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

Abstract

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.