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# ! 3 has 3 bidirectional vectors & 1 dimensionless operator.

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We identify specific properties of the complex plane that allow functions of a continuous n-dimensional (Hilbert) measure space to be transformed into a well ordered counting sequence. We discuss proof strategies for problems in number theory (Goldbach Conjecture) and topology (Poincare ´ Conjecture) that suggest correspondence between the physical...

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## Citations

... Identity inversion in the most natural terms, then, 2 4 = 4 2 , informs us that transitivity implies identity. 5 So we take for granted that our 2-dimension measures adjoin 4 dimensions, where the "boundary of a boundary is zero." § Yet that's not what we experience. ...

... Thus, gravity is universal negative feedback, assured by recurrence of the discrete source state. This source is everywhere local and nowhere extended-a consequence of the choice function-which as shown in (Ray, 2006) is independent of observer choice. Local meta-stability is dependent on global instability in the form of random oscillations. ...

... g 00 g 01 g 02 g 03 g 10 g 11 g 12 g 13 g 20 g 21 g 22 g 23 3 Bekenstein and Mayo 9 made the 1-dimension case for equivalent past and future entropy, for if black holes are 1-dimensional, past and future information enters and leaves randomly, on one channel only. Cf. the Chaitin Ω 10 random result. ...

... l infinitely orientable metric on a self-avoiding random walk, a network of random-output computers ( ! " calculating machines) corresponds to quantum time intervals randomly orienting in an infinite dimensional (Hilbert) space-in which the principle of self-similarity forces an ordered direction of continuous time in the limits of the 3-manifold. [Ray. 2006] 3.3 In other words, the output of each machine is sensitively dependent on the machine's orientation in space-the idea being that the 3-space time vector at any node is uniquely rectified in the process of the n-space self avoiding random walk. One models a folding sequence on the continuously changing rectilinear path disclosed by the ...

... The sticking point is that if, as we claim, time is a physical agent driving change, the discontinuous and random reorienting of that complex metric on a scale of measure zero generates random information in an information-rich future state (n-dimensional Hilbert space [Ray, 2006]) 4.1.3 The present state, information-poor, is shaped uniquely by random n-dimensional motion. ...

... 2.1 In an earlier paper [Ray, 2006] we explored the physical consequences of a mathematical model of dimension self-organization, driven by the summation of cardinal points of discrete dimension sets: ...

... (*) 2.2 Figure 1 implies why this is so-the 24-1 summation of cardinal points up to non-redundant points of the tensor metric field leaves one 0-dimension point-because this singularity seeds the expansion of n-dimensional sets of cardinal points, it must recur infinitely often in the history of n-dimension evolving space, and because it is a singularity, it can only be treated by complex analysis, where the continuous n-dimension function length 1 term returns to zero (returns to itself) as a result of rotation in the complex plane. [Ray, 2006] We suppose that the cardinality of dimension D n contains, in each of its points D n!1 , the cardinality of the lower dimension. The warrant for this idea is from projective geometryduality between spaces that for dimension N with subspaces R is given by N- 2.3 Given this interpretation of time, we also see that d ! 4 (hyperspace) is the least ! ...

... These characterizations of background independence suggest an organic continuation of physics with the analytical mathematical model, the entire theme of [Ray, 2006]. ...

If time is n-dimensional continuous, n > 4, time has a physical meaning independent of space. We construct a theory, precisely supported by WMAP observational evidence and consistent with special relativity and quantum field theory, that gives time an active role in the cosmology and self- organized sustainability of the universe. Our complex system model finds that a precise dimension limit, d=10, supports string field theory in principle, and compels imaginary time and negative mass.

Can gravity be derived from first principles, independent of quantization? Because gravitation is intimately tied to dimensionality, we find a path to gravitational emergence ground-up, from a set theory model of self-organized spacetime. As a consequence, gravity has its origin in complex systems science.