{"ID":2883921,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.08397","arxiv_id":"2508.08397","title":"Anchored Implication \u0026 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.","short_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 $B$ i...","url_abs":"https://arxiv.org/abs/2508.08397","url_pdf":"https://arxiv.org/pdf/2508.08397v1","authors":"[\"Faruk Alpay\",\"Bugra Kilictas\",\"Taylan Alpay\"]","published":"2025-08-11T18:40:50Z","proceeding":"math.FA","tasks":"[\"math.FA\",\"math.LO\",\"math.OC\"]","methods":"[]","has_code":false}