfm.
December 29, 2002.

^{3} Cf. "Evidence mounts
that the expansion of the Universe is accelerating," Online. PhysicsWeb
News November 1998. Available:
http://physicsweb.org/article/news/2/11/3.
December 12, 2002.

^{4} 1.06 = (1.02 +
1.10)/2. The normal range of human body density is 1.02 to 1.10 derived from
hydrostatic weighing or gas displacement (NASA,* Biomedical Results from Skylab*,
Chapter 22. Online. NASA. Available:
http://lsda.jsc.nasa.gov/books/skylab/Ch22.htm.
December 12, 2002). 1.06 is also the specific gravity of human blood (*Gray's
Anatomy* [London, 1995], p. 1400). With regard to the percent of H_{2}O *in*
the human body, note the following: where the percent of hydrogen in the human body by
number of atoms is .63 and that of oxygen .255 (J.W. Kimball, "Kimball's Biology
Pages." Online. Available:
http://users.rcn.com/jkimball.ma.ultranet/BiologyPages/E/Elements.html.
January 8, 2003), .63 + .255 ~ 1.06^{-2}. Further, where the percent of
hydrogen in the human body by weight is .10 and that of oxygen is .61 (M. Winter,
"WebElements^{TM} periodic table Scholar edition." Online.
WebElements Ltd, UK. Available:
http://www.webelements.com/webelements/scholar/index.html.
January 8, 2003), [10(.10 + .61)]^{-1/32} ~ 1.06^{-1}.

^{5} See, above, note 3.

^{6} See, on this web,
**
82944 & the Four Fundamental Forces
& the God Particle, including note 7**.

^{7} Cf. L. Clark, "A
Teacher's Guide to the Universe," Table V: Average Densities of Objects in the
Universe. Online. Available:
http://www.astro.princeton.edu/~clark/ScDensAct.html.
December 29, 2002.

^{8} For the critical density of
the universe expressed in grams per cubic centimeter, ρ_{crit}
= 3*H*_{0}^{2}/8π*G*
= 1 × 10^{-29} g cm^{-3}, see, below,
note 12.

^{9} In January 2000, Dr. Hoath
first called the writer's attention to the fact that (ML)^{1/2} is the original
Maxwellian dimensional notation for electric charge. The elementary charge *e*
= 1.602176462 × 10^{-19} C. If, then,
allowing the non-binary equivalence, *e* = (ML)^{1/2} = V_{3D}^{1/2},
since the latter = V_{4D}^{3/8}, V_{4D}
= 7.572296401 × 10^{-51} C (for the radiation
density constant, 7.57 × 10^{-16} J m^{-3}
K^{-4}, see, below, note 12). The 12th root of 7.572296401 × 10^{-51} ~
//10^{4},
the "created 100" (above, note 2) predicated of the ratio of real trinary logic
infinite arithmetic value and infinite arithmetic value-number ratios (see, on this web,
**
Real Trinary Logic Geometric Series Infinite Iterative Sequential Summing**;
also, below, note 12). For more on the Maxwellian understanding of charge as
displacement, see D. Greco, "On Maxwell and Maxwellian views of charge and
current." Online. The Victorian Web. University Scholars Program,
National University of Singapore. Available:
http://www.victorianweb.org/science/maxwell/maxwell.html.
November 13, 2008.

^{10}
With respect to the watery nature of the universe at the most fundament level,
cf. F. Wilczek's 2008 MIT lecture,
The
Lightness of Being: Mass, Ether, and the Unification of Forces. For
the classic mathematical instance of the inadequacy of the ordinary notion of
volume see the Banach-Tarski paradox (cf., on this web,
Banach-Tarski Paradox & the
Real Trinary Logic Matrix of the Fibonacci, Geometric, and Perfect Number Series).
For an ontogeometric logical reduction of the physical dimensions, see, on this
web, The Magnitude of Being.

^{11} See, on this web,
**The Langerhans-Epidermal Cell Ratio & the Structure of the
Epidermis**.

