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

Abstract

This work investigates whether a compact set of local scalar-geometric quantities is sufficient to characterize structural classes in the framework of Universal Collapse. The candidate signature Σ* = (C, X², type(X), I₂, J) is tested constructively through the geometry of its fibers. Rather than inferring sufficiency from the absence of random collisions, the analysis fixes the candidate signature and searches explicitly for admissible configurations that remain distinct under the structural identifications used in the construction. Continuous kernel-based deformations, exact fiber constructions, rank and inertia analysis, and population-level tests show that the same Σ*-fiber can contain operationally distinct configurations. 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 Σ*-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 K₃ = Tr[(Q^μ_ν)³] is the first additional quantity found to separate the previously indistinguishable configurations. This motivates the tested extension Σ_min = (C, X², type(X), I₂, J, K₃). 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 Σ* + K₃ was found in that population. These results do not establish global or universal sufficiency of Σ_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: Σ* is insufficient within the tested domain, while Σ* + K₃ 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 reserved for subsequent work.