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Correlations between the Invariant Set Postulate and the Codex 5 Model

Fritz Hoffecker, February 2010

1.0 Abstract........................................................................................................................ 2

2.0 Correlations between the ISP and the Codex 5 Model ................................................ 23.0 Summary of the Codex 5 Model.................................................................................. 33.1 Major Phases of the Model ...................................................................................... 3

3.2 Details of the Codex 5 Model .................................................................................. 8

4.0 Description of a Fractal State-Space Geometry......................................................... 10

5.0 Dynamical Systems.................................................................................................... 136.0 Invariance................................................................................................................... 22

6.1 Comparing the ISP and Codex 5 Model ................................................................ 22

6.2 Codex 5 Model Evolution of DC5 and the Physical World ................................. 237.0 Asymptotic Behavior ................................................................................................. 26

8.0 Gravity ....................................................................................................................... 28

8.1 Gravity: ISP and Codex 5 Model........................................................................... 288.2 Gravity and Dark Matter: Other Observations...................................................... 32

9.0 Quantum Theory....................................................................................................... 32

10.0 Conclusions.............................................................................................................. 33

References......................................................................................................................... 33

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1.0 Abstract

The purpose of this paper is to show correlations between the ideas and hypotheses

described in two different papers:1) The Invariant Set Postulate: a new geometric framework for the foundations of

quantum theory and the role played by gravity, by T. N. Palmer, first published online on29 July, 2009, Proceedings of The Royal Society.

2) The Codex 5 Model: Describing the Origin & Nature of Dark Energy & Dark Matter

and Predicting the Distribution of Energy / Matter in the Universe, by Fritz Hoffecker,2008.

Note that the authors did not collaborate in writing their respective papers, and havenever met, spoken or communicated in any way.

The Invariant Set Postulate [ISP] states:

A new law of physics is proposed, defined on the cosmological scale butwith significant implications for the microscale. Motivated by nonlinear

dynamical systems theory and black-hole thermodynamics, the Invariant

Set Postulate proposes that cosmological states of physical reality belongto a non-computable fractal state-space geometry I, invariant under the

action of some subordinate deterministic causal dynamicsDI .

The Codex 5 Model proposes a new approach that describes the origin and development

of dark energy, dark matter, and baryonic matter. In doing so, the Codex 5 Models

statements about, and descriptions of the physical universe correlate with the ISPs in

some interesting areas.

This paper will show correlations between the ISP and the Codex 5 Model parallels that

may strengthen and add depth to the concepts about and descriptions of physical reality

found in both papers.

2.0 Correlations between the ISP and the Codex 5 Model

This section summarizes correlations between the ISP and the Codex 5 Model. Forreaders unfamiliar with Codex 5, a summary is presented.

The papers show correlations in the following key areas:

1) Description of a fractal state-space geometry see Section 4.0 below.2) Use of dynamical systems theory, with reference to and description of a

dynamical systems deterministic, causal effects on a state-space geometry see

Section 5.0.

3) Proposal that an invariant set lies at the heart of physical reality - see Section 6.0.4) Description of asymptotic behavior under certain conditions - see Section 7.0.

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5) View that gravity is related to and/or a manifestation of variations (orheterogeneities) in the geometry of state space (including the state space of the set

Physical Reality, PR, per Table 3-1 below) - see Section 8.0.

6) Both papers lead to a questioning of some basic precepts of Quantum Theory.This paper will only summarize the relationship between the Codex 5 Model and

Quantum Theory - see Section 9.0. Work has begun on another paper that willinvestigate this relationship in more detail.

3.0 Summary of the Codex 5 Model

The Codex 5 Model paper (Hoffecker, 2008) presents the Model in detail; this section

summarizes key aspects of the Model.

3.1 Major Phases of the Model

Figures 1a and 1b below present a summary of the phases of the Codex 5 Model. The

diagrams describe the following phases of the universes evolution:

[Note: For reference purposes, Figure 2 below presents 2-dimensional, generic views of

the entities of the Codex 5 Model: Basic Pairs (BP), Large Pairs (LgP), Very Large

Quads (VLQ), Very Large Octets (VLO), and Extra Large-12s (XL-12). These are

referred to in the Phase descriptions below.]

Phase 0 (aka Phase Zero)Absolute Nil and Non Nil must exist, and can only exist as entities called Basic Pairs

(BPs).

Phase 1: BPs interact.

Results of these interactions: Some BPs remain whole (i.e., do not change), while othersmerge to form Large Pairs (LgPs). While the term some is used here, the Codex 5

paper provides exact percentages. Also see Table 5-3 below for a summary.)

Phase 2: BPs and LgPs interact.

Results of these interactions: Some BPs and LgPs remain whole (i.e., do not change),

while some LgPs merge to form Very Large Quads (VLQs).

Phase 3: BPs, LgPs and VLQs interact.

Results of these interactions: Some BPs, LgPs and VLQs remain whole (i.e., do not

change or merge), while some VLQs merge to form Very Large Octets (VLOcts).

Phase 4: BPs, LgPs, VLQs and VLOcts interact.

Results of these interactions: Some BPs, LgPs, VLQs and VLOcts remain whole (i.e., donot change or merge), while VLQs and VLOcts merge to form Extra Large 12s (XL-

12s).

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Subsequent phases involve the merger of VLOcts and XL-12s.These interactions are described in the Codex 5 paper, Appendix 1, Section 2, which

includes rough diagrams of the interactions. This section of the paper doesnt provide

details of these interactions, but proposes that they facilitate subsequent particlesacquisition of mass, and thus lead to the creation of baryonic matter.

Evolution of the Universe.After the creation of baryonic matter, the universe evolves to, for example, its current

state, about 13.7 billion years after the Big Bang.

Devolution of the Universe.The Codex 5 Model proposes that in subsequent phases, all types of matter except Dark

Energy (which is made of Basic Pairs) will unwind. That is, baryonic and Dark Matter

will devolve until only one Large Pair is left among a universe-wide sea of Basic Pairs.

The Next Singularity.

The final Large Pair eventually unwinds, and this marks the location where thesingularity of the next Big Bang will occur. See Codex 5 Model paper, Section 4.0, Part

2 - Death and Rebirth of the Universe, and details in Appendix 1, Section 10.

At this point, it is important to note that the next iteration of the universe (i.e., after thenext Big Bang) will not evolve to become an exact copy of the current universe. This is

because initial conditions at the next Big Bang will be different from the last Big Bang:

The distribution of Basic Pairs will not be precisely the same as the distribution was atthe time of the most recent Big Bang.