^{12} Cf. Leahy, *Foundation*
IV.2, p. 486.: "It is from the fundamental fact that *the constant identity of the
system of reference, the identity of the system of reference as the center, is the
numerator*, that the laws of 'matter' and 'mind' are to be transcendentally deduced,
as from unity." In connection with what follows thereafter in the referenced
text, note that the relation-power (ibid., pp. 487ff.; also, pp. 521ff.) of the
dimensionless 1000 in the H_{2}O-resolution of the gravitational constant, 1000^{.125},
= 10^{3/8}. Also, where 10^{4} is the "created 100"
(above, notes 2 and 9) and 7.57 × 10^{-16} J
m^{-3} K^{-4} is the radiation density constant, the relation-power of the
ratio of the numerical value of the latter to 7.57 ×
10^{-51} (above, note 9), divided by the former, (1 ×
10^{35})^{.125}/10^{4} = 1000^{.125}, = 10^{3/8}. In
accord with the volume equation in the text above, 10^{3/8} = V_{3D}^{1/2}.
In this case V_{3D }is a cube whose
face-diagonal is the diameter of a circle whose circumference is that *exact* part
of the circumference of the face of the absolute dead center cube (cf. ibid., pp. 524f.,
n. 98)––whose face-diagonal is that *exact* part of that of the absolute
dead center cube––whose denominator is the fourth root of the tenth part of the
four-dimensional volume whose base is the absolute dead center cube: 10[144π/(2^{1/2}[(2.371373706^{2})^{1/3}]π)]^{4} = 10368^{2} (cf. ibid., pp. 433ff., for the
basic dimensions of the dead center cube). Where 2.371373706^{2} is the
volume of this cube whose face-diagonal is the aforementioned part of that of the absolute
dead center cube, and (7.57 × 10^{-16}
J)/(7.57 × 10^{-51} C) = 1 × 10^{35} V, the last multiplied by the elementary
charge, *e*, = 1 × 10^{35} eV =
1.602176462 × 10^{16} J ~ 2.371373706^{-2}
kg. (Indeed, the conversion factor from eV to kg = *e*/*c*^{2}
~ 1/V_{3D} × 10^{-35}
= 2.371373706^{-2} × 10^{-35}.)
In addition to its role in the volume equation, V_{4D}^{3/8}
= V_{3D}^{1/2} = (ML)^{1/2} = (L^{4})^{1/2},
3/8 is the numerical factor in the cosmological critical density formula, ρ_{crit} = 3*H*_{0}^{2}/8π*G* (G. Woan, *The Cambridge Handbook of Physics Formulas*
[Cambridge, 2000], p. 185). (Note here that the relation-power of the dimensionless
1000 in the H_{2}O-resolution of the gravitational dimensions, 1000^{.125},
is the immediate building block of cosmological critical density, indeed, the base factor
of the Hubble constant: where ρ_{crit} = 3*H*_{0}^{2}/8π*G* = 1 × 10^{-29}
g cm^{-3} [cf. "The Physics Factbook: A Scientific Encyclopedia," ed. G.
Elert, "Density of the Universe." Online. Available:
http://hypertextbook.com/facts/2000/ChristinaCheng.shtml. January 13, 2003], and V_{3D}^{1/2} =
1000^{.125} = 10^{3/8} = 2.371373706, 3/8π*G
~ *2.371373706^{-2} × 10^{7} g cm^{-3}
and *H*_{0}^{2} ~ [1000^{.125} × 10^{-18}]^{2} = [2.371373706 × 10^{-18}]^{2}. The critical density
of the universe, ρ_{crit} = 3*H*_{0}^{2}/8π*G*, ~ [1000^{.125} ×
10^{-18}]^{2} × [1000^{.125}]^{-2}
× 10^{7} g cm^{-3} = 1 × 10^{-29} g cm^{-3}. The Hubble
constant, *H*_{0}, ~ 73172.95674 m s^{-1} Mpc^{-1}
= 1000^{.125} × 10^{-18} s^{-1}
= 2.371373706 × 10^{-18} s^{-1}.)
The inverse of 3/8 is the numerical factor in the formula for the mean distance of
a random walk <*r*> = (8/3π)^{1/2} *N*^{1/2}
*l* (ibid., p. 59), and in that for the Thomson electron scattering cross section
(ibid., p. 155) σ_{T} = 8π/3 × *r*_{e}^{2}
= 6.652 × 10^{-29} m^{2} (the base
factor of this last ~
/, the ratio of real trinary logic infinite
arithmetic value and infinite arithmetic value-number ratios [see, above, note 9]).
Note the appearance in the Thomson cross section of the 1 ×
10^{29} order of magnitude of the molecular ratio cited in the text Note above and
of the critical density of the universe expressed as g cm^{-3} (above, note 8 and
this note). Where 10^{4} is the "created 100," the relation-power
of the water/universe molecular/critical density ratio, divided by the former, (1 × 10^{29})^{.125}/10^{4} = 1/1000^{.125},
= 1/10^{3/8} (cf. above, this note, [1 × 10^{35}]^{.125}/10^{4}
= 1000^{.125}, = 10^{3/8}). The further analogy: 1 × 10^{29} eV = 1.602 176 462 × 10^{10} J ~ 2.371373706^{-2} × 10^{-6 }kg (cf. above, this note, 1 × 10^{35} eV = 1.602176462 × 10^{16} J ~ 2.371373706^{-2} kg).
Finally, 3/8 is the inverse of the rational product of the foundation/minimum order (4/1 × 2/3 = 8/3) of the fourfold turning of the trinary logic
cornerstone (Leahy, *Foundation* III.1, pp. 268ff.). Note the following
controversial finding by Kopeikin and Fomalont:

1.06*c* – *c* ~ (1000^{.125}/10^{4})^{-2}
m s^{-1} = (10^{3/8}/10^{4})^{-2} m s^{-1}.

(1.06*c*)^{1/2} × 10^{-5}
~ (1000^{.125})^{-2} m s^{-1} = (10^{3/8})^{-2} m
s^{-1}.