A Model for Creation: Part I Paul Bernard White1 March 4, 2019 (Updated: March 12, 2019)2 Abstract Four initial postulates are presented (with two more added later), which state that construction of the physical universe proceeds from a sequence of discrete steps or "projections" --a process that yields a sequence of discrete levels (labeled 0, 1, 2, 3, 4). At or above level 2 the model yields a (3+1)-dimensional structure, which is interpreted as ordinary space and time. As a result, time does not exist below level 2 of the system, and thus the quantum of action, h, which depends on time (since its unit is time•energy), also does not exist below level 2. This implies that the quantum of action is not fundamental, and thus e.g. that the physical universe cannot have originated from a quantum fluctuation. When the gravitational interaction for the model is developed, it is seen that the basic ingredient for gravity is already operating at level 1 of the system, which implies that gravity, too, is not fundamentally quantum mechanical (since, as stated, h only kicks in at level 2) --perhaps obviating the need for a quantum theory of gravity. Further arguments along this line lead to the conclusion that quantum fluctuations cannot be a source of gravity, and thus cannot contribute to the cosmological constant --thereby averting the cosmological constant problem. Along the way, the model also provides explanations for dark energy, the beginning and ending of inflation, quark confinement, and more. Although the model dethrones the quantum, it nevertheless elevates an idea in physics that was engendered by quantum mechanics: the necessary role of "observers" in constructing the world. Contents 1 Introduction....................................................................................................................3 2 Levels, projections, and relations: the structure and basic properties of system P. 4 2.1 Some properties of system P....................................................................................5 3 Constructing spaces in system P...................................................................................7 3.1 Constructing a (3+1)-dimensional structure at level 2.............................................7 3.2 The spaces of levels 3 and 4.....................................................................................9 3.3 The number of dimensions at levels 2, 3, and 4.....................................................10 3.4 Isotropy and homogeneity of ordinary space.........................................................11 3.5 Rapid expansion of ordinary space within the first instant of time........................12 1 pbwx@att.net / ORCID: 0000-0002-2681-3670 (BSc Physics) 2 There may be a newer/updated version of this paper at: http://www.paulbwhite.org/papers/ A much shorter /abridged version of this paper is: A Derivation of Space and Time (10 pages) 1 4 Constructing particles in system P.............................................................................12 4.1 Relations between energies....................................................................................13 4.2 A quantum of action...............................................................................................14 4.3 Energy distribution within ordinary space.............................................................15 4.3.1 Large-scale distribution of energy within ordinary space..............................15 4.3.2 Small-scale distribution of E≥2p energy...........................................................16 4.3.3 Further aspects of E1p energy..........................................................................16 4.3.4 Further aspects of E3p energy distribution......................................................17 4.3.5 Substructure at level 4....................................................................................18 4.4 Some particles of system P....................................................................................18 4.4.1 Associating particle properties with projections.............................................19 4.4.2 Partitioning of properties/charges within level-3 objects...............................21 4.5 Analogy of system P with biological genetics........................................................21 4.6 Real and virtual particles........................................................................................21 4.7 Fundamental interactions.......................................................................................22 4.7.1 The electromagnetic interaction.....................................................................23 4.7.2 The strong interaction.....................................................................................24 4.7.3 The weak interaction, and electroweak unification........................................25 4.7.4 The gravitational interaction...........................................................................26 4.7.5 Other possible fundamental interactions........................................................28 4.8 Interaction examples..............................................................................................28 4.8.1 Electromagnetic..............................................................................................29 4.8.2 Weak...............................................................................................................29 4.8.3 Strong.............................................................................................................30 4.8.4 Gravitational...................................................................................................30 4.9 Matter-antimatter asymmetry.................................................................................30 4.9.1 Primary and secondary projections................................................................32 4.10 Postulates 5 and 6.................................................................................................34 4.11 Constructing position, velocity, acceleration, and spin properties for objects/particles in system P................................................................................34 4.12 Revisiting mass....................................................................................................39 4.13 Revisiting gravity.................................................................................................39 4.14 Backward projection onto level 0........................................................................41 5 The physical universe is a meaning circuit................................................................42 6 The construction of the physical universe is a type of logical derivation...............43 6.1 Systems without postulate 3...................................................................................44 7 Making sense of the projection types.........................................................................45 8 Conclusion....................................................................................................................45 Acknowledgments........................................................................................................46 References.....................................................................................................................46 2 1 Introduction Systems that are based on information typically contain a basic information element and a basic information structure. In Biological systems, for example, the basic information element is the nucleotide molecule, and the basic information structure is a sequence of nucleotides (e.g. a codon, or a gene). Likewise, for computer systems the basic information element is the bit, and the basic information structure is a sequence of bits (e.g. an 8-bit byte). And in natural language the basic information element is the letter or phoneme, and the basic information structure is a sequence of letters or phonemes (e.g. a word or a sentence). Such systems must also have a way of translating or computing the information elements and structures into meaningful output. In biology this is accomplished by the operations of ribosomes, enzymes, etc., acting on the nucleotide strings. For computers, the operations of logic gates on the bit strings typically perform this function. And in natural language the operations of lexical analysis, parsing, and context translate a string of letters/phonemes into meaning. Likewise, if the physical universe is based on information (as many have speculated, e.g. [1], [2], [3]), then the following questions arise: (a) What is the basic information element for this system?; (b) what is the basic information structure for the system?; and (c) how are these elements and structures translated (or computed) into the meaningful output that we call the physical universe? In answer to questions (a) and (b) above, I propose the following two postulates: 1. For creation of the physical universe, the basic information element is a type of projection --more specifically, a projection from a prior level. 2. The basic information structure is a sequence of such projections. With respect to the first postulate, we may refer to both projections and levels as "elements" (or basic elements) of the system, but will reserve the term "basic information element" for the projections alone. We now add two more postulates: 3. Each such projection is a one-dimensional vector, constituting a different, but related, one-dimensional space. (The basic relations between these projections/vectors are stated in the next postulate.) 4. Prior things (e.g. projections, levels, and constructions from them) are independent of subsequent things; and, conversely, subsequent things are dependent on prior things. (The terms prior, subsequent, dependent, and independent denote here logical/ontological relations. See e.g. [4].) Using these four postulates (and two more that will be stated later), we develop a model for the basic construction of the physical universe --including the construction of ordinary space and time themselves, a quantum of action, fundamental particles and 3 interactions, inflation, dark energy, the cosmological constant, etc. As development of the model progresses, the essential role of "observers" in the construction process becomes more and more clear. With respect to question (c) above, it will be shown that a method for translating sequences of projections into physical meaning is by taking into account the relations between projections --specifically, their dependence and independence relations (i.e. postulate 4). In particular, such relations will allow us to derive a (3+1)-dimensional structure which, in the context of the model, is best interpreted as the ordinary space and time dimensions of our experience. From now on, I will often refer to the model for constructing the physical universe, developed herein, as system P, and the world so constructed from it as world P. 2 Levels, projections, and relations: the structure and basic properties of system P To construct our model for the physical universe (i.e. system P), we must begin with a state at which the things of the universe do not exist (otherwise our construction would be circular), i.e. a state that is absent the energy, elementary particles, and even space and time, as we know them. We will call this state level 0 of system P (or "losp", which comes from "level 0 of system P", where we pronounce the zero as the letter 'o'); or, when it is clear that we are talking about system P, simply level 0. We do not, however, presume that level 0 is a state of nothingness, or that nothing exists at level 0. We merely claim that nothing that comes into being with the construction of the physical universe exists at level 0; for level 0 is by definition a state that is immediately prior to the construction of the physical universe. Recalling our first three postulates, we say that a projection from level 0, to be denoted as p0, generates a new state, which we call level 1. Likewise, a projection from level 1, denoted as p1, generates another new state, which we call level 2. And a projection from level 2, denoted as p2, yields level 3; and so on. (Note that the choice of the letter "p" in our denotation here refers not to "projection" per se, but to system P; later, especially in Part II [5], the sequel to the present paper, we will use different letters to denote projection in different systems.) So, in general, the projection pk represents a sort of displacement from level k that generates level k + 1 (for k = 0, 1, 2, ...); thus, relative to each other, level k is prior, and level k + 1 is subsequent; also, relative to each other, pk is prior, and pk+1 is subsequent. (Again, the terms "prior" and "subsequent" refer to logical/ontological priority and subsequence.) In Fig. 1, where levels are represented by horizontal lines, and projections are represented by vertical arrows from a prior level to the next subsequent level, we illustrate the construction of levels 1 through 4 via projections p0 through p3. To the right of each level in Fig. 1 is shown the sequence of projections that is required to construct that level (the round brackets indicate a sequence, as is common in mathematics). Thus, the sequences of projections that are needed to create levels 0, 1, 2, and 3 are ( ), (p0), (p0, p1), and (p0, p1, p2), respectively; and the sequence (p0, p1, p2, p3) constructs all of the 4 levels (above level 0) in Fig. 1. Fig. 1 Construction of levels 1 through 4 of system P via the projection sequence (p0, p1, p2, p3). The projection sequence that is required to construct a given level is shown to the right of that level. As just described, the order of construction in system P starts with level 0 at the bottom of Fig. 1 and proceeds in the upward direction. Thus, level 0 is prior to all other elements (levels or projections) in system P, and subsequent to none; p0 is subsequent to level 0, but prior to level 1, p1, level 2, etc.; and so on. So, in general, a given element x in system P is subsequent to everything below it in Fig. 1, but prior to everything above it. By postulate 4, this means that element x is dependent on everything below it in the Figure, but independent of everything above it. Thus, for example, level 0 is independent of all other elements in system P, and dependent on none. Since level 0 is our starting point (or starting state) for constructing system P, then we must say that it is a nonconstructed element of that system, whereas the subsequent projections and levels (p0, level 1, p1, level 2, etc.) are constructed elements of system P. So anything subsequent to level 0 is a constructed entity of the system. 