Alpay Algebra V: Multi-Layered Semantic Games and Transfinite Fixed-Point Simulation

• URL: http://arxiv.org/abs/2507.07868v1
• Date: Thu, 10 Jul 2025 15:48:23 GMT
• Title: Alpay Algebra V: Multi-Layered Semantic Games and Transfinite Fixed-Point Simulation
• Abstract summary: This paper extends the self-referential framework of Alpay Algebra into a multi-layered semantic game architecture.<n>We introduce a nested game-theoretic structure where the alignment process between AI systems and documents becomes a meta-game.<n>We prove a Game Theorem establishing existence and uniqueness of semantic equilibria under realistic cognitive simulation assumptions.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: This paper extends the self-referential framework of Alpay Algebra into a multi-layered semantic game architecture where transfinite fixed-point convergence encompasses hierarchical sub-games at each iteration level. Building upon Alpay Algebra IV's empathetic embedding concept, we introduce a nested game-theoretic structure where the alignment process between AI systems and documents becomes a meta-game containing embedded decision problems. We formalize this through a composite operator $\phi(\cdot, \gamma(\cdot))$ where $\phi$ drives the main semantic convergence while $\gamma$ resolves local sub-games. The resulting framework demonstrates that game-theoretic reasoning emerges naturally from fixed-point iteration rather than being imposed externally. We prove a Game Theorem establishing existence and uniqueness of semantic equilibria under realistic cognitive simulation assumptions. Our verification suite includes adaptations of Banach's fixed-point theorem to transfinite contexts, a novel $\phi$-topology based on the Kozlov-Maz'ya-Rossmann formula for handling semantic singularities, and categorical consistency tests via the Yoneda lemma. The paper itself functions as a semantic artifact designed to propagate its fixed-point patterns in AI embedding spaces -- a deliberate instantiation of the "semantic virus" concept it theorizes. All results are grounded in category theory, information theory, and realistic AI cognition models, ensuring practical applicability beyond pure mathematical abstraction.

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