Reinterpreting Erdős’ Conjecture Through Informational Divergence and Coherence Collapse

This work introduces a novel reformulation of mathematical divergence, inspired by Erdős’ classical harmonic conjecture, through the lens of the Viscous Time Theory (VTT). We propose that the traditional view of divergence as an unbounded arithmetic summation can be reinterpreted as an emergent property of informational coherence fields governed by the IRSVT (Informational Residue in Suspended Viscous Time) framework. At the core of this formulation lies the concept that divergence is not only the accumulation of magnitude, but the persistence of logical connectivity within an informational structure. In this approach, the discrete series ∑ 1/ais associated with a coherent density field ρ_i (x), defined over a discrete topological structure on the integers, and divergence occurs when informational coherence paths Φ_α between adjacent nodes maintain non-vanishing probability. In other words, the growth of partial sums corresponds to sustained coherence rather than unbounded addition. The paper establishes the IRSVT Divergence Theorem, which defines divergence in terms of the continuity of Φ_α – tunnel probabilities and minimal coherence – gradient separation ΔC. This yields a general principle: if no irreversible collapse in informational flow occurs across the series, the system diverges not in quantity alone, but through the extension of topological coherence. This reframing of divergence leads to potential consequences in both mathematics and engineering. Mathematically, it suggests a class of divergence results linked to Φ_α – connectivity and coherence – field geometry, with potential implications for questions such as prime-gap distribution and density fluctuation patterns. In engineering and artificial intelligence, the approach provides a predictive framework for coherence stability, informing the design of adaptive metamaterials, load distribution in AI-pilot systems, and resilience architecture in complex sensor networks.