The ΔΦ Transport Invariant: A Topology-Dependent Collapse Law in Conserved Systems

This work introduces a unified transport framework based on the flux structure ΔΦ = ρv, where ρ represents density and v represents velocity. While this form is standard in conservation laws, it is typically treated as a descriptive quantity rather than a governing principle. Using a continuity-based engine with mode-specific geometric penalties, this study demonstrates that systems governed by ρv exhibit a bifurcation between two regimes: a globally coupled regime characterized by stabilization and a locally unconstrained regime characterized by finite-time concentration (collapse). Across multiple topologies (grid, small-world, and random), a critical coupling threshold p* is observed, above which collapse emerges. This threshold is shown to depend on topology, indicating that while transport is universal, collapse behavior is not. The results are consistent with known nonlinear transport equations, including porous medium and Keller–Segel-type systems, and support the interpretation that collapse arises as a symmetry-breaking event within conserved flux dynamics. This work does not introduce a new conservation law, but identifies a unifying structural role of flux within existing conservation frameworks. The framework is fully falsifiable and provides testable predictions regarding stability, collapse, and topology-dependent thresholds in distributed systems.