The DNA double helix presents three quantitative features that have been described independently without a common structural explanation: the integer quantisation of topoisomerase action (ΔLk = ±1 for Type I, ΔLk = ±2 for Type II), the antiparallel counter-rotation of the two strands generating B-form helical winding of 10.4–10.5 base pairs per turn, and the bipartite toroidal megadomain organisation of the inactive X chromosome documented by Hi-C, contrasted with the palindromic directional structure of the Y chromosome. We propose that these three features share a common geometric origin: the structure of a principal Z/2Z-bundle over the helical axis, generated by the counter-rotational dynamics of the N–Z axis. The N–Z axis describes the two complementary poles of DNA torsional orientation: N (the convergent, centripetal pole, instantiated by the 3'→5' strand) and Z (the divergent, centrifugal pole, instantiated by the 5'→3' strand). The oscillation between N and Z generates the torsional operator X with the involutive property X² = −Id, which determines the Z/2Z structure group of the associated principal bundle. Within this framework: topoisomerases are gauge transformations predicting ΔLk = ±1 (Type I) and ±2 (Type II), verified with 0% deviation; the relaxed connection curvature F∇ = 0 corresponds to helical winding 2π/10.5 rad bp⁻¹, deviation less than 1%; and the telomere shortening formula ΔT(n) = −α·τ(n) + β·D(n) is calibrated against documented shortening rates. The inactive X chromosome and Y chromosome are identified as closed and open bundle configurations respectively, consistent with Hi-C data.
Andrea Succi (Sat,) studied this question.