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In macromolecular self-avoiding random walk, movement of each chain-particle accompanies an instantaneous spin system with de Gennes n = 0 that provides extra energy, extra vacancy volume and relaxation time needed for chain-particles co-movement. Using these additional and instantaneous spin systems not only directly yields the same Brownian motion mode in glass transition (GT) and reptation-tube model, but also proves that the entangled chain length corresponding to the Reynolds number in hydrodynamics and the inherent diffusion - delocalization mode of entangled chains, from frozen glass state to melt liquid state, is a chain-size solitary wave with transverse ripplon-like soft wave. Thus, the order parameter of GT is found. The various currently available GT theories, such as Static Replica, Random First-Order Transition, Potential Energy Landscape, Mode-Coupling and Nanoscale Heterogeneity, can be unified using the additional and instantaneous spin system. GT served as an inspiration and continues to serve as the paradigm in the universal random delocalization transitions from disorder to more disorder until turbulence.

The existence of fixed point in self-similar Lennard-Jones (L-J) potentials has been proved based on the mosaic geometric structure theory of glass transition (GT) [J. L. Wu, Soft Nanoscience letters, 1, 3–86 (2011)]. A geometric local-global mode-coupling recursive equation, different from the current Mode-Coupling Theories, has been introduced to find out the non-integrable induced potential structure of boson peak at GT. The recursively defined variable in reduced recursive equation is the potential fluctuation of reduced L-J potentials associated with reduced geometric phase potentials. A series of results have been deduced directly at GT. (i) There are only 8 orders of molecule-clusters. (ii) Two orthogonally fast-slow reduced phase potentials, 3/8 and 5/8, are accompanied with density fluctuation and clusters hop-delocalization along 8 geodesics. (iii) The stability condition of potential fluctuation is the Lindemann ratio. (iv) A new reduced attractive potential of –17/16, lower than reduced potential well energy –1, occurs.