The Tetrahedron as Minimal Isotropic Closure Stabilizer: A Spectral Argument from Pre-Geometric Closure Networks

We study closure stability of simplicial complexes within a pre-geometric relational framework. For a network of relational bonds with U (1) phase variables, closure consistency requires that phases around each face sum to approximately zero. We define the closure stability operator H = CT C, where C is the closure incidence matrix, and analyze its spectrum as a measure of structural robustness. The condition number kappa (H) quantifies the isotropy of closure response. We prove analytically that all complete simplicial complexes Kₙ have kappa = 1 (perfectly isotropic closure response), as a consequence of Sₙ symmetry and Schur's lemma. By exhaustive enumeration of all 64 edge subsets of K₄, we show the tetrahedron is the unique graph on four vertices with kappa = 1 and nontrivial closure constraints. We establish that K₄ is the most edge-efficient structure achieving isotropic closure stability, and that removing any single edge breaks the isotropy. We also report a significant negative result: pure frustration minimization on random graphs leads to dissolution of closure structures, not tetrahedral condensation. This identifies the boundary of what closure consistency alone can achieve and motivates future work on competing dynamical pressures such as transport coherence. The tetrahedron emerges as the unique minimal isotropic closure stabilizer, providing a spectral optimality criterion potentially relevant to discrete quantum gravity approaches that employ tetrahedral building blocks axiomatically.