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 *x*^{4}
is the absolute dead center hypercube volume 107495424 (cf., **
on this web, **
The Deep Epidermal Surface**,
**
Dali Crucifixion, and
Epidermal
Immunity Structure & the Absolute Dead Center Hypercube
Volume/Boundary Ratio), and *i*(*x*^{4}) its integral product
1016064,

*x*^{4}/*i*(*x*^{4})
= 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 10^{84}.
At 10^{84 }and 10^{85} the maximum integral product (9^{43})^{2}
= 1.16106307035309 × 10^{82} which is < (10^{85} × 784/82944) and > (10^{84}
× 784/82944). Beyond 10^{84} all *i*(10^{x})
< (10^{x} × 784/82944). This hypercubic limit 10^{84
}= (10^{21})^{4} is particularly neat since all *i*(*x*)
≠ *x *begins at 10^{21}* *(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 10^{18} × 784/82944 m ~ 9.46053 × 10^{15 }m = 1 light year
(Gregorian). Where 829.6656 m = 1000 megalithic yards (cf. Leahy, *Foundation*, p. 525, n.
99), 10^{18} × 784/82944 × (829.6656 m/784)^{1/64} = 9.46053 ×
10^{15 }m = 1 light year (Gregorian) accurate to 1 part in 1 million.
Note also that Julian/Gregorian light year = 9.460730 × 10^{15 }
m/9.460528 × 10^{15 }m = (*v _{maxwell}/c*)

**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
*x*^{4} = 107495424, (8*x*^{3}/*x*^{4})^{2}/[(*x*/6)^{4}/*x*^{3}]^{2}
= 1 (see, on this web,
Theorem & Proof:
The Uniqueness of the Absolute Dead Center Cube). Where *x*^{4}
= 1016064 (integral product of 107495424), (8*x*^{3}/*x*^{4})^{2}/[(*x*/6)^{4}/*x*^{3}]^{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}/*x*^{3} × 8*x*^{3}/*x*^{4}
= 162^{-1}. With *x* = 1, the value of (*x*/6)^{4}/*x*^{3}
= 36^{-2}. The value of (*x*/6)^{4}/*x*^{3 }
increases as the value of *x* increases to infinity, at the rate of 36^{-2}
per unit increase. With *x* = 1, the value of 8*x*^{3}/*x*^{4}
= 8. The value of 8*x*^{3}/*x*^{4} decreases as the
value of *x* increases to infinity, at the rate equal to the 8^{-1 }
per unit increase in its inverse value. (8*x*^{3}/*x*^{4})^{2}/[(*x*/6)^{4}/*x*^{3}]^{2}
= 1 will then be true once, when *x* = 107495424^{1/4 }–– when (*x*/6)^{4}/*x*^{3}
= 107495424^{1/4 }× 36^{-2} = 8*x*^{3}/*x*^{4
}= (107495424^{1/4}/8)^{-1}. Likewise (8*x*^{3}/*x*^{4})^{2}/[(*x*/6)^{4}/*x*^{3}]^{2}
= 82944/784 will be true once, when *x* = 1016064^{1/4
}–– when (*x*/6)^{4}/*x*^{3} =
1016064^{1/4
}× 36^{-2} and 8*x*^{3}/*x*^{4 }=
(1016064^{1/4}/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
(8*x*^{3}/*x*^{4})^{2}/[(*x*/6)^{4}/*x*^{3}]^{2} (see, on this web,
Theorem & Proof:
The Uniqueness of the Absolute Dead Center Cube, where uniquely for
the absolute dead center cube (8*x*^{3}/*x*^{4}) × (*x*/6)^{4}/*x*^{3
}= (8*x*^{3}/*x*^{4})^{2} = [(*x*/6)^{4}/*x*^{3}]^{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 *w*^{1/4} = *x*, (8*x*^{3}/*x*^{4})* ^{y}*/[(

*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
*x*^{4}
= 107495424 to the integral product of the hypercube whose base is the
face diagonal of the base cube of the hypercube *i*(*x*^{4}) =
1016064 that is the integral product of the absolute dead center hypercube is:

*i*(2^{1/2}*x*)^{4}/*i*(2^{1/2}[(*i*[*x*^{4}])^{1/4}])^{4}
= 107495424/82944.

But so also:

*x*^{4}/(*x*/6)^{4}
= 107495424/82944.

The relation of the
absolute dead center hypercube
*x*^{4}
= 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*(*x*^{4}) = 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
*x*^{4} = 107495424 is the hypercube volume
*i*(*x*^{4})
= 1016064,
the integral product of the hypercube whose base is the diagonal
of the square of the latter's fourth root

*i*([1016064^{1/4}
× 2^{1/2}]^{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*([4064256^{1/4}
× 2^{1/2}]^{4}) = *i*(16257024) = 784,

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

(16257024^{1/4}
× 2^{1/2})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}/*x*^{3}
× 8*x*^{3}/*x*^{4} = 162^{-1}, only in the
case of the absolute dead center cube when *x* = 10368^{1/2} and *
x*^{4} = 107495424 does (*x*/6)^{4}/*x*^{3}/(8*x*^{3}/*x*^{4})
= 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 *x*^{4} hypercube volume,

(82944/784)* = x*^{4}*/y*
= *x*^{4}/1016064,

that is, *y*
= 1016064. Among all natural numbers there are only three *x*^{4 }
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 6^{4} 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 6^{4}, 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 6^{4
}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] =
1^{0}, 1 = *i*[0]
= 0^{0} (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] = 1^{2},
784 *= i*[784] = 784^{1}, 82944 *= i*[82944] = 82944^{1},
and similarly the single digit natural numbers other than 1, e.g., 9 = *i*[3]
= 3^{2}, but 1016064 = *i*[107495424] ≠ a natural power of
107495424, and, e.g., 1 = *i*[13] = 13^{0} ≠ 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}/*x*^{3} × 8*x*^{3}/*x*^{4}
= 162^{-1}, but only for the absolute dead center cube (*x*/6)^{4}/*x*^{3}/(8*x*^{3}/*x*^{4})
= 1, just so, while it is the case that for all natural number hypercubes the V/BSA hypercube =
hypercube/6^{4} = 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/6^{4}
= 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 *x*^{4} 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 6^{4 }hypercube is unique among cubes as being
the only cube whose volume equals its surface area 6^{3} = 6^{2}
× 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 6^{4} 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 × 10^{30}) and το σωμα του Χριστου
(4.29981696 × 10^{40}) (cf. Leahy, *Foundation*, Sections III.7 and
IV.2; also, on this web, Dali Crucifixion),
while 23887872 is the base factor of 2.3887872 × 10^{34} 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!) = 829440.
*φ*(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
*v _{maxwell}*

5. *φ*(107495424)/*φ*(82944)
= 107495424/82944 = 6^{4} the perfect hypercube.

See also, on this web, The Diagonal Logic of the Triple Absolute Dead Center Cube & Certain Fundamental Mathematical & Physical Constants.