MEASURE BEYOND BEYOND REACH

In the infinite series of natural numbers there are only three numbers identically their integral products: 1 = i(1), 784 = i(784), and 82944 = i(82944) (cf. D.G. Leahy, Foundation: Matter the Body Itself [Albany, 1996], Section III.6, and, on this web, Transdecimal Calculation of Number Identity: A Note on Integral Product & Related Terms).  Their product is a function of relations of the 9 foundational natural numbers,

1 × 784 × 82944 = (9!/ i).

Where x4 is the absolute dead center hypercube volume 107495424 (cf., on this web, The Deep Epidermal Surface, Dali Crucifixion, and ), and i(x4) its integral product 1016064,

x4/i(x4) = 82944/784/1.

This relation of the hypercube volume to its integral product is unique in the infinite series of natural numbers.  The order of magnitude limit to the possible appearance of this relation among the natural numbers is at 1084.  At 1084 and 1085 the maximum integral product (943)2 = 1.16106307035309 × 1082 which is < (1085 × 784/82944) and > (1084 × 784/82944).  Beyond 1084 all i(10x) < (10x × 784/82944).  This hypercubic limit 1084 = (1021)4 is particularly neat since all i(x) ≠ x begins at 1021 (cf. Leahy, Foundation, Section III.6).  A computer search of the first 2.147 billion natural numbers by Mr. J. DePompeo so far confirms that no other instance of this relation occurs.

[The writer notes that 1018 × 784/82944 m ~ 9.46053 × 1015 m = 1 light year (Gregorian).  Where 829.6656 m = 1000 megalithic yards (cf. Leahy, Foundation, p. 525, n. 99), 1018 × 784/82944 × (829.6656 m/784)1/64 = 9.46053 × 1015 m = 1 light year (Gregorian) accurate to 1 part in 1 million.  Note also that Julian/Gregorian light year = 9.460730 × 1015 m/9.460528 × 1015 m = (vmaxwell/c)1/32 accurate to 1 part in 1 million (cf., on this web, Maxwell's Constant & the Trinary Logic Triple-cube, et passim).]

Theorem and Proof 1

Theorem: There is no natural number other than the absolute dead center hypercube volume 107495424 in the ratio 82944/784/1 to its integral product.

Proof:  In the series of natural numbers there are only three numbers identically their integral products: 1 = i(1), 784 = i(784), and 82944 = i(82944).  Since the hypercubic (x/6)4 V/BSA ratio of the absolute dead center cube which is integral to its unique rational hypercubic constitution = 82944 = i(82944), the relation of a natural number to its integral product in the ratio 82944/784/1, should such exist, must be hypercubically rationally constituted.  Where x4 = 107495424, (8x3/x4)2/[(x/6)4/x3]2 = 1 (see, on this web, Theorem & Proof: The Uniqueness of the Absolute Dead Center Cube).  Where x4 = 1016064 (integral product of 107495424), (8x3/x4)2/[(x/6)4/x3]2 = 82944/784.  This rationally constituted hypercubic relation 107495424 : 1016064 :: 82944 : 784 is unique in the infinite series of cubes.  For all cubes (x/6)4/x3 × 8x3/x4 = 162−1With x = 1, the value of (x/6)4/x3 = 36-2.  The value of (x/6)4/x3 increases as the value of x increases to infinity, at the rate of 36-2 per unit increase.   With x = 1, the value of 8x3/x4 = 8.  The value of 8x3/x4 decreases as the value of x increases to infinity, at the rate equal to the 8−1 per unit increase in its inverse value.  (8x3/x4)2/[(x/6)4/x3]2 = 1 will then be true once, when x = 1074954241/4 –– when (x/6)4/x3 = 1074954241/4 × 36-2 = 8x3/x4 = (1074954241/4/8)−1.  Likewise (8x3/x4)2/[(x/6)4/x3]2 = 82944/784 will be true once, when x = 10160641/4 –– when (x/6)4/x3 = 10160641/4 × 36-2 and 8x3/x4 = (10160641/4/8)−1.

