Combinatorial Rigidity in Circumstellar Envelopment Lattices: Deriving Hyper-Structural Resilience via the Intersection Numbers of Resolvable Quasi-Symmetric Designs

Mathematical Applications of Science Fiction We present a rigorous derivation of the structural stability conditions for Type-II Kardashev envelop-ment arrays (Dyson Swarms) modeled as geometric lattices derived from Resolvable Quasi-SymmetricDesigns (RQSDs). By mapping the block intersection numbers (x, y) of a design D(v, k, λ) to thestress coefficients of a tensegrity framework, we establish a direct isomorphism between the spectralgap of the design’s Bose-Mesner algebra and the infinitesimal rigidity of the resulting circumstellarlattice. We prove the Central Rigidity Theorem, demonstrating that for a specific class of RQSDs,the resulting lattice LΩ exhibits Hyper-Structural Resilience—a state where the kernel of the rigiditymatrix is trivial modulo rigid body motions, even under anisotropic gravitational shear. We furtheranalyze the homological stability of these structures, showing that the first homology group overcoefficients in R vanishes, implying a lack of localized failure modes. This monograph serves as afoundational blueprint for the construction of mega-structures capable of withstanding stellar radiationpressure and tidal forces.