2.1 Some properties of system P Let x be a thing of system P (e.g. x is a level, a set of one or more projections, or something constructed from them). By postulate 4, things that are subsequent to x are (logically/ontologically) dependent on x. Such dependence implies that x is in effect, effective, operative, or operant at those subsequent things; or, alternatively, we say that those subsequent/dependent things are within the scope of x. Conversely, since things that are prior to x are independent of it, we say that x is not in effect or operant at those prior things; or, alternatively, we say that those prior/independent things are not within the scope of x. All of this is summarized in what will be called the scope rule for system P, stated as follows: 5 p 1 p 0 p 2 p 3 0 1 2 3 4 ( ) (p 0 ) (p 0 , p 1 ) (p 0 , p 1 , p 2 ) (p 0 , p 1 , p 2 , p 3 ) A given thing in system P is in effect/operant at (i.e. contains within its scope) those things which are subsequent, and is not in effect at (does not contain within its scope) those things which are prior. From this we may deduce the following corollary to the scope rule: A given element in system P (i.e. a projection or level) is in effect/operant at (contains within its scope) those elements that are above it in Fig. 1, and is not in effect at (does not contain within its scope) those elements that are below it in Fig. 1. Thus, for example, since all of the constructed elements of system P (i.e. p0, level 1, p1, level 2, etc.) are subsequent to level 0 (or, conversely, level 0 is prior to them), then level 0 is in effect/operant at all of those things; or, all of those things are within the scope of level 0. Likewise, p1, level 2, p2, level 3, etc., are within the scope of level 1; but level 0 is not within the scope of level 1. And so on. Since pk is not in effect at level k, but is in effect at level k + 1, then level k + 1 represents the state at which the projection pk first comes into effect; by the scope rule, pk then stays in effect for all subsequent levels. Thus, the projection p0 first comes into effect at level 1, and stays in effect for levels 2, 3, and 4; likewise, p1 first comes into effect at level 2, and stays in effect for levels 3 and 4. Let us say that the level at which a projection first comes into effect is its native level. Thus, level 1 is the native level for p0; level 2 is the native level for p1; and so on. That is, the native level for pk is level k + 1. Moreover, the concept of native level can be extended to things that are constructed from projections; thus, for example, something that is constructed using p0 and p1 (and no other projections) is native to level 2, since those two projections are first jointly in effect at that level. We note also that the projections that are in effect/operant at a given level are the same as the ones that are required to construct that level (as described earlier, and as listed in the sequences to the right of each level in Fig. 1). In constructing the sequence of projections (p0, p1, p2, p3), since any projections that are in effect at level k are also in effect at the subsequent level k + 1, then we can think of the latter level as inheriting all of the projections that are in effect at the former level. And since this is true of projections, then it is also true of anything that is associated with or constructed from them. This aspect of system P --whereby that which is in effect at one level (or, if you will, generation) is passed on to the next subsequent level (and thus, by extension, to all subsequent levels) --will be called the inheritance rule. As described above, the scope of a given element contains the scope of the next subsequent element, and is contained (or nested) within the scope of its immediately prior element. Thus, the scopes of the sequence of elements in system P can be pictured as a (matryoshka-doll-like) structure of concentric spheres; the outer sphere being the scope of level 0, the first inner sphere being the scope of p0, the next inner sphere being the scope of level 1, and so on. Again, since this applies to elements of system P, it also applies to things that are associated with or constructed from them. This leads to the following general statement, which will be referred to as the nesting rule for system P: 6 Subsequent things (e.g. levels, projections, and any constructions derived from them) are internal to or nested (and thus embedded, contained, encapsulated, confined) within the scope of prior things; and, conversely, prior things contain, encapsulate and confine (within their scope) subsequent things. Thus the nesting rule may also be called the confinement rule. Since the sequence of projections, (p0, p1, p2, p3), that constructs system P springs out of level 0 of that system, then we can say that level 0 (or losp) is the origin of system P. And since, by the nesting/confinement rule, all of the constructed elements/entities of that system are nested/confined within level 0, then we can also say that level 0 is the boundary of system P. Thus, level 0 is the origin and boundary of system P; so, in that sense, although level 0 is a necessary part of the ontology of system P, it is not technically within system/world P. Finally, when a set of things in system P (i.e. elements, or things constructed from them) is arranged from prior to subsequent, then we will say that the things are arranged in their order of priority. If a thing is neither prior nor subsequent to another thing, then we say that they are of the same order, or coordinate. 3 Constructing spaces in system P Following postulate 3, let us model each projection as a one-dimensional vector; i.e. we model each pk (k = 0, 1, 2, ...) as a one-dimensional vector going from level k to level k + 1. Thus, p0 is a one-dimensional vector from level 0 to level 1; p1 is a onedimensional vector from level 1 to level 2; and so on. These vectors are represented graphically by the vertical arrows in Fig. 1. Moreover, each pk constitutes a different one-dimensional space. Though they are different in this respect, the pk are nevertheless related by the dependence and independence relations that have been postulated and discussed. 3.1 Constructing a (3+1)-dimensional structure at level 2 Since p0 is the only projection in effect at level 1, and since (by postulate 3) it is one dimensional, then it is fair to say that system P is one dimensional at level 1. Since both p0 and p1 are in effect at level 2, and since (by postulate 3) each of these constitutes a different one-dimensional space, then it might seem --at first glance --that system P should be two dimensional at level 2. But this would be wrong. To get the correct dimensionality at level 2, we must take into account the relations between p0 and p1, as per postulate 4 --i.e. the fact that p0 is independent of p1, and that this relation is asymmetric (p1 is dependent on p0). Since p0 and p1 are vectors, we interpret that these relations imply a kind of (asymmetric) linear independence, with the following property: from the perspective of p1, the vector p0 may be collinear with p1, but 7 is also free to be noncollinear with p1. With these considerations in mind, we ask the question: What is the direction of p0 with respect to p1? Or, in other words, how does p0 "look" relative to p1? Since p0 may be both collinear and noncollinear with p1 (from the latter's perspective), then p0 may have a component parallel to p1, and may also have a component perpendicular/orthogonal (i.e. at 90 degrees) to p1. But, by symmetry, the perpendicular component can be anywhere in a two-dimensional plane orthogonal to p1. The two dimensions of this orthogonal plane, plus the one dimension parallel to p1, makes three dimensions. Thus, from the viewpoint of p1 (and from the perspective of level 2), p0 has three dimensions; i.e. p0 constitutes a three-dimensional space (whereas, recall that p0 has only one dimension at level 1). We might say, therefore, that the view of p0 from the perspective of p1 "bootstraps" the former from a one-dimensional vector into a threedimensional space. In summary, to construct its interpretation of p0, we can think of p1 as applying postulates 3 and 4 in succession: first, by postulate 3, p0 is a one-dimensional vector; second, by postulate 4, p0 is independent of p1 --which allows the former to have a component that is orthogonal to p1, with the result that p1 sees p0 as three dimensional. Conversely, we can ask, how does p1 "look" relative to p0? Since p1 is dependent on p0, then the former is not free to have a component that is orthogonal to the latter, and so p0 sees p1 as being collinear; or, more simply, p0 sees p1 strictly as per postulate 3: as a one-dimensional vector. So, at level 2 we have the three dimensions of p0, plus the one dimension of p1, for a total of four dimensions. Since system P is a model for constructing the physical universe, we interpret that the three dimensions of p0 are just the three dimensions of ordinary space, and the one dimension of p1 is the dimension of time; thereby yielding at level 2 the signature 3+1 space and time dimensions of our experience. The dimension of time, therefore, being a consequence of p1 (and p0), does not exist at levels 0 and 1, but only comes into existence at level 2; likewise, since ordinary, three-dimensional space is a consequence of p0 and p1, it also does not exist at levels 0 and 1, but only comes into existence at level 2. Note that, although p0 itself is independent of p1, the triple dimensionality of p0 at level 2 is not independent of p1. That is, in the process described above, p0 only manifests as three dimensional when it is related to, or juxtaposed with, p1. Thus, the triple dimensionality of p0 at level 2 (i.e. the triple dimensionality of ordinary space) is in fact dependent on p1. Conversely, both p0 and p1 are prior to, and thus independent of, ordinary space. We have shown, among other things, that p0 manifests differently at levels 1 and 2. At level 1 it is one dimensional. But when juxtaposed with p1 at level 2 it manifests as a three-dimensional space. Note that p0 itself does not change from level to level: it represents a projection from level 0 to level 1 wherever it appears (i.e. wherever it is in effect). This is analogous to e.g. the G nucleotide in biology, which is always the same 8 molecule wherever it appears, but yields a different output (i.e. amino acid) depending on what other nucleotides/letters it is juxtaposed with in a sequence. In other words, like the letter G in a DNA sequence, the meaning of p0 is context dependent; which is just what we might expect for an element of a language, thus supporting our earlier notion that the basis of the physical universe is, to some degree at least, informational in nature. We might say that level 2 has two dimensions as input (one dimension for p0, plus one for p1), but has four dimensions as output --three for p0, and one for p1. Which brings us back to question (c) in the introduction: How are the basic information elements of the model (which at level 2 are the inputs p0 and p1) translated (or, if you will, computed) into the meaningful output that we call the physical universe? We now see that at least a partial answer is that the relations between prior and subsequent elements are what translate them into meaningful output. In the present case, the independence relation between p0 and p1 at level 2 translates/transforms the manifestation of the former from a one-dimensional entity into a three-dimensional space. We can thus say that the construction of each space at level 2 requires the participation of an observer, in the sense that p1 "observing" p0 constructs ordinary, three-dimensional space, and p0 "observing" p1 constructs one-dimensional time. With ordinary space itself constructed by an observation of sorts, it becomes more plausible that e.g. the position of an object within ordinary space might also be constructed by some type of observation, as seems to be the case in quantum mechanics (more about that in section 4.11). We have just described how the spaces and dimensionalities at level 2 are constructed. Now we will do the same for levels 3 and 4. 3.2 The spaces of levels 3 and 4 At level 3, the projections p0, p1, and p2 are in effect. The relations between p0 and p1 at level 3 are the same as they are at level 2 (i.e. p0 is independent of p1, but not the converse). Thus, at level 3 --as at level 2 --p0 will appear to p1 as a three-dimensional space (i.e. ordinary space), and p1 will appear to p0 as a onedimensional space (i.e. time). In other words, the spaces that exist at level 2 also exist at level 3. Indeed, as per the inheritance rule, we might say that level 3 inherits these spaces from level 2; or, more precisely, level 3 inherits p0, p1, and the relations between them from level 2, and uses them to construct ordinary space and time. Let us denote ordinary, three-dimensional space as S01p 3 , where the superscript indicates the number of dimensions; the 0 followed by 1 in the subscript indicates that this is the space of p0 as seen by p1; and the "p" in the subscript reminds us that we are talking about a space in system P. Likewise, let us denote time as S10p 1 , where, again, the superscript indicates the dimension of this space; and the 1 followed by 0 in the subscript indicates that this is the space of p1 as seen by p0. Focusing again on level 3, we start by discussing the relations between p0 and p2 at 9 that level. The former is independent of the latter, so p0 is three dimensional from the viewpoint of p2. We thus have the generation of a new three-dimensional space at level 3, which we denote as S02p 3 . Likewise, p1 is independent of p2; so the latter will see p1 as three dimensional (whereas, as already stated, p1 has only one dimension from the viewpoint of p0). This yields yet another three-dimensional space at level 3, which we denote as S12p 3 . In addition, at level 3 we also have the one-dimensional space, S20p 1 , of p2 as seen by p0; and the one-dimensional space, S21p 1 , of p2 as seen by p1. We now ask, where are these spaces located relative to each other? The three-dimensional spaces at level 3 have the order of priority S01p 3 , S02p 3 , S12p 3 ; 3 so, by the nesting rule, S12p 3 is nested/contained/confined within S02p 3 , which in turn is nested/confined within S01p 3 (ordinary space); or, to put it another way, S01p 3 (ordinary space) contains S02p 3 , which in turn contains S12p 3 . The one-dimensional spaces, S20p 1 and S21p 1 , being modes of p2, are nested/contained within S02p 3 , and possibly also within S12p 3 . Now let us move up to level 4, where the projections p0, p1, and p2 are still in effect. So the spaces of level 3 also exist at level 4. In addition, the projection p3 is in effect at level 4, so the relationship between it and the prior projections px (x = 0, 1, 2) generates three new, three-dimensional spaces, which we denote collectively as Sx3p 3 ; and we note that these spaces will be internal to, or nested within, S12p 3 . Lastly, there are the three, onedimensional spaces of: p3 as seen by p0, denoted as S30p 1 ; p3 as seen by p1, denoted as S31p 1 ; and p3 as seen by p2, denoted as S32p 1 -which may be denoted collectively as S3yp 1 (y = 0, 1, 2).4 3.3 The number of dimensions at levels 2, 3, and 4 At level 2 the four dimensions of S01p 3 and S10p 1 are in effect. At level 3, the same four dimensions at level 2 are in effect, plus the eight additional dimensions of S02p 3 , S12p 3 , S20p 1 , and S21p 1 , which makes a total of twelve dimensions at level 3. And at level 4, the same twelve dimensions at level 3 are in effect, plus the 12 additional dimensions of Sx3p 3 (x = 0, 1, 2) and S3yp 1 (y = 0, 1, 2), for a total of 24 dimensions at level 4. The union of spaces at levels 2 and 3 is the twelve-dimensional set { S10p 1 , S01p 3 , S02p 3 , S12p 3 , S20p 1 , S21p 1 }, 3 Obviously, S01p 3 is prior to the other two, since it is native to level 2, whereas the others are native to level 3. S02p 3 is prior to S12p 3 since the former is a manifestation, or mode, of p0, whereas the latter is a mode of p1 (and the former projection is prior to the latter). 4 Note that, for a space Sxyp z , the superscript, z, is actually redundant information, since it is always equal to 3 if x < y, and is equal to 1 if x > y. 10 and the union of spaces at levels 2, 3, and 4 is the twenty-four-dimensional set { S10p 1 , S01p 3 , S02p 3 , S12p 3 , S20p 1 , S21p 1 , Sx3p 3 S3yp 1 }. And we note that the 10 "extra" spaces (or 20 extra dimensions) of these sets (i.e. the spaces/dimensions that are native to levels 3 and 4) are nested/confined --and, indeed, as will be concluded in section 4.3.4, compacted --within ordinary space, S01p 3 . The dimensionalities of system P thus bear some similarity to dimensionalities in string theory [6]. Does this suggest that the projections of the present model have some relation to strings? Possibly --but we will not specifically expand on the string idea any further in this paper. As alluded to, more will be said about the nested spaces of levels 3 and 4 in section 4, where we construct the particles of system P. 3.4 Isotropy and homogeneity of ordinary space Recall that S01p 3 (ordinary, three-dimensional space) is created when p0 is viewed from the perspective of p1. So it follows that (a) the creation/construction of S01p 3 is dependent on p0 and p1; and (b) p0 and p1 are prior to, and thus (by postulate 4) independent of, S01p 3 . Suppose now that an outcome of constructing S01p 3 is that p0 (or p1) manifests with a particular orientation or direction within that space. Since this would make p0 (or p1) functionally dependent on S01p 3 , and thus contradict (b) above, we conclude that the construction of S01p 3 (ordinary space) cannot result in p0 (or p1) having a particular direction/orientation within that space. Presumably, then, there is no way for the process that constructs S01p 3 to establish a distinctive (i.e. special, preferred, or favored) direction within that space. We thus conclude that, as constructed above, ordinary space is perfectly isotropic. Now suppose that an outcome of constructing S01p 3 is that p0 (or p1) manifests with a particular position within that space. This, again, would make p0 (or p1) functionally dependent on S01p 3 , and thereby contradict (b) above; and so we conclude that the construction of S01p 3 cannot result in p0 (or p1) having a particular position within that space. Presumably, then, the process that constructs S01p 3 cannot establish a distinctive (i.e. special, preferred, or favored) position within that space. We thus conclude that, as constructed above, ordinary space is perfectly homogeneous. In addition, the construction of S01p 3 cannot result in either p0 or p1 manifesting as vectors, or vector fields, within that space; for if they did, then these projections/vectors would be functionally dependent on S01p 3 , which would again contradict (b). Given that vector fields have been ruled out, it seems we have little choice but to assume that p0 and p1 manifest within S01p 3 as uniform scalar fields --uniform, because any nonuniformity 11 would make the manifestations of p0 or p1 functionally dependent on S01p 3 , which would, again, violate/contradict their independence from that space. Presumably, the uniform scalar field for p0 is just (raw, unstructured) ordinary space itself, and the uniform (onedimensional) scalar field for p1 is just proper time. Lastly, let us recall that p0 sees p1 as a one-dimensional vector. This, presumably, would impart some directionality to p1 --which, as we have concluded, could not manifest as a direction within ordinary space. Since p1 has been associated with time, we interpret that this directionality of p1 (with respect to p0) is just the "arrow" of time. 3.5 Rapid expansion of ordinary space within the first instant of time Recall that p0 at level 1 is one dimensional --having, let us say, a length of p0. The time dimension, being a result of p1, does not exist at this level/stage. Given that a onedimensional object has zero volume, then the physical universe at this stage of development has a volume of zero. Since the time dimension comes into existence with the projection p1, then the advent of p1 defines the time point t = 0, at which point p0 has the value p0(t = 0), which may be denoted as p0,0. So, at exactly t = 0, or within the first instant after it, the existence/perspective of p1 causes p0 to manifest as the three-dimensional space S01p 3 , with a volume that should be proportional to p0,0 3 . Thus the volume of S01p 3 (ordinary space) goes from zero to around p0,0 3 within a time interval of zero, or near-zero, length --which constitutes a potentially very large, perhaps infinite, rate of spatial expansion. I propose, therefore, that this rapid spatial expansion, triggered by the advent of p1 at t = 0, is the process known as inflation [7]. Note that, under the above mechanism, inflation has a natural beginning: the advent of p1 at t = 0. And it also has a natural ending: it ends when the volume of ordinary space is around p0,0 3 . So inflation only lasts for the time (if any) that it takes (from the perspective of p1) for the one-dimensional space of length p0,0 to become the three-dimensional space of approximate volume p0,0 3 . 