The Closed-System Self-Update Paradox

This paper proves a fundamental theorem on the impossibility of self-memory in finite closed systems. Let the system state be represented by a finite-dimensional vector, where each storage cell is an information atom (carrying a unique, indecomposable minimal semantic unit at any given instant). If the system undergoes a non-trivial update without invoking any external storage, then after the update it cannot simultaneously retain the old value, the new value, and the directed temporal order. The proof employs a minimal-update reduction method: starting from the inevitable logical contradiction of a single-component change, the result is rigorously extended to arbitrary large-scale updates. The theorem directly yields the corollary that, in a closed atomic storage system, the only realizable form of non-trivial update is destructive overwrite—the old state information is permanently discarded at the instant of update.