MATHEMATICS: The Fourth Mode as Phenomenology of Logos

This work completes the systematic construction of Metamonism by revealing its fourth fundamental mode—mathematics, understood not as a science of abstract objects, but as formal phenomenology of the operations diff (differentiation) and fix (fixation). It is shown that these operations, being the ontological origins of Logos as the operational hypostasis of Monos, constitute the foundation of mathematical reality preceding the concepts of number, set, and proof. Within the proposed axiomatics, basic mathematical concepts are derived, and G¨odel’s incompleteness theorems receive ontological proof as direct consequences of the prohibition of selfidentity (t ̸= t). Mathematics emerges as the pure domain of Logos realization, and its foundational crises as evidence of its deep connection with the principle of incomplete fixability. The work integrates this mode into the completed architecture of Metamonism, where physics, consciousness, and mathematics constitute the threefold phenomenology of one non-substantial Principle.