4 Constructing particles in system P In constructing the sequence (p0, p1, p2, p3) for system P, let us assume that energy is needed to create each of the projections pk (for k = 0, 1, 2, ...). We can think of this energy as being stored along the length of pk, and/or as being stored in the level that is created by pk. So we can speak of "pk energy", and/or we can speak of the energy, E[k+1]p, that pk inputs into level k+1 (the "p" part of the subscript merely indicates that this is energy for system P). We can think of the projection p0, then, as a process through which energy E1p is input into level 1 of system P. Likewise, p1 is a process that inputs energy E2p into level 2; p2 is 12 a process that inputs energy E3p into level 3; and so on. Let us collectively refer to the energy that is input into level 2 and above as E≥2p ; thus E≥2p = E2p + E3p + E4p + .... The total energy, Etp, that is input into system P is therefore Etp = E1p + E≥2p. We assume that all of these energies are nonzero and positive, so the energy of system P at level 1 and above, due to contributions from the sources mentioned, is positive. 4.1 Relations between energies Now that we have labeled the various energies that are involved in constructing the sequence (p0, p1, p2, p3) for system P, we discuss some properties of, and relations among, these energies, as follows: • Due to the scope rule, pk energy is in effect, or operant, at level k+1 and above; so level k+1 is the native level for pk energy. Thus, p0 energy is native to level 1 (i.e. it is operant at level 1 and above); p1 energy is native to level 2 (operant at level 2 and above); p2 energy is native to level 3 (operant at level 3 and above); and so on. • In constructing the sequence (p0, p1, p2, p3) for system P, all of the needed energy presumably enters (i.e. is input or "piped" into) that system via the projection p0; in other words, all of that energy begins its life in the system as p0 energy, at level 1. Some of this p0 energy is then projected up to level 2, via p1, thereby making it also "p1 energy"; and some of this p0/p1 energy at level 2 is further projected up to level 3, via p2, thereby making it also "p2 energy"; and so on. Alternatively, in constructing the said sequence, we can say that pk energy is also pk-1 energy, pk-2 energy, ..., p0 energy. Thus, all of these pk energies are essentially also "p0 energy". The central role of p0 as the portal or "pipe" through which the energy that constructs the sequence (p0, p1, p2, p3) comes into the system, merits the following description/definition of "p0 energy": p0 energy is energy (a) that enters system P at level 1, and so is native to level 1; (b) whose entry into the system is thereby dependent only on the projection p0 ; and (c) whose entry into the system is thus independent of p1, p2, p3, etc. We note that, as already alluded to above, the subsequent forms of energy that are involved in constructing the sequence (p0, p1, p2, p3) --i.e. p1 energy, p2 energy, etc. --also , essentially, satisfy this definition of p0 energy. That is, in constructing the said sequence, these subsequent forms of energy share the provenance of being energy that entered system P via the p0 portal --and so, by the definition above, are essentially "p0 energy". Figuratively, we might say that, due to their common provenance in the projection p0, those subsequent forms of energy inherit the nature, likeness, or "mantle" of p0 energy (a result which might thus be attributable to the inheritance rule). 13 4.2 A quantum of action Recall that the dimension of time is associated with p1. Since p1 does not exist at levels 0 and 1, then time also does not exist there; i.e. all time intervals are zero at those levels. Indeed, we can say that levels 0 and 1 are independent of time. But p1 does exist at level 2 and above; so time exists there, and all time intervals at those levels are nonzero (and presumably positive). Thus, at level 1, energy is nonzero, but time is zero. At level 2 (and above), however, both energy and time (intervals) are nonzero. Consequently, at level 2 and above, the product of energy and time --the quantity known as action --is nonzero, and thus has a positive lower bound; i.e. at level 2 (and above) the action is quantized. We thus have the derivation of an action quantum, which we interpret to be the basis for the empiricallyknown "quantum of action", commonly referred to as Planck's constant, and denoted as h. In the present model, therefore, the quantum of action, h, depends on both p0 and p1, and so does not exist at levels 0 and 1, but only comes into being at level 2. Thus, quantum mechanics, which is based on h, also comes into being at level 2 of system P. And therefore, due to the scope rule, both h and quantum mechanics are operant at level 2 and above; i.e. they are native to level 2. Note that the advent of h at a later stage (level 2) in the construction of system/world P is in contrast to the usual notion in which the quantum of action is assumed to (magically) operate at all phases in the construction of the physical universe. The late advent of h in system P has the following immediate consequences: • Because h is not operant at levels 0 and 1, then system P is not quantum mechanical at those primal levels, and so system P is not fundamentally quantum mechanical. Since system P is our model for constructing the physical universe, we conclude that the physical universe is not fundamentally quantum mechanical, and thus cannot have originated from a quantum effect (e.g. a "quantum fluctuation") --because the source/origin of system P is at level 0, and h does not exist there. • The energy of quantum vacuum fluctuations (or zero-point energy) necessarily depends on h, which (as we have found) depends on both p0 and p1, and is native to level 2; that is, quantum vacuum energy enters system P at level 2 or above, and its entry into the system is dependent on p0 and p1. Now recall our recent definition of "p0 energy": It enters system P at level 1; and its entry into the system depends only on p0, thereby making that entry independent of p1. Clearly, then, the energy of quantum fluctuations is not p0 energy (essentially or otherwise); rather, due to their dependence on p1, such fluctuations can presumably only produce pk energy forms that are native to level 2 or above --i .e. p1 energy, p2 energy, p3 energy, and so on. This result will later be shown to have important implications for the cosmological constant problem. 14 4.3 Energy distribution within ordinary space We know how the different terms in Etp are distributed among the levels (i.e. Ekp is input into level k), but how are they distributed initially and presently on large and small scales within ordinary space, S01p 3 ? The following six factors of the model likely dominate the distribution of energy within S01p 3 : 1. The projection p0 is independent of S01p 3 (ordinary space). 2. In addition, p0 is independent of p1. 3. The projections p1, p2, p3, etc., are dependent on p0. 4. At levels 0 and 1, neither time nor h exist. 5. At level 2 (and above), both time and h exist, and are thus in effect/operant. 6. At level 3 the space S12p 3 is nested/confined within S02p 3 , which in turn is nested/confined within S01p 3 (ordinary space). The role of these factors in shaping energy distribution is elaborated in the next few subsections. 4.3.1 Large-scale distribution of energy within ordinary space If the p0 process were to distribute its energy E1p nonuniformly within S01p 3 , then p0 would be favoring particular directions or positions within that space. But, as concluded in section 3.4, the independence of p0 from S01p 3 (factor 1) implies that it cannot favor particular directions or positions within that space. We conclude, therefore, that the p0 process must distribute its energy E1p uniformly throughout the volume p0 3 of S01p 3 (ordinary space). Moreover, given our earlier conclusion that the construction of S01p 3 cannot result in p0 manifesting as a vector field within that space, then presumably its associated energy (E1p) also cannot manifest as a vector field in ordinary space. Consequently, it seems that E1p must manifest within S01p 3 as a uniform scalar field. Since, in constructing the sequence (p0, p1, p2, p3), the projections p1, p2, and p3 are dependent on p0 (factor 3), and their associated energies all enter the system via the p0 portal, then they presumably "follow the lead" of p0 and also distribute their energies uniformly (on the large scale) throughout S01p 3 . Note that the uniform distribution of E1p energy throughout S01p 3 (ordinary space) might strictly apply only to that energy which comes into the system after the advent of p1 (with the concomitant advent of time and S01p 3 itself). Prior to the advent of p1, the E1p energy is distributed (perhaps nonuniformly) along the line of p0 (as described earlier). Given its 15 priority, this linear distribution might manifest at the advent of S01p 3 , and remnants of it might survive the inflation process, thereby imprinting some large-scale anisotropy on the cosmic microwave background (CMB) --perhaps yielding e.g. the so-called "axis of evil" [8], [9]. We can thus divide the production and distribution of E1p energy into two logical phases, or epochs: (1) the epoch prior to the advent of p1, in which E1p is distributed along the line of p0, and (2) the epoch after the advent of p1, in which E1p is distributed uniformly throughout the volume p0 3 of ordinary space, S01p 3 . 4.3.2 Small-scale distribution of E≥2p energy Recall that both time and h exist at level 2 (factor 5). The presence of time means that the input of energy E2p into level 2 from p1 can be, and we assume is, time-dependent and time-limited --and thus finite. So p1 inputs a finite amount of energy E2p into level 2. Furthermore, we expect the presence of h to partition this energy into smaller bits or chunks, yielding a multiplicity of what we will generically refer to as level-2 entities, objects, or particles. Given their dependence on p1, these objects will see p0 from the perspective of p1, and so they will see themselves as nested/enveloped/contained/embedded within the three-dimensional space of p0 (as seen from the perspective of p1) --that is, they will see themselves as embedded within S01p 3 (ordinary space). Lastly, since time exists at level 2, we assume (as per special relativity) that the level-2 objects possess mass. The same considerations apply at level 3. The presence of time means that p2 can (and, we assume, does) input a finite amount of energy E3p into level 3, and the presence of h partitions this energy into a multiplicity of level-3 objects/particles. And, as with the level-2 objects, their dependence on p1 will cause these level-3 objects to see themselves as embedded within S01p 3 (ordinary space). Moreover, the presence of time at level 3 means that the level-3 objects possess mass. And likewise for the energies E4p, E5p, etc. (if any) at levels 4 and above. In general, then, the presence of h causes the distribution of the energies E≥2p to be granular or chunky on the small scale. 4.3.3 Further aspects of E1p energy Unlike the situation just described for level 2 and above, the absence of h at level 1 (factor 4) means that the energy E1p cannot be broken into chunks; and so the energy E1p at level 1 constitutes a single, continuous entity. In addition, since time does not exist at level 1, we assume that the single entity at level 1 is massless. Recall now that p0 is native to level 1, but time is native to level 2 (factor 5). Thus, p0 is prior to time. By postulate 4, this means that the p0 process, which pumps energy E1p into level 1, is independent of time, and is therefore a continual process --i.e. it never 16 stops, and so it must be happening right now. Consequently, the quantity E1p is always increasing. Moreover, since E1p is the energy of p0 at level 1, and since p0 (as seen by p1) is ordinary space, then it is clear that E1p is just the energy of space itself; hence, an always-increasing E1p should yield a continual expansionary pressure on space. Indeed, an increase in E1p may produce an increase in the length of p0 , and thus an increase in p0 3 (the size/volume of the physical universe). As concluded in section 4.3.1, the p0 process must (in the second epoch) distribute its energy E1p uniformly throughout space. Since this process is also independent of time, then it is constant in time. So the continual influx of E1p energy into the system via the p0 process yields an input of energy per unit volume of space that is uniform throughout space, and constant in time; in other words, E1p yields a cosmological constant. Taken all together, the above results suggest that we interpret E1p to be the phenomenon known as dark energy [10]; i.e. dark energy = E1p . Moreover, since the p0 process and E1p are level-1 phenomena, but h only becomes operant at level 2 (factors 4 and 5), then dark energy/E1p is prior to --and thus independent of --h and quantum mechanics, and so is not a zero-point energy. 4.3.4 Further aspects of E3p energy distribution As stated in section 4.3.2, the presence of h at level 3 partitions the energy E3p into a multiplicity of smaller chunks --the level-3 objects/particles. Furthermore, we determined that these particles have mass. Let us say that a generic level-3 object has the energy ε3 p . In section 3.2, we found that the spaces S02p 3 and S12p 3 come into existence at level 3, and that the latter is nested within the former, which in turn is nested within S01p 3 (ordinary space) (factor 6). Thus, S12p 3 is a subset of S02p 3 , which is a subset of S01p 3 . We now ask: with respect to a level-3 object/particle, where is the S02p 3 space located? In section 4.3.2, we found that a level-3 object sees itself as being embedded within S01p 3 (ordinary space). Since S01p 3 is prior to, and thus independent of, S02p 3 , then the space that a level-3 object sees itself as embedded in is independent of, and thus outside the scope of, the (subsequent) S02p 3 space. This means that the S02p 3 space must be internal to, confined, or compacted within the level-3 object. And since the S12p 3 space is nested within S02p 3 , then it too is confined/compacted within a level-3 object. A level-3 object thus partitions S01p 3 (ordinary space) into two zones: the outside or exterior zone, which it sees itself as embedded in; and an inside zone, which corresponds to the restricted scope of the S02p 3 space, and forms the interior of the object. Since S01p 3 and S02p 3 are three-dimensional spatial modes of the projection p0, which is 17 independent of h, then we assume that (within their respective scopes) each of those spaces manifests as a single, continuous, three-dimensional space (i.e. having no discontinuities or chunkiness). On the other hand, since S12p 3 is a spatial mode of the projection p1, for which h is operant, then we will assume that S12p 3 manifests (in some ways, at least) as three, discrete, one-dimensional spaces. It is thus reasonable to assume that at least some of the energy, ε3 p , of a level-3 object will be distributed into each of these three, discrete, one-dimensional spaces of S12p 3 , resulting in a triplet of energy nodes or particles within the object. Consequently, each level-3 object will have internal structure that includes a triplet of subparticles that are confined within the object. We interpret, therefore, that level-3 objects are the particles known as baryons, and their three subparticles are the objects known as quarks. In this way, the present model accounts for the existence of baryonic quarks, and also their confinement within the baryons. 