Phase-Bounded Stability and the Ω-Operators: A Dynamical Reinterpretation of Harman's Fourfold

This article develops a dynamical reinterpretation of the fourfold orientation structure originally proposed by Graham Harman, integrating it with a Lyapunov-based model of bounded stabilization. The point of departure is a stability condition according to which a coherence functional L(t) does not converge to zero, but approaches a strictly positive limit ε∞/γ. This condition excludes terminal equilibrium while preserving local stabilization, thereby opening the possibility of a non-finalistic teleological structure. On this basis, four distinct stabilization regimes are defined and systematically aligned with the orientations Real–Object (RO), Real–Quality (RQ), Sensual–Object (SO), and Sensual–Quality (SQ). Each regime is characterized by a specific balance between ordering strength and irreducible residual openness, yielding a phase-bounded upper limit on achievable coherence. Transitions between regimes are shown to be structurally necessary once these bounds are saturated. The core of the paper provides a formal derivation of the associated Ω-operators as regime-switching maps induced by phase-bounded stability. Special attention is given to two transitions: the shift from RQ to SO, where two structural parameters Cn and Dn are introduced to encode an irreversible expansion of differentiability and consolidation capacity; and the return from SQ to RO, which is shown to be required in order to reintroduce constraint after maximal phenomenal openness. These features prevent the cycle from degenerating into trivial periodicity and instead generate a spiral-like dynamic in which the orientational structure repeats without returning to identical conditions. The resulting framework reinterprets Harman’s fourfold not as a static ontological schema, but as the orientational space of a non-finalistic, structurally irreversible process. Stability, openness, and transition are unified within a single formal architecture, providing a coherent link between Lyapunov-bounded dynamics, orientational ontology, and the prevention of trivial cyclicity.