A Unified Theory of Reconstruction Defects from Networks to Cellular Sheaves

Many descriptions of reconstruction failure compress a richer system into an obstruction space, a spectral quantity, a cohomological invariant or a persistence module. Such invariants can discard information needed for stronger reconstruction questions. We identify and quantify two such losses. First, we introduce a relative-resolution reconstruction problem on a finite-dimensional Hilbert space. Its effective obstruction metric is the classical shorted operator, equivalently a Schur complement, and its logarithmic jet defines a hierarchy of local response spaces. For filtered systems, projected obstruction transport need not be functorial; we derive the exact composition defect and a sufficient functoriality condition. Under functorial transport, response directions generate a historical response persistence module. An explicit pair of filtered systems has identical stagewise Hodge spectra, cohomological dimensions and obstruction-quotient persistence, yet non-isomorphic order-two historical response modules. Ordinary filtered obstruction data therefore do not determine historical response data. Second, we address a structural Cheeger question raised by Hansen and Ghrist: how spectral information relates to the distance from a cellular sheaf to the nearest sheaf with a nontrivial global section. Under a degree-zero fixed-stalk Frobenius repair model, we give an exact variational formula for this distance RX(F), show that it depends only on the 1-skeleton, and recover directly their lower bound λ1(LF0) ≤ RX(F)2. A scalar triangle family with λ1 → 0 and RX(F)2 → 2 shows that there is no universal function f with f(t) → 0 as t → 0+ such that RX(F)2 ≤ f(λ1(LF0)). The counterexample exhibits local visibility collapse, and under quantitative visibility/noncollapse conditions we prove two-sided structural Cheeger estimates.