The N4π Closure Invariant and Effective Boundary Length on a Growing Hyperspherical Time Front

This work introduces two distinct geometric quantities associated with compact closure structures on a growing hyperspherical time front. The first quantity is the closure invariant (N4π), which measures the number of complete (4π) orientation-return cycles required for a compact closure to recover its original internal state. The analysis suggests that stable compact closures are characterized by the condition (N4π=1). The second quantity is the effective boundary length (Leff). Unlike the closure invariant, which characterizes topological completion, the effective boundary length determines the metric scale associated with the closure and provides the corresponding mass scale through a Compton-type relation. A central result of the paper is the separation between topological stability and mass generation. The closure invariant governs whether a compact closure state is topologically complete, whereas the effective boundary length governs the characteristic mass scale. The framework is applied to electron-like, proton-like, antiproton-like, and neutron-like closure structures. The resulting mass scales are shown to be consistent with the observed electron and proton mass scales while maintaining the common topological closure condition (N4π=1). The work extends previous studies of SU(2)-locked delay closures and three-branch closure structures on a growing hyperspherical time front.