Derivative-Induced Topology (DIT) Vol. III: Historical Sedimentation and the Selective Persistence of Structural Traces

Vol. I of this series established the axiomatic foundation of Derivative-Induced Topology (DIT), introducing the perception scale λ = (δ, Δt), the all-order discernibility spectrum S_λ^{(n)}(x,y), and the four universality classes of dynamic separation. Vol. II operationalized DIT as a computational engine, replacing the abstract variational operator Δ_λ^{(n)} with a parameterized bank of convolution kernels κ_{θ^{(n)}}^λ, and establishing the equivalence between the topological quotient X/∼_λ and density-based clustering in a Sobolev feature space. Vol. III extends the DIT recognition program from spatial and morphological differentiation to historical sedimentation: the identification of structural traces left by past events. Unlike Vol. I and Vol. II, the objects of recognition here are no longer directly observable; they are obscured by subsequent modifications, partial forgetting, and adversarial rewriting. Vol. III therefore addresses the central question: How is identification still possible when history itself has been edited? The conceptual foundation for this extension is provided by the DNS (Dynamic Number System) framework, where structured zeros 0^{(n)} encode historical costs, and the non-invertibility theorem establishes the fundamental irreversibility of formation-decomposition. DIT III translates this algebraic irreversibility into a topological identification problem: how to read historical traces from current structures, even when those traces have been partially modified or overwritten. We introduce three contributions. First, we define a three-tier model of historical sedimentation: the sedimentary layer (selectively archived patterns), the active layer (currently influential patterns), and the ghost layer (dormant but recoverable patterns). A sedimentation threshold Θ_sed ensures that only events of sufficient duration, energy, and structural stability enter the archive, avoiding infinite noise accumulation. Second, we establish the Selective Persistence Theorem: while the sedimentary layer is monotonic (once selectively deposited, never deleted), the active layer is subject to pruning and rerouting, governed by a competition criterion between crystallization and erosion rates. Third, we reformulate the recognition task in the limit of zero driving as the approach to a Historical Topological Ground State — a frozen fixed point where all remaining traces are locked into topological invariants, analogous to a structural glass rather than a perfect crystal. Vol. III thus completes the DIT trilogy: from separation (Vol. I), through computation (Vol. II), to the recognition of selectively persisted history in the face of irreversible modification.