We present an algorithmic framework for incremental maintenance of first sheaf cohomology $H^1(X; \mathcal{F})$ on dynamically evolving 1-dimensional cellular complexes equipped with finite-dimensional cellular sheaves. The classical computation of $H^1$ via factorization of the coboundary matrix requires $O(n^3)$ time; when the complex evolves with a stream of $m$ edits, full recomputation after each edit costs $O(mn^3)$. Under a bounded local geometry assumption -- bounded cell size $v_{\max}$, 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 $v_{\max}$, $d$, 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 $V = 10^6$). 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 ($d \geq 2$, non-identity restriction maps) do not admit the same locality. Experiments on Barabasi-Albert graphs with up to $5 \times 10^6$ vertices and $1.7 \times 10^7$ streaming edits show 35 $μ$s 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.

翻译：我们提出一种算法框架，用于动态演化的一维胞腔复形（配备有限维胞腔层）上的一阶层上同调 $H^1(X; \mathcal{F})$ 的增量维护。经典计算通过余边界矩阵分解求 $H^1$ 需要 $O(n^3)$ 时间；当复形经历 $m$ 次编辑的流式演化时，每次编辑后完全重算代价为 $O(mn^3)$。在边界局部几何假设（有界胞体大小 $v_{\max}$、有基茎维数 $d$、有界神经度 $D$）下，每次编辑（顶点插入、边插入、限制映射更新）仅影响有界的局部余边界块集合。因此，该算法以相对于复形总规模 $n$ 的 $O(1)$ 时间处理惰性流式编辑（代价与局部几何参数 $v_{\max}$、$d$、$D$ 呈多项式关系，这些参数视为与 $n$ 无关的常数），并将局部特征值求解和Mayer-Vietoris全局组装推迟到同步点（Flush）。在同步时，维护状态与对应分片层模型的批处理组装结果一致；在所有批处理验证的实验中（通过 $V = 10^6$）观测到零测量漂移。我们还给出胞腔分解的摊销 $O(|E|)$ 流式构造，并讨论了对抗性代数RAM障碍：该障碍表明非平凡层（$d \geq 2$，非恒等限制映射）在非分片情形下不具备相同局域性。在顶点数达 $5 \times 10^6$、流式编辑达 $1.7 \times 10^7$ 的Barabasi-Albert图上的实验显示，惰性单次编辑更新中位延迟为35微秒（不含flush）；查询时间（同步时的全局组装）在所实现的全遍历路径中为每次flush的 $O(n)$。精确同步代价另行报告。