At these instants of Big Bangs, it is estimated that Basic Pair distribution will be fairlyeven throughout the universe; however, there will be heterogeneities, which will account

differences in the eventual emergence of stars, galaxies and other baryonic matter. So, it

is anticipated that in the next iteration of the universe (i.e., after the next Big Bang),matter will evolve in much the same way as in the current universe (e.g., Hydrogen will

form in the same way), but distribution of instantiations of Energy / Matter (from atoms

to galaxies) will be different. In addition, the locus of the next singularity (relative to therest of the universe) will not be exactly the same as last singularitys, and given the

different distributions of BPs just before the Big Bangs, then it can be concluded that

stars, galaxies, planets and other matter will not evolve in exactly the same trajectories

and locations for both iterations of the universe.

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Figure 2: Codex 5 Model - Entities

In line with the phases and Codex 5 entities outlined above, Section 5.0 below will

describe the Codex 5s Dynamical System,DC5, as causal and deterministic, but non-computable. For example, theDC5 that has followed the last Big Bang has the samedeterministic rules as theDC5 that will follow the next Big Bang, but the nextDC5 will

not lead to an exact replication of the physical reality that has existed and evolved in the

past 13.7 billion years.

Basic Pair:

71.43%

Large Pair:

21.43%

V Lg Octet:

3.57%

XL-12:

1.07%

V Lg Quad:

2.50%

Light DarkMatter

Heavy

Dark Matter

~73% of

Matter is DarkEnergy

Dark

Matter -~23%

~4% of all

Matter is

Baryonic4.64%

VLOs &

XL12's are

transformed

into

Baryonic

Matter

Dark Energy

consists of BasicPairs

(secondary

possibility:

-About half of V Lg

Quads becomeNeutrinos. Other half

migrates to baryonic

and/or Heavy DM.-Or 1.5% of neutrinos

not detected yet.)

V Lg Octets & XL-

12s are transformed

into Baryonic matter.

Hot DM:

Neutrinos may

be ~

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3.2 Details of the Codex 5 Model

As detailed in the Codex 5 paper, Phase 0 begins with the following statements:Absolute-Nil is defined as a Basic Entity that is complete Nothingness. For

example, it is not composed of baryonic matter, does not emit or reflect any

energy in the electro-magnetic spectrum. Figuratively, it can be thought of as the

tiniest possible bit of nothingness in the Universe.Non-Nil is defined as a Basic Entity that consists of matter that cannot be

further decomposed. Figuratively, it can be thought of as the tiniest possible bit

of matter in the Universe.

Figure 3 below shows Phase 0, which illustrates the Models statements:

1) that Absolute Nil and Non-Nil must exist, and

2) furthermore, that they can exist only as a Basic Pair:

Figure 3: Codex 5 Model - Phase 0: A Basic Pair

Figures 1a and 1b (above) and Table 3-1 (below) show that the Models set structure is:1) A set consisting of all physical reality, PR, over the entire life-span of the

Universe, including baryonic matter, Dark Matter and Dark Energy. While thedefinition of physical reality is debatable, in this case, certain abstractions are not

included, such as imaginary numbers and hypothetical musings like If Planet

Earth were 40 million miles from the sun instead of 93 million, then life as weknow it would never have evolved. One could argue that the brains imagining

such things are made of baryonic matter, and that therefore, such things are part of

physical reality. However, this avenue of reasoning isnt useful in this paper, e.g.,

a person can imagine that a Helium atom is more massive than a Carbon atom,that the South won the Civil War, etc. but such an approach is irrelevant here.

2) A set, here called C5, of all instantiations of the entities described in Phases 0through 4 of the Codex 5 Model: Basic Pairs, Large Pairs, Very Large Quads,Very Large Octets, and Extra Large-12s. C5 is a subset ofPR.

3) A set, here calledAN-NN, which consists of any instance of Absolute Nil (AN)and Non-Nil (NN). The Codex 5 Model proposes that any pair of one instance ofAbsolute Nil and one instance of Non-Nil can exist only as a Basic Pair. AN-NN

is a subset ofC5, and therefore also a subset ofPR.

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Table 3-1: Set Structure of Codex 5 Model

Set AN-NN Set C5 Set PRPhases of Codex 5s Dynamical

System (DC5)

Phase 0, #1Ab-Nil / Non-Nilinstances asBasic Pairs Phase 1, Initial State

Phase 1, Next State

Phase 2, Next State

Phase 3, Next State

All entities inPhase 0 thruPhase 4

(BPs, LgPs,VLQs,

VLOcts, XL-12s)

Phase 4, Next State

Emergence of Baryonic matter

Recombination: 1st stable H & He

atoms formStelliferous era starts

Stelliferous era continues, e.g., topresent day:

End of stars (burned out)

Ongoing decay, unwinding of allBaryonic matter

Codex 5: Black Holes, stars, allBaryonic matter asymptote to Set

C5, then backwards thru thefractal progression (Items 1 thru 4bbelow), until Set AN-NNisreached.

1) XL-12s first devolve into VLOcts& VLQs.

2) VLOcts devolve into VLQs

3) Decay of VLQs into Large Pairs

4a) Decay of Large Pairs into BasicPairs

4b) Last LgP unwinds into 2 BPs

Phase 0, #2 (Set AN-NN)Invariant Ab-Nil/ Non-Nil as

BPs

Set of allphysical

reality: DarkEnergy, Dark

Matter,Baryonicmatter

Phase 1, Initial State, #2

Reviewing the entire Codex 5 Model paper will enhance understanding of Models

phases and set structure, but this summary has shown key points.

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4.0 Description of a Fractal State-Space Geometry

The Invariant Set Postulate (ISP):

proposes that cosmological states of physical reality belong to a non-

computable fractal state-space geometry I, invariant under the action ofsome subordinate deterministic causal dynamicsDI.

This single sentence marks several correlations between the ISP and the Codex 5 Model,

but Section 4.0 will focus on fractal state-space geometry.

The ISP makes many other references to a fractal state-space geometry, such as:

The Invariant Set Postulate posits the existence of a fractionallydimensioned subset I of the state space of the physical world (i.e., the

universe as a whole).

In summary, the Codex 5 Model describes a state-space geometry that is fractal in that all

the entities that evolve through Phase 4 of the Model (i.e., all the entities in Set C5) have

self-similar structures. Using the Codex 5 Models terminology, the Model describes theevolution of the physical entities in Set C5: Basic Pairs (BP), Large Pairs (LgP), VeryLarge Quads (VLQ), Very Large Octets (VLOct) and Extra-Large 12s (XL-12).

Figure 4 (below) shows each of these entities on the left; their self-similar structures areevident. Note that while the diagram shows flat, 2-dimensional shapes, each entity exists

in three dimensions, e.g., approximately as spheres, not circles. (The sphere is selected

for conveniences sake and because it represents a fairly even distribution about a centerpoint. However, additional work is being done on the harmonic oscillation aspects of

each Codex 5 entity.)The entities are different in that each different type of entity has a different level of

Energy/Mass (E/M). The base unit of E/M is the BP. Since a LgP is formed by themerging of two BPs, the E/M of a LgP is twice as large as the E/M of a BP.

