Formalization of the ${Ddpr}{(Dupertation})$ Operator

Abstract

Classical combinatorics employs the permutation operator $P(n, r)$ to measure ordered selections by treating elements as abstract, invariant multipliers. This approach neglects the physical state transitions of the background depository ($n-r$). We propose the \textbf{$\mathrm{Ddpr}$ operator} (\textbf{D}ynamic \textbf{D}u\textbf{p}e\textbf{r}tation, derived from $Dpr$), introducing the principle of \textbf{Physical Extraction and State Completion}. Under the $\mathrm{Ddpr}$ framework, extracting an element from the background consumes its residual microstate probability ($1/n$), decoupling it from background multiplicative scaling and transitioning it into an independent foreground state container.