We resolve the long-standing open problem of Boolean dynamic data structure hardness, proving an unconditional lower bound of $Ω((\log n / \log\log n)^2)$ for the Multiphase Problem of Patrascu [STOC 2010] (instantiated with Inner Product over $\mathbb{F}_2$). This matches the celebrated barrier for weighted problems established by Larsen [STOC 2012] and closes the gap left by the $Ω(\log^{1.5} n)$ Boolean bound of Larsen, Weinstein, and Yu [STOC 2018]. The previous barrier was methodological: all prior works relied on ``one-way'' communication games, where the inability to verify query simulations necessitated complex machinery (such as the Peak-to-Average Lemma) that hit a hard ceiling at $\log^{1.5} n$. Our key contribution is conceptual: We introduce a 2.5-round Multiphase Communication Game that augments the standard one-way model with a verification round, where Bob confirms the consistency of Alice's simulation against the actual memory. This simple, qualitative change allows us to bypass technical barriers and obtain the optimal bound directly. As a consequence, our analysis naturally extends to other hard Boolean functions, offering a general recipe for translating discrepancy lower bounds into $Ω((\log n / \log\log n)^2)$ dynamic Boolean data structure lower bounds. We also argue that this result likely represents the structural ceiling of the Chronogram framework initiated by Fredman and Saks [STOC 1989]: any $ω(\log^2 n)$ lower bound would require either fundamentally new techniques or major circuit complexity breakthroughs.

翻译：我们解决了长期存在的布尔型动态数据结构难度公开问题，证明了 Patrascu [STOC 2010] 所提多相问题（以 $\mathbb{F}_2$ 上的内积实例化）的无条件下界为 $Ω((\log n / \log\log n)^2)$。该结果与 Larsen [STOC 2012] 为带权问题建立的著名屏障相匹配，并填补了 Larsen、Weinstein 和 Yu [STOC 2018] 所获 $Ω(\log^{1.5} n)$ 布尔型下界遗留的空白。先前的障碍是方法论层面的：所有现有工作均依赖于“单向”通信博弈，其中验证查询模拟的不可行性迫使研究者使用复杂机制（如峰均比引理），而这最终在 $\log^{1.5} n$ 处遭遇硬性上限。我们的核心贡献是概念性的：引入一个 2.5 轮多相通信博弈，该博弈在标准单向模型基础上增加一轮验证环节，Bob 借此确认 Alice 的模拟结果与实际内存的一致性。这一简单定性变化使我们能绕过技术障碍，直接获得最优界。因此，我们的分析可自然扩展至其他布尔型困难函数，为将差异性下界转化为 $Ω((\log n / \log\log n)^2)$ 动态布尔型数据结构下界提供通用方案。此外，我们认为该结果很可能代表了 Fredman 与 Saks [STOC 1989] 开创的 Chronogram 框架的结构性上限：任何 $ω(\log^2 n)$ 的下界都需要根本性新技术或电路复杂性的重大突破。