Likewise, when two LgPs merge to form a VLQ, the VLQ has E/M equivalent to that of

two LgPs (or four BPs). Two VLQs can merge to form a VLOct (which has an E/M of

eight BPs). A VLQ and a VLOct can merge to form an XL-12 (E/M of 12 BPs), but theModel stops short of detailing any mergers of entities after Phase 4 ends. Instead, after

Phase 4, the Codex 5 paper proposes that the vector associated with any further merging

among VLOcts and XL-12s leads to the evolution of known subatomic particles and thecreation of baryonic matter. For details, see Section 3.0 and Appendix 1, Section 2 of the

Codex 5 paper.

The Codex 5 Model does not estimate the magnitude of a BPs E/M, though work in this

area has begun and is ongoing. However, note that Seven-Year Wilkinson Microwave

Anisotropy Probe (WMAP1) Observations: Cosmological Interpretation, Table 2,

indicates that:

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.using WMAP+BAO+H0, [the limit on the total mass of massiveneutrinos is] mv < 0.58 eV (95% CL).

Footnote g states that For WMAP+LRG+H0, mv < 0.44 eV.

Calculations in the Codex 5 paper indicate that a BP has less mass than a neutrino. Some

correlation is proposed between the Energy/Mass of a neutrino and that of a VLQ. UsingSeven-Year WMAP observations (referenced above), and if a VLQ and the referencedneutrino have about the same mass, then since a BP has 25% of the mass of a VLQ, then

the mass of a BP would be < 25% of 0.58 eV (i.e., < 0.1475 eV). Per Footnote g, for

WMAP+LRG+H0, the mass of a BP would be < 25% of 0.44 eV (i.e., < 0.11 eV).

Further work in this area is ongoing.

Note that in addition to showing the fractal aspects of the Codex 5 entities, Figure 4 also

maps the types of entities to types of matter (ref WMAP observations and data, 2003 2009).

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Basic Pair:71.43%

Large Pair:21.43%

V Lg Octet:3.57%

XL-12:1.07%

V Lg Quad:

2.50%

Light Dark

Matter

HeavyDark Matter

~73% of

Matter is DarkEnergy

DarkMatter -

~23%

~4% of all

Matter isBaryonic4.64%

VLOs &

XL12's are

transformed

into

Baryonic

Matter

Dark Energyconsists of Basic

Pairs

(secondary

possibility:-About half of V Lg

Quads becomeNeutrinos. Other half

migrates to baryonicand/or Heavy DM.

-Or 1.5% of neutrinosnot detected yet.)

V Lg Octets & XL-12s are transformedinto Baryonic matter.

Hot DM:Neutrinos may

be ~

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Mandelbrot (1982) has defined fractals as a rough or fragmented geometric shape thatcan be split into parts, each of which is (at least approximately) a reduced-size copy of

the whole.

As can be seen in Figure 4, the entities described in the Codex 5 Model are 3-dimensionalfractals, in that all the structures of the different types of entities are self-similar. For

example, a BP and a LgP have the same basic structure, with the only difference beingthat the Energy/Mass (E/M) of the LgP is twice as large as that of the BP. Likewise, BPs,LgPs, VLQs, VLOcts and XL-12s all have a self-similar 3-D structure, the only

difference being in measurements of E/M. (The evolution from BPs to LgPs, and so on

through XL-12s, is described in detail in the Codex 5 paper, Section 3.0.)

In summary, Section 4.0 has shown the set structure of the Codex 5 Model, including the

fractal aspects of the sets involved. Sections 5.0 and 6.0 below will show relationships

between the Codex 5s fractal geometry and, respectively, a) the Codex 5s DynamicalSystem, and b) Invariance.

5.0 Dynamical Systems

As noted above, the ISP states:

the Invariant Set Postulate proposes that cosmological states of physicalreality belong to a non-computable fractal state-space geometry I,

invariant under the action of some subordinate deterministic causal

dynamicsDI.The Invariant Set Postulate is framed in terms of invariance, a concept that

forms the very bedrock of physics, and conjectures that states of physical

reality are defined by a fractal geometry I, embedded in state space and

invariant under the action of some subordinate causal dynamicsDI.

The Codex 5 Model describes a dynamical system,DC5, that operates according to certain

deterministic, causal rules, but as noted above at the end of Section 3.1:

it is important to note that the next iteration of the universe (i.e., afterthe next Big Bang) will not evolve to become an exact copy of the current

universe

In line with the phases and Codex 5 entities outlined above, Section 5.0below will describe the Codex 5s Dynamical System, DC5, as causal and

deterministic, but non-computable. For example, the DC5 that has

followed the last Big Bang has the same deterministic rules as the DC5that will follow the next Big Bang, but the next D

C5will not lead to an

exact replication of the physical reality that has existed and evolved in the

past 13.7 billion years.

The Codex 5 paper describesDC5 in detail, and a summary is presented below. DC5

drives the evolution of entities from BPs to LgPs, and so on, such that a fractal geometry

evolves from the smallest entities (BPs, at Phase 0) to the largest (XL-12s, at the end ofPhase 4). Note that given the initial state described in the Model, and the effects ofDC5

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throughout various phases, certain aspects of the next states are predictable, i.e., thedynamical system described in the Codex 5 Model is deterministic and causal through its

Phase 4. See details in Section 3.0 of the Codex 5 paper. However, the Codex 5 Model

also shows non-computable properties, in thatDC5 is non-halting: It can be shown whereit starts (at Phase 0) and many aspects of subsequent phases can be predicted but many

properties (such as the properties of molecules resulting from combinations of atoms) cannot be calculated before they occur.

The Codex 5s dynamical system is symbolic in that the BPs and other entities are not

intended to be completely accurate graphical representations of entities in physical reality,

as no BPs, LgPs, et al. have actually been observed. However,

1) The diagrams intend to show key properties of actual entities, and2) The interactions shown between BPs, LgPs, et al. are a generalized, but integrated

graphical representation of actual interactions and transformations.

In summary, the diagrams provide symbolic representations of BPs, LgPs and the otherentities described in the Codex 5 Model. The Codex 5s dynamical system,DC5, aligns

fairly closely with the ISPs references to symbolic dynamics:

One key technique to represent the evolution of states on an invariant setis that of symbolic dynamics... Given the non-computability of fractal

invariant sets, the analysis here suggests that the most likely approach to

finding a robust mathematical representation of the invariant set of theuniverse is through the symbolic approach; as discussed in 1a, this

approach can define the invariant set to topological equivalence. Indeed,as remarked by Bohr himself, the very fact that a quantum theoretic state

has the form | A + | B suggests that quantum theory is itselfprofoundly symbolic.

The Codex 5 Models state space as used here to illustrate the Models Dynamical

System,DC5 - can be described as a tuple [E, T, IS, NS], where:

E = a set of entities (such as in the Codex 5 Model Absolute Nil and Non-Nil,which can only exist paired together in Basic Pairs)

IS = an Initial State, which describes which entities are available to participate ina subsequent Interaction

IA = Interaction between entities in E

NS = Next State, which describes the situation after the Interaction (e.g., Do 2different Basic Pairs merge to form a Large Pair or do they stay separate?)