Note 1: The logic of hypercubicity is manifestly related to the logic of certain integral products insofar as in the proof of the uniqueness of the absolute dead center cube (ibid.) hypercubic (x/6)4 = 82944 = i(82944), and insofar as the integral product of the absolute dead center hypercube volume is such that its hypercubic (x/6)4 ratio = 784 = i(784), and, finally, insofar as the ratio of any hypercubic volume to its hypercubic (x/6)4 ratio is itself the hypercubic volume 1296 whose hypercubic (x/6)4 ratio = 1 = i(1).  This specific relationship of hypercubicity to the three unique numbers is simply a fact (see, below, "Elements in the Dialogue").  The question whether 107495424 and its integral product 1016064 are, as such, uniquely related in the ratio 82944/784/1 is a distinct question pertaining to the relation of a natural number to its integral product when the latter is other than itself, which, since hypercubicity here involves at once the integral relation between cubic volumes and between natural numbers and the squared products of their odd-numbered digits (see Note 2, below), is provable by employing the foundational hypercubic ratio (8x3/x4)2/[(x/6)4/x3]2 (see, on this web, Theorem & Proof: The Uniqueness of the Absolute Dead Center Cube, where uniquely for the absolute dead center cube (8x3/x4) × (x/6)4/x3 = (8x3/x4)2 = [(x/6)4/x3]2), at once the measure of the rational hypercubic constitution of all cubes and of the rational constitution of the natural numbers identically their integral products that are the elements of the ratio under consideration.  The question at hand pertains, precisely, to such natural number integral product relationships (should there be more than one) involving the specified relationship between the three unique numbers that are their integral products and that are, as described, essentially hypercubic.  While any number of natural numbers might possibly be related to one another in the ratio 82944/784/1, it is evident from the foregoing that for any natural number other than 107495424 to be related to its integral product in that ratio the relationship of rational identity must be hypercubically constituted.  But no other such relationship exists.  For, to imagine two cases contrary to the proof, if, when y ≠ 2 is the integrating function, and when w1/4 = x, (8x3/x4)y/[(x/6)4/x3]y = 82944/784, and when z1/4 = x, (8x3/x4)y/[(x/6)4/x3]y = 1, since in this case it must always be that z = 107495424 and w ≠ 1016064, it is not and cannot be that w : z :: 82944/784 : 1, and also where y = 2, but u ≠ 1, when w1/4 = x and (8x3/x4)y/[(x/6)4/x3]y = 82944u/784 and when z1/4 = x and (8x3/x4)y/[(x/6)4/x3]y = u, w cannot be the integral product of z.  In both cases the sought for integral hypercubic relationship is rationally constitutionally non-existent.

The writer thanks Mr. DePompeo for calling attention to the first equation immediately following.

Note 2: The ratio of the integral product of the hypercube whose base is the face diagonal of the base cube of the absolute dead center hypercube x4 = 107495424 to the integral product of the hypercube whose base is the face diagonal of the base cube of the hypercube i(x4) = 1016064 that is the integral product of the absolute dead center hypercube is:

i(21/2x)4/i(21/2[(i[x4])1/4])4 = 107495424/82944.

But so also:

x4/(x/6)4 = 107495424/82944.

The relation of the absolute dead center hypercube x4 = 107495424 to its base cube V/BSA hypercube ratio (x/6)4 = 82944 is thus demonstrably at once the rational hypercubic constitutional relation of the natural number to its integral product, i(x4) = 1016064.  Thus demonstrated, discretely so, the integral relationship of natural number integral product relations to constitutional hypercubicity.

Mr. DePompeo brings to the writer's attention the following further evidence of

the beautiful deep structure of integral product constitutional hypercubicity relations.

Note 3: Where the integral product of the absolute dead center hypercube x4 = 107495424 is the hypercube volume i(x4) = 1016064, the integral product of the hypercube whose base is the diagonal of the square of the latter's fourth root

i([10160641/4 × 21/2]4) = i(4064256) = 82944,

and the integral product of the hypercube whose base is the diagonal of the square of this last hypercube's fourth root

i([40642561/4 × 21/2]4) = i(16257024) = 784,

and the hypercube whose base is the diagonal of the square of this last hypercube's fourth root

(162570241/4 × 21/2)4 = 1 × 784 × 82944 = (9!/ i)2.

Theorem and Proof 2

The writer thanks Mr. DePompeo for his contributions to the dialogue whose fruit was the following proof.

Theorem: There is no natural number other than the absolute dead center hypercube volume 107495424 in the ratio 82944/784/1 to its integral product.

