Topology of World Space

The idea of a closed space-time manifold with an involution has been mooted on the basis of nuclear synthesis (figure 2.6), number theory (figure 2.8), historical argument (4.4), absorber theory (figure 4.8) and chirality (5.9.3). All of these schemes can now be combined into a single construct based on curved Thierrin space-time. 

An imperfect lower-dimensional analogue of the envisaged world geometry is the Mobius strip. It is considered imperfect in the sense of being a two-dimensional surface, closed in only one direction when curved into three-dimensional space. To represent a closed system it has to be described as either a one-dimensional surface (e.g. following the arrows of figure 7) curved in three, or a two-dimensional surface (projective plane) closed in four di- 

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On weighing the impartial evidence it becomes obvious that standard cosmology underestimates the age of the world by the same margin that it overestimates the size of the universe. Both factors relate to the Doppler interpretation of redshifts, which should be re-examined, and to geometry and the topology of space-time. The obvious alternative to an infinite Euclidean universe is a world closed in space-time. As originally pointed out by Weyl (1922)(p.278) ... 

The findings reported here provide new evidence for the unity of micro- and macrophysics and refute the perception of separate quantum and classical domains. The known universe exists as a four-dimensional space-time manifold but is observed in local projection as three-dimensional Euclidean tangent space that evolves in universal time. The observable world, at either micro- or macroscale, can be described in either four-dimensional (nonclassical) or in classical three-dimensional detail. The descriptive model may change, but the reality stays the same. This realization is at the root of self-similarity between large and small. The symmetry operator, which reflects the topology of space-time, is the golden logarithmic spiral. 

The same applies to protons, neutrons and more complex matter. In the physical world annihilation of matter and anti-matter is prevented by separation on an interface in the vacuum. Without such an interface matter cannot exist and this mandates a space-time topology that allows the appearance of this interface. 

A limitation of the scope of the method is that each equation is structure-class specific. In real-world examples that do not readily fall into a previously studied class, model selection is a problem. Two methods have been described. In one approach, all the unique atom-based fragments obtained from all structure classes studied are projected into n-dimensional space (six and seven dimensions) based on topological environment vectors. A vector is then calculated for each carbon atom in the query compound. That model equation which corresponds to the atom fragment in n-dimensional space which is closest to the query atom fragment (Euclidean distance) is selected for predicting its chemical shift. Alternatively, a neural network is trained to relate the chemical environment of the set of carbon... 

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