Emergence of Three-Branch Closure Structures from a Growing Hyperspherical Time Front

This paper develops a geometric framework in which the expanding universe is represented as a growing hyperspherical front S^3. A local increment of front growth is decomposed into three equivalent tangent directions associated with the three-dimensional tangent space TpS^3.

Starting from isotropic delay redistribution, the unique symmetric state

(1/3, 1/3, 1/3)

emerges naturally from the three-dimensional tangent geometry. The first symmetry-breaking mode is then obtained as

(1, 1, -2),

representing the lowest traceless axisymmetric redistribution of the isotropic state.

The analysis establishes the geometric chain

S^3 -> three tangent channels -> (1/3,1/3,1/3) -> (1,1,-2) -> SU(2) -> S^3 -> S^2.

Within this framework, the characteristic three-branch closure structure appears as a direct consequence of localized delay redistribution on an expanding hyperspherical front. The resulting fractions are interpreted as geometric projection and orientation structures rather than as electric charges or Standard Model quantities.

The paper provides the geometric foundation for subsequent investigations of localized closure states, orientation spectra, and emergent particle-like structures.