Proof: While it is the case for all cubes that (x/6)4/x3 × 8x3/x4 = 162−1, only in the case of the absolute dead center cube when x = 103681/2 and x4 = 107495424 does (x/6)4/x3/(8x3/x4) = 1 (see, on this web, Theorem & Proof: The Uniqueness of the Absolute Dead Center Cube).  Where x is the edge of the absolute dead center cube, and y the hypercube volume in the ratio 784/82944 to its x4 hypercube volume,

(82944/784) = x4/y = x4/1016064,

that is, y = 1016064.  Among all natural numbers there are only three x4 hypercube volumes, 107495424, 1016064, and 1296, whose corresponding (x/6)4 hypercube volumes are numbers identically their integral products, respectively, 82944, 784, and 1.  The V/BSA hypercube corresponding to any hypercube volume is a function of the division of the primary hypercube by the volume of the perfect hypercube 64 whose relation to its V/BSA hypercube is to 1 identically the latter’s integral product.  While all hypercubes are related to their V/BSA hypercubes as a function of a division by the perfect hypercube 64, only those hypercubes related to V/BSA hypercubes that share the property of the V/BSA hypercube of the perfect hypercube of which they are a function—being a number identically its integral product—only such constitutionally perfect hypercubes can possibly be related as primary number and integral product in the ratio of the numbers that are their integral products, 82944/784/1.  When so related, the primary number, qua perfectly hypercubic, is not related to a number beyond itself, but rather to a number that is, as its integral product, beyond beyond itself.  This intimately rational relation is possible only in the case of constitutionally perfect hypercubes.  Constitutionally imperfect hypercubes related to V/BSA hypercubes that do not share the property of the V/BSA hypercube of the perfect hypercube of which they are a function—not being a number identically its integral product—although possibly related in the ratio of the numbers that are their integral products, 82944/784/1, cannot possibly be so related as primary number to its integral product, since an imperfectly hypercubic primary number, as such, can only be so related to a number beyond itself—to a number not its integral product.  Other than the perfect hypercube 64 only the constitutionally perfect hypercubes 107495424 and 1016064 are related to V/BSA hypercubes that are numbers identically their integral products, and only these natural numbers are related one to the other as primary number and integral product in the ratio 82944/784/1.

Lemma 1: If natural number hypercube volumes other than the constitutionally perfect were imagined to be related as primary number to integral product in the ratio 82944/784/1, the distinction between them and the constitutionally perfect would be reduced to the fact that their V/BSA hypercubes, unlike those of the latter, would not be identically their integral products, i.e., to the fact that the constitutionally imperfect would not be constitutionally perfect!  Obviously a distinction without a difference, i.e., not a real distinction.  The real distinction restricts the 82944/784/1 ratio of primary number to integral product to the constitutionally perfect hypercubes whose V/BSA hypercube volumes are identically their integral products.

Lemma 2: The integral product of a natural number is the square product of the latter’s odd numbered digits.  Its production involves no number other than these digits.  Essentially the relation of a natural number to its integral product may be to a number other than itself, indeed, intrinsically more so than a power of itself, but, qua square product of its odd numbered digits, not beyond itself, a number beyond beyond itself.

Lemma 3: The natural power of a number is not second to the number.  The integral product of a number is properly second to the number.  Other numbers are related as improperly second one to another.  A number not second to another is not beyond that number.  A number improperly second to another is beyond that number.  A number properly second to another is beyond beyond that number.

Lemma 4: The integral product is in mathematics the imperfect reflection of a relation pertaining properly to the primary elements of real trinary logic.  In the latter (where 0 ≠ nothing, i.e., = a natural digit, and 0 and   are sorts of 1) at least one natural power of any logical digit or array of digits (e.g.,  0 = 1 or [01]0  = 1, cf. Leahy, Foundation, p. 256; also, on this web, Real Trinary Logic Geometric Series Matrix of the Numeric Geometric Series & the Series of Perfect Numbers)  is always properly second and not second to it, properly beyond beyond it: 1 = i[] =  0, 1 = i[1] = 10, 1 = i[0] = 00 (cf. ibid).  Qua mathematical, or logically derivative, the integral product of a natural number may be improperly beyond beyond that number.  So 1= i[1] = 12, 784 = i[784] = 7841, 82944 = i[82944] = 829441, and similarly the single digit natural numbers other than 1, e.g., 9 = i[3] = 32, but 1016064 = i[107495424] ≠ a natural power of 107495424, and, e.g., 1 = i[13] = 130 ≠ a natural power of 13.