4.3.5 Substructure at level 4 Since everything at level 3 is inherited by level 4, then level-4 objects would, like level-3 objects, also have a substructure of three discrete particles or quarks. However, in section 3.2 it was stated that level 4 generates three new, three-dimensional spaces, which were denoted as Sx3p 3 (x = 0, 1, 2), and that these spaces are internal to S02p 3 and S12p 3 . It seems likely, therefore, that the three subparticles/quarks within level-4 objects would themselves contain internal structure (whereas the quarks at level 3 would have no internal structure). 4.4 Some particles of system P We have identified the objects at level 3 of system P as the baryons, and the single entity at level 1 as dark energy. What about the objects at level 2? We first note that there are no internal spaces at level 2 (in contrast to level 3), so level-2 objects have no internal structure; i.e. they are structureless. Second, as concluded in section 4.3.2, the level-2 objects have mass. We therefore identify the objects at level 2 as the leptons. Fig. 2 illustrates the identification of objects at different levels of system P with known (or, in the case of dark energy, suspected) object/particle classes. 18 Fig. 2 Identification of the level-1 object as dark energy, the level-2 objects as leptons, and the level-3 objects as baryons. 4.4.1 Associating particle properties with projections We have already established that, in constructing the sequence (p0, p1, p2, p3), it is the projection p0 that inputs energy into the system. We therefore associate p0 with energy. The main difference between a lepton and a baryon is that the latter has the property known as baryon number (or baryon charge), and the former does not. In system P, the main difference between a level-2 object and a level-3 object is that the latter has the projection p2, whereas the former does not. We therefore associate baryon charge with p2. Likewise, a main difference between dark energy and leptons is: The latter possess the properties known as lepton number (or lepton charge), mass, and possibly electric charge; whereas the former presumably possesses none of these properties. In system P, the main difference between the level-1 and level-2 entities is that the latter have the projection p1, whereas the former does not. We therefore associate lepton charge, electric charge, and mass with p1. However, for reasons given in section 4.12, we will more specifically associate mass with the conjunction of p1 and p0, which we denote as p1p0 .5 This conjunction is native to level 2, since that is the level at which p0 and p1 are first jointly operant. Given that the conjunction of p1 and p0 might be meaningful, then at level 3 the conjunctions of p2 with the other projections that are in effect (p0 and p1) might also be meaningful. There are three such conjunctions at level 3: p2p0, p2p1 and p2p1p0 (which may collectively be abbreviated as simply p2x, where x stands for each of p0, p1, and p1p0). Since we have identified the level-3 objects as baryons, and since baryons are believed to possess a triplet of color charge (one for each quark), then it might be useful to associate color charges with the three conjunctions p2x at level 3. Fig. 3 shows the projections and conjunctions that are in effect at levels 0 through 3. In Fig. 4, we translate the projections and conjunctions of Fig. 3 into the properties/attributes/charges that we have associated with them. 5 As in logical conjunction, the order of the conjuncts does not matter here; so p1p0 ≡ p0p1. 19 p 1 p 0 p 2 0 1 2 3 dark energy leptons baryons Fig. 3 The projections and conjunctions of projections that are in effect at levels 0 through 3 of system P. The x in p2x (at level 3) stands for each of p0, p1, and p1p0 . Fig. 4 The projections and conjunctions of Fig. 3 translated into the properties/attributes/charges that we have associated with them. Of course, electric, lepton, and baryon charges can have values that are positive, negative, or zero. At level 3 in Fig. 4, if we set lepton charge to zero, and baryon charge to positive or negative, and let electric charge vary among its possible values, then we get bundles of properties that fit with the baryons (and antibaryons); which reinforces our identification of the level-3 objects as baryons. At level 2 in Fig. 4, if we restrict lepton charge to nonzero values, and let electric charge vary among its possible values, then we get bundles of properties that fit with the various leptons; which reinforces our identification of the level-2 objects as leptons. Thus, we say that the set of projections and conjunctions that are operant at a given level yield a class of objects/particles at that level --i.e. at level 1 the set {p0} yields the class of object known as dark energy; at level 2 the set {p0, p1, p1p0} yields the lepton class; and at level 3 the set {p0, p1, p1p0, p2, p2x} yields the baryon class of objects. From this association of projection/conjunction sets with object/particle classes, we see that the presence of p1 is necessary but not sufficient for the presence of (nonzero, i.e. +/-) electric and lepton charges; e.g., although p1 is in effect at level 2, electric charge can be zero there (as for neutrinos); and although p1 is in effect at level 3, lepton charge is always zero there. From this pattern, we presume also that p2 is necessary but perhaps not sufficient for the presence of finite (nonzero) baryon charge; that is, while baryon charge 20 p 1 p 0 p 2 0 1 2 3 { } {p 0 } {p 0 , p 1 , p 1 p 0 } {p 0 , p 1 , p 1 p 0 , p 2 , p 2 x} p 1 p 0 p 2 0 1 2 3 { } {energy} {energy, electric & lepton charges, mass} {energy, electric & lepton charges, mass, baryon charge, three color charges} seems to always be finite at level 3, we reserve the possibility that it may be zero at higher-numbered levels. 4.4.2 Partitioning of properties/charges within level-3 objects I now propose that, at level 3, not only is energy partitioned/distributed among the three, one-dimensional spaces of S12p 3 (as asserted in section 4.3.4), but so too are the other charges and properties at level 3 --electric charge, baryon charge, the three color charges (with, we assume, one color charge going to each of the one-dimensional spaces), and spin6 --thereby forming (as already noted) a triplet of baryon subparticles with those properties, i.e. the quarks. 4.5 Analogy of system P with biological genetics By analogy with biological genetics, we might think of the individual projections p0, p1, p2, etc. as "codons". Different combinations of these codons yield "genes" for particle properties/attributes/charges; e.g. p0 by itself is the gene for energy; p1 by itself is the gene for electric and lepton charge; p2 is the gene for baryon charge; the conjunction p1p0 is the gene for mass; and the conjunctions p2x are the genes for three color charges. The set of genes at a given level yields the "genome" for an object/particle class or "genus" at that level. Thus, the genome at leve1 0 is { }; the genome for dark energy at level 1 is {p0}; the genome for leptons at level 2 is {p0, p1, p1p0}; and the genome for baryons at level 3 is {p0, p1, p1p0, p2, p2x}7. The object classes or genomes at each successive level constitute "generations" of entities. Thus, dark energy at level 1 is the first-generation entity; leptons at level 2 are entities of the second generation; and baryons at level 3 are entities of the third generation. Each such generation inherits the genes (indeed, the complete genome) of the prior generation. Moreover, epigenetic analogies may also apply. For example, we may think of the lepton-charge characteristic of p1 as being "turned on" at level 2, allowing its expression there, but "turned off" at level 3, suppressing its expression. 4.6 Real and virtual particles Let us define a real object/particle as one whose energy entered system P in the "normal" way --i.e. via the projection p0, native to level 1. As discussed earlier, in constructing the sequence (p0, p1, p2, p3) for system P, some of this "p0 energy", as we have called it, is further projected up the levels, thereby also becoming p1 energy, p2 energy, etc. --but, due to its provenance, always retaining its essential nature as "p0 energy". This process 6 How spin is derived in the model will be described below. But for now we just take it as given. 7 If, however, it turns out that the conjunctions p1p0 and p2x are not meaningful, then the genome for the leptons at level 2 would be simply {p0, p1}, and the genome for baryons at level 3 would be {p0, p1, p2}. But, for the present, we will assume that such conjunctions might be meaningful, and thus will include them in the genomes. 21 yields real entities/particles at each level: dark energy at level 1, with genome {p0}; real leptons at level 2, with genome {p0, p1, p1p0}; and real baryons at level 3, with genome {p0, p1, p1p0, p2, p2x}. The other way that energy enters system P is (as described earlier) via the quantum of action, h, which depends on p0 and p1, and is thus native to level 2. This is the quantum vacuum energy, or energy of quantum fluctuations, which produces virtual particles. Since the entry of this energy into the system bypasses the normal p0 process, then it is not "p0 energy", and cannot produce dark energy, {p0}, at level 1. Rather, such quantum vacuum energy can only produce pk -energy forms (and corresponding virtual particles) that are native to level 2 and above --i.e. p1 energy (virtual leptons), p2 energy (virtual baryons), etc.8 Although virtual particles do not contain p0 energy, they are still dependent on p0, and so it seems appropriate to include the p0 component in their genomes. As such, the genomes for virtual leptons and virtual baryons will be denoted in the same way as their real counterparts; i.e. as {p0, p1, p1p0} and {p0, p1, p1p0, p2, p2x}, respectively. 4.7 Fundamental interactions The projections that we have discussed so far may be described as forward projections, since (as described in postulate 1) they go from a prior level to a subsequent level (i.e. they are directed away from level 0). We associated these projections with the input of energy into higher/subsequent levels. We now want to introduce the concept of reverse or backward projection, which goes in the opposite direction: from a subsequent level to a prior level (i.e. toward level 0). Specifically, I propose that each forward projection p0, p1, p2, etc., has a corresponding backward projection, which we denote as -p0, -p1, -p2, etc., respectively. Basically, for an object that has pk in its genome, we define -pk as "the backward projection of pk energy onto level k" (for k = 0, 1, 2, ...). That is, such a backward projection, -pk, takes pk energy --which is above level k --and places it at level k, yielding an object/particle at that level. Note that, if pk energy were to produce its own backward projection onto level k, it would be a violation of the scope rule, because level k is prior to (and thus outside the scope of) pk energy. How, then, does backward projection occur without violating the scope rule? The short answer is that pk energy does not perform the backward projection by itself. The longer answer is as follows: Since the forward projection sequence (p0, p1, p2, p3) springs from level 0, then there must exist a projection-generating engine of sorts, operant at level 0, which produces that sequence of forward projections, and which, we assume, is also capable of producing backward projections. By the scope rule, this engine (operant at level 0) is then operant throughout system P, and thus contains both level k and pk energy within its scope, and is thereby able to cause the backward projection of pk energy onto level k without violating the scope rule. The nature of this projection engine is developed further in Part II [5]. Backward projection in the present model is in some ways similar to "projection" as 8 The origin of photons, gluons, gravitons, etc., is explained in sections 4.7, 4.13, and 4.14. 22 we know it from ordinary geometry. Consider a three-dimensional sphere for example: it becomes a two-dimensional circle when projected onto the x-y plane; and the circle becomes a one-dimensional line when projected onto the x-axis. So, with each "backward" projection the space dimensions of the object are reduced by one. Similar things happen with backward projection in the present model. In addition to forward and backward projections, let us also define a lateral or intralevel projection of pk energy, which places that energy onto pk's native level (which, it may be recalled, is level k + 1), thereby yielding an object/particle at that level. Backward and lateral/intralevel projections may be classified together as nonforward projections.9 We assume that all projections of system P, including the lateral type, are performed by the projection engine described above. 4.7.1 The electromagnetic interaction Objects at level 2 or above (e.g. leptons and baryons, with genomes {p0, p1, p1p0} and {p0, p1, p1p0, p2, p2x}, respectively), whether real or virtual, have the projection p1 in their genome, and have p1 energy. We now ask, what happens when p1 energy, associated with these objects, is backward projected onto level 1 (the operation that we have denoted as -p1)? We could denote the genome for the resulting object as {p0}, but that would confuse it with the genome for dark energy, and also obscure the object's/energy's history (of, inter alia, being the product of a backward projection). A better way to denote its genome is therefore {p0, -p1p1}. This object has energy, but no electric charge (due to the net absence of p1 in its genome), and (due to the absence of p1p0) is massless. Its "source", if you will, is p1, which we have associated with electric charge. We interpret, therefore, that the object denoted by the genome {p0, -p1p1}, resulting from backward projection of p1 energy onto level 1, is the photon. So, in this sense, we say that the photon is a level-1 object that is produced by backward projection (in contrast to dark energy, which is a level-1 entity produced by forward projection). Furthermore, we interpret that the backward projection of p1 energy onto level 1 (i.e. -p1) is just the electromagnetic interaction (actually, it is the emission phase of that interaction; the absorption phase will be discussed later). In general, then, it is suggested that backward projection is the basis for at least some of the fundamental interactions. At level 2 the dimensionality of p0 (with respect to p1) is 3, and the dimensionality of p1 (with respect to p0) is 1 (i.e. the dimensions of ordinary space and time, respectively). When p1 energy is backward projected onto level 1, the dimension of time (associated with p1) vanishes to zero. Likewise, the component of p0 parallel to p1 also vanishes, leaving only two space dimensions (which may thus be called the residual dimensions, or the residual 2-space, of p0 with respect to p1), both of which are orthogonal to p1. In other words, the backward projection of p1 energy onto level 1 produces an object for which the time dimension does not exist, and whose space dimensions are reduced from three to two --which agrees with the result implied by special relativity for the photon, further supporting our interpretation that photons are the result of backward projection of p1 9 These new types of projection will later be incorporated into a fifth postulate. 23 energy onto level 1. Light (i.e. electromagnetic radiation) is thus energy that starts out at level 2 (or above) and is backward projected onto level 1 (where time does not exist). Its origin at level 2 or above (where time does exist) endows this energy with a history, an aspect of which is its exposure to h, which (I propose) endows light energy with its quantum properties. Dark energy, in contrast, is a strictly level-1 entity/phenomenon, so we can say that it is energy without a history. Its lack of a history at level 2 or above means that dark energy has never been exposed to h, and so it is not quantized, and has no quantum properties (e.g., as determined earlier, it has no quantum chunkiness). It seems, then, that the quantum properties and two-dimensional space of light/photons may all be residual aspects (or, if you will, residues) that stem from the energy's history at level 2 or above. 