Closed System Self-Update Paradox

Abstract

This paper proves a fundamental theorem concerning the impossibility of self-memory in finite closed systems. Suppose the total information storage capacity of a system consists of a finite number of information atoms (indivisible minimal semantic units). We prove that any non-trivial update (change of system state) cannot be completely recorded by the system itself. The proof proceeds in two steps: Step 1 (Single-Step Overwrite Theorem): Since historical recording requires simultaneously carrying the "old value," the "new value," and the "temporal direction," while the sole information atom that undergoes change can only assume a single value after the update, it cannot bear a triple information load. Therefore, a single update necessarily ends in irreversible overwrite. Step 2 (Historical Degeneracy Collapse Theorem): Even if we assume, hypothetically, that the system could somehow "record" history in a single step, after sustained updates, the number of possible historical paths grows exponentially with the number of updates, while the system's state capacity remains fixed. By the pigeonhole principle, infinitely many historical paths will eventually be compressed into a finite state space, resulting in complete historical degeneracy — the system cannot distinguish whence it came. Combining these two steps, this paper rigorously derives the corollary: in a finite closed atomic storage system, the only sustainable mode of evolution is Irreversible Overwrite. The old state is permanently overwritten at the moment of overwriting and cannot be reversibly recovered. This theorem provides a discrete mathematical model for information irreversibility, the statistical origin of the Second Law of Thermodynamics, and the philosophy of "presentism," without relying on physical assumptions.