Phase 0 of the Dynamical SystemDC5:The Codex 5 paper begins with the following state space:

E0 = {Absolute Nil0, Non-Nil0} Note that in the totality of physical reality (refthe set named PR in Table 3-1), E contains more than one such set. In fact, it

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contains many billions, though the calculation of an estimated number will beaddressed in a future paper.

IS0 = In the Codex 5 Model, the Initial State explains that Absolute Nil and Non-Nil must exist.

IA0 = according to the Codex 5 Model, the existence of the set {Absolute Nil 0,

Non-Nil0} can lead to one and only one Next State: NS0 = {Basic Pair0}. This means that neither a real instantiation of Absolute Nil

nor a real instantiation of Non-Nil can exist alone in PR; they can exist in physical

reality only as a Basic Pair.

(Note that NS0 is not a next state from the point of view of the forward arrow of

time, because Absolute Nil and Non-Nil can only exist as in a Basic Pair. In otherwords, Ab-Nil and Non-Nil dont exist first, and then a Basic Pair exists. Instead,

NS0 is a next state in terms of the logical sequence used to describe them: That

is, once the existence and properties of Ab-Nil and Non-Nil are understood, thenthe next state in understanding is to see that the can exist only as a Basic Pair.

Phase 1 of the Dynamical SystemDC5:Given that many sets exist where E0 = {Absolute Nil0, Non-Nil0}, e.g., {Absolute Nil2,

Non-Nil2}.. {Absolute Niln, Non-Niln}, then many NSA = {Basic Pair1} will also exist

after Phase 1 of the Dynamical State occurs many millions of times. However, given that

there are only four possible arrangements for Basic Pairs, the Codex 5 Model addressesand focuses on those four alternatives.

For the dynamic system described in the Codex 5 Model,DC5, the Next State thatresults from Phase 0 becomes the Initial State that Phase 1 begins with. Therefore, in

Phase 1, the initial view of E (E1 for the Initial State of Phase 1) is:

E1

= {Basic Pair1}, {Basic Pair2}, {Basic Pair3}, {Basic Pair4}

This will be extended shortly, but first, here are some notes about E1:

The four types of BPs in E1 are distinguished by variations in the following properties:1) Direction of rotation: Clockwise or Counter-clockwise2) Placement of the larger (in the diagram) part (head) of the Non-Nil part of the

BP: top or bottom

3) Placement of the larger part (head) of the Absolute-Nil part of the BP: top orbottom

[Note that the terms large part and head are just wording used to align with the

symbolic states shown in the diagrams. For example, work is being done to comparethe current representations and properties of BPs, LgPs, et al. with aspects of quantum

harmonic oscillation. This will lead to additional ways to represent the Codex 5

entities and their properties.]

Table 5-1 below shows, for any BP, the only possible combinations of these three

properties. Note that some combinations that look mathematically or categorically

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possible are actually counter-factual; specifically, the following cannot exist in thephysical reality described in the Codex 5 Model:

1) There cant be a BP with both the Absolute Nil and Non-Nil head (i.e., thewider part of Ab-Nil or Non-Nil) at the top, or both at the bottom.

2) There cant be a BP that has 2 Absolute Nils (and zero Non-Nils) or 2 Non-Nils

(and zero Absolute Nils).3) Both the Absolute Nil and the Non-Nil parts of a BP have to rotate in the samedirection.

Table 5-1: Codex 5 Model - All Possible BP Properties

Direction ofRotation

Location of Non-Nil Head

Location of Ab-NilHead

Clockwise = 1 Top = 1 Top = 1

CCWise = -1 Bottom = -1 Bottom = -1

BP1 -1 -1 1

BP2 -1 1 -1

BP3 1 -1 1

BP4 1 1 -1

In order to show all the possible Interactions (IAs) of this state space, E needs to beexpanded to include additional BPs BP1b, BP1b, BP3b and BP4b which, respectively,

have the same properties as BP1, BP2, BP3 and BP4.

Therefore, for Phase 1:

E1 = {Basic Pair1}, {Basic Pair2}, {Basic Pair3}, {Basic Pair4}, {Basic Pair1b},{Basic Pair1b}, {Basic Pair3b}, {Basic Pair4b}

IS1 = In the Initial State, all 8 BPs are separate, but are poised to interact in everypossible combination.

IA1 = Interactions in this step of this dynamical system take into consideration thefact that every possible interaction between any two unique BPs will occur. In

Table 5-2 below, the first 2 columns show the interactions that take place.

NS1 = See Column 3 of Table 5-2 below (Results of Interaction), which showsthe Next States resulting from the interactions.

Figure 5 below shows typical Initial States that involve different BPs. (Additional details

and graphics are provided in the Codex 5 paper, Section 3.0.)

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Interaction 1 (IA1)

x

y

b

a

IA2

x

y b

a

IA3

x

y

b

a

IA4

Figure 5: Codex 5 Model Possible Interactions (IAs) of Initial States

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The first two columns of Table 5-2 (below) show the 28 possible Initial States thatinvolve different pairings of BPs, e.g., in the first Initial State, the 1st Basic Pair is

shown as 1 (i.e., BP1), and the 2nd Basic Pair is shown as 2 (i.e., BP2).

For example, Row 1, Column 3 (Results of Interaction) shows the Next State that

resulted from the interaction of BP1 and BP2 (BPs merge).

Table 5-2: Interactions and Next States

1st BasicPair

2nd BasicPair

Result ofInteraction

1 2 BPs merge

1 3 BPs stay separate

1 4 BPs stay separate

1 1b BPs stay separate

1 1b BPs merge

1 3b BPs stay separate

1 4b BPs stay separate

2 3 BPs stay separate

2 4 BPs stay separate

2 1b BPs merge

2 1b BPs stay separate

2 3b BPs stay separate

2 4b BPs stay separate

3 4 BPs merge

3 1b BPs stay separate

3 1b BPs stay separate3 3b BPs stay separate

3 4b BPs merge

4 1b BPs stay separate

4 1b BPs stay separate

4 3b BPs merge

4 4b BPs stay separate

1b 1b BPs merge

1b 3b BPs stay separate

1b 4b BPs stay separate

1b 3b BPs stay separate

1b 4b BPs stay separate

3b 4b BPs merge

In Table 5-2, when the Result of Interaction (3rd

column) is BPs stay separate, thismeans that the BPs retain their existence as individual entities and do not merge to form a

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Large Pair. For example, in Row 2, when BP1 and BP3 interact, the result is BPs stayseparate, which means that BP1 and BP3 remain separate entities.

