In this research investigation, the authors present some important applications of the rigor presentment in the universal representation theory, rigor I, II, and III. These include—outline solution to the Poincare’s topological connectedness conjecture (which states that all points of an n dimensional space volume are connected (through all the dimensions from 1 through n) by one curve given by a metric of universal nature, special curves connecting any two positive real numbers, net slope or cadence change optimization along the special curve, generic representation of special curves analysis, an example—restrictions imposed on native continuum field manifestation in lieu of special curves-based compatibility condition connecting any two discretization (finite element mesh) nodes, quantum set of constitutive deformations, model validation, minimizing sample bias, centroid of a set of positive real numbers in primes basis, computation of a representative future average for a time series type of sequence of positive real numbers, sample design with minimum bias, flux emission extremization analysis based on the constraint of zero reversed time entrainment of the flux complement, universal recursive wave equation of the universe, star engineering and management, creation of a star, universal permittivity handler, static and dynamic worm-hole engineering, gravity interaction, perception window size, permittivity of free space for a graviton, universal permittivity modulator, design of the space–time potential well functions, design of a quantum tunnel, invisibility engineering, constructing special wave functions of space–time—an example of wave function of a photon, space–time bubble transportation and extension of the same to a worm-hole, some notes on time causality architecture, hyper speed time causality engineering, the modified Dirac type commutator that can explain the parallax error effects, and hyper-causal manifestation engineering.