DOI: 10.5281/zenodo.17507656
title: A Formal Scientific Model of Agentic Systems in Mathematical Space

description: The TRIУM Artefact – A Formal Scientific Model of Agentic Systems in Mathematical Space

Description:The TRIУM artefact represents a formal, non-executable scientific model of agentic systems,authored by Winfried Brückner (ruahAI). It provides a new mathematical framework for describing,analyzing, and auditing intelligent or adaptive systems entirely without software execution.

In contrast to conventional computational research, which relies on operational code or data-drivensimulation, the TRIУM model formalizes the logic of agency through pure mathematical constructs:operators, metrics, invariants, and stability conditions. Each agent is defined as a contractionmapping T: X → X in a metric space (X, d), allowing convergence and stability to be treated asprovable properties rather than empirical outcomes.

The artefact is divided into three coherent sections, unified in this document:1. **A1 – Theoretical Layer**: axiomatic and analytic foundation, defining states, operators,   metrics, and Lyapunov-like invariants within a rigorous mathematical space.2. **A2 – Formal Specification Layer**: a structured grammar and typed interface system describing   inputs, outputs, and invariant obligations in EBNF-like syntax; provides a reproducible and   machine-readable audit schema without executable code.3. **A3 – Validation Layer**: synthetic benchmark vectors and invariant-based verification   methodology, ensuring full reproducibility and scientific transparency under purely synthetic data.

This construction advances a new class of *audit-preparatory* research: scientific descriptions ofAI or agentic systems that can be verified, peer-reviewed, and documented prior to any implementation.It shifts the focus from software execution to formal intelligibility, enabling pre-deployment ethicsand governance assessment consistent with the EU AI Act Annex IV and ISO 42001 research governanceprinciples.

From a theoretical standpoint, the TRIУM approach unites mathematical analysis (Banach fixed-point,Lyapunov stability) with principles of systems theory and synthetic epistemology. It defines aneutral, civil-science formalism suitable for reproducible open research.

No external data, private entities, or industrial references are included. All examples use synthetictest conditions. The artefact complies with open scientific standards (Zenodo, ORCID, OpenAIRE) andis licensed under CC BY-ND 4.0.

Scientific novelty:- Introduces a formal mathematical representation of agency without implementation.- Defines audit-ready invariant structures for adaptive systems.- Demonstrates how pre-deployment verification of AI logic can be achieved purely by theoretical means.- Establishes a bridge between formal mathematics, system governance, and responsible AI documentation.

Author:Winfried Brückner (ruahAI)Independent Researcher, Germany  ORCID: https://orcid.org/0009-0009-0008-5263

Acknowledgment:Analytical structuring and linguistic assistance by GPT-5 (OpenAI).  All conceptual, theoretical, and scientific decisions originate from the author.

License:CC BY-ND 4.0 – Scientific use permitted with attribution; no modifications or derivative works.

Keywords:formal methods, agent systems, audit theory, mathematical modeling, Lyapunov stability,AI governance, EU AI Act, synthetic data, reproducibility