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Self Organization in Real & Complex Analysis

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  • independent researcher, complex systems

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We identify specific properties of the complex plane that allow functions of a continuous n-dimensional (Hilbert) measure space to be transformed into a well ordered counting sequence. We discuss proof strategies for problems in number theory (Goldbach Conjecture) and topology (Poincare ´ Conjecture) that suggest correspondence between the physical principle of least action and the mathematical concept of well ordering. The result implies a deeply organic connection between physics and mathematics. 1.0 Counting & Measure 1.1 Counting is so fundamental that even Kindergartners have little trouble grasping the concept of ordering objects into discrete sets. The concept of a continuous measurement function has been formalized only in recent centuries, however. Newton’s calculus of the rate of change of the rate of change, e.g., is not intuitively obvious. 1.1.1 The growth of generalization in geometry (topology) in the last century, and the proliferation of sophisticated techniques in number theory (Wiles ’ proof of Fermat’s Last Theorem, e.g.) leave clues to subtle relations between continuous functions and discrete
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1
Self Organization in Real & Complex Analysis
T.H. Ray
ICCS 2006
Abstract
We identify specific properties of the complex plane that allow functions of a continuous
n-dimensional (Hilbert) measure space to be transformed into a well ordered counting
sequence. We discuss proof strategies for problems in number theory (Goldbach
Conjecture) and topology (Poincare´ Conjecture) that suggest correspondence between
the physical principle of least action and the mathematical concept of well ordering. The
result implies a deeply organic connection between physics and mathematics.
1.0 Counting & Measure
1.1 Counting is so fundamental that even Kindergartners have little trouble grasping the
concept of ordering objects into discrete sets. The concept of a continuous measurement
function has been formalized only in recent centuries, however. Newton’s calculus of the
rate of change of the rate of change, e.g., is not intuitively obvious.
1.1.1 The growth of generalization in geometry (topology) in the last century, and the
proliferation of sophisticated techniques in number theory (Wiles’ proof of Fermat’s Last
Theorem, e.g.) leave clues to subtle relations between continuous functions and discrete
objects, as one correlates pattern to number.
1.2 Where problems of physics and mathematics overlap, in Chaos Theory, Quantum
Theory, and Self Organized Criticality, et al, critical correspondences between pattern
and number are accompanied by empirical results. Yet, mathematical theories (string
theory, e.g.) that breach the limit of empiricism address the unity of nature in
mathematical terms that are both rigorously true and rooted in natural phenomena.
1.2.1 One is compelled to ask, therefore, whether – or in what sense – phenomenological
order becomes mathematical order. Is the fundamental notion of counting something that
one is born with the knowledge of, as Leopold Kronecker assumed (“God created the
natural numbers, all else is the work of man”) or even deeper, what nature itself is born
with, an order programmed into our brains (as Kant believed) that we translate into
abstract language.
1.3 If nature appears to be self-organized, and mathematics appears to be the language of
nature, what can we mean when we suggest that mathematics (at the level of Analysis) is
itself self-organized? Suppose we mean: the self-organization of spatial dimensions. We
are obliged to show, then, how dimensions are self-similar and self-limiting. And we are
obliged to show how the physical quality of time relates to the mathematical quantity of
number. We measure motion by changes in relative position of points in space; if these
motions are random and self-avoiding, the self-similarity produced by such a system may
make it appear classically that “time flows equably” as Newton believed – or that time
has no reality independent of space, as Einstein asserted – while time is actually chaotic.
I.e., there is no actually smooth function in nature corresponding to our mathematically
smooth models. The model is not the theory. Nevertheless, we will show how such a
disordered system organically produces a well ordered sequence, with no appeal to the
Axiom of Choice.
2
2.0 The Geometry of Counting
2.1 Let us introduce some interrelated tools that differentiate our continuous experience
of time, from the discrete metric that imposes a moment of time onto our experience:
2.2 Suppose time is an independent physical quantity, a 0-dimensional point on a random
self-avoiding walk in n dimensions. The singularity imposes the limit. (Therefore, just
as in discrete counting we say that the set is not empty, we say that the point is not
dimensionless. An important distinction, since a series of zero-dimensional points is not
self-limiting.) Suppose the point is self-similarly extended on a 1-dimensional metric
whose (arbitrarily chosen) endpoints define relative positions of the evolving point and its
complex conjugate on the complex (Riemann) sphere.
