Universal Collapse III: Structural Sufficiency, Equivalence Fibers, and Minimal Structural Extension

The structural regime framework introduced in the preceding stages of the Universal Collapse program requires a stronger question than the construction of a regime classifier alone: whether the proposed structural representation contains sufficient information to distinguish configurations that are physically or structurally inequivalent within the admissible domain. In the present work, this question is investigated constructively. We consider the candidate local structural signature\( \Sigma^{*} = \left( C,X^{2},\operatorname{type}(X),I_{2},J \right) \), where \( X_{\mu}=\nabla_{\mu}C \), \( \qquad X^{2}=X_{\mu}X^{\mu} \), and \( Q_{\mu\nu} = \nabla_{\mu}\nabla_{\nu}C -\frac{1}{4}g_{\mu\nu}\Box C \), with \( I_{2} = Q_{\mu\nu}Q^{\mu\nu} = \operatorname{Tr}\!\left(Q\eta Q\eta\right) \) and \( \qquad J=X^{\mu}Q_{\mu\nu}X^{\nu} \), where the index contraction is performed with the background metric used in the computational construction. In the matrix representation adopted in the numerical tests, this contraction is represented by \( \operatorname{Tr}(Q\eta Q\eta) \). Rather than searching only for accidental collisions among independently generated configurations, we construct and analyze fibers of fixed \( \Sigma^{*} \). This reverses the usual classification procedure: instead of asking whether different configurations happen to share a signature, we fix the signature and actively search for structurally distinct configurations inside the corresponding fiber. The resulting computational campaign shows that configurations sharing \( \Sigma^{*} \) can nevertheless differ in tensorial rank, kernel structure, inertia class, spectral information, and local response. These distinctions persist under the admissibility and compatibility tests implemented in the study, including Hessian realizability, local scalar dynamics, conformal metric construction, representative gravitational constraints, and selected higher-jet consistency conditions. Population-level tests further show that the same \( \Sigma^{*}\ \)-fiber can contain multiple operationally separated classes. We then investigate a minimal extension of the candidate signature through higher spectral moments of the trace-free Hessian. In the tested extension sequence, the cubic spectral moment \( K_{3}=\operatorname{Tr} \left[ \left(Q^{\mu}{}_{\nu}\right)^{3} \right] \ \) is the first additional quantity found to separate the previously indistinguishable configurations. This motivates the tested extension \( \Sigma_{\min} = \left( C,X^{2},\operatorname{type}(X),I_{2},J,K_{3} \right) \). A subsequent constructive attack on this extended signature involved 22.5 million random deformation attempts distributed equally among timelike, spacelike, and non-null sectors. No collision of \( \Sigma^{*}+K_{3} \) was found in that population. These results do not establish global or universal sufficiency of \( \Sigma_{\min}\ \). The final collision search is a finite sampling procedure and does not constitute an exhaustive parametrization of the corresponding fiber. The principal result of this work is therefore more restricted: \( \Sigma^{*}\ \) is insufficient within the tested domain, while \( \Sigma^{*}+K_{3}\ \) survives the strongest constructive attack performed in the present computational campaign. The unresolved question of whether this extension is globally sufficient, and how its fibers should be characterized constructively, is left for subsequent work.