The merging of two different BPs creates a new type of entity, called a Large Pair (e.g.,Table 5-2, Row 1). See Figure 6 below:

x

y

x/a

y/b

InitialState 2

NextState 2

x

y

{x/y} and {b/a}Force

Intersection

b

a

b

a

Figure 6: Codex 5 Model - Two Basic Pairs Merge to Form a Large Pair

The structure of the Large Pair is the same as the structure of a BP, so they can be

described as 3-dimensional fractals (see Section 4.0 above). As the dynamic stateDC5

continues through subsequent steps, this self-similar fractal structure is maintained,

through Phase 4, as other entities emerge from the interactions described inDC5.

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After Phase 2, the Codex 5 Model describes subsequent phases ofDC5. Table 5-3 (below)shows that after Phase 4, the initial total of Energy/Mass in the first phase (i.e., the 56

BPs shown in Table 5-2), has become distributed among BPs, LgPs, VLQs, VLOcts and

XL12s.

After Phase 4, as shown above in Figures 1a and 1b, the geometry of state space evolvesas follows, under the dynamical system DC5:

The remaining (post-Phase 4) BPs, LgPs and VLQs continue to exist, and are nottransformed further (until the Devolution of the Universe phase noted in Section

3.1).

[Note, however, that other than the interactions described in DC5, there may be

cases in the universe where some interactions among LgPs and/or VLQs occur.

For example, the Codex 5 Model proposes that LgPs and VLQs make up Dark

Matter. Observations of Abell 520 (Mahdavi, et al., 2007) may be explained by

weakly self-interacting dark matter. In Codex 5 terms, this would mean that

LgPs and/or VLQs were interacting; note that in the case of Abell 520, this would

have occurred only after a complicated collision of galaxy clusters, some of themost massive objects in the Universe.

The Codex 5 Model maintains that after Phase 4, the remaining BPs, LgPs and

VLQs do not normally interact; however, it appears that events such as this

collision could allow for some additional interactions possibly because of the

high levels of energy involved. Further analysis has started and will be addressed

in the future, e.g., explaining the apparent differences between the behavior and

properties of Dark Matter in Abell 520 and the Bullet cluster (1E 0657-56).]

Post-Phase 4, VLOcts and XL-12s continue to be transformed through the actionof theDC5. However, one could also say that they become subject to the action ofa new epoch ofDC5. This is a semantic issue that doesnt change what actually

happens in physical reality (the set PR). In these subsequent transformations,

VLOcts and XL-12s interact and new entities are formed as mass is acquired,until subatomic particles are created. Physical reality continues to evolve, e.g., to

our current universe until the final unwinding of the last Large Pair as shown

in Figure 1b and describes in Section 3.1.

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Table 5-3: Dynamic System after Phase 4

Phase 4Type of

Entity

Quantity of each

Entity

Units of

Energy / Mass

per Entity/

Total Units of

Energy per

Entity Type

End of Phase 1: %

of Total E/M in each

type of Entity

End of Phase 2: % of

Total E/M in each

type of EntityThroughout Phase 4, 71.4%

of E/M remains in BasicPairs BPs 40 1 40 71.4% 71.4% 71.4%

Throughout Phase 4, 21.4%of E/M remains in Lg Pairs LgPs 6 2 12 21.4%

21.43% - (0.75 *

.286): means after Ph

2, 21.4% of total E/Mis in LgPs

Phase4

Calcs

A = % of

7.14% in

VLQ, VLOct

or XL12:

B = E/M not in BPs

and LgPs = 7.14%: A x BVLQ (each has 4 Units of

E/M) VLQs 14 = gross # 4 35.00% 2.50%VLOct (each has 8 Units of

E/M) VLOcts 10 = gross # 8 50.00% 3.57%

XL12 (each has 12 Units ofE/M) XL12s 2 = gross # 12 15.00% 1.07%

7.14%

Phase4

Results

Phase4Results

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6.0 Invariance

6.1 Comparing the ISP and Codex 5 Model

The ISP states:

The Invariant Set Postulate is framed in terms of invariance, a concept that formsthe very bedrock of physics, and conjectures that states of physical reality are

defined by a fractal geometry I, embedded in state space and invariant under the

action of some subordinate causal dynamicsDI.Specifically, the Invariant Set Postulate subordinates the notion of the differential

equation and elevates as primitive, a dynamically invariant fractal geometry in the

state space of the universe. This geometry is used to define the notion of physical

realitystates of physical reality are precisely those on the invariant set.

Codex 5 Model and Invariance

The Codex 5 Model aligns with the concept of invariance in that an invariant set is one that

evolves to itself under the transformations of some Dynamical System. In the Codex 5 Model,the Dynamical SystemDC5 starts action on an invariant set,AN-NN(consisting only of Absolute

Nil and Non Nil in the form of a Basic Pair): See details above in Section 3.0 and Table 3-1.

DC5 runs through multiple transformations, which eventually lead back to a state that, again,

consists only of instances of Absolute Nil and Non Nil, which can exist only as Basic Pairs. See

Figures 1a and 1b, and Table 3-1 above for a summary of the phases ofDC5.

If only the first four Phases ofDC5 are considered (i.e., Set C5), then at the end of Phase 4:

1) Most instances of Absolute Nil and Non Nil (which exist as Basic Pairs) - remainunchanged from the beginning of Phase 0; Most here means that the unchanged BPstotal 71.43% of the total Energy/Mass (E/M) of the universe.

2) Many instances of Large Pairs (21.43% of the universes E/M) and a few Very LargeQuads (2.5% of the total E/M) also remain unchanged from when they first emerged

during the actions ofDC5.

3) Baryonic matter begins to emerge, as VLOcts and XL-12s eventually evolve into entitiesthat acquire mass. The VLOcts and XL-12s make up 4.64% of the universes total E/M.

After further transformations, VLOcts and XL-12s evolve to form baryonic matter, which

also accounts for 4.64% of the total E/M (as is seen in comparable WMAP observations,which estimate Baryonic matter at ~4.6%).

4) After tens of billions of years (a better estimate will be addressed in the future),DC5ultimately leads to an end state where the universe once again consists only of instancesof Absolute Nil and Non Nil in the form of Basic Pairs. At this point, the Codex 5 Model

proposes that the event where the last LgP unwinds into two BPs becomes the

Singularity where the next Big Bang occurs (also see Codex 5 paper, Section 4.0, Part 2,

and Appendix 1, Section 10).

In summary, the initial setAN-NNin the Codex 5 Model can be considered invariant if viewed

from a long enough time period (i.e., all the time from just before the last Big Bang to just beforethe next Big Bang), but during that time period, the Dynamical SystemDC5:

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1) Allows for the emergence of some variance (e.g., for the emergence of LgPs, VLQs,VLOcts, XL-12s and baryonic matter; though the last item does not evolve until after

Phase 4 ofDC5), while

2) Concurrently creating a subset ofC5 consisting of: a) Absolute Nil and Non Nil as BasicPairs, b) LgPs and c) VLQs. After the noted phases ofDC5, these entities make up

percentages of all the universes E/M that dont vary until the universes devolutionbegins (see Tables 3-1 and 5-3 above).3) The overall set, PR, is invariant in that after billions of years, it evolves back to its initial

set, E0, which includes Absolute Nil and Non-Nil instantiated in the form of Basic Pairs

(see Section 4.0 above, Phase 0 of the Dynamical System).