2.3 The Euler equation,
ei
!
="1
, therefore describes the least path such a point travels in
the complex plane, as the Euler Identity:
ei
!
+1=0
. (The unit radius is therefore self-
similar to a point; we will explain, in 5.11, this fact of analysis.)
2.4 The complex sphere is a 2-dimensional, 1-point compactification of the complex
plane, with a point at infinity. [Mathworld, “Riemann Sphere.”]
2.5
! !
=1
(quantum unitarity, i.e., the wave function and its complex conjugate),
describes the evolution of a state vector of length 1, on a sphere of radius 1. The equation
preserves the probability of finding the wave state of a given wave function in that same
state.
3.0 The Poincare Conjecture & Self Organization of Dimensions
3.1 Consider how the continued sum of cardinal points of dimensions 0 – 4 corresponds
to prime numbers. (Table *)
Dimension Cardinal Points
!d
0 1
1 2 + =
(1-sphere)
S1
2 4 + =
(2-sphere)
3 6 + =
(3-sphere)
4 10 + =
Table *
The continued sum of the sequence of cardinal points in dimensions 0-4
Suppose length 1 is preserved in this prime number correspondence. Then the 0-
dimensional point on a random self-avoiding walk in n dimensions (3.2) not only returns
1
3
7
13
23
23
3
to itself on the 1-sphere, as (2.3) implies, but again on the 2-sphere, supported by the
trivial extension of Euclidean geometry to 3 dimensions, and again on the 3-sphere where
the identity is complete.
3.2 The continued sum of the sequence of cardinal points implies that four spatial
dimensions sufficiently describes an n-dimensional (Hilbert) space. Every point of the
line, plane and 3-sphere is itself a 3-sphere; empirical dimensions differentiated from the
0-measure point of d = 0 and the infinite-measure 3-sphere, d = 4.
3.3 Implied is an arithmetic proof strategy of the Poincare Conjecture, consistent, so
far as we can tell, with Grisha Perelman’s geometric strategy. [Perelman, 2006]
Insofar as the Ricci Flow describes the time evolution of a Riemannian metric on a
closed manifold of positive curvature, our presentation of the problem finds that all
time evolution in d
!
4 is over such a closed manifold; i.e., a discrete step in time is
defined by an exchange of continuous curves for discrete points. Therefore, if every
point of n-dimensional space is homeomorphic to a 3-sphere, and time is a physical
quantity (of zero measure) on an n-dimensional self-avoiding random walk, a move in
time of any magnitude bridges an infinite gulf (Continuum Hypothesis) by avoiding
infinity – “counting” is over the manifold of a 3-sphere embedded in the Hilbert
space; continuous curves are exchanged for discrete points in a complex network of
randomly oriented self-similar metrics, in a self-limiting system projected between
and
S1
.
3.4.1 Counting leaves us with the impression of well ordered linearity, because the
random walk of time in hyperspace only follows its path of least action (call it “meta-
action” if you wish) as the network allows. That such points are continuously, and
randomly, re-orienting, shows up in our empirical experience as the effects of
quantum experiments as fundamental as Thomas Young’s two-slit experiment (1804).
Ricci Flow is perfectly compatible with this view, in that Perelman’s proof introduces
a technique of Ricci Flow with surgery that exists for infinite time on the half open
interval
0, !
[
)
. [Anderson, 2004]. The positivity condition that Perelman’s technique
demands guarantees counting as we have defined it, on the line of positive real
integers, as a result of the proof that shows only finitely many surgeries are
performed in any finite time interval – the surgery times are locally finite, even while
the action and orientation of the time metric is infinite. As Perelman shows that there
are only finitely many times at which singularities form, our network of 0-
dimensional randomly oriented self-similar points has currency.
