Incremental Sheaf Cohomology on Cellular Complexes: O(1)-in-n Lazy Edit Processing under Bounded Local Geometry

We present an algorithmic framework for incremental maintenance of first sheaf cohomology H1(X; F) on dynamically evolving 1-dimensional cellular complexes equipped with finite-dimensional cellular sheaves. The classical computation of H1 via factorization of the coboundary matrix requires O(n3) time; when the complex evolves with a stream of m edits, full recomputation after each edit costs O(mn3). Under a bounded local geometry assumption -- bounded cell size vmax, bounded stalk dimension d, and bounded nerve degree D -- each edit (vertex insertion, edge insertion, restriction map update) affects only a bounded set of local coboundary blocks. The algorithm therefore processes lazy streaming edits in O(1) time with respect to the total complex size n (with cost polynomial in the local geometry parameters vmax, d, and D, which are treated as constants independent of n), deferring local eigensolves and Mayer-Vietoris global assembly to synchronization points (Flush). At synchronization, the maintained state agrees with the corresponding batch assembly of the partitioned sheaf model; we observe zero measured drift in all batch-verified runs (through V = 106). We also give an amortized O(|E|) streaming construction for the cellular decomposition and discuss an adversarial algebraic-RAM barrier arguing that unpartitioned non-trivial sheaves (d ≥ 2, non-identity restriction maps) do not admit the same locality. Experiments on Barabasi-Albert graphs with up to 5 × 106 vertices and 1.7 × 107 streaming edits show 35 μs median lazy per-edit update latency (excluding flush); query time (global assembly at synchronization) is O(n) per flush in the implemented full-traversal path. Exact synchronization costs are reported separately.