TRANSARCHIMEDEAN ONTOLOGY UNDER MODAL DISCIPLINE An expanded critical–propositional analysis of The Omega Number System: Toward a Graded Transarchimedean Extension of Complex Analysis, by Olav Mitchell Underdal, in confrontation with the Theory of Ob

This article offers an expanded critical–propositional analysis of Olav Mitchell Underdal’s The Omega Number System: Toward a Graded Transarchimedean Extension of Complex Analysis in confrontation with the Theory of Objectivity (TO). The study examines the possible compatibilities and tensions between the graded transarchimedean architecture of the system and the modal-ontological discipline of the TO, especially in light of its Seven Absolute Truths, its foundational bibliography, its recent developments, and its supporting and dialogical literature. The paper argues that Underdal’s proposal is philosophically and mathematically significant insofar as it recognizes the insufficiency of the classical domain for dealing with zero, infinity, singularity, divergence, and multiple scales. In this respect, the system is read as a fertile formal structure for reflecting on boundaries, stratified magnitudes, composition by anteriority, and non-classical regimes of intelligibility. At the same time, the article argues that the proposal remains in tension with the TO because it does not derive its categories from modal necessity, does not explicitly formulate an ontology of being, does not satisfy the requirement of minimal relational observability, and cannot replace the cosmogonic theorem of the Theory of Objectivity. Beyond the critical comparison, the article develops a constructive interpretation of the system as a possible auxiliary formalism for future operational bridges within the TO. In particular, it explores how graded transarchimedean structures may contribute to the modeling of phenomenic thresholds, Inductive Effects, scale reflections, convergence zones, and informational gradations. In accordance with the objectivist framework adopted here, the transcendent element is interpreted as the knowledge or information produced in atomic relations and equivalent to atomic radiations, allowing the system to be reread as a potential mathematical grammar for manifestation, intelligibility, and relational memory. Rather than treating the Theory of Objectivity as a substitute for contemporary mathematics or physics, the article reaffirms that the TO should be understood as a necessary logical, ontological, and scientific discipline for any model that seeks to coherently describe a possible universe. Under this perspective, Underdal’s work is not rejected, but received under modal discipline as a promising formal interlocutor within a broader ontological cosmology. Keywords:Theory of Objectivity; Omega Number System; transarchimedean analysis; modal ontology; infinity; zero; phenomenic thresholds; Inductive Effects; convergence zones; relational information; ontological cosmology; operational bridges.