Anchored Implication & Event-Indexed Fixed Points in Hilbert Spaces: Uniqueness and Quantitative Rates
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arXiv
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| Formato: | Preprint |
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2025
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| author | Alpay, Faruk Kilictas, Bugra Alpay, Taylan |
| author_facet | Alpay, Faruk Kilictas, Bugra Alpay, Taylan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_08397 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Anchored Implication & Event-Indexed Fixed Points in Hilbert Spaces: Uniqueness and Quantitative Rates Alpay, Faruk Kilictas, Bugra Alpay, Taylan Functional Analysis Logic Optimization and Control 47H10, 47H09, 47H05, 03G12, 06C15, 81P10 F.4.1; G.1.6; I.2.3 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. |
| title | Anchored Implication & Event-Indexed Fixed Points in Hilbert Spaces: Uniqueness and Quantitative Rates |
| topic | Functional Analysis Logic Optimization and Control 47H10, 47H09, 47H05, 03G12, 06C15, 81P10 F.4.1; G.1.6; I.2.3 |
| url | https://arxiv.org/abs/2508.08397 |