Given the preceding description of aspects of the Codex 5 Model, it is useful to loop back to the

Invariant Set Postulate paper, which states:

The Invariant Set Postulate posits the existence of a fractionally-dimensionedsubset Iof the state space of the physical world (ie the universe as a whole). Iis

an invariant set for some presumed-causal (ie relativistic) deterministic dynamical

systemDI ; points on I , hereafter referred to as world states, remain on Iunderthe action ofDI. World states of physical reality are those, and only those, lying

precisely onI.

Rewriting this sentence from the point of view of the Codex 5 Model, and referring to Table 3-1,would lead to:

The Codex 5 Model indicates that the fractionally-dimensioned subset of the state space of the

physical world (i.e., the universe as a whole) can be seen as the set C5, which includes BPs,LgPs, VLQs, VLOcts and XL-12s. These entities have self-similar structures and represent a

fractal subset of the Codex 5s set called Physical Reality (PR) see Table 3-1. The Modelwould also say that World states of physical reality are those, and only those that are in the Set

PR. Since C5 is a subset ofPR, world states of physical reality also lie on C5. Note that like the

ISP, the Codex 5 Model allows for consideration of counterfactual entities, such as imaginarynumbers, but states that these entities do not lie on PR. (See Section 5.0 for a discussion of

counterfactual data related to Basic Pairs.)

6.2 Codex 5 Model Evolution of DC5 and the Physical World

The Codex 5 Model defines the SetAN-NNas invariant, in that:Absolute-Nil is defined as a Basic Entity that is complete Nothingness. For

example, it is not composed of baryonic matter, does not emit or reflect any

energy in the electro-magnetic spectrum. It can be thought of as the tiniestpossible bit of nothingness in the Universe.

Non-Nil is defined as a Basic Entity that consists of matter that cannot be

further decomposed. It can be thought of as the tiniest possible bit of matter in

the Universe.

In the Codex 5 Model, neither Ab-Nil nor Non-Nil vary during all the transformations described

in the Codex 5 Models Dynamical SystemDC5, except when two BPs merge to form a LgP.Also, note that each BP that doesntmerge to form a Large Pair is also invariant. Subsequently,

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LgPs and VLQs that dont merge to form other entities are also invariant throughout all phases ofthe Dynamical SystemDC5 (until the Devolution of the Universe phase begins).

The Codex 5 Model correlates this persistence of invariant Basic Pairs, Large Pairs and VLQs -in the current universe - by proposing that:

71.43% of the Energy/Mass in the current universe consists of Basic Pairs (aka, DarkEnergy)

21.43% of the Energy/Mass in the current universe consists of Large Pairs (aka, LightDark Matter)

2.5% of the Energy/Mass in the current universe consists of Very Large Quads (aka,Heavy Dark Matter).

The remaining Energy/Mass starts as Very Large Octets (3.57 %) and XL-12s (1.07 %), twoentities that:

a) Initially evolved during Phases 3 and 4 of the Codexs Dynamical System, and

b) Eventually evolved into what today is baryonic matter (i.e., the Codex 5 Model proposesthat baryonic matter makes up 4.64 % of the total Energy/Mass in the universe). SeeTables 6-1 and 6-2 below for details.

Note that if an initial state of Phase n ofDC5 is defined as starting to exist well after the last BigBang (e.g., in 2009), it could be defined to include all entities existing in the world space (the

current instantiation of Set PR), including Dark Matter, Dark Energy, and all baryonic matter

(from Codex 5 entities to subatomic particles to atoms to molecules to stars).Only real physical phenomena that have resulted from the ongoing actions of the Codex 5

Models Dynamical SystemDC5 would be included in this instantiation ofPR. Other entities -such as imaginary numbers,10th dimensions, etc. would be considered counter-factual, i.e., not

lying on PR, nor have they ever lain on SetsAN-NNor C5. See Section 5.0 (Phase 1 of the

Dynamical SystemDC5) above for further discussion of the Models view of counter-factualentities and states.

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Table 6-1 - Energy / Mass Distribution in the Universe

Final Distribution

TypeofEntity

End of Phase 4:% of Total E/M ofuniverse

CurrentNomenclature

Throughout Phase 4, 71.43% of E/Mremains in Basic Pairs BPs 71.43% Dark Energy

Throughout Phase 4, 21.43% of E/Mremains in Large Pairs LgPs 21.43%

"Light" DarkMatter

VLQ (each has 4 Units of E/M) VLQs 2.50%"Heavy" DarkMatter

VLOct (each has 8 Units of E/M) VLOcts 3.57%

XL12 (each has 12 Units of E/M) XL12s 1.07%

Baryonic Matter

The Codex 5 Model also notes that these results align well with WMAP observations and

analysis. Table 6-2 below combines the Codex 5 Models predictions aligned with WMAPsobservations:

Table 6-2: Distribution of Energy / Matter Types in the Universe

Codex 5 Model:Codex 5Model: Variation:

>Gross Difference of Codex

vs. WMAP: "xx%"

Final Distribution

% of TotalE/M in each

type of EntityWMAP

CategoryWMAP %

distribution>% Diff (=Gross % / Codex%): "yy%"

Gross Diff: 0.57%

Dark Energy 71.43% Dark Energy 72.0% % Diff: 0.8%

"Light" DarkMatter 21.43%

"Heavy" DarkMatter 2.50%

Gross Diff: 0.93%

Total Dark Matter 23.93% Dark Matter 23.0% % Diff: 3.89%

Gross Diff: 0.04%

Baryonic matter 4.64%

Baryonic

matter 4.6% % Diff: 0.87%Total 100.00% 100%

To put these percentages in terms of quantities of the different types of entities:

Table 6-3 below shows that for every 1120 BPs that the Dynamical System DC5 begins

operations on, at the end of Phase 4, there will be 120 LgPs, 7 VLQs, 5 VLOcts and 1 XL-12.

After the end of Phase 4, the BPs, LgPs and VLQs no longer change (i.e., remain invariant),while - under the action ofDC5 - the VLOcts and XL-12s go on to form baryonic matter.

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Table 6-3: Distribution of Entities after Phase 4

Type of Entity Quantity of Entities at

End of Phase 4

Basic Pair 800Large Pair 120

VLQ 7

VLOct 5

XL-12 1

7.0 Asymptotic Behavior

The ISP includes references to how a state-space geometry, under the effects of a dynamical

system, will eventually asymptote. For example:

Lorenz (1963) proposed a very different type of chaotic motion, associated withforced nonlinear dissipative dynamical systems, = f(X). In contrast withHamiltonian systems, the states X(t) of such systems evolve asymptotically to

fractionally dimensioned (fractal) attractors. If X is initialized on an attractor, X

stays on it forever; the attractor is a dynamically invariant subset of state space.Fractal attractors reveal some of the most beguiling of geometries known to

physics, and form the basis of discussion in this paper. However, the dynamical

systems which generate these geometries are usually considered

phenomenological rather than fundamental, since they are explicitly dissipative.In 1b, an inherently relativistic reason for supposing these geometries to be

fundamental will be proposed.