3.4.2 All real functions are continuous. [Weyl, 1918; Dedekind, 1901] A
mathematical “move in time” is fundamental. [Brouwer, 1981] The mathematical
properties of limit and function, therefore, suggest a relation between the physical
principle of least action (Fermat) and a self-limiting sequence of arbitrary magnitude.
3.5 The Poincare Conjecture.
!
(The conjecture for the remaining, formerly open
case of three dimensions is: Any connected, simply connected 3-dimensional
manifold is homeomorphic to the 3-sphere.)
4.0 The Goldbach Conjecture, Part 1
4.1 We mean to suggest as strong an arithmetic congruence between two pairs of
terms on the complex plane, as the geometric congruences defined by Euclid. (Euclid
4
used the term “congruence” as we today use “equals.”) We needn’t actually find
complex terms to suit the case, however. For if all real functions are continuous and
if arithmetic and geometry are self-similar on the complex plane, we should find that
arithmetic terms on the real line also “move in time” to this same beat; i.e., the
sequence is self-limiting. An odd prime, as in Table* is indifferent to magnitude –
one cannot differentiate self-similar indecomposable quantities. As physics informs
us that things that are not differentiable are identical (e.g., the ether is not
differentiable from the vacuum), whatever move in time that an odd prime makes
should not be differentiable in principle from the least action of a point particle
tracing a metric in a vacuum. We do find such a self-limiting arithmetic case. We
cite the reformulation of the Goldbach Conjecture and the Twin Primes Conjecture.
[Popper, 1983] Karl Popper called the Goldbach Conjecture true if, G: For every
natural number x > 2, there exists at least one natural number y such that
x+y
and
(2 +x)!y
are both prime. Popper called the Twin Primes Conjecture true if H: For
every natural number x > 2, there exists at least one natural number y such that
x+y
and
(2 +x)+y
are both prime.
4.1.1 G is demonstrable by iterated arithmetic calculation. A program to test the
conjecture potentially halts when it comes on a counterexample. H is not, in
principle, testable at all. Popper used these formulations only to illustrate his
philosophy of hypothesis testing (neither G nor H is verifiable, but G is falsifiable).
We find deeper implications.
4.2 Consider that the pair of terms separated by “and” in G share characteristics in the
same important way that identical 90° angles do, in Euclid’s fifth (parallel) postulate.
The pairs are judged independent of each other (hence the presence of the minus sign)
as identical angles are also correlated independently.
4.2.2 The sign change in H makes the critical difference between the conjectures.
The Twin Primes Conjecture is not testable because the pairs are not and cannot be
judged independently. H therefore corresponds to non-Euclidean geometry
(Lobachevsky, Bolyai) in that to say “An infinite number of lines may be drawn
through a point …” is tantamount to saying that an iterated computation cannot halt.
4.3 (4.2) provides the motivation by which we sum cardinal points of space in (3.1).
Does a 0-dimensional point generate self-similar sets? Classical addition (Dedekind-
Peano) reveals a result, but not a property; properties (primeness and evenness, e.g.)
are judged independently of the language that conveys them. Meaning is independent
of language. [Ray, 2002] Were this not true, mathematics would not be a useful
language for making true statements, because all quantitative meaning would be
assigned by fiat, thus only qualitative and never subject to the closed theoretical
judgment by which one determines truth content. Chaitin’s incalculable
!
is an
example of the independence of language and meaning.
!
is defined in formal
language, with meaning assigned by theory and tested against result. The self-limiting
(Chaitin says “self delimiting”) nature of
!
is a property independent of a result.
[Chaitin, 2005] I.e., we can make a closed judgment linguistically, but the meaning of
nature – a scientific judgment – is always open. [Popper, 1983, pp. 110-111]
5
4.4 We conclude that the Goldbach Conjecture is not provable by classical methods,
i.e., by summing integer terms on the positive real line. A continuous property of
evenness is geometric, and independent of results on the real line.
5.0 The Goldbach Conjecture, Part 2
5.1 Consider that the unity of the ordered real line
!
with the non-ordered set
z
, the
universal set of complex numbers, is self-organized and self-limiting.
