标题： 带锚定蕴含和事件索引不动点的希尔伯特空间：唯一性和定量速率 显示英文标题

标题： Anchored Implication & Event-Indexed Fixed Points in Hilbert Spaces: Uniqueness and Quantitative Rates

摘要： 我们开发了正交模逻辑（投影作为命题）与希尔伯特空间中算子不动点理论的综合方法。 首先，我们引入了一个锚定的蕴含联结词$A \Rightarrow^{\mathrm{comm}}_{P} B$，从语义上定义，只有当$A$为假，或者$A$为真且$B$在由固定非零投影$P$指定的“对易”上下文中为真时，该联结词才为真。 这个联结词通过添加一个侧条件$[E_B,P]=0$（$B$与锚点的对易性）来改进物质蕴含，并在布尔（对易）情况下退化为经典蕴含。 其次，我们研究了事件索引收缩下的不动点收敛性。 对于单个非扩张（不一定线性）映射$T$，我们证明事件索引条件等价于经典的断言，即某个幂$T^N$是严格压缩；因此在该设置中，“不规则事件”的表述并不增加一般性。 我们随后介绍真正更一般的变分算子（切换/随机）情况：如果演化的复合块具有有界的事件间间隙并且存在公共不动点，我们得到唯一性以及显式的包络速率。 最后，带有与$T$可交换的锚$P$，同样的推理确保在该子空间上事件索引压缩条件下在$PH$上的收敛性。 我们包含精确的作用域条件、示例和可视化解释。 显示英文摘要

摘要： We develop a synthesis of orthomodular logic (projections as propositions) with operator fixed-point theory in Hilbert spaces. First, we introduce an anchored implication connective $A \Rightarrow^{\mathrm{comm}}_{P} B$, defined semantically so that it is true only when either $A$ is false or else $A$ is true and $B$ is true in a ''commuting'' context specified by a fixed nonzero projection $P$. This connective refines material implication by adding a side condition $[E_B,P]=0$ (commutation of $B$ with the anchor) and reduces to classical implication in the Boolean (commuting) case. Second, we study fixed-point convergence under event-indexed contractions. For a single nonexpansive (not necessarily linear) map $T$, we prove that the event-indexed condition is equivalent to the classical assertion that some power $T^N$ is a strict contraction; thus the ''irregular events'' phrasing does not add generality in that setting. We then present the genuinely more general case of varying operators (switching/randomized): if blocks of the evolving composition are contractive with bounded inter-event gaps and a common fixed point exists, we obtain uniqueness and an explicit envelope rate. Finally, with an anchor $P$ that commutes with $T$, the same reasoning ensures convergence on $PH$ under event-indexed contraction on that subspace. We include precise scope conditions, examples, and visual explanations.