4.7.2 The strong interaction Objects at level 3 or above (e.g. baryons, with genome {p0, p1, p1p0, p2, p2x}), whether real or virtual, have the projection p2 in their genome, and have p2 energy. We now ask, what happens when p2 energy, associated with these objects, is backward projected onto level 2 (the operation that we have denoted as -p2)? As stated earlier, when p1 is related to p2 the dimensionality of the former is 3 (i.e. S12p 3 ), and the dimensionality of the latter is 1 (i.e. S21p 1 ). When p2 energy is backward projected onto level 2, the one dimension associated with p2 vanishes to zero, and the three dimensions associated with p1 are reduced to two. Since we earlier interpreted that, at level 3 (or above), the three, one-dimensional spaces associated with p1 (from the viewpoint of p2) were responsible for the triple-quark substructure of baryons, I now propose that the reduction of these dimensions down to two --when p2 energy is backward projected onto level 2 --is responsible for the production of objects with a double-quark substructure. Consequently, we interpret that the objects produced by backward projection of p2 energy onto level 2 are the mesons (which, indeed, are thought to have a double-quark --actually, quark-antiquark --substructure). So, in this sense, the mesons are level-2 objects, whose genome may be denoted as {p0, p1, p1p0, -p2p2} --which comports with the mesons possessing energy, electric charge, mass, and two opposite baryon charges (i.e. p2 and -p2, for a net of zero baryon charge). Since the p2 and -p2 components of the meson are presumably distributed/separated among its double-quark substructure, then, in this sense, we can think of mesons as level-2 objects that contain remnants of level-3-type substructure (i.e. the two quarks). This is probably allowed by the nesting rule, since the (subsequent) level-3-type substructure is embedded/contained within the (prior) level-2 structure, not the other way around. As long as the p2 and -p2 components are separated within the meson substructure, we will assume that energy associated with them can behave like any other p2 energy; e.g. it can be backward projected onto level 2, yielding other mesons. In the above description (of backward projection of p2 energy onto level 2), we 24 utilized the residual 2-space of p1 with respect to p2 (to account for the double-quark substructure of mesons), but we neglected to mention another such space that arises in this process: the residual 2-space of p0 with respect to p2; i.e. the 2-space that remains when, due to backward projection of p2 energy onto level 2, the component of p0 in the direction of p2 vanishes. We have thus identified three residual 2-spaces: the residual 2space of p0 with respect to p1, which may be referred to as the first residual 2-space; the residual 2-space of p0 with respect to p2, which may be referred to as the second residual 2-space; and the residual 2-space of p1 with respect to p2, which may be referred to as the third residual 2-space. The numbering/ranking of these residual spaces reflects their order of priority. Given such priority relations, together with the nesting rule, we conclude that, where applicable: the third residual 2-space is nested/encapsulated/confined within the second, and the second is nested/encapsulated/confined/compacted within the first. When p2 energy is backward projected onto level 2, only the second and third residual 2-spaces are applicable (since p1 is operant at level 2, and thus the space of p0 with respect to p1 --i.e. ordinary space --is fully three dimensional). By the priority relations established above, the third residual 2-space, which yields the double-quark substructure of mesons, would be confined within the second --a result which might explain the confinement of quarks within mesons. Consider now a process in which p2 energy is backward projected onto level 2, and then immediately backward projected onto level 1 --a sequence of backward projections which may be denoted by (-p2, -p1). The result of this process is an object at level 1 whose genome may be written as {p0, -p1p1, -p2p2} --which therefore is energetic but massless (due to the presence of p0, but absence of p1p0), and possesses zero electric, lepton, and baryon charges. I propose that this object is the gluon. In this gluon-production process, the third residual 2-space would cease to exist (since p1 is not operant at level 1). So whatever is left over from the backward-projection sequence (double color charge?) would be confined within the second residual 2-space, which would be confined within the first. This result might explain the confinement of gluons and their color charges. In summary, we interpret that the strong interaction as mediated by mesons is the result of backward projection of p2 energy onto level 2; and the strong interaction as mediated by gluons is the result of a sequence of backward projections, (-p2, -p1), of p2 energy onto level 1. 4.7.3 The weak interaction, and electroweak unification The weak interaction is known to be mediated by the W and Z bosons. Given that the W and Z possess mass, have zero baryon number, and have no internal structure (as far as we know), it follows that they are level-2 objects in the present model. Thus the weak interaction involves the projection of energy onto level 2, which thereby yields objects/particles at that level (i.e. the W and Z). We know that leptons participate in the weak interaction, and (according to the present model) that they are also level-2 objects, 25 having the genome {p0, p1, p1p0}, thereby possessing p1 energy and (for a real lepton) p0 energy. Since leptons and the W/Z are all level-2 objects, then (unlike the interactions described above) the weak interaction cannot be attributed to backward projection --of either the leptons' p0 energy or its p1 energy (since such backward projections would yield objects at levels 0 and 1, respectively --i.e. not at level 2, where the W and Z reside). Rather, the weak interaction must be attributable to a lateral/intralevel projection, of either p0 or p1 energy. But p0 is native to level 1, so a lateral projection of p0 energy would place the energy at level 1, not level 2. So it seems that we must attribute the weak interaction to a lateral/intralevel projection of p1 energy. Since the electromagnetic interaction has also been attributed to a nonforward projection of p1 energy, then we can say that the electromagnetic and weak interactions have the same source --p1 (or p1 energy). Consequently, as we have associated electric charge with p1, it seems that we must also associate the weak charge with p1. As such, the present model offers an explanation for the known "electroweak unification": the electromagnetic and weak interactions are related or "unified" because they both stem from the projection of p1 energy --where the former interaction is attributed to backward projection of such energy (i.e. onto level 1), and the latter interaction is attributed to lateral projection of such energy (i.e. within level 2). In addition, the lateralness (or, if you will, handedness) of an intralevel projection might explain why the weak interaction violates left-right symmetry, or parity. Although it may be useful at some future time to develop a special notation to indicate in their genomes that the W and Z particles are products of lateral projection of p1 energy, for now we will just denote them as having the same genome as level-2 objects that are produced by forward projection --i.e. {p0, p1, p1p0}. 4.7.4 The gravitational interaction Following the pattern of the interactions above, we should attribute gravity to a nonforward projection of one of the pk energies (k = 0, 1, 2, ...). Since dark energy is a source of gravity, and is a level-1 phenomenon, then the energy type to which we attribute gravity must be present/available/operant at level 1 --a condition that only p0 energy satisfies. So it follows that we should attribute gravity to a nonforward projection of p0 energy: either backward projection onto level 0, or lateral/intralevel projection onto level 1. But which one? For reasons to be provided in section 4.13, we choose the latter --that is, we attribute the gravitational interaction to the lateral/intralevel projection of p0 energy onto level 1. This means that the "graviton" (in common with the photon and the gluon) is a level-1 object/particle that is produced by nonforward projection. Thus, the strong, electromagnetic, and gravitational interactions all utilize nonforward projection onto level 1 (the strong interaction also utilizes nonforward projection onto level 2, as described above). Our model for gravity yields the following results and explanations concerning that interaction: 26 Cosmological constant: Since the source of gravity is p0 energy, and since, as determined earlier, the quantum vacuum (or zero-point) energy is not p0 energy, then the energy of quantum vacuum fluctuations is not a source of the gravitational interaction (i.e. it does not gravitate). Therefore, quantum fluctuations cannot contribute to the cosmological constant, Λ. Or, putting this argument more figuratively, we might say: Energy that enters system P through the "front door" (i.e. via p0, at level 1) inherits the mantle of "p0 energy", and is thereby a source of the gravitational interaction; but energy that sneaks in through a "side door" (i.e. via h, at level 2 or above) does not inherit the mantle of "p0 energy", and therefore is not a source of the gravitational interaction --and so cannot contribute to the cosmological constant. As far as we can tell from the present model then, the only contribution to Λ comes from the (non-quantum-mechanical, non-zero-point) dark energy/E1p that the p0 process continually inputs into level 1, and which we have associated with the creation and expansion of ordinary space (see section 4.3.3). By eliminating contributions to Λ from quantum fluctuations, the present model thereby averts the cosmological constant problem [11], and may thus explain the observed small value of that constant. Note that, although quantum fluctuations do not produce p0 energy (which is native to level 1), and thus are not a source of gravity, our earlier results indicated that such fluctuations can produce pk-energy forms (and corresponding virtual particles) that are native to level 2 and above; i.e. p1 energy (virtual leptons), p2 energy (virtual baryons), etc. These pk-energy forms can then be nonforwardly projected, yielding (for p1 energy) the electromagnetic and weak interactions, and (for p2 energy) the strong interaction. The electromagnetic interaction, due to the backward projection of p1 energy for such virtual particles, can presumably then yield the Casimir effect and the Lamb shift. In summary, we might say that quantum vacuum energy (and thus virtual particles) can emitate10, weakitate, and strongitate --but cannot gravitate. Nonquantization (or, why gravity seems to defy attempts at quantization): Since the source of gravity is p0 energy (which is native to level 1), and since the quantum of action h is native to level 2, then the basis for the gravitational interaction is prior to, and thus independent of, h; and so gravity is not fundamentally quantum mechanical.11 Or, in more figurative language, we might say: The basic ingredient for gravity (p0 energy) is "up and running" at level 1, before quantum mechanics (i.e. h) "gets its boots on" at level 2. Furthermore, given our result that quantum fluctuations cannot contribute to Λ, then Λ appears also to be independent of h, and thus independent of quantum 10 Where "emitate" might also have been written as EMitate, with obvious meaning in this context. 11 It should thus be apparent that use of the term "graviton" in this paper does not imply any bias towards a quantized conception of gravity. 27 mechanics. So, as far as we can tell from the present model, gravity is completely independent of h and quantum mechanics --perhaps obviating the need for a quantum theory of gravity.12 Universality (almost): Recall that, in constructing the sequence (p0, p1, p2, p3) for system P, all of the energy that is involved --in whatever form it might take (p0 energy, p1 energy, p2 energy, etc.) --shares the provenance of entering the system via the p0 portal, and is therefore (essentially, at least) "p0 energy". Consequently, the "lateral/intralevel projection of p0 energy within level 1" is a universal characteristic of all these energy forms and their associated objects/particles (e.g. real leptons, real baryons, etc). In other words, all of the energy associated with these things will laterally project onto level 1, thereby making them sources of gravity. This would seem to account for the "universal" nature of the gravitational interaction. Of course, this "universality" of gravity comes with the one exception, described above, that quantum vacuum energy, due to its provenance of entering the system at level 2 or above, and thereby bypassing the p0 process, is not p0 energy, and is therefore excluded as a source of gravity. So provenance matters. 4.7.5 Other possible fundamental interactions Given our characterization of fundamental interactions as being the result of "nonforward projection of pk energy", then the case for which k = 3 presents the possibility for new interaction types, via (a) backward projection of p3 energy, and (b) lateral/intralevel projection of p3 energy (onto level 4). Likewise, for k = 2, there might be an interaction type that is due to lateral/intralevel projection of p2 energy (onto level 3). However, due to the confinement that occurs for p2 energy, p3 energy, etc., it is possible that such interactions would not manifest outside of the compacted spaces of level 3 and above, which could explain why such novel interactions (if they actually exist) have not yet been identified. And, for the supposed interaction resulting from lateral projection of p2 energy, it is possible that any of its effects that do manifest have been confused/conflated with aspects of the strong interaction. Finally, there is the question of "backward projection onto level 0": might this, too, produce a new type of interaction? This question will be taken up in section 4.14. 4.8 Interaction examples In our descriptions of fundamental interactions above we actually only provided an account of the first half of some interactions: emission of the mediating particle from its source, by either backward or lateral projection. In the examples below, we give more complete descriptions by also including (where appropriate) the absorption of the mediating object. 12 Some articles that question the need for quantum gravity are [12] and [13]. 28 4.8.1 Electromagnetic Consider the electromagnetic interaction between an electron and a proton, in which the electron emits a photon, and the proton absorbs it. Our description of the electromagnetic interaction in section 4.7.1 accounted only for the first half of that interaction: emission of the photon by the electron, which we described as the backward projection, onto level 1, of p1 energy. The second half of the interaction --absorption of the photon by the proton --can be accounted for as follows: the photon energy, at level 1, is forward projected13 from level 1 (i.e. along the line of p1), thereby becoming associated with the proton's p1 component. (Of course, this means that only objects having a p1 component in their genome can participate in the electromagnetic interaction; which, again, indicates that electric charge is associated with p1.) The converse interaction, in which the proton emits the photon, and the electron absorbs it, is described by exactly the same language above, except we replace the word "electron" with the word "proton", and vice versa. 4.8.2 Weak Consider the decay of a neutron into a proton via the weak interaction, mediated by emission of a W  particle (so-called beta decay). Since the neutron is a level-3 object, and the W  is a level-2 object, doesn't this process involve backward projection onto level 2 (i.e. not intralevel projection, within level 2 --the modus operandi that we determined above for the weak interaction)? No; as with the leptons at level 2, we interpret that the neutron's participation in the weak interaction is due to intralevel/lateral projection of p1 energy. But, to understand this, we must recall that p1 is native to level 2, so that a lateral projection of p1 energy associated with the neutron will be placed onto level 2 --not level 3 (the level of the neutron itself). Beta decay is thus described as follows: p1 energy associated with the neutron is laterally projected onto/within level 2, as the W  particle --thereby transforming the neutron into a proton. As another example, the weak interaction of a proton (emitter) and electron (absorber), mediated by exchange of a Z particle, is described as follows: p1 energy associated with the proton is laterally projected onto level 2 (as the Z particle), which then becomes associated with the electron's p1 component. And vice versa for an electron emitter and proton absorber. 