5.2 We will prove the strong Goldbach Conjecture, by weakening it.
Theorem*: N** on the Riemann sphere congruent (mod 2) to arbitrary P, odd
prime, on the positive real line, implies the Goldbach Conjecture.
!
(Strong
Goldbach Conjecture: Every even integer
>
4 may be expressed as the sum of two
odd primes.
5.3 We invent the term, N** (read “N-double-star”), to represent a hypercomplex
number that performs the same function – i.e. the function of preserving the property
of evenness (parity) – as the unique prime integer 2.
5.3.1 A move in time, i.e. a real continuous function – on the surface of a closed
manifold
(S2)
– makes no differentiation between a closed loop and a continuous
line, but accommodates both. E.g. the Euler Network Formula for a flat plane,
V – E + F = 1, becomes V – E + F = 2, for a closed (compact) manifold. If we were
to speak of “hypertime,” we would find that what Brouwer took as a fundamental
analytical fact of mathematical “twoity” [Brouwer, 1981] is in hyperspace a “fourity”
of terms. (Indeed, the “hyper” classifications of numbers – quaternions, octonions and
their extensions – due to W.R. Hamilton, Cayley et al – follow. We shall not need
these.)
5.4
a!b(mod m)
for
a!b(mod m)
, where
a!b
is a multiple of m
5.4.1 For twin primes and all odd primes:
P
1!P
2(mod 2)
5.4.2 Over complete N**:
N** !P(mod 2)
(1)
5.5 Because we can fix the condition for twin primes, we allow the inductive
generalization to (1) for the reason that any discrete value in the entire odd prime
sequence, being
>
4 when two odd primes are summed, is indifferent to the magnitude of
the primes; one looks only for the property of evenness (as the congruence of equal 90°
angles is indifferent to the length and orientation of the sides).
5.5.1 The deep reason is that the metric diameter of a unit sphere in n dimensions
!4
is
always odd, and – like the integer 2 on the real line of positive integers – as a least
element, does not decompose into primes. This only holds, of course, in that – as we
have found – every point of the 3-sphere is itself a 3-sphere, so that the network of 3-
spheres in n-dimensional Hilbert space is self-similar and self-limiting and thus self-
organized.
5.5.2 Then, as the n-dimensional line evolves two even metric radii in real time, we find a
metric radius defined by the linear transformation map
!A"B(0) mod 2
[ ]
to the even
part of
B
and the odd part of
A
, and
!A"B(1)mod 2
[ ]
to the odd part of
B
and the
even part of
A
.
6
5.5.3 In other words, when we speak of a unit diameter of a 3-sphere
(S3)
, we should
find that the center – or equilibrium – point is not 0-dimensional, yet 0-valued. We shall
explain.
5.6 What is the center point of a space that has no center? Or, what is the median prime
number? Because we know that the primes are infinite (Euclid), we know that the
question has no answer. On the other hand, an arbitrary choice of endpoints in an ordered
prime sequence, or in a finite set of primes, allows us to answer from Zorn’s Lemma, or
the Axiom of Choice (which are equivalent). [Mathworld, “Zorn’s Lemma”] Suppose we
do not wish to appeal to this axiom. One would ask, is nature well ordered in principle?
We know that it is not. Quantum events are discrete and random.
5.6.1 The point is not dimensionless. If every point is identical to
(S3)
we say that “0-
valued radii” means that the property of evenness is preserved in
S3
and in every
dimension with fewer cardinal points than
S3
. Insofar as a metric radius implies a metric
diameter, we may speak of a kind of “continuous parity” on the n-dimensional unit
diameter, by congruence (mod 2) (5.5.2). Einstein asked whether physics has to give up
continuous functions to be complete. [Einstein, 1955] It does not. Continuous functions
take discrete values in higher dimensions.
5.7 Because a well ordering on the positive real line has least element 0 or 1, we find that
Theorem* supports the construction of every even integer
!
4 by the sum of two odd
primes in that domain. This is necessarily true, in that every pair of discrete integers
!
