GEOMETRY AS A RELATIONAL PHASE: a critical–propositional analysis of Matthew D. Lehman's When Geometry Breaks Down in confrontation with the axioms, phenomenic elements, Inducer Effects, cosmogonic theorem, and cosmological Eras of the Theory of Obje

This article presents a critical-propositional analysis of Matthew D. Lehman’s “When Geometry Breaks Down: A plain language account of four papers on relational structure, spectral collapse, and what lies beyond” (Zenodo, 2026, https://doi.org/10.5281/zenodo.20148024), in dialogue with the Theory of Objectivity developed by Vidamor Cabannas and Denivaldo Silva. The analysis examines Lehman’s proposal that geometry should not be treated as an absolute given, but as an admissible relational phase dependent on coherent propagation in finite weighted networks, measured by the spectral gap λ₁. When λ₁ > 0, a system may sustain a geometric description; when λ₁ → 0, geometry collapses, although relational structure may persist in post-geometric phases. In confrontation with the Theory of Objectivity, the article discusses possible compatibilities and points of tension between Lehman’s theoretical framework and the modal axioms, phenomenic elements, Inducer Effects, cosmogonic theorem, and cosmological Eras of TO. Special attention is given to the TO thesis according to which the transcendent element corresponds to knowledge or information produced in atomic relations, equivalent to atomic radiations. From this perspective, Lehman’s post-geometric phase is interpreted as a relevant operational bridge for thinking the persistence of relational information beyond ordinary spatial representation. The article argues that Lehman’s work does not replace the complete cosmogonic theorem of the Theory of Objectivity, since it does not explain the origin of elements, the primordial mathematical Nothing, the antagonistic Tempus, the initial perfect sphere, or the modal genesis of universal space. Even so, it offers a valuable conceptual and mathematical bridge for dialogue between contemporary relational models of emergent geometry and the modal ontology of TO. This analytical text counted on the analytical support of ChatGPT. Keywords Theory of Objectivity; Vidamor Cabannas; Denivaldo Silva; Matthew D. Lehman; When Geometry Breaks Down; emergent geometry; relational structure; spectral gap; spectral collapse; post-geometric phase; finite weighted networks; modal ontology; phenomenic elements; Inducer Effects; cosmogonic theorem; cosmological Eras; relational information; atomic radiations; philosophy of physics; ontology of space