It can therefore be supposed that volumes V(t) will shrink to zero and asymptoteonto one of the following: a fixed point, a limit cycle or a fractal attractor. Since

fractal attractors are generic, we will assume henceforth that the zero-volume

asymptotic limit is a fractal set. Clearly it will take an infinite time for a genericvolume in the state space of the Hawking Box to evolve precisely onto its

invariant set.

In summary, the Codex 5 Model describes a dynamical systemDC5 that over billions of years(from one Big Bang to the next) evolves to a fractal attractor. Based on the Sections above,

Absolute Nil and Non-Nil (which can exist only as a Basic Pairs) can be considered fundamental

(not phenomenological) fractal attractors, because the Codex 5 Model proposes:

1) That they are the most fundamental entities in the physical world (ref the set PR, Table 3-1), and

2) That the entire universe will eventually asymptote to a state space geometry where itconsists of only Basic Pairs, which are the smallest fractals (i.e., have the least

Energy/Mass) in the set of all fractals described in the Codex 5 Model.

It appears that these characteristics of the Model are analogous to the type of dynamical systems

the ISP refers to above, i.e.:

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In contrast with Hamiltonian systems, the states X(t) of such systems evolveasymptotically to fractionally dimensioned (fractal) attractors. If X is initialized

on an attractor, X stays on it forever; the attractor is a dynamically invariant

subset of state space.In the Codex 5 Model, the dynamical system asymptotes to a dynamically invariant subset of

state space, namely, a state-space geometry consisting only of Absolute Nil and Non-Nil in theform of Basic Pairs. See Figure 7 below (from Codex 5 paper, Appendix 1, Section 10):

x/a

y/b

The LastLarge Pair

x

y

b

a

x

y

b

a

x

y

b

a

Singularityis here

Force of surrounding ocean of

Basic Pairs starts to pull the Last

Large Pair apart.

10 Where the Singularity Came From

Hypothesis: The Singularity where theBig Bang occurs is at the spot where theLast Large Pair unwinds into 2 Basic

Pairs.

Correlation:

...the farther back the fluctuationhappened, the lower the entropy it wouldhave had to attain (entropy starts to rise

after any dip to low entropya small

fluctuation early ona modest jump to

the favorable conditions, within a tinynugget of spaceinevitably yields thehuge and ordered universe we are

aware of.

B. Green, The Fabric of the Cosmos.

Figure 7: Codex 5 System Asymptotes back to Phase 0: All Basic Pairs

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Though the Codex 5 environment is different from what is mentioned in the reference above, theresult is similar: In both cases, the systems asymptote to a fractal attractor. Also, as in the ISP,

the Codex 5 Model proposes that the asymptotic evolution to a fractal attractor (Ab-Nil and Non-

Nil as BPs, in the case of the Codex 5 Model) is fundamental, not phenomenological.

In summary, in the Model, the entire universe will asymptote to a physical reality consisting onlyof Basic Pairs, i.e., to the most basic fractals described in the Codex 5 Model. Note that whenthe universe consists only of BPs, it has asymptoted down to its most fundamental entities,

Absolute Nil and Non-Nil, which can exist only as BPs. Thus the Dynamical SystemDC5 has

returned state-space geometry to the Initial Set, E0. (see Section 5.0 above).

8.0 Gravity

Both the ISP and the Codex 5 Model present views that gravity is related to, or a manifestation

of, variations / heterogeneities) in the geometry of state space.

8.1 Gravity: ISP and Codex 5 Model

Palmers Invariant Set Postulate paper makes several references to gravity, such as:

It is proposed that heterogeneities in the fractal geometry ofIare manifestationsof the phenomenon of gravity.

Indeed this leads to the following rather radical suggestion. If the geometry ofIisto be considered primitive, then the geometric properties of the invariant set

which lead to certain regions being relatively stable and other regions unstable

should be considered a generalization of the notion introduced by Einstein that thephenomenon we call gravity is merely a manifestation of some more primitive

notion of geometryhere the geometry of a dynamically invariant subset of statespace.

if gravity should be seen as a manifestation of the heterogeneity in the

geometry of the invariant set, then attempts to quantize gravity with theframework of standard quantum theory will also fail. As such, it is misguided to

assume (as almost all serious attempts have so far done) that theories of

everything can be formulated within conventional quantum theory.

As indicated in Tables 6-1, 6-2 and 6-3, the Codex 5 Model addresses gravity by proposing that:

Dark Energy consists of Basic Pairs; Dark Matter is made up of Large Pairs and Very Large Quads;

Baryonic matter evolves from VLOcts and XL-12s.

[To use some terminology from the Invariant Set Postulate, The Codex 5 Model views gravity as

a manifestation of heterogeneities in the geometry of instantiations of:

1) the invariant set AN-NN (which the paper proposes as fundamental to physical reality),and

2) the invariant subset of C5 that consists of the BPs, LgPs and VLQs that never transform(see).

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Note also that the fractal nature of C5 equates to variations in the E/M of the sets entities (BPs,

LgPs, VLQs, VLOcts and XL-12s).]

In summary, Basic Pairs are Dark Energy and have the gravitational properties observed in DarkEnergy. Large Pairs and VLQs are Dark Matter and have the gravitational properties observed in

Dark Matter.

As previously noted, BPs, LgPs and VLQs are all part of the fractal invariant set C5 of the Codex

5 Model. The differences between these entities (e.g., BP vs. LgP vs. VLQ) and the variations

in their distribution throughout the universe correlate to heterogeneities in the state space

described in the Codex 5 Model, and likewise, correlate to the uneven distribution of DarkMatter and Dark Energy gravity in the universe. Thus, the Codex 5 geometry aligns with the

ISPs reference to Einsteins speculation that the phenomenon we call gravity is merely a

manifestation of some more primitive notion of geometryhere the geometry of a dynamically

invariant subset of state space.

Taking the insights of the Invariant Set Postulate, and seeing that the Codex 5 Model describesinvariant fractal subsets and geometry of state space (in the form of BPs, LgPs, etc.), another

view of gravity might emerge:

On a macro level (planets, asteroids, stars, etc.), gravity can be seen as curvature (i.e., hascharacteristics that can be described using geometric terms) of the space-time fabric. On the

micro-level of BPs, LgPs and VLQs, gravity can be viewed as a fundamental, geometric aspect

of the state-space, as seen in variations in:1) the distribution of BPs, LgPs and VLQs, and2) the differences in Energy/Mass of these different entities.