4,
that is not a pair of primes, decomposes further into prime integers. Insofar as such a pair
is not (therefore) discrete, the pair of primes that forms a truly discrete unit is –
topologically – an even metric radius
!A"B(0) mod 2
[ ]
, preserving the unity of two
odd primes
!A"B(1)mod 2
[ ]
as an odd (unit) diameter.
5.7.1 (“Continuous parity” means that the inverted relationship of diameter and radius
(diameter odd, radius even), in hyperspace, forces the real function (radius odd, diameter
even) projectively between
S1
and
S3
, backward and forward.) (5.9)
5.7.2 How a point becomes a metric unit diameter follows, as we move from the real line
to complex function analysis. If
N* *
is a hypercomplex term that shares properties with
the even prime, 2, it does so in terms that in both cases are realized only by projection of
the line onto the plane. Consider a measure space projected on the complex plane in
which
N* *
is an analog for
2
in the Euclidean plane.
5.8 As
( 2 ) 2=2,
(N**)2=N* *
(2)
( 2 ) 2=2,
is a 2-dimensional identity.
N* *
already lives in 2 dimensions (a complex
number of the form
a+bi
). So as
( 2 ) 4=4
(N**)4=2N* *
(3)
5.8.1 The geometric terms of (2) and (3) correspond to metric identities. This would be
true for any value of
n
,
(N**)n
because the properties of the indecomposable
N* *
cannot differ from a unit. (We have a clue here – not in the scope of this paper – to
7
deep theoretical support for Abel’s Impossibility Theorem. [Hamilton, 1839] The
theorem limits the acquisition of roots of polynomial equations, by methods of radicals
and rational functions, to degree 4 and lower. (This connection arises again in 5.11.1.)
5.9 It should now be more or less obvious that coherent well-ordering must allow a
backward-forward (and so, not arithmetically commutative) projection between
S1
and
, because all points being self-similar to
S3
where n-dimensional (Hilbert space) unit
diameters are odd and corresponding unit radii are even, a necessary “move in time” that
creates “continuous parity” creates, as well, a continuous well-ordering.
5.10 We speak axiomatically and colloquially of the natural numbers as if they were
dimensionless and therefore independent of time. In real terms of complex function
analysis, however, no positive counting term
<
4 can exist. The move in time is critical,
and the point which is extended self-similarly along the radius must return to its origin at
least
!
times, in order to sweep a diameter in the complex plane.
5.11 Let us return to Euler’s geometric interpretation of the complex plane, and fulfill our
promise to explain the terms in which the unit radius in the Euclidean plane is self -
similar to a point, as projected to hyperspace (2.3). Euler’s equation is derived from
eix =cos x+isin x
, which expressed in logarithmic terms,
x=
!
, is
ln(!1) =i
"
.
5.11.1 We know (5.8) that
N* *
is a complex number. We know that in the complex
plane,
i2=!1
. If we want to find a least element, in real continuous function terms, we
are going to have to find a number that is greater than
!
and less than 5, and that obeys a
positivity condition. We can guarantee this only by employing the radical (
i=!1
).
Classical mathematics in which well ordering is axiomatic would inform us,
(4 +0i)
4
=1
Thinking classically, we would proceed by iteratively assigning values, by
the successor operation, to the real part of the complex number. If we claim a continuous
function, though, in that
N* *
is a complex plane analog to the plane quantity,
2
(5.7.2), we don’t want to assign values by iteration – we want a positive real algebraic
quantity, a true analog to the Euclidean plane measure. Then,
(4 +i2)
4
=3
4
(4)
In decimal notation, 0.8660254 … then (5.10), adding
!
= 4.00761805… (5)
5.12 By existence: because the property of evenness can be reified on the positive real
line only by transitive reference to the complex plane, as demonstrated, Theorem*. The
proof is therefore complete for the Goldbach Conjecture.
6.0 Discussion: Are Dimensions Self Organized in String Theory?
Sometimes, there are only subtle clues to the difference between coincidence and true
correspondence. For example, why is String Theory (in at least one modern synthesis of
the theory) necessarily formulated in 10 dimensions? What is the special significance of
this number, if any?
