Quantum Structural Theory of Harmony (QSTH X.C) — Rotationally Inequivalent States and Transition Paths in a Nonlocal Model on the Sphere

Describtion This study examines structural distinguishability and transition paths in a finite-dimensional, rotationally invariant model of a real scalar field on the unit sphere. The functional combines a nonlocal quadratic kernel with a positive local quartic term. A spherical-harmonic truncation at L = 24 gives 621 real coordinates. Two numerically obtained, sign-related minima have equal energy but belong to different SO(3) orbits. Their rotational inequivalence follows from a nonzero cubic field moment J₃, which is invariant under spatial rotations and changes sign under field inversion. A numerically constructed transition network connects them through index-one saddles and a nonzero intermediate local minimum that is rotationally equivalent, within numerical precision, to its sign reversal. The constructed path reaches an excess energy of approximately 0.031949 dimensionless model units, compared with 0.396997 for the specified radial path through the zero field. A constrained Hessian audit further shows that the known saddle remains unstable at fixed J₃. Thus, the invariant distinguishes the endpoints but does not fully characterize the transition geometry. The result is a model-specific example, not a general theory of phase transitions. The manuscript separates analytic statements, numerical evidence, and physical interpretation. It does not establish a global minimax barrier, convergence as the harmonic cutoff increases, a physical lifetime, or a memory read/write mechanism. Physical realization remains open. The manuscript does not derive the structural field from a specific interface. It identifies the missing microscopically derived, interface-conditioned carrier functional discussed in X.C-B. The separate X.C-DC effective-entropy construction is presented conditionally; no map from the model field to that construction is derived here. Building on QSTH X.INTRO and X.0, this work examines how rotationally inequivalent structures can be connected through shape reorganization with substantially lower excess energy than the specified radial path through zero. The accompanying files provide reproduction materials for the transition network, rotational-orbit checks, and fixed-J₃ Hessian analysis. Their checksums, dependencies, and verification scope are documented in Appendix A. The audits reuse the supplied implementation; they are not an independent reimplementation of the complete study.