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)

S2

3 6 + =

(3-sphere)

S3

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

S3

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

S3

, 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 (

S3

) 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.

S3

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.

S3

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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