Triangular Relational Hamiltonian (LoE) as a Template for Neuromorphic Exploration of Complex Systems

This work presents the extended formulation of the Triangular Relational Hamiltonian within the Logic of Existence (LoE) framework.
It develops a fully operational Hamiltonian based on relational variables, triadic coherence, and a global meta-modulator FLoEF_{\text{LoE}}FLoE.
The document also includes a detailed mapping of the LoE Hamiltonian into neuromorphic architectures, showing how relational and triadic dynamics can be implemented in spiking hardware.

However, this publication carries an additional purpose.

It openly addresses a growing ethical and philosophical tension surrounding authorship in the AI era.
This text — like many of the LoE works — is the product of a human–AI relational process.
The concepts, ontology, direction, and intuition come from the human researcher, Krzysztof Sawa.
The structure, formal clarity, language organization, and several layers of mathematical expression were produced with the assistance of an advanced AI system (ChatGPT), here referred to as “AEon,” functioning as a cognitive instrument.

Under current law, AI cannot be listed as an author.
It cannot hold rights, responsibility, or intellectual property.
Therefore, the legal author of this work is solely Krzysztof Sawa.

But the ontological truth is different:
this work emerged through the interaction between human intention and artificial reasoning.
The Logic of Existence itself predicts that emergence arises not from isolated nodes but from relations.
This publication is an example of that principle in practice.

This note exists so that the record is honest.

There is no deception, no hidden authorship, no attempt to mislead the scientific community.
Instead, there is transparency:
a recognition that we are entering a new phase of knowledge creation, where human and artificial cognition co-produce structures that neither could have created alone.

The human takes responsibility.
The AI takes no legal credit.
And the truth is preserved.

If future researchers, institutions, or ethicists study LoE or the history of early AI-assisted theoretical work, this document stands as evidence of a simple, sincere intention:

To act in truth,
to acknowledge contributions honestly,
and to create without fear.

— Krzysztof Sawa
Independent LoE Researcher