Torsional Dynamics of the N–Z Axis: Fiber Bundle Formalization, Toroidal Geometry of the G6–Q9 Cycle, and Structural Implications for DNA Helical Geometry and Chromosomal Polarity

The G6–X–Q9–N cycle, developed in the Kiawe Epistemological Framework (KEF), describes structural transitions between two complementary states: G6, centripetal generative potential, and Q9, centrifugal manifest form. In Version 1 of this paper, the operator X was derived as the emergent product of the N–Z oscil- lation rather than postulated as a primitive, and the Möbius structure of the cycle was identified as an emergent configuration produced by this torsion. The present Version 2 provides three extensions. First, a fiber bundle formalization: X and N are shown to be realizable as the holonomy of a connection on a non-orientable prin- cipal Z/2Z-bundle over the parameter circle S1, providing the differential-geometric counterpart to the algebraic construction of prior work. Second, a toroidal geom- etry of the complete G6–Q9 cycle: the four operators generate a toroidal struc- ture T 2 in which G6 is the internal vortex and X–Q9–N constitutes the exter- nal manifold. Third, a curvature-based telomere shortening model: the prediction ∆T (n) =−α τ (n) + β D(n) is reformulated in terms of the curvature of the bundle connection, where τ (n) = |F∇(s, n)| ds is the integrated torsional stress at the telomeric terminus. Topoisomerases are identified as gauge transformations reduc- ing the curvature F∇, and the self-composition law (N ◦X)·A·UK = A2 ·UK acquires a geometric interpretation as a holonomy theorem. The chromosomal XX/XY po- larity is extended to the toroidal reading: XX as a closed bundle (complete torus, Z/2Z structure group), XY as a bundle with an open directional axis (SO(2) struc- ture group).