The GUT Group from Recursion Closure Geometry

The running couplings paper identified grand unification as the density thresholdDGUT at which the three Standard Model recursion mode frequencies become equal.This paper identifies the grand unified group from the phase rotation symmetry ofthe recursion medium at DGUT. The argument proceeds from three geometric factsestablished in this series: (1) neutrinos are incomplete n = 6 closures with leakageparameter ϵ ≪ 1 at the electroweak scale; (2) at DGUT, the closure leakage ϵ → 1and partial closures become equivalent to full closures; and (3) the complete set ofrecursion closure modes at DGUT therefore includes both full closures (quarks, chargedleptons) and formerly-partial closures (neutrinos, now promoted to full status). Thiscomplete set maps precisely onto the 16-dimensional spinor representation of SO(10) —the unique irreducible representation that accommodates all Standard Model fermionsplus one right-handed neutrino per generation. SO(10) is therefore identified as thenatural GUT group of the Cohesion UFT: it is the full phase rotation symmetry ofthe recursion medium at DGUT. The seesaw mechanism for neutrino mass emergesgeometrically from the relationship between the closure leakage at the electroweak scaleand the full closure at the GUT scale. Baryon minus lepton number B − L is identifiedas the slip phase orientation. The proton decay predictions of SO(10) are consistentwith experimental lower bounds, unlike the minimal SU(5) which is experimentallyexcluded.