Provenance trees are append-only directed acyclic graphs of artifact registrations anchored on a public blockchain, recently introduced as the data substrate of operator-gated provenance infrastructure. Their defining data-structural pattern is a parent-hash directed acyclic graph (PHDAG), in which each append performs a constant number of storage writes to previously-untouched slots. This pattern has not previously been isolated as a standalone primitive, formally bounded with explicit constants, or benchmarked against the standard alternative, the incremental Merkle tree (IMT). We formalize PHDAG append as O(1) in gas cost, independent of registry size and tree depth, and develop a stochastic cost model for IMT in which per-insert cost is a random variable over the leaf index, deriving closed-form expressions for its mean and variance. We validate both analyses empirically on Base Sepolia across tree depths 1 to 25. PHDAG is observed to be depth-invariant at 76,276 gas (standard deviation about 6 gas), while IMT cost grows linearly with depth. The crossover below which IMT is cheaper falls far beneath the depths of every production registry surveyed. We further establish trustless registry reconstruction from public event logs in linear time with no off-chain dependency.

翻译：溯源树是锚定在公共区块链上的工件注册的仅追加有向无环图，最近被引入作为操作者门控溯源基础设施的数据基底。其定义性数据结构模式是父哈希有向无环图（PHDAG），其中每次追加操作对之前未触及的存储槽执行恒定数量的写入。该模式此前从未被独立抽象为基元形式、从未被显式常数界定边界，也从未与标准替代方案——增量默克尔树（IMT）进行基准测试。我们形式化证明PHDAG追加操作的Gas成本为O(1)，与注册表规模和树深度无关；同时为IMT开发了随机成本模型，其中每次插入成本是叶子索引上的随机变量，并推导出其均值和方差的闭式表达式。我们通过在Base Sepolia网络上对1至25层树深度的实证分析验证了两个模型。观测结果显示PHDAG成本保持深度不变性，为76,276 Gas（标准差约6 Gas），而IMT成本随深度线性增长。IMT成本更低的深度交叉点远低于所有被调查生产注册表所采用的深度。我们进一步建立了从公共事件日志中以线性时间实现无信任注册表重构的方法，且无需链下依赖。