13 This kind of "forward" projection is different from the forward projections that create the sequence (p0, p1, p2, p3) for system P. This will be clarified as we go along, and then summarized/formalized in postulate 5. 29 4.8.3 Strong The strong interaction between nucleons A (emitter) and B (absorber), mediated by exchange of a meson, can be described in the following way: p2 energy associated with A is backward projected onto level 2, as a meson. This meson energy is then forward projected along the line of p2, thereby becoming associated with B's p2 component. The strong interaction between quarks A and B, mediated by exchange of a gluon, can be described as follows: p2 energy associated with A is backward projected onto level 1 (as a gluon), via the sequence (-p2, -p1). This gluon energy is then forward projected along the lines of p1 and p2 --i.e. via the sequence (p1, p2) --thereby becoming associated with B's p2 component. 4.8.4 Gravitational Consider the gravitational interaction between A and B, which are two objects/particles/entities at level 1 or above (so that each has a p0 component), and where A is the emitter, and B is the absorber. First, p0 energy associated with A is laterally projected onto level 1 (as the graviton), which then becomes associated with B's p0 component. Note that, as worded above, the criterion for emitting a graviton (the presence of p0 energy), and thereby being a source of gravity, seems to be stricter than the criterion for absorbing a graviton (the mere presence of the p0 component in an object's genome). It might be the case that we need to increase the criteria on the absorber; i.e. we might insist that it has not just a p0 component in its genome, but that it meet the stricter requirement of having p0 energy, as defined earlier. This stricter requirement would not only prevent virtual particles (born out of quantum vacuum energy) from being emitters of gravitons (and thus sources of gravity), but would also prevent them from being absorbers of gravitons --with the result that virtual particles would not participate at all in the gravitational interaction. On the other hand, if we keep the looser requirement for absorbing gravitons, then we would have a situation where the virtual particles of quantum fluctuations cannot be a source of gravity, but can absorb gravitons, and thus be a gravitational sink. Could such a gravitational-sink effect be an explanation for the weakness of gravity with respect to the other forces? That is, with virtual particles acting as a gravitational sink, could the tensof-orders-of-magnitude difference in the energy densities between virtual particles and ordinary particles account for the tens-of-orders-of-magnitude difference in strength between gravity and the other forces? 4.9 Matter-antimatter asymmetry In constructing the sequence (p0, p1, p2, p3) for world P, we have found that the p0 process inputs (dark) energy into level 1, and also acts as a portal for inputting energy into 30 subsequent levels. Likewise, the projection p1 can input electric charge at level 2, and also at subsequent levels; and p2 can produce baryon charge at level 3 and above. For p1 and p2, let us now assume the following, more specific, pattern of electric and baryon charge production: At the first level for which a given of these projections (p1 or p2) is in effect (i.e. its native level), its associated charge (electric or baryon, respectively) takes on a particular sign, either positive or negative (+/-). At the second level for which a given of those projections is in effect, its associated charge takes on the opposite sign that it had at the first level. And, at the third level or above there are equal amounts of positive and negative, so the net charge is zero. Or, in short: The charge associated with a given of these projections (p1 or p2) alternates in sign for the first two levels that the projection is in effect, and is net zero at later levels. Note that this charge-production scheme is nonconservative at each of the first two steps (but might be conservative overall). Thus, the electric charge (associated with p1) that is produced at level 2 has, let us say, negative sign; the electric charge produced at level 3 has positive sign; and the electric charge produced at level 4 and above has a net value of zero. Likewise, the baryon charge (associated with p2 ) that is produced at level 3 has, let us say, positive sign; the baryon charge produced at level 4 has negative sign; and is net zero at level 5 and above. The result of this pattern of charge production is leptons with negative electric charge at level 2 (which yields electrons); baryons with positive electric charge and positive baryon charge at level 3 (which yields protons); and objects at level 4 that have zero electric charge and negative baryon charge. And let us suppose that the level-4 objects do not participate in the strong interaction --perhaps because the extra internal structure of those objects (i.e. the spaces Sx3p 3 that are internal to, or nested within, S12p 3 ; see section 3.2) interferes with the pathway for that interaction. Such level-4 objects, having zero electric charge, and not interacting strongly, are thus candidates for dark matter.14 These results are shown in Fig. 5. Levels 1 through 4 in Fig. 5 constitute a world of dark energy and "matter", with a conspicuous absence of antimatter (i.e. matter-antimatter asymmetry), which comports with the world that we find ourselves in right now. And if, as shown in Fig. 5, the projection sequence for system P ends with p3 placing energy E4p into level 4 (i.e. that Etp = E1p + E2p + E3p + E4p), and if we assume that E4p ≈ 5(E2p + E3p), then we would have the situation that we observe today in which the dark matter has about five times the mass of ordinary matter (i.e. matter at levels 2 and 3). 14 Having negative baryon charge, one could think that these dark matter candidates --call them transbaryons --might annihilate on contact with baryons. However, following the paradigm that we have described, the transbaryons would likely have a new type of charge --call it transbaryon charge --of a certain sign, say positive, associated with their p3 component. So their annihilation might require contact with an antitransbaryon (having negative transbaryon charge), which might only rarely exist, if at all. Some of the transbaryons could then exist as stable particles within galaxies, mingling with baryons without annihilating, and thereby play the role of dark matter. 31 In conclusion, our model for constructing the physical universe via a sequence of discrete, ordered, steps that may be nonconservative of charge --in contrast to the nondiscrete standard big bang model (where charge conservation seems to always hold) --enables us to generate the observed pattern of matter-antimatter asymmetry, while also yielding a new candidate for dark matter. Fig. 5 The spectrum of particle types that the sequence (p0, p1, p2, p3) for system P would yield, given the pattern of alternating-sign-then-zero charge production that we have assumed for the first three steps/levels at which a projection is operant (excepting p0). This particle spectrum comports with observed matter-antimatter asymmetry, and level 4 yields a new candidate for dark matter. 4.9.1 Primary and secondary projections In contrast to the charge-production scheme described above, which yields matterantimatter asymmetry, our present-day experience is that all electricand baryon-charge production is fully conservative. Consideration of this, as well as our model of the fundamental interactions developed above, in which some projections propagate "along the lines of" other, already-existing projections, suggests the existence of two different projection types. These projection types will be called primary and secondary --defined as follows: The primary projections are the ones that construct the infrastructure of system/world P, the sequence for which we have denoted as (p0, p1, p2, p3). That is, starting at level 0, they construct the subsequent levels (1, 2, 3, etc.) and the connections/projections between them; and relations between these projections construct the different spaces of the system. In addition, the primary projections input energy into the levels that they construct, which is partitioned among (and within) the spaces, thereby constructing what may be called the (initial population of) primary entities/particles of the system (i.e. dark energy at level 1, leptons at level 2, baryons at level 3, and --possibly --dark matter at level 4). In our scheme of matter-antimatter asymmetry for world P, the primary projections p1 32 p 1 p 0 p 2 p 3 0 1 2 3 4 dark energy electrons protons dark matter? and p2 (at least) are nonconservative of charge in each of their first two steps. The primary projections are all forward projections. The secondary projections may be forward, backward, or lateral/intralevel. These are the projections that happen in world P once the infrastructure of that world has been constructed/established by the sequence of primary projections. Among these secondary projections are the ones that we have described as being responsible for the fundamental interactions, the processes of which construct --let us say --secondary entities/particles of the system (i.e. photons, mesons, gluons, gravitons, and W/Z particles). In essence, the secondary projections utilize the existing infrastructure (of levels, and the primary projections which connect the levels) as a kind of interaction network --moving energy (and information) in all directions (forward, backward, and lateral). In this respect, the levels and primary projections of system P act as nodes and edges, respectively, in a communication network. In world P, the secondary projections (unlike the primary projections) are, evidently, fully conservative of charge. To denote the secondary forward projections, we will utilize the same notation that is used for the primary projections; i.e. p0, p1, p2, etc. When using these symbols, either the context or explicit statements will tell us whether they represent the primary or secondary type. Likewise, for the backward projections (which are always secondary), we may use the notation -p0, -p1, -p2, etc. (as we have already done above); or we may state it in words, e.g. "the backward projection of pk energy onto level k" (as we have also done above). And for lateral/intralevel projections (which, too, are always secondary), there is no special notation yet developed --so we simply state it in words, e.g. "the lateral/intralevel projection of pk energy within level k + 1" (as we have done above). In summary, primary projections are about constructing the infrastructure of a new system or world, and priming it with energy; secondary projections are about what happens within that already-existing infrastructure. Both primary and secondary projections are forms of communication, since they both communicate energy and information among levels. However, the primary projections only communicate in one direction --forward; whereas the secondary projections may communicate in all directions: forward, backward, and lateral. So, to be clear, the primary projections do not just disappear after their initial creation; rather, they remain operant, and form a system of communication channels along which the secondary projections can propagate. We said above that the secondary projections include those that are responsible for the fundamental interactions. An example of a secondary projection that is not a fundamental interaction per se would be the creation of an electron-positron pair from a photon --which can be described as the (conservative) secondary forward projection of the photon energy, along the line of p1, from level 1 to level 2. Similarly, the creation of a protonantiproton pair from a photon can be described as the (conservative) secondary forward projection of the photon energy, along the sequence (p1, p2), from level 1 to level 3. 33 4.10 Postulates 5 and 6 Recent developments in the model suggest the need for two more postulates: Postulate 5: Within the infrastructure of "nodes and edges" that is constructed by a sequence of primary projections, secondary projections may be propagated in all directions (backward, forward, and lateral/intralevel). Postulate 6: A projection of any type, to a given level, involves the input/communication of energy onto that level, which constitutes an object/entity (e.g. a "particle", real or virtual) at that level. 4.11 Constructing position, velocity, acceleration, and spin properties for objects/particles in system P We have already described how the projection p0 "looks" from the perspective of p1: as an isotropic, homogeneous three-dimensional space (i.e. ordinary space, or S01p 3 ). We also found that the process of constructing ordinary space cannot leave p0 (or p1) manifesting as a vector, or having a favored position, within that space. So, in the latter sense, p0 (p1) is located everywhere in ordinary space. We now ask, how does p0 look from the perspective of (an observer at) level 0? The answer (by postulate 3) is simply that p0 is a one-dimensional vector. Furthermore, by the scope rule, this interpretation/meaning of p0 (from the perspective of level 0) is available/operant everywhere within system/world P (whereas the three-dimensional meaning from the perspective of p1 is only operant at level 2 and above, since p1 itself is only operant at those levels). Consider now an object/particle, A, of world P; in particular, let A be an object at level 2 or above (e.g. an electron or proton), which thus has, at minimum, p0 and p1 in its sequence/genome. Consider also an object, B, of level 1 or above; i.e. an object that has, at minimum, p0 in its genome. In general, A's p1 component sees p0 as ordinary, isotropic, homogeneous three-dimensional space; and, specifically, it sees B's p0 component (and thus B itself) as being uniformly distributed throughout this space --and thus, in this sense, having every possible position, but no particular position within ordinary space. By the scope rule, however, the meaning of B's p0 component from the perspective of level 0 is also available to object A; to wit, from this perspective, A sees B's p0 component as a one-dimensional vector. We interpret that this meaning of p0 is just the position vector of B with respect to A. Since A is any object at level 2 or above, then every object at level 2 or above generates a position vector for B in the same way. And since B is any object at level 1 or above, then position vectors are generated in this way for every object at level 1 or above. In general, a position vector so constructed may be denoted by the 34 symbol rp 0, where the "p0" in the subscript indicates that its construction depends on the presence of the projection p0 in the object B. We note the following about the position vector, rp 0, described above: • Since it is the position vector of "B with respect to A", then rp 0 is a relative position vector. • The process of constructing an object's position in ordinary space involves two different perspectives of the p0 projection: (1) The perspective of p0 as seen from p1, which creates the ordinary space itself, and sees objects as being "smeared out" over all of this space; and (2) the perspective of p0 as seen from level 0, which creates the object's rp 0 position vectors with respect to other objects. It is the combination of these dual perspectives, I propose, that yields the "waveparticle duality" of objects. And we might further say that the "wave" aspects of an object come mainly from (1), and the "particle" aspects come mainly from (2). • Since p0 is independent of ordinary space (as established earlier), then an object within system P is not inherently associated with (or bound to) any particular position within that space. So the magnitude and direction of an rp 0 have no inherent values, and therefore have a range of possible values; that is, the magnitude and direction of an rp 0 are variables. Combining this with the first bullet point above, we conclude that the magnitude and direction of an rp 0 position vector are relative variables. • The process of constructing a relative position vector (rp 0) is dependent on the perspective of the "observer" at level 0. Consequently, an object in system/world P (the physical universe) has no rp 0 position vectors independent of this observer. • The construction of an rp 0 position vector for an object depends on that object having a p0 component; thus, for objects that lack a p0 component (i.e. objects at level 0), construction of rp 0 relative position vectors does not occur. That is, the relative position vector, rp 0, is only supported for objects at level 1 or above (where p0 is operant). It follows from the last bullet point that an object at level 0 either has no position property at all, or it has a position property that is different from rp 0. We will now see that the latter is the case. Let us use the symbol r0 to refer to the hypothetical position vector for an object at level 0. To sort out the nature of r0, we first note that, due to the scope rule, an object at level 0 is operant throughout system/world P; so the "position" of such an object is everywhere within world P. Thus, from the perspective of an object at level 2 or above, the direction of r0 is every direction within ordinary space. So, in that sense, the direction of r0 is (like that of rp 0) a relative variable. On the other hand, since an object at level 0 is operant everywhere within world P, then the "distance" to that object is always exactly zero. So, in that sense, the magnitude/length of r0, being always zero, is an absolute constant. Thus we might say that r0 is the zero vector, i.e. r0 = 0. 