THEOREM FOR A FLAT UNIVERSE:
THE ρcrit = 3H02/8πG BUILDING BLOCK 1000.125

Background:  In correspondence on November 2, 2002, SB Hoath1 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 m3 kg-1 s-2 would be simplified to 1000/s2.2  The writer proposed the following theorem.

Theorem:  To demonstrate that the universe is flat from the premise of the H20-resolution of the dimensions of the gravitation constant m3 kg-1 s-2 = 1000/s2.  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 1.062 = 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 = 1.062 = 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 H20-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 × 1022 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 × 1029, reinforcing the substantive analogy between the critical density of water and that of the universe.8  More precisely, the ratio is 1.0035 × 1029, and where 1000 is the dimensionless number in the H2O-resolution of the gravitational constant, [1.0035 × (1029)/1000]1/1024 ~ 1.06, the specific gravity of the human body.  For fundamentally binary thinking a contradiction is introduced by the H2O-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––V4D3/8 = V3D1/2 = (ML)1/2 = (L4)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 universe10) 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 H2O-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 H20-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 H2O 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 H2O 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, "WebElementsTM 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, ρcrit = 3H02/8πG = 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 = V3D1/2, since the latter = V4D3/8, V4D = 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 ~ //104, 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 H2O-resolution of the gravitational constant, 1000.125, = 103/8.  Also, where 104 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 × 1035).125/104 = 1000.125, = 103/8.  In accord with the volume equation in the text above, 103/8 = V3D1/2.   In this case V3D 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/(21/2[(2.3713737062)1/3]π)]4 = 103682 (cf. ibid., pp. 433ff., for the basic dimensions of the dead center cube).  Where 2.3713737062 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 × 1035 V, the last multiplied by the elementary charge, e, = 1 × 1035 eV = 1.602176462 × 1016 J ~ 2.371373706-2 kg.  (Indeed, the conversion factor from eV to kg = e/c2 ~ 1/V3D × 10-35 = 2.371373706-2 × 10-35.)  In addition to its role in the volume equation, V4D3/8 = V3D1/2 = (ML)1/2 = (L4)1/2, 3/8 is the numerical factor in the cosmological critical density formula, ρcrit = 3H02/8πG (G. Woan, The Cambridge Handbook of Physics Formulas [Cambridge, 2000], p. 185).  (Note here that the relation-power of the dimensionless 1000 in the H2O-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 ρcrit = 3H02/8πG = 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 V3D1/2 = 1000.125 = 103/8 = 2.371373706, 3/8πG ~ 2.371373706-2 × 107 g cm-3 and H02 ~ [1000.125 × 10-18]2 = [2.371373706 × 10-18]2.  The critical density of the universe, ρcrit = 3H02/8πG, ~ [1000.125 × 10-18]2 × [1000.125]-2 × 107 g cm-3 = 1 × 10-29 g cm-3.  The Hubble constant, H0, ~ 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 N1/2 l (ibid., p. 59), and in that for the Thomson electron scattering cross section (ibid., p. 155) σT = 8π/3 × re2 = 6.652 × 10-29 m2 (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 × 1029 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 104 is the "created 100," the relation-power of the water/universe molecular/critical density ratio, divided by the former, (1 × 1029).125/104 = 1/1000.125, = 1/103/8 (cf. above, this note, [1 × 1035].125/104 = 1000.125, = 103/8).  The further analogy: 1 × 1029 eV = 1.602 176 462 × 1010 J ~ 2.371373706-2 × 10-6 kg (cf. above, this note, 1 × 1035 eV = 1.602176462 × 1016 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:

1.06cc ~ (1000.125/104)-2 m s-1 = (103/8/104)-2 m s-1.

(1.06c)1/2 × 10-5 ~ (1000.125)-2 m s-1 = (103/8)-2 m s-1.

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