The Archilogon is a philosophical research project that aims to reconstruct the ancient Pythagorean diagram designed to represent the logical structure of reality. This logic was studied and preserved by Plato but never fully transmitted, due to the premature loss of the diagram that conveyed it. Tradition has left us scattered and often contradictory fragments, interpreted at times as esoteric symbols or theological doctrines. Modern academia, despite its refined tools, has lost the systemic perspective that animated the first philosophers and the true purpose of authentic Metaphysics, which lies in the study of the geometric forms of the archetypal diagram. The Archilogon seeks to bridge this gap, avoiding any dogmatic assertions and instead proposing a new rational perspective. Elias Artista has structured the project as a path marked by key words. The work alternates between moments of historical reconstruction (from Pythagoreanism to the Mathesis Universalis, from the Kabbalah to German Idealism) and theoretical sections, dedicated to defining the two guiding principles of 'Synthetic Logic': the Principle of Reciprocal Implication and the Principle of Relativity.