35 In summary: In terms of their directions, both rp 0 and r0 are relative variables. In terms of their magnitudes, however, rp 0 is a relative variable, but r0 is an absolute constant (with a value of zero). In the above analysis, the relative position vector (i.e. the position vector of B with respect to A) is constructed when A references the (perspective of the) observer at level 0 to interpret the meaning of B's p0 component. Likewise, object B can (without help from A) reference the observer at level 0 to interpret the meaning of its own p0 component. The result, again, is that p0 is a one-dimensional vector. However, since this interpretation of p0 (as a one-dimensional vector) is not relative to another object (A) within system P, then it will yield a property that appears to be intrinsic to B. To wit, it should yield an intrinsic or internal axis (or orientation) for object B. Since B is any object at level 1 or above, then every object at level 1 or above will have such an internal axis. We interpret this internal axis to be the property known as spin. I have not as yet worked out why, according to the present model, some objects/particles should have half-integral spin (i.e. fermions), and others should have integral spin (i.e. bosons), but the following is likely to be an important clue: In the present model, all of the fundamental fermions are created by forward projection, and all of the fundamental bosons are created by non-forward projection. Let us now proceed to the succeeding projection, p1, and ask: what is the meaning of that projection from the perspective of (an observer at) level 0? The answer is that (like p0) it is a one-dimensional vector, and (due to the scope rule) this meaning of p1 is available/operant throughout system/world P. Now consider two objects, A and B, both at level 2 or above, which thus have (at minimum) p0 and p1 components in their genomes. By the scope rule, the meaning of B's p1 component from the perspective of level 0 is available to object A; to wit, from this perspective, A sees B's p1 component as a one-dimensional vector. But what property should be associated with this p1 vector? Since p1 is subsequent to p0, then, in this sense, the observer at level 0 (and thus also object A) may see p1 as being "attached" to the end of the position vector associated with p0 (i.e. rp 0) --which evokes the image from classical mechanics of the velocity vector of an object being attached to the end of its position vector, thereby suggesting that the property of velocity might be associated with p1. This association is further supported by the following argument. Recall that we earlier associated p0 with energy, and p1 with time, the product of the two being action. Since we have just associated p0 also with the position variable, then (in order for the product of the properties associated with p0 and p1 to be action) p1 must be associated with momentum, and thus with velocity. We therefore interpret that the meaning of B's p1 component, as seen by A (applying the perspective of level 0), is just the velocity vector of B with respect to A --to be denoted as vp1, where the "p1" in the subscript indicates that the construction of this velocity vector depends on the presence of the projection p1 in the object B. Since A is any object at level 2 or above, then every object at level 2 or above generates a velocity vector (vp 1) for B in the same way. And since B is any object at level 2 or above, then 36 velocity vectors/vp 1s are generated in this way for every object that is at level 2 or above (where p1 is operant). We note the following about the velocity vector, vp 1, described above: • Since it is the velocity vector of "B with respect to A", then vp 1 is a relative velocity vector. • By applying the pattern that was established by our interpretation of rp 0, we interpret that both the direction and magnitude/speed of the velocity vector vp 1 are relative variables. And we note that this is indeed the case for objects that have p1 in their genomes (e.g. leptons and baryons). • The process of constructing a vp 1 velocity vector is dependent on the perspective of the "observer" at level 0. Consequently, an object in system/world P (the physical universe) has no vp 1 velocity vectors (nor corresponding momentum vectors) independent of this observer. • The construction of a vp 1 velocity vector for an object depends on that object having a p1 component; thus, for objects that lack a p1 component (i.e. objects at levels 1 or 0), construction of a vp 1 relative velocity vector does not occur. That is, the relative velocity vector, vp 1, is only supported for objects at level 2 or above (where p1 is operant). It follows from the last bullet point that an object at level 1 either (a) has no velocity property at all, or (b) it has a velocity property that is different from vp 1. But an object at level 1 has the position vector rp 0, which is a variable; and variable position implies a velocity property. So we rule out (a), and conclude that (b) must be the case. Let us use the symbol vp 0 to refer to the hypothetical velocity vector for an object at level 1. If we now apply the pattern that was established by our interpretation of r0, we get the following result: for objects at level 1 (e.g. photons, gluons, and gravitons) the direction of the velocity vector vp 0 should be a relative variable, but the magnitude/speed should be an absolute constant. Of course, we know that this is actually the case for photons, and is supposed to be the case for gluons and gravitons, for all of which the constant speed is just the speed of light, c. However, for dark energy (a level 1 entity created by forward projection) this constant speed is likely zero.15 Let us now use the symbol v0 to refer to the velocity vector for an object at level 0. If we apply the patterns above, we get the following result: for an object at level 0, the magnitude of its velocity (i.e. its speed, |v0|) should be an absolute constant. Indeed, this is supported by the following analysis: Due to the scope rule, an object at level 0 is in effect/operant/available at all locations/positions within ordinary space. So, in this sense, 15 That is, since dark energy is a single entity spanning all of ordinary space, then it cannot really move within that space; and so its velocity is zero. But even if we divide the dark energy into "elements", then an element of it that might be moving at velocity v with respect to a given point in space is always matched by an element of it that is moving with velocity -v ; the sum of the velocities of all such elements thereby being zero. 37 the position of an object at level 0 is all of space. Since its position is everywhere in space, and this position is a constant, then such an object does not "travel" or propagate from one point in space to another --for, it is already there. In this sense, its speed is zero. Yet, because an object that is created (at a particular point in ordinary space) by backward projection onto level 0 becomes instantly operant/available everywhere in space, then, in this sense, its speed is infinite. Thus, an object at level 0 has an absolute constant speed of zero, but an object that is backward projected onto level 0 has an absolute "constant" speed of infinity.16 We can summarize the basic kinematic results above in the following statement: From the perspective of an observing object A that has p1 in its genome, for k = 0, 1: Wherever pk is present/operant (e.g. in an observed object B), the magnitude of the kinematic property associated with it is a relative variable; conversely, wherever pk is not present/operant, the magnitude of the kinematic property associated with it is an absolute constant. (Where the kinematic property associated with p0 is position, whose magnitude is distance; and the kinematic property associated with p1 is velocity, whose magnitude is speed.) At this point we recall that projections can be either forward or non-forward, and that the latter type is the source of fundamental interactions or forces, with accompanying accelerations. Since the indented statement above deals only with the presence or nonpresence of forward projections (p0 or p1), it tacitly assumes the absence of nonforward projections and/or their effects, which means that it assumes the absence of their resultant forces or accelerations. Making this assumption more explicit, for the case of k = 1 (p1) we could rewrite the indented statement above as follows: In the absence of any acceleration of the observing object A (having p1 in its genome), an observed object B at level 1 (thereby lacking p1 in its genome) has an absolute constant speed. Since photons are level-1 objects, and a nonaccelerating observer is an inertial observer, this statement yields the second postulate of special relativity. Now, for the succeeding projection, p2, we ask: what is the meaning of that projection from the perspective of (an observer at) level 0? As usual, from that perspective, p2 is a one-dimensional vector; and, following the pattern above, it is suggested that we associate this meaning of p2 with an acceleration vector, for which both the direction and magnitude are relative variables. Apparently, then, an object can have a (physical) relative acceleration property simply by virtue of having a p2 component, i.e. without the involvement of nonforward projections. Consider now an object A at level 2, and an object B at level 3. Object A, applying the perspective of level 0, will then see object B's p2 component as a relative acceleration vector/variable. However, as indicated earlier, p2 is confined within the internal spaces of 16 Note that backward projection onto level 0 may yield an exception to the following statement from section 2: "...nothing that comes into being with the construction of the physical universe exists at level 0". 38 objects at level 3 and above (e.g. baryons), or within the residual/internal spaces of mesons at level 2. It follows that, outside of those internal spaces, i.e. within the ordinary space of our experience, where p2 is not operant, such a relative acceleration variable is not supported, and so physical acceleration is an absolute constant there.17 As before, since this result deals only with the presence or nonpresence of the forward projection p2, it tacitly assumes the absence of nonforward projections and/or their effects, which means that it assumes the absence of (net) forces caused by those nonforward projections. This allows us to make the following statement: In the absence of (net) forces, an object within the ordinary space of our experience has an absolute constant acceleration. If we then assume the value of that constant to be zero, this statement becomes simply the law of inertia, also known as Newton's first law of motion. In summary, physical relative acceleration is confined to the internal spaces of objects such as baryons and mesons; outside of those internal spaces, i.e. within the ordinary space of our experience, acceleration is absolute. This result comports with the fact that the physical acceleration of an object within ordinary space (as measured by an accelerometer) is independent of external observers, and is thus absolute (not relative). (In contrast, the position and velocity of an object in ordinary space can, as described above, be absolute or relative, depending on the absence or presence of the projections p0 and p1 in its genome.) 4.12 Revisiting mass Suppose we define mass as "energy that has relative position and relative velocity". Since we have associated the former kinematic property with p0, and the latter with p1, then it seems appropriate to associate mass with the conjunction of p0 and p1 --as we did earlier (in section 4.4.1), denoting it as p1p0 . This conjunction and its relation to mass might also be interpreted in the following way: Since p1 is dependent on p0, we could say that p1 is coupled to p0; or, rather, we could say that p1 couples to the scalar energy field (E1p /dark energy) that we have associated with p0 (sections 4.3.1 and 4.3.3) --and that this coupling yields mass for objects at level 2 and above (where p1 is in effect). In this way, the present model might account for some aspects of the Higgs mechanism. 4.13 Revisiting gravity Recall our earlier result that (the emission phase of) gravity should be attributed to a nonforward projection of p0 energy --either backward projection onto level 0, or lateral/intralevel projection onto level 1. However, if gravity were attributed to backward 17 The indented statement above for the cases k = 0 and 1 can then be extended to the case k = 2, where the kinematic property associated with p2 is acceleration. 39 projection of p0 energy (onto level 0), then (as concluded in section 4.11) the graviton would have a speed of infinity. But, of course, gravity actually has a finite speed --indeed , all indications are that it propagates at the speed of light, c [14]. Apparently, then, gravity cannot be the result of backward projection onto level 0; and so it should be the result of lateral/intralevel projection of p0 energy onto level 1. This means that the graviton is a level-1 object, with only a p0 component, thereby making its position a relative variable, and its speed an absolute constant. Accordingly, we ascribe the following aspects/properties to the graviton: • It has only a p0 projection, which may be denoted as p0,g. Since it comes from p0 energy, and retains its p0 component in the lateral (i.e. nonbackward) projection, then it (the graviton/p0 ,g) also consists of p0 energy. • It (i.e. its p0 ,g projection) is subsequent to, and thus dependent on, p0 --since it is a secondary projection of p0 energy, which is subsequent to p0 itself. • It (i.e. its p0 ,g projection) is prior to, and thus independent of, p1 (since it springs from intralevel projection onto level 1 --not backward projection from level 2 or above, as with the photon and gluon). Note that, even if p1 and level 2 did not exist, intralevel projection of p0 energy onto level 1 (as the graviton/p0 ,g) would still occur at level 1 (obviously without any help from, or dependence upon, p1). • We thus have the following order of priority: p0, p0,g, p1. The first, third, and fourth bullet points indicate that, from the perspective of p1, the graviton's p0 ,g projection will "look" much like the p0 projection; i.e. the graviton/p0 ,g will manifest to p1 as a form of p0, albeit with less priority (as indicated in the last bullet point). And we have already determined that p0, from the perspective of p1, manifests as ordinary, three-dimensional space, S01p 3 . Thus, due to its likeness with p0, it is proposed that the presence of a p0 ,g projection (i.e. the presence of a graviton) will also make a contribution to the construction of ordinary space. That is, the p1 projection will combine the inputs of p0 and (if present) p0,g to construct a hybrid space, to be denoted as S01p ,g 3 , which may be considered as just the regular S01p 3 space modified by the presence of the p0 ,g projection (i.e. the graviton). In what way does the presence of the graviton/p0 ,g modify the regular S01p 3 space? Recall from section 3.4 that the basic construction of ordinary space leaves p0 and p1 without position, orientation, and direction properties within that space, which thereby leaves no way to establish any special/preferred position or direction --with the result that (as basically constructed) ordinary space is perfectly isotropic and homogeneous. In contrast, a graviton/p0 ,g does have position, direction, and orientation within ordinary space: its initial position is the same as the position of the object (e.g. proton or electron) that emits it; upon emission, the graviton travels in some direction within ordinary space; and it also has spin. So, from the perspective of p1, the graviton/p0 ,g is a form of p0 that has position, direction and orientation within ordinary space, and thus does establish a 40 special position, direction, and orientation; that is, the presence of a graviton/p0 ,g has the effect of modifying space from isotropic and homogeneous S01p 3 into anisotropic and inhomogeneous S01p ,g 3 . Moreover, recall that the p1 projection, as seen from the perspective of p0, yields S10p 1 , or time. Since the p0 ,g projection of the graviton will have its own perspective of p1, then it is proposed that the combination of this perspective with that of p0 will produce a time, S10p ,g 1 , that is different from S10p 1 . The present model thereby complements general relativity by explaining why energy (with its emission of gravitons) affects the properties of space, time, and spacetime. The results above allow the following description of a gravitational interaction (which may be taken as an update of, or alternative to, the one given in section 4.8.4). Consider an observer A at level 2 or above, i.e. an observer for which the projections p0 and p1 are operant. As usual, p1 sees p0 as regular, isotropic, homogeneous, threedimensional space, S01p 3 . Consider also an object/particle B at level 1 or above, having p0 energy, which emits a graviton in the direction of A. When this graviton arrives at A, it will be processed as follows: from the perspective of p1, A will combine the p0 projection with the graviton's p0 ,g projection to construct a resultant, hybrid three-dimensional space S01p ,g 3 . In contrast to S01p 3 , the directional/positional aspects of the graviton's p0 ,g projection within that regular space will cause the resultant S01p ,g 3 space to be anisotropic and inhomogeneous. In addition, A will combine the views of p1 from the perspectives of both p0 and p0 ,g, thereby transforming from the regular time S10p 1 to the altered time S10p ,g 1 . 