8
Does our approach to the Poincare Conjecture reveal anything but coincidence? As we
claim that every point is a 3-sphere, all the cardinal points of
S3
(10) either correspond to,
or coincide with, the dimensions of String Theory modulo 10. But more than that, the
Ramanujan function that appears in string theory as 24 modes of string vibration is the
same number that includes the sum of the cardinal points, plus the 0-dimensional point.
[Kaku, 1994] Further,
3!13(mod10)
;
13 !23(mod10)
; and if the inverse modulo 10
relation
3!7"1(mod10)
supports backward-forward projection of points of the plane to
the line, then the same inverse relation
7!23 "1(mod10)
supports the projection of
hyperspatial points in (
) to the plane (
S1
). If string theory could be explained as the
self-organization of dimensions, we now have the arithmetic to do so.
7.0 Discussion: The Riemann Hypothesis as an Equilibrium Function on the
Complex Sphere?
The positivity requirement (5.11.1) and the infinite variety of state spaces made available
by the Hilbert space – along with the constructed continuity that unites real and complex
analysis in a backward-forward projection between
S1
and
S3
suggests the preservation of
equilibrium on the intervals
(0,1)
and
[0,1]
in a true transformation of an indefinite and
continuous measure space to a discrete counting function symmetrical about the complex
plane axis. [Ray, 2002]
What follows, is a narrative explanation of how the concepts we have introduced support
integration of the complex function of the Riemann Hypothesis with the arithmetic
concept of discrete counting values geometrically identified with a self-organized
network of randomly oriented points of
S3
.
We allowed (3.1) non-redundant cardinal points of a tensor metric, the characteristic
number of a given dimension, to linearly sum as
!d
. We find that
!d
3
+1=D+1
, to
allow the central point of 3-dimensional empirical space to preserve the cut characteristic
D!1=Dn
in
n
dimensions. 3-space embedded in hyperspace has 1 bidirectional (or in
n-dimensions, 1 infinitely orientable) central point value of +2. This is not controversial;
we simply prefer to explain it in a non-conventional way, in order to support the
observational consequences. Graphically and intuitively, one realizes that 7 points
describe a 3-dimensional Euclidean space
!3
with a central operator, represented as a
dimensionless point, to capture the six empirical vectors:
Figure 1.
!3
has 3 bidirectional
vectors & 1 dimensionless operator.
Because the central point is dimensionless, the cut principle (i.e., 7 - 6) is preserved. A
hyperspatial extension of this 3-space accommodates one line orthogonal to each
+2
+2
+2
+1
+2+1
+2+1
+2+1
+2
Figure 2.
has 10 non-
redundant cardinal points of
a 4-dimensional tensor field & 1
dimensionless bidirectional
operator.
9
cardinal direction of 3-space, plus a bidirectional operator. What we’re saying is, that if –
as we have established (2.2) – the zero dimensional point of time is n- dimensional
infinitely orientable, the net effect of random n- dimensional motion will appear +1
positive, backward or forward.
!d
3
=1
is precisely the observational consequence we
should expect when length 1 is preserved on a sphere of radius 1 even when we cannot be
sure of the status of the metric diameter, the true state vector, until we measure the result.
! !
=1
The unitary measurement is local. That makes quantum mechanics, from a
theoretical viewpoint, profoundly boring. That is,
!d
3
"1=0
.
The proof of the Poincare Conjecture [Perelman, 2006], frees us to actually measure
space in non-empirical terms. The global measurement space does not allow
simultaneous access to antipodal endpoints; in principle, however, there remains the
singularity to fix the local measure on a central limit, while globally time remains random
and chaotic.
Hermann Weyl [1918] used the term “betweenness” to describe geometrical constraint.
