Surmounting the Cartesian Cut: Klein Bottle Logophysics, The Dirac Algebra and the Genetic Code

Abstract

We reintroduce the Klein Bottle (KB) logophysics at the foundations for the unification of quantum geometry, cell biology, embryology and evolution, and stereochemistry, to extend it to the genetic code and allosteric recognition. We find that the genetic code has three possible topologies: two different fractal families of KBs embedded in HyperKBs, or still a 2-torus, depending which pair of three subalphabets for the genetic letters is chosen. We derive this from the 4-state KB Logic (KBL), and construct a digital representation for the genome. We complete the genetic codification of embryological differentiation introduced in an accompanying article. We discuss the Hadamard-KB representation of this codification and alternatively in terms of the golden mean, and discuss its robustness with respect to environmental noise. We discuss the relation with the double helix model, and in particular with the isomorphic representation of the 64 codons in terms of the Dirac Algebra of Quantum Mechanics, in the framework of the Nilpotent Universal Computational Rewrite System (NUCRS), a self-referential syntax that generates much of mathematics and the fundamental symmetries of physics in terms of hypernumbers. We show that the double helix can be obtained from the duplication of the KB associated with the bistable perception of the tetrahedron representation of the four fundamental genetic letters. Thus, the Dirac Algebra and its double tetrahedron codification introduced in NUCRS, is associated with a pair of KBs, yielding two remarkable structures: In one hand the double 3d space (space coordinates and momentum), and in the other hand the double helix, which thus appears related to a pair of KBs, being the case that only one is necessary to construct the genome; we show that the depth-time variable materializes as space Thus, the Dirac Algebra is derivable from two KBs, and the purported dualism of physics as presented in NUCRS, is rendered to be based in the non-dualistic KBL. We discuss the relations with the theory of evolutions, and in particular, with the possible relations between biological periodicities and the KB of the periodic table of elements including the stable nuclides, due to Boeyens. We present ideas on evolution in terms of time waves defining in toto the different structures. We present the basis for the ontoepistemology of the Hyper KB Logic and discuss its relations with the Anthropic Principles.