Alpay Algebra V: Multi-Layered Semantic Games and Transfinite Fixed-Point Simulation
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908444799270912 |
|---|---|
| author | Kilictas, Bugra Alpay, Faruk |
| author_facet | Kilictas, Bugra Alpay, Faruk |
| contents | 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 $ϕ(\cdot, γ(\cdot))$ where $ϕ$ drives the main semantic convergence while $γ$ 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 $ϕ$-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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_07868 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Alpay Algebra V: Multi-Layered Semantic Games and Transfinite Fixed-Point Simulation Kilictas, Bugra Alpay, Faruk Computation and Language Artificial Intelligence 68T50, 68T07, 03G30, 18C10 I.2.7; I.2.6; F.4.1 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 $ϕ(\cdot, γ(\cdot))$ where $ϕ$ drives the main semantic convergence while $γ$ 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 $ϕ$-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. |
| title | Alpay Algebra V: Multi-Layered Semantic Games and Transfinite Fixed-Point Simulation |
| topic | Computation and Language Artificial Intelligence 68T50, 68T07, 03G30, 18C10 I.2.7; I.2.6; F.4.1 |
| url | https://arxiv.org/abs/2507.07868 |