Twisted Shell Topology and Spectral Closure: An Eight-State Phase-Return Construction

Abstract

This preprint develops a reduced geometric model in which a four-phase compact progression undergoes an orientation reversal after one complete traversal and returns to its full original state after a second traversal. The construction produces an eight-state phase-return orbit from four phase quadratures combined with two return orientations. Removing the uniform phase mode leaves a three-dimensional active sector, and projecting onto the return-restored sector produces a rank-three subspace within the eight-state shell. This gives a normalized active-shell fraction of three-eighths directly from the geometry rather than by fitting. The same three-dimensional active structure is represented through a tetrahedral tight frame and connected to the first nontrivial spherical harmonic sector. This sector has three active components and supplies a natural degree-one map on the sphere, giving a common geometric and spectral origin for several normalization factors used in later shell-based constructions. The paper also studies spectral selection on the twisted compact loop. Once the inactive periodic zero mode is excluded, the first twisted mode becomes the lowest admissible nonconstant branch and has one quarter of the quadratic gradient energy of the first nonzero periodic mode. This provides a spectral distinction between twisted and ordinary periodic closure within the reduced construction. A further result is the separation of distinct primitive closure classes. The paper shows that repeated powers of a single projector cannot generate all of the different coefficients appearing at different perturbative orders. The second-order return contribution and the fourth-order curvature contribution therefore represent distinct reduced structural classes rather than repeated copies of the same operation. The work is deliberately structural rather than phenomenological. It does not derive a measured coupling constant or claim that the reduced topology alone fixes a unique physical stability value. Instead, it establishes the finite-state architecture, projector structure, angular sector, topological degree, and spectral mode relations, while leaving the canonical observable bridge and continuous modulus as open problems for subsequent work.