Derivative-Induced Topology (DIT): A Foundational Framework for Dynamic Discernibility and Scale-Dependent Spaces

This paper establishes the foundational framework of Derivative-Induced Topology (DIT), a novel topological framework designed to govern systems characterized by diachronic flux, non-local history dependence, and scale-adaptive transitions. Classical point-set topology, predicated on static open sets and disjoint neighborhoods (Hausdorff T0–T4 separation), exhibits fundamental limitations when describing dynamic boundary formation and continuous phase transitions. While the recent Condensed Mathematics paradigm successfully translates topological structures into algebraic condensed sets via Grothendieck topologies, it remains inherently synchronic and lacks a natural mechanism to incorporate explicit time-memory kernels and dynamic scales. To address these gaps, we introduce the joint perception operator λ = (δ, Δt), defining the spatial and temporal resolution boundaries of an observer. Under this framework, open neighborhoods are not axiomatically pre-established but spontaneously emerge from the behavioral equivalence of a state field's all-order derivative tower. We classify dynamic separation into four distinct universality classes (Class I through Class IV), establishing rigorous algebraic conditions for progressive flux, structural memory (including the D_{II-∞} Structured Zero Phase), resonant synchronization, and scale-dependent hierarchical sedimentation. The framework is shown to unify disparate phenomena across physics (fluid-vapor interfaces, skin effect, coupled oscillators), signal processing (Gibbs phenomenon, frequency modulation), statistical monitoring (fatigue accumulation, the May anomaly), cognitive science (novice-to-expert transition), social dynamics (consensus drift, the Red-Blue Eye paradox), and organizational diagnostics (integrity monitoring). A companion numerical study (Reset Dynamics Plus 4) provides falsifiable evidence, revealing a characteristic +18.9% topological relaxation overshoot with monotonic relaxation and no Gibbs oscillations. This paper establishes the conceptual and operational core of DIT. A full functional-analytic formalization (embedding in Sobolev spaces, strict topological axiomatization, invariant theory) is deferred to subsequent work. We invite collaboration from mathematical physicists, functional analysts, and topologists to complete these foundational extensions.