Background and global literature gap. All existing spectral geometry studies on the three-sphere—including the submitted T2-1 and T2-3, and the forthcoming T2-2—are confined to smooth metric deformations within a single isometry class. The entire theoretical framework presupposes full \ (SO (4) \) isometry preservation and globally smooth spectral evolution. No complete rigorous theory exists for supercritical deformations crossing distinct isometry classes, symmetry group degradation, jump discontinuities of spectral derivatives, or geometric topological phase transitions. Existing results remain at the level of physical/numerical conjectures; no purely analytic necessary-and-sufficient criteria, no discontinuous evolution theory, no global topological phase classification, and no quantitative entropy jump theory have been established. 2. Four independent core theorems. (i) A necessary and sufficient spectral criterion for detecting isometry-class geometric phase transitions on \ (S³\), with rigorous proof of global existence, uniqueness, and smooth monotonicity of the phase boundary curve. (ii) Rigorous proof that the order parameter \ ( (a, ) \) exhibits a first-order jump discontinuity in its first derivative at the phase boundary, with complete higher-order regularity analysis and explicit numerical counterexamples distinguishing smooth subcritical from discontinuous supercritical behavior. (iii) Complete topological classification of the full \ ( (a, ) \) parameter plane, extending the simplified T2-2 three-region partition to include post-transition broken-symmetry subregions, triple points, singularities, and bifurcations, with structural stability proofs. (iv) A first-order jump theorem for spectral entropy at the geometric phase transition, with explicit analytic upper and lower bounds for the jump magnitude, non-degeneracy proof excluding zero-jump degeneracy, and a revised full-coverage non-overlapping three-layer multi-scale stratification theory. 3. Bidirectional complementarity within the T2 series. This manuscript completes the final missing piece of the complete theory of metric deformations on compact 3-manifolds. The full logical chain is: static spectral foundation (T2-1) → general tensor perturbation framework (T2-3) → smooth subcritical conformal scaling analysis (T2-2) → supercritical topological phase transition and discontinuous spectral dynamics (T2-4, this work). This paper retroactively delimits the strict applicability boundaries of all three preceding theories. The four papers form a bidirectionally supportive, non-overlapping, independent closed-loop system with a unified research thread: characterizing the evolution of Laplace spectral invariants on the 3-sphere under arbitrary Riemannian metric deformations. 4. Applicability boundary declaration. This work provides pure mathematical mapping to three geometric models: closed FLRW cosmology, compact topological horizons, and Skyrmion topological solitons. It does not construct complete physical dynamical equations, perform observational numerical simulations, or include quantum corrections or matter field coupling.
Q Zhao (Fri,) studied this question.