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Figure 9: Homogenous vs. Heterogeneous Geometries in State-Space

In Figure 9, some entity (e.g., a Basic Pair, an electron) is shown passing through the following:

1) Phase 1, Initial State, is shown as a fairly homogenous field of 56 BPs. Since the field ishomogenous (BPs distributed more or less evenly within the indicated subset of state

space), the path of the moving entity is fairly straight. [Note that the term fairly is

admittedly far from ideal. As mentioned above Section 4.0, further work in this area is in

progress.]

Note that the BP distributions relative homogeneity may align with the ISPs reference

to regions of relative stability (small local Lyapunov exponents).

2) Phase 4, Next State, is diagrammed as a field that has many heterogeneities, due to thepresence of many different types of entities (BPs, LgPs, etc.), each type with a different

level of Energy/Mass. These heterogeneities cause the path of the moving entity to varymore than when it passes though the more or less homogenous field that consists only ofBPs.

As a converse corollary to the note in Item 1 immediately above, these regions of higher

heterogeneity may align with the ISPs reference to regions ofrelative instability(large local Lyapunov exponents).

As noted above, the ISP states:

If the geometry ofIis to be considered primitive, then the geometric properties ofthe invariant set which lead to certain regions being relatively stable and other

regions unstable should be considered a generalization of the notion introduced by

Einstein that the phenomenon we call gravity is merely a manifestation of somemore primitive notion of geometryhere the geometry of a dynamically invariant

subset of state space. As such, a challenge for the future will be to try to unify thenotions of pseudo-Riemannian geometry for spacetime, and fractal geometry for

state space. This is a very different perspective on quantum gravity than

suggested by any existing approaches to the subject. [italics added]

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The Codex 5 Model has some implications regarding unify[ing] the notions of pseudo-Riemannian geometry for spacetime, and fractal geometry for state space. As noted elsewhere

in this paper, the entities described in the Codex 5 Model exist in space-time (i.e., are not just

abstract concepts, do not require 6 or 8 or 10 dimensions, etc.). For example, the Modelproposes that BPs make up Dark Energy; that LgPs and VLQs make up Dark Matter; and that

VLOcts and XL-12s are key to the ultimate emergence of baryonic matter. Furthermore, theseentities play a key role in how the last Big Bang came about, and in the ongoing and futureevolution of the universe, including the eventual path to the next Big Bang.

Concurrently, the Model describes a comprehensive fractal geometry for state space, and a

dynamical system that affects the elements of this geometry.

8.2 Gravity and Dark Matter: Other Observations

Note also that observations have indicated that Dark Matter clusters around galaxies. For

example, Canada-France-Hawaii (CFH) Telescope analyses of Dark Matter clustering state:

1) Figure 2, Deflection of light rays crossing the universe, emitted by distant galaxies: Thedark matter is concentrated into a web-like distribution of filaments that intersect at densenodes where great clusters of galaxies are expected to form and become visible.

2) Figure 3, Image of the distant galaxies lensed by the dark matter of the universe: Bymeasuring the systematic distortion in the images of distant galaxies, one can see the

dark matter.

However, observations of Abell 520 (Mahdavi, et al., 2007) show areas where Dark Matter existsbut is not clustered around galaxies:

We consider possible mechanisms for separating the dark matter from the

galaxies. Two possibilities stand out: (a) the galaxies originally in the dark core

could have been ejected through a multiple-body interaction within the merging

system; or (b) allowing for weakly self-interacting dark matter, the dark peak wasdeposited as a result of dark matter collisions during the merger impact;

From the point of view of the Codex 5 Model, further work is ongoing in this area. The initial

working hypotheses are that Dark Matter, in the form of LgPs and VLQs:

1) Does in fact cluster around galaxies.2) Can also exist apart from galaxies, such that Dark Matter entities (LgPs and VLQs) can

and do interact amongst themselves, as indicated in the Codex 5 paper. This supports

Item b (above) from the Abell 520 paper, i.e., allowing for weakly self-interacting darkmatter.

9.0 Quantum Theory

Section 6 of Palmers Invariant Set Postulate (ISP) paper addresses how the precepts of the ISP

impact Quantum Theory. The Codex 5 Model also has implications for Quantum Theory andphysics, but these will be addressed in a future paper.

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For example, the ISP states:In the context of the invariant set postulatey, the superposed quantum state

| A + | B has no fundamental ontological significance; it indeeddescribes a probability of ignorance, here ignorance of the intricate structureof the invariant set based on a sample space of trajectory segments in some

neighbourhood on the invariant set. From this perspective, Schrdingers catis alive or dead, and not both.

As in the ISP, the Codex 5 Model postulates invariant fractal sets and, in fact, aspects of physical

reality (ref the setsAN-NNand C5) that Quantum physics is ignorant of. For example, BPs,

LgPs and the other Codex 5 entities are never mentioned in Quantum physics. Given theproperties of the Codex 5 Models invariant set (BPs, LgPs, et al.), superposition has no meaning

from the Models point of view; as with the ISP, the cat is not both alive and dead it must be

one or the other only.

10.0 ConclusionsIn summary, this paper has presented a summary of the Codex 5 Model and shown its

correlations with Palmers Invariant Set Postulate, specifically in areas such as: Fractal state-space geometry. Dynamical systems theory, with reference to and description of a dynamical systems

deterministic, causal effects on a state-space geometry. Existence of invariant sets that describe physical reality. Description of certain types of asymptotic behavior. View that gravity is related to / a manifestation of heterogeneities in the geometry of state

space.

Quantum Theory is only briefly mentioned, but is the subject of ongoing work.

Exploration of the Codex 5 Model will continue, as it already provides potentially usefulcorrelations with WMAP observations and aligns with many aspects of the Invariant Set

Postulate. During the next few years, it is anticipated that:

1) WMAP and other surveys will make further observations and attendant analyses, whichwill be considered in any future papers.

2) As the ISP paper states, Future papers will attempt to provide the mathematical detailrequired to develop this exploratory analysis into a rigorous physical theory.

As new data, studies, analyses and observations emerge, their relationships to the Codex 5 Model

will be considered and will open up new areas for analysis and exploration.

References

Andisheh Mahdavi, Henk Hoekstra, Arif Babul, David D. Balam, Peter L. Capak, A DARKCORE IN ABELL 520. Accepted June 18, 2007 for publication in The Astrophysical Journal.

Hoffecker, F.S. The Codex 5 Model: Describing the Origin & Nature of Dark Energy & Dark

Matter and Predicting the Distribution of Energy / Matter in the Universe. January 2009.

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E. Komatsu, J. Dunkley, M. R. Nolta, C. L. Bennett, B. Gold, G. Hinshaw, N. Jarosik, D. Larson,M. Limon, L. Page, D. N. Spergel, M. Halpern, R. S. Hill, A. Kogut, S. S. Meyer, G. S. Tucker, J.

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