Betweenness that is not arbitrary, is probablilistic; i.e., just as one cannot locate an odd
metric diameter, yet can prove its existence in hyperspatial terms, the central limit in a
fair contest of odds and evens assures infinite betweenness locally, as infinite self-
similarity. Going back to the cut principle [Dedekind 1901], we can draw an analog of 4-
dimensional betweenness in a 2-dimensional graph where four brackets define four
orders of differentiation:
3
2
The interval described by 3 { is open. The closed intervals [0,+1] and [0,-1] make a two-
valued measure space. Four half open intervals complete the probabilistic measure of
odd-even betweenness for 7 points of 3 dimensions including a zero-dimensional
operator, which is the open interval. The projection to hyperspace follows as we have
described, forming a complete space of n-dimensional continuous functions (Hilbert
space). The complex plane is rigorously defined at the zero intersection between points
of all 2-dimensional cardinal directions. Because two of the directions are imaginary,
measured real functions demand positivity and vectorization. Riemann’s complex plane
interpretation of Euler’s zeta function,
!
s=1
ns
n=1
"
#
recognizes that ordering in complex
terms describes 2-dimensional objects by existence. That is, although we may reduce one
part of a complex term to zero to describe a point on a line, this point is not technically
1
1
10
dimensionless. To say,
a+0i=a
, is to reduce the identity but not the dimension. We
mean, by letting the
b
term (of
a+bi
) go to zero, we do not extinguish the dimension
represented in
i
. i.e.,
0+i=i
is independent of this identity. The Euler Identity (2.3)
further shows how 2-dimensional arithmetical calculus separates addition and
multiplication, unifying arithmetic terms in geometric terms. The additive identity of
zero preserves the dimension 2 property of the radical (
i2=!1
;
i=!1
) while the
multiplicative identity of
a+1i
preserves the dimension 1 property of the line. This is
how we confidently add terms in eqn. (5),
3
4
+
!
to describe the curvilinear property by
which a point traces a 1-dimension line on an n-dimension plane. Suppose we multiply
3
4
!
"
#$
%
&
'
. We find a familiar looking number, 2.720699 …, familiar for differing only
slightly from
e
, and given that
e
and
!
are transcendental, may not differ at all. We say
this, because the multiplicative identity, 1, is the number for which
e
is the root of the
natural logarithm. The logarithmic expression of the Euler Identity (5.11) preserves unity
because the additive identity has been transformed from a complex term (0i) to an
arithmetic term. In other words, the null-valued multiplicative property in which 0i = 0,
is replaced by an arithmetic additive property that preserves the identity of the quantities
added:
ei
!
=0"1; ei
!
=0+("1)
. This expression is more powerful than when we force
complex terms to obey our idea of arithmetic well ordering, because it orders geometric
least elements into a coherent whole, a unity that is not dimensionless in that 2-
dimensional analysis transfers smoothly to 3 and 4 dimensions.
is a well defined topological space. Historically, the point has not been well defined in
any other terms than a point set. To have a discrete point, as a member of a continuous
set, implies complex analysis at the most fundamental level. Consider that
n
4
=n
2
in
positive real terms secures a smoothly continuous function not possible on the real line
without incorporating negative integers; for n = 4:
!4
4
=!1"!4
2
. When positive 4
is substituted for n, the result is unity and when n is unity, the result is
1
2
. Thereby, the
condition of Riemann’s zeta function for n to be positive 1,
!
s=1
ns
n=1
"
#
, because the least
element (of
N*
, the positive integers absent of zero) mediates positive order in negative
terms. What we mean by that, is that the trivial zeros of the zeta function, the negative
even integers, are ordered on the negative real line by a
1
2
monotonically increasing
order,
!n
4
=m
:
n=!2, !4, !6, !8...
m=!0.5, !1, !1.5, !2..
11
While on the positive side of the complex plane, the equivalent expression
n
2
suggests
no such order for
n
4
=n
2
. The floor value of the result (when converted to a linear
expression, the equation is
2n
on both sides) will always be even, like linear array of
the trivial zeros on the negative real line. We ignore the decimal expansion of the non
square-integrable values of
n
, because we are concerned with only the case of mapping
evens to evens on the entire complex plane. And because we know that process – i.e., in
the self-limiting case of continuous real numbers, all odd values of n are converted to
evens – we know the corollary, that only in the complete and universal set of complex
numbers do we find a non-arbitrary way to differentiate odd and even properties. In this,
the Riemann Hypothesis (RH) plays, literally, the central role.
