Projective space-time

All evidence points at a multi-dimensional, non-orientable structure, topologically equivalent to projective space-time. It is of interest to note that the same conclusion has been reached before on the basis of astronomical observation [229]. In two instances has the same pattern, defined by a cluster of quasars, been observed as distorted multiple images at different positions in the sky and interpreted in terms of multiply connected projective space. 

Analysis of the periodicity of atomic matter therefore guides us to a projective model of a closed imiverse in the double cover of four-dimensional projective space-time. Transport across the interface, or along the involution, results in the inversion of CPT symmetry. 

In all of these statements the emphasis is on unified fields and not on cosmology. As a matter of fact, the equivalence of the projective unified model to five-dimensional spaces in general, and to that of Einstein and Mayer in particular, was first demonstrated by Veblen himself (Monograph Chapter Vlll). The crucial observation is that this equivalence mapping is done in the tangent space, without implying the equivalence of the Einstein-Mayer five-dimensional construct with four-dimensional projective space-time. The five-dimensional spaces are not projective, but affine spaces. 

This observation is no proof of the topology of space-time but it shows how matter and antimatter can coexist, without mutual annihilation, in projective space-time. 

Gauge invariance, as formulated here, represents the elusive link between general relativity and quantum theory. It does not appear natmally in either theory and had to be introduced by special assumption. The amazing truth is that this link was in fact discovered many years ago and described in Veblen s (1933) monograph. It emerges naturally from relativity theory formulated in projective space-time. 

The imphed wave nature of elementary matter furthermore clarifies their mode of interaction through standing waves generated by the interference between advanced and retarded wave components. The negative-energy solutions of relativistic wave equations first indicated the existence of antimatter, as later confirmed experimentally. To avoid the annihilation of matter and antimatter on a cosmic scale an involuted structure of the vacuum, consistent with projective space-time, is inferred. 

It all hangs together. To account for such consilience, Plichta [6] conjectured that numbers have real existence in the same sense as space and time. A more conservative interpretation would link numbers, through the golden ratio, to the curvature of space-time. A common inference is that the appearance of numbers as a manifestation of the periodicity of atomic matter is due to a spherical wave structure of the atom. A decisive argument is that the fiiU symmetry, implied by the golden ratio, incorporates both matter and antimatter as a closed periodic function with involution, as in Fig. 9, in line with projective space-time structure. 

At this point, the utility of this property with respect to (P2) deserves attention. A careful look at P2 reveals that the shaded region in the projected space (for example, the X -X space) is exactly the projection on the X -X space of the feasible region of P2. The concave PFR projection defines the concentrations in segregated flow, and the interior is a convex combination of all boundary points created by the residence time distribution function. This gives a new interpretation to the residence time distribution as a convex combiner. For any convex objective function to be maximized, the solution to the segregated flow model will always lead to a boundary point of the AR. 

This makes the physical body a kind of vessel which "contains" higher dimensional constructs. In three-dimensional terms it is logically absurd for a vessel to be smaller than its contents, yet this is not an inevitable conclusion if other dimensions are factored into the equation. Another way to conceptualize this is to think of space/time as the "outer" projection of an inner infinity. Imagine what it would be like to be a two-dimensional entity living on one of the six faces of a cube. 

The assumptions on which the model rests are too crude to be realistic. In particular, the assumption of a universal time coordinate directly contradicts the basis of general relativity. To avoid the problem de Sitter repeated the calculation based on relativistically curved space-time, with the surprising result of an empty universe with variable radius. The traditional interpretation of this result as an expanding universe is not unique. It could just as well imply space-time with continuous curvature, characteristic of projective space. In the event, both solutions were soon superseded by an expanding-universe cosmology based on a Doppler interpretation of galactic redshifts. 

The predicted periodicity at Z/N = r corresponds with the empirically derived periodic table of the elements, naturally associated with the uniform curvature of low-density space-time. By inference Z/N = 0.58, the value at which the periodicity based on the quantum-mechanical energy spectrum of hydrogen is projected out, corresponds to flat empty space. 

Theories like those of Lemaitre or Friedmann, which predict an expanding universe, are all based on forcing an affine metric, such as the Robertson-Walker metric, on the projective geometry of space-time. This has the effect of splitting local Minkowski space into separate space and time coordinates, without the natural complex relationship that ties space and time together. 

The foregoing is interpreted to mean that the projective model of space is closed by a single surface that corresponds to the ideal plane at infinity. In Euclidean geometry this plane appears curved. If we therefore assume that the structure of the cosmos is subject to mathematical analysis and that the mathematics applies without exception, it is a logical necessity that the geometry of space-time be projective. 

As the total field is inferred closed in both the Z, as well as the Z/N directions, the Mobius model is incomplete and should be expanded into a projective plane, which cannot be embedded in 3-dimensionaJ space. Like the physical imiverse, the cosmic distribution of matter should then also be specified in fom--dimensional space-time. The reconstruction of Figures 5.4 and 5.7 can therefore, at best, be seen as a three-dimensional caricatme of the actual fom--dimensional distribution in the curved Minkowski space of general relativity. 

The coordinate system in which this metric is expressed is obtained by parallel projection along the a 4-axis in R". Topologically, this space-time could be seen as a 4-dimensional cylinder in Minkowskian R . 

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