Note 1: In the case of the 9 foundational natural numbers the second power of a number is its integral product—a number not second to a number properly second to that number.  But then the number properly second to a foundational natural number is not without qualification not second to the latter, being so—except where the latter is 1—only as the latter’s second power.  Not unqualifiedly properly second and not second : unqualifiedly properly second and not second :: 9 foundational natural numbers excepting the number 1 : real trinary logic elements together with the number 1 :: square produced by external multiplication : ‘square essence’ of the infinite lattice of 0’s, ’s, and 1’s (see, on this web, Quantum Gravitational vs. Quantum Logic: Virtually Left-handed Real Trinary Logic) :: intermediate third logical element : absolute third logical element :: quantum indeterminateness : absolute placedness :: relative mediation : absolute mediation (see, on this web, The Simplicity & Syntax of the Concepts, Immediacy, Mediation, Omnipotence, & Beginning).

Note 2: Just as, for all cubes (x/6)4/x3 × 8x3/x4 = 162−1, but only for the absolute dead center cube (x/6)4/x3/(8x3/x4) = 1, just so, while it is the case that for all natural number hypercubes the V/BSA hypercube = hypercube/64 = a natural number, only the absolute dead center hypercube together with the hypercube that is its integral product shares with the perfect hypercube the property that its V/BSA hypercube = hypercube/64 = a natural number identically its integral product.  Absolute dead center cube : all cubes :: absolute dead center hypercube together with the hypercube that is its integral product : all natural number hypercubes :: proportionate division in extreme and mean ratio : proportionate division in all other ratios.

Elements in the Dialogue
(October 2−17, 2007)

Mr. DePompeo called attention to the fact that in the infinite series of natural numbers there are by definition only three x4 hypercube volumes whose V/BSA hypercube volumes (x/6)4 (see, on this web, The Deep Epidermal Surface and Theorem & Proof: The Uniqueness of the Absolute Dead Center Cube) are identically their integral products:

1.  The absolute dead center hypercube volume 107495464 whose V/BSA hypercube volume is 82944 = i(82944),

2.  The hypercube volume of the integral product of the absolute dead center hypercube 1016064 whose V/BSA hypercube volume is 784 = i(784),

3.  The hypercube volume 1296 whose V/BSA hypercube volume is 1 = i(1).

Mr. DePompeo noted that the base cube of the 64 hypercube is unique among cubes as being the only cube whose volume equals its surface area 63 = 62 × 6.  The writer notes that 6 is also uniquely the perfect number the sum of whose factors = their product, 1 + 2 + 3 = 1 × 2 × 3 = 6.  Qua V/BSA hypercube volume 64 is the perfect hypercube.

The three unique hypercube volumes are related to the three unique natural numbers that are their integral products as follows:

1.  1296/1296 = 1

2.  1016064/1296 = 784

3.  107495424/1296 = 82944

The volume 429981696 of the hypercube whose integral product 107495424 equals the volume of the absolute dead center hypercube has a base cube of volume 2985984 the integral product of which equals 331776 the volume of its V/BSA hypercube, while its hypercube boundary volume is 23887872.  429981696 and 2985984 are the base factors of the integral products of alternate forms of the Greek for Corpus Christi, το σωμα Χριστου (2.985984 × 1030) and το σωμα του Χριστου (4.29981696 × 1040) (cf. Leahy, Foundation, Sections III.7 and IV.2; also, on this web, Dali Crucifixion), while 23887872 is the base factor of 2.3887872 × 1034 the linear product of Genesis 1:1, בראשית ברא אלהים את השמים ואת הארץ, “In the beginning God created the heaven and the earth," (see, on this web, The Deep Epidermal Surface, et passim; also Leahy, Foundation, p. 505).

Mr. DePompeo also reported the following:

1.  Where Euler’s totient function φ(n) is the number of positive integers less than or equal to n that are coprime to n, that is, whose greatest common
denominator is 1, [
(
9!/[1 + 2 + . . . 9])/(φ[1] + φ[2] + . . . φ[9])]2 = 82944.

2.  The integral product of the Euler functions of the 9 foundational natural numbers taken together as one number, 112242646, = 82944, and the
square of their linear sum = 784.

3.  φ(9!) = 82944.  φ(10!) = 829440φ(11!) = 8294400.

4.  In the range 1 through 9! there are 288 + 1 (= √82944 + 1 = 9!/[1 + 2 + . . . 9]/[φ(1) + φ(2) + . . . φ(9)] + 1) occurrences of φ(n) = 82944.  Cf., on this web, The vmaxwell1/16 × 1 s/m Cube, Human Body Surface & BSA, & the Infinitely Flat Structure of the Universe.

5.  φ(107495424)/φ(82944) = 107495424/82944 = 64 the perfect hypercube.

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