4.14 Backward projection onto level 0 From the results above, it appears that the gravitational interaction does not utilize backward projection onto level 0. What, then, might that form of projection produce? In sections 4.7 and 4.9.1 we found that backward projection is about interaction/communication between existing entities/particles. Thus we should expect backward projection onto level 0 to yield a new type of interaction or communication. What can be said about this new kind of interaction? In section 4.11, we concluded that an object created (at a point within ordinary space) by backward projection onto level 0 becomes instantly operant/available everywhere in space; so, in this sense, its speed is infinite. It is proposed, therefore, that backward projection onto level 0 produces instantaneous interaction/communication within system/world P, i.e. the physical universe. Recalling our previous result that the observer at level 0 plays a key role in constructing properties such as position, velocity/momentum, and spin, it is suggested that backward projection onto level 0 is a means by which this observer can also act as a mediator for performing instantaneous, system-wide "bookkeeping" in regard to the 41 values of those properties --as exhibited in, e.g., EPR/entanglement-type experiments. Indeed, the object that is backward projected onto level 0 is an observer at level 0, which might thus by itself play the role of mediator in this respect. Stating all of this in an alternative way, we might say that, since level 0 is the origin and boundary of system/world P, then backward projection onto level 0 might be a way of setting boundary conditions that are instantaneously operant throughout that world. Such boundary conditions might then instantaneously "collapse" a family of possibilities/solutions into a particular outcome. Of course, it is just something like this that appears to be happening in the "quantum measurement phenomena". Just as all energy forms that are native to level 2 or above (i.e. p1 energy, p2 energy, etc.) may backward project onto level 1, so also we might expect that all energy forms that are native to level 1 or above (which is to say, all energy forms within system P --i .e. p0 energy, p1 energy, p2 energy, etc.) backward project onto level 0. In this respect, instantaneous communication via backward projection onto level 0 might be a truly universal mode of interaction, performed by all energy types within system P, including quantum vacuum energy (in contrast to gravity, which, as we found, excludes quantum vacuum energy as a source). Backward projection of p0 energy, p1 energy, and p2 energy onto level 0 may be denoted by -p0, (-p1, -p0), and (-p2, -p1, -p0), respectively. A complete interaction via backward projection onto level 0 may thus be described as follows: Energy associated with object A backward projects onto level 0, producing an object at that level. Due to the scope rule, this object is then forward projected throughout all of ordinary space, to every object B. All of this happens instantaneously. Since level 0, and the object at level 0, likely do not (or cannot) distinguish between objects A and B, then the object at level 0 is also forward projected to A, producing a kind of echo or feedback effect. However, because the whole interaction is instantaneous, there is no time delay between emission of the object by A, and feedback to A. 5 The physical universe is a meaning circuit The construction of meaning in system P seems to always involve some application of the postulates, including (as derived from postulate 4) the scope, nesting, and inheritance rules. Recall, for example, that p1 constructs the meaning "p0 is a three-dimensional space" by first applying postulate 3, which says that "p0 is a one-dimensional vector", and then applying postulate 4, which says that "p0 is independent of p1". Moreover, it was shown that the isotropy and homogeneity of that same space are properties/meanings that come from the application of postulate 4. To delve deeper into the semantic process for system P we therefore ask: where do the postulates themselves come from? Given their overarching scope in the construction of system P, the postulates must be prior to the sequence (p0, p1, p2, p3), and so must be operant/present at level 0 itself. The postulates thus come into system/world P via level 0. 42 So, application of the postulates to construct meaning in system P is an indirect reference to level 0. And recall that the position, velocity, and spin properties of objects (and their values) are meanings that are constructed through direct reference to (the perspective of) level 0, or the observer at level 0. For the construction of meaning in system P, therefore, all roads apparently lead to level 0. We can thus summarize the basic construction process for system P as follows: a) The projection sequence (p0, p1, p2, p3) springs from level 0. b) Each entity/observer which is constructed by that sequence --e.g. each projection, composite of projections, or object/particle --refers back to level 0 (directly, or indirectly via the postulates) to construct meaning. System/world P (the physical universe) thereby constitutes a Wheelerian meaning circuit [15], which begins at level 0 and then, within the system, via the meaning-producing operations of each entity/observer constructed by the sequence (p0, p1, p2, p3), loops back to level 0. Clearly, then, there are many "observers" in the system-P meaning circuit: every object/particle/entity, and each of its component projections, is an observer that plays a role in constructing world P, and is thereby in the loop of its meaning circuit. But the central role (or linchpin, if you will) in the system-P meaning circuit is played by level 0, or the observer at level 0, since every subsequent entity in system P has in common that it refers back to that level to construct meaning. (The nature of "the observer at level 0 of system P" is developed further in Part II [5].) Since level 0 is also the origin and boundary of system P, then to reference level 0 (directly or indirectly) may be tantamount to applying/invoking initial and boundary conditions on the system to construct meaning. 6 The construction of the physical universe is a type of logical derivation Consider a derivation in the logical system known as natural deduction (ND)18, containing a sequence of several steps. The derivation begins with a state that may be called level 0, at which the inference rules of the system are operant and a set of zero or more initial propositions/assumptions/premises are stated. The first logical step of the derivation then generates level 1; the second step generates level 2; and so on. Each successive step in the sequence contains propositions that are inferred (via the inference rules) from those that are prior. A proposition at a given level can be asserted at (i.e. has within its scope) all subsequent levels. So, the initial premises (if any) can be asserted at level 1 and above; a proposition at level 1 can be asserted at level 2 and above; and so on. 18 Natural deduction is (since the 1950s) the type of logic most often taught in introductory logic books, where the modifier "natural" is said to indicate that it follows the method/pattern/paradigm by which our minds perform deductive reasoning [16, pp. 2-3], [17], [18]. 43 In this way, subsequent levels inherit everything from, and are nested within the scope of, prior levels. The scopes of the premises and derived propositions can thus be pictured as a matryoshka-doll-like set of concentric spheres, with each successive scope embedded/contained within the scopes of its priors. Thus the scope, inheritance, and nesting rules that are properties of system P are also properties of system ND. In other words, systems P and ND have many similarities in structure and function, with projections in system P corresponding to logical steps in ND. For this reason, we say that the construction of system/world P (i.e. the physical universe) is a type of logical derivation. 6.1 Systems without postulate 3 Our development of system P included the use of postulate 3, whereby the projections are considered to be vectors. In section 4.11, it was shown that the characterization of p0 as a vector was crucial for establishing the property of relative position (i.e. the position of one object relative to another). Since the relative-position property is presumably necessary for there to be geometry, then we might say that the incorporation of postulate 3 endows the projections, and their corresponding world, with a geometric character. Given the results above regarding the logical character of system P, we can say that a system created with postulate 3 is both logical and geometric in character. Conversely, in a system that omits postulate 3, or at least omits the requirement that the projections be vectors, objects will have no relative position vectors, and so the system will presumably be lacking in geometry. Consequently, such nonvectorial projections, and the systems or "worlds" that they generate, may be described as nongeometric. Notwithstanding this lack of geometry, the nonvectorial projections in a sequence still obey the scope, nesting and inheritance rules (which, after all, stem from postulate 4, not postulate 3). If we then include the rules of inference for propositional and predicate calculus, and identify some projections as "hypotheses/suppositions"19, then such a system will be formally similar to, or possibly the same as, the system of natural deduction (ND). We therefore say that such nonvectorial/nongeometric systems, formed by the omission (or stated modification) of postulate 3, have a purely-logical character. As with their vectorial counterparts, we assume that the nonvectorial projections come in primary and secondary versions, and that the former type (primary-nonvectorial) produces new infrastructure (of levels, and connections between them), whereas the latter type (secondary-nonvectorial) utilizes existing infrastructure as "nodes and edges" on which to propagate --in the backward, forward, and lateral/intralevel directions. A sequence of primary-nonvectorial projections may be denoted as e.g. (u0, u1, ..., um), where the nonbolding of the component projections indicates that they are not vectors. We can think of this sequence as creating the system U, consisting of m+1 levels, where the nonvectorial projection u0 creates level 1 of the system, the projection u1 creates level 2, and so on. Similar to their vectorial counterparts, secondary-forward-nonvectorial 19 The nature of projections that we identify as "hypotheses/suppositions" is specified in Part II [5]. 44 projections (within system U) may be denoted as u0, u1, u2, etc.; and secondarybackward-nonvectorial projections may be denoted as -u0, -u1, -u2, etc. Such nonvectorial/nongeometric/purely-logical projections show their relevance in Part II of this paper [5]. 7 Making sense of the projection types A projection type in system P can be designated by selecting one term from each of the following three groups of projection characteristics (with some restrictions, noted below): 1. Primary; secondary. 2. Forward; backward; lateral/intralevel. (Called the direction.) 3. Vectorial; nonvectorial. Thus, the designation of a specific projection type is given by a triplet of these terms, such as: primary-forward-vectorial (which is the type that constructs the infrastructure of world P); or secondary-backward-vectorial (which is the type that the present model holds to be responsible for the emission processes in the strong and electromagnetic interactions, and the emission process of instantaneous entanglement interactions); or secondary-forward-vectorial (which is the type that is responsible for the absorption processes in the same interactions); or secondary-lateral-vectorial, which is the type that is responsible for both emission and absorption in the weak and gravitational interactions. Note, however, that the type designation "primary-forward-vectorial" is actually redundant, since, as stated above, all primary projections are forward (i.e. there is no such thing as a backward primary projection); so this designation can be stated equivalently as "primary-vectorial". Likewise, all backward projections are necessarily secondary projections (since backward projections do not create new infrastructure of edges/lines and nodes; rather, they propagate along existing lines, and onto existing levels/nodes); so, for example, the designation "secondary-backward-vectorial" is equivalent to "backwardvectorial". And all lateral projections are secondary; so, for example, the designation "secondary-lateral-vectorial" is equivalent to "lateral-vectorial". Lastly, we may omit the hyphens when writing these designations, and change the order of their components. 8 Conclusion From six very simple postulates, we have managed to provide basic explanations for many hitherto unexplained physical phenomena: the (3+1)-dimensional structure of space and time; inflation (its beginning and ending); the quantum of action; dark energy; the small value of the cosmological constant; quark construction and confinement; the construction of position, velocity, and spin properties; etc. In addition, as quantum physics seems to demand, the model elevates observers to key roles in constructing the world, and provides a means for instantaneous communication/influence across all of 45 space, by which the values of particle properties might be regulated. In the sequel to this paper, titled "A Model for Creation: Part II" [5], the present model is generalized and applied to the construction of a hypothetical system that is prior to system P, the physical universe. Once the form of this prior system is worked out (its structure is essentially identical with that of system P), an extensive interpretation of it, and its entities, then follows. Acknowledgments I would like to acknowledge the following people for enabling the research for this paper to take place: My late mother and late father, Margean A. White and Phillip B. White; my brother, Jeffrey C. White; John P. A. Higgins; and the late Louise L. Hay (The Hay Foundation). References [1] J. A. Wheeler, "It From Bit", in his At Home in the Universe (AIP Press, 1994). [2] J. A. Wheeler, "The Computer and the Universe", Int. J. Theor. Phys. 21, Nos. 6/7, (1982). [3] P. Davies, N. H. Gregersen (eds.), Information and the Nature of Reality (Cambridge University Press, 2010). [4] J. 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