Consider that the self-limiting case of the continuous real line corresponds to a local
measurement, and the extension of measure to the entire complex plane and sphere, to
global measurement.
Then we can show that the global complex property begs inversion of the hyperspatial
metric in real terms. (5.7.1) As a real unit metric radius,
1
2
is the point that divides a
diameter into 2 even parts, in 1 dimension – an n-dimensional radius that integrates odd
terms cannot go to equilibrium. I.e., let O = odd integer;
n
O
!n
2(O)
, lacking the self-
limiting completeness of
2n=2n
.
Given the projection between
S1
and
S3
, therefore,
n
4
=n
2
!RH
[Ray, 2002].
Reciprocal inversion in the most natural terms,
24=42
, informs us that transitivity
implies identity.
The identity element (zero) is then transitively reflected symmetric about the axis of the
positive real line because of very small values
<1
(the critical strip), as the hyperspatial
metric seeks equilibrium on the strip. Eqn 1 answers why primes, rather than all
composite odd numbers, have a special place on the critical line. Because
N* *
provides
a complete interpretation of the distribution of primes, the infinite correspondence (mod
2) between primes of any magnitude > 2 is identical to the inversion
1
2
on the positive
real line – which only works, of course, due to Riemann’s positivity requirement, n =1.
The zeta function that results in the RH is therefore the most important zeta function
because it is the most elementary. In fact, elementary correspondence between the well
ordering of the positive real line and the non-ordering of the complex plane and Riemann
sphere begs the transitivity of quantum least action. Transitivity implies identity.
12
True but uncertain results in real functions, e.g., Chaitin’s
!
[Chaitin 2005] and our
result (eqn 5) do not prevent us from knowing the true state of a measured metric within
the tiny limits of quantum least action. Note that the result of eqn 5 differs in the 4th root,
4.00761805...
4!1.41489...
only slightly from
2
. As a first order estimate of the
length of hyperspatially projected points to Euclidean space,
!2
, the value suggests a
future algebraic ability to predict the differentiated origin of events in hyperspace that by
present methods (tensor calculus, e.g.) are considered undifferentiable from any
arbitrarily chosen point of 4-dimensional space-time. (Our technique gives time an
independent physical role.)
**
With grateful thanks to Dr. Patrick Frank of the Stanford University Synchrotron
Radiation Laboratory, for years of kind support and good humored encouragement.
**
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... 2.1 In an earlier paper [Ray, 2006] we explored the physical consequences of a mathematical model of dimension self-organization, driven by the summation of cardinal points of discrete dimension sets: ...
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... These characterizations of background independence suggest an organic continuation of physics with the analytical mathematical model, the entire theme of [Ray, 2006]. ...
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Geometrization of 3-Maniolds via the Ricci FlowZorn's LemmaRiemann Sphere
  • G Perelman
  • W Eric
  • Weisstein
Perelman,G. [2006] " The entropy formula for the Ricci flow and its geometric applications. " ArXiv.math.DG\0211159 v1 Anderson, M.T. [2004] " Geometrization of 3-Maniolds via the Ricci Flow. " Notices of the AMS (vol. 51. No. 2, February) Eric W. Weisstein. "Zorn's Lemma." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/ZornsLemma.html Eric W. Weisstein. "Riemann Sphere." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/RiemannSphere.html Einstein. A. [1955] The Meaning of Relativity, Princeton University Press. Brouwer, L.E.J.[1981] Brouwer's Cambridge Lectures on Intuitionism, Cambridge University Press.
Essays on the Theory of Numbers. Dover Books English translation by Realism and the Aim of Science
  • R Dedekind
Dedekind, R. [1901] Essays on the Theory of Numbers. Dover Books English translation by Beman, W. 2001. Popper, K. [1983] Realism and the Aim of Science; Routledge.
When science becomes mathematics: a new demarcation problem
  • T Ray
Ray, T. [2002] "When science becomes mathematics: a new demarcation problem," contributed paper, Karl Popper Centenary Congress, University of Vienna.