Sheaf Constraint Networks L: Spectral Flow of Cellular Sheaf Constraint Networks

Abstract

Constraint network theory models constraint systems with geometric structureas sheaf-theoretic objects on hypergraph skeletons, and one of its core goals is todetect connectivity phase transitions of networks through zero-mode jumps in theLaplacian spectrum. Within the framework of finite CW cellular sheaves, this paperdistinguishes the deletion model of failed edges from the stalk-preserving accountingmodel, and proves that the correspondence among zeroth-order spectral multiplicity, branch structure, and zeroth cohomology requires only that cross-branch edgesfail completely, without relying on any continuous analytic assumptions. All-ordercohomological splitting and sheaf isomorphism require the combined conditions ofthe deletion model and clean cuts; in the stalk-preserving accounting model, highercohomology exhibits residual terms carried by failed-edge stalks and cross-cut cellstalks, whose cross-differentials need not vanish in a general CW complex. Asfailure density increases, the multiplicity of the zero eigenvalue is a nondecreasinginteger-valued step function, and the jump caused by the failure of a bridge edgeequals the codimension of the gluing condition between the two end sections. Forthe penalty-parameter spectral flow, the multiplicity of the zero eigenvalue in a oneparameter family is concentrated in a single jump at λ = 0, and the jump amplitudeequals the dimension of flat but constraint-violating sections. Positive definitenessof the Cartan matrix and Brauer’s elementary divisor theorem respectively ruleout the possibility of zero determinant, and the determinant of the four-vertex deformation family is exactly computed as 4(1 − t). The core algebraic assertionscan be checked pointwise by exact rational rank computation; random instances inthe machine check are generated from fixed seeds and serve only as reproducibleverification; the whole paper does not rely on external datasets.