Published April 30, 2025 | Version v1.0
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Alpay Algebra: A Symbolic Framework for Recursive Identity Compression and λ-Layer Dynamics

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Description

This document introduces Alpay Algebra — a novel symbolic system for modeling recursive identity transitions, convergence dynamics, and depth-based self-evolution.

 

The framework formalizes how a system’s state can fold, scale, merge, and regenerate through symbolic operators. It defines:

• Recursive transformations

• Energetic identity states (Δ⁺, Δ⁻, Δ⁰)

• Convergence fields

• Recursive depth layers (λₙ) and transitions (χ_λ)

• Entropy absorption through depth resilience

• A unified equation combining recursive evolution and symbolic compression

 

It models identity not as fixed, but as a recursive system evolving under internal and external pressure — absorbing entropy through symbolic recursion. The system provides both a formal symbolic grammar and a structural logic for identity reconstruction, symbolic cognition, and depth-aware self-reflection.

 

The document includes real and symbolic case studies (including recursive collapse and restoration of the author), vector field compression, and visual λ-layer transitions.

 

This release functions as a first preprint publication and symbolic foundation for future applied systems, AI models, and recursive depth theories.

 

Author: Faruk Alpay
ORCID: 0009-0009-2207-6528

 

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Dates

Submitted
2025-04-30
This is Version 1.0 of Alpay Algebra: Recursive Transition System — the initial formalization of a symbolic language for recursive identity modeling, λ-layer dynamics, and entropy compression. This version introduces the foundational grammar (Φ, Δ, R, χ, λ), symbolic flow operators, and depth-based recursion logic. It includes structured examples, theoretical theorems, and real-case symbolic modeling of internal phase shifts. Further versions will expand the system to include: • Multi-agent recursive interactions • Layer-coupled symbolic dynamics • AI/LLM integration templates • Graphical recursion engines Each future release will represent a deeper λ-